‘3 *U my; z w t: H. K 23; E3 Y E u. 5 L1.. I I! ,‘E'I mam Q" ... gv. L .\ IEGQESTu 7‘ 1"? .. '- ‘5 AQAN 1‘ \ wqr A QGQEL. 7",}? n} :_L L !’ hi, “- n [b.- n- I v avg? #1 .0 _ .. L4t-\‘-‘&-¢ .; r9 ¥ -T-r-..-‘r‘e 5%."??9, . . ‘ .~-.. , ,- x w.«.4 .4 A (.4 «can; ; «he I T“. g .-' I; “-75 U L-AHLA.’ 7537fijV .. pr‘Lu'L “334$ '. ,_._ > “-9 .- x.‘ 8 Date 0-7639 This is to certify that the thesis entitled INTRA-URBAN RESIDENTIAL MOBILITY IN LANSING- EAST LANSING: THE CONSTRUCTION , VALIDATION , AND APPLICATION OF A VACANCY CHAIN MODEL presented by S. Charles Lazer has been accepted towards fulfillment of the requirements for Ph. D. degree in October 24, 1975 Sociology Major professor LIBPARY Mich-i3: n 5“” Univctn'ty VITAE/3i!» +_ , 4“. y. um 7' w k ‘ h ”anemia! “if a "C‘ r. ' '1‘ wty (TH-"9 A ' .r ‘1 3.1; sob-Ivar - - l."- fhu' 73L!" rm hut t ,. 2-] . '. gin-Ln! ‘-“ T I _ ., “.51“ , =1, ,, ; . E. “ .1',f.l~ -15 3 a ii) I ABSTRACT INTRA-URBAN RESIDENTIAL MOBILITY IN LANSING—EAST LANSING: THE CONSTRUCTION, VALIDATION, AND APPLICATION OF A VACANCY CHAIN MODEL By S. Charles Lazer This study employs a vacancy chain model to examine intra-urban residential mobility. The purpose of the study is two— fold: to probe the utility of the vacancy chain model and to use the model to analyze the filtering process within a housing system. The study area consists of the contiguous cities of Lansing and East Lansing, Michigan. Vacancies are inferred from changes in successive occupancies as reported in the Lansing City Directory, and a weighting procedure is used to derive an unbiased sample of vacancy chains. This procedure resulted in the selection of 1268 chains for 1969—1970, and 707 chains for 1964-1965. The model fits the data extremely well in each time period, and in each of the housing sub-systems, though vacancy chains are extremely short. The excellent fit of the model lends support to the notion of the independence of vacancy transitions. The model indicates that some filtering—down of housing does occur in almost all sub-systems in both time periods. However. this is overshadowed by the fact that vacancy chains within Lansing— East Lansing are very short. If the length of chains is a function S. Charles Lazer 1131:“ size, as it appears to be, then the major bene— fnf' housing vacancy creations in Lansing-East Lansing are WSW“ of the study area, But the residents of the larger --£‘-. within which.Lansing—East Lansing is embedded. V M3 a]. - INTRA-URBAN RESIDENTIAL MOBILITY IN LANSING—EAST LANSING: THE CONSTRUCTION, VALIDATION, AND APPLICATION OF A VACANCY CHAIN MODEL By S. Charles Lazer A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Sociology 1975 (9 “Wright by 3. CHARLES LAZ 1975 ER ACKNOWLEDGMENTS The support and assistance of many people facilitated the completion of this work. In particular, I would like to thank my major professor, Thomas L. Conner, whose advice and encouragement were invaluable. The additional comments and suggestions of James Zuiches, S. F. Camilleri and Hans Lee individually and in committee were greatly appreciated. Thanks are also due to Barbara Vold and Carol Codling for their help in the preparation of the several drafts of this dissertation. Lastly, I would like to thank Jacque Lazer, who not only advised, encouraged, and typed, but who also displayed incredible patience while this work was in progress. 11 TABLE OF CONTENTS List of Tables List of Figures List of Maps 1. Residential Mobility 1.1: Characteristics of Movers 1.2: The Migration Process 1.3: The Aim of the Dissertation Models of Mobility 2.1: Markov Processes 2.2: Refinements of the Simple Markov Model The Vacancy Chain Model 3.1: The Vacancy Chain Model 3.2: The Mathematics of Vacancy Chains 3.3: Mean First Passage Times 3.4: Testing the Model Estimation of the Parameters of the Vacancy Chain Model 4.1: The Study Area 4.2: Classification of Occupancy States 4.3: Sub-Areas of Lansing-East Lansing 4.4: Estimation of the Transition Probabilities iii viii ix 12 18 18 20 26 32 34 34 35 41 60 5. 6. 7. Vacancy Chains in Lansing—East Lansing The General Model, 1969—1970 The general Model, 1964-1965 The Basic Models -- Discussion The Housing Sub-Systems, 1969-1970 The Housing Sub-Systems, 1964—1965 The Housing Sub-Systems —- Discussion Summary of Findings Vacancy Chains and the Filtering of Housing 6.1: The Filtering Process 6.2: Vacancy Chains and Filtering Effects 6.3: Findings 6.4: Filtering Effects in Lansing-East Lansing —- Discussion Conclusions Appendix A, Data Matrices Appendix B, Maps Bibliography iv 68 68 77 84 96 103 108 113 119 119 124 125 134 140 147 212 215 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 5.6 5.7 LIST OF TABLES Mobility by Race and Tenure Average Value of Housing Average Monthly Rent Average Value Levels Average Rent Levels Proportion of Housing Units Owner Occupied Proportion Owner Occupied Levels Characteristics of Housing Sub—Areas Distribution of Unweighted Vacancy Chains by Length, 1969-1970 Unweighted Vacancy Chain Arrivals and Departures by Stratum, 1969-1970 Distribution of Weighted Vacancy Chains by Length, 1969-1970 Weighted Vacancy Chain Arrivals and Departures by Stratum, 1969—1970 Comparison of Observed and Predicted Chain Length Distributions for the Complete Model, 1969-1970 Comparison of Observed (L) and Predicted (A) Mean Chain Lengths by Stratum of Origin, 1969-1970 Distribution of Unweighted Vacancy Chains by Length, 1964—1965 40 43 45 47 47 48 so 52 68 69 72 75 75 77 -'t ’ehgn Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Unweighted Vacancy Chain Arrivals and Departures by Stratum, 1964—1965 Distribution of Weighted Vacancy Chains by Length, 1964-1965 Weighted Vacancy Chain Arrivals and Departures by Stratum, 1964-1965 Comparison of Observed and Predicted Chain Length Distributions for the Complete Model, 1964—1965 Comparison of Observed (L) and Predicted (A) Mean Chain Lengths by Stratum of Origin, 1964-1965 Predicted Chain Length Distributions, 1969—1970 and 1964—1965 Comparison of Observed and Expected Chain Length Distributions, Using N(j)=NqJ‘lp, 1970 Comparison of Observed and Expected Chain Length Distributions, Using N(j)=NqJ'1p, 1965 Multipliers and Housing Area Pepulations Distribution of Unweighted Chains by First Unit in Chain, 1969-1970 Distribution of Unweighted Pure Chains by First Unit in Chain, 1969-1970 Distribution of Weighted Vacancy Chains by First Unit in Chain, 1969-1970 Distribution of Weighted Pure Chains by First Unit in Chain, 1969-1970 Observed and Predicted Chain Length Distributions, Mixed Sub—Systems, 1969—1970 Observed and Predicted Chain Length Distributions, Pure Chains, 1969—1970 vi 78 79 81 82 82 84 86 87 95 97 97 98 99 100 101 Table Table Table Table Table Table Table Table Table Table Table Table 5.23 5.24 L_and A, Mixed Sub—Systems, 1969-1970 Distribution of Unweighted Chains by First Unit in Chain, 1964—1965 Distribution of Unweighted Pure Chains by First Unit in Chain, 1964—1965 Distribution of Weighted Vacancy Chains by First Unit in Chain, 1964—1965 Distribution of Weighted Pure Chains by First Unit in Chain, 1964—1965 Observed and Predicted Chain Length Distributions, Mixed Sub—Systems, 1964—1965 Observed and Predicted Chain Length Distributions, Pure Sub—Systems, 1964-1965 L and A, Mixed Sub—Systems, 1964—1965 Pure Chains as a Proportion of All Chains of Length 2 or More, 1969—1970 Pure Chains as a Proportion of All Chains of Length 2 or More, 1964-1965 Filtering-Down Ratio by Type of First Unit in Chain and Year Selected Characteristics of Housing Sub-Areas 102 104 104 105 106 107 108 110 110 130 137 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure {missile-.3. " LIST OF FIGURES Size of migrant stream and distance separating the origin and destination of moves The vacancy chain model Smallest space analysis: characteristics of housing units and population Average value of housing in thousands of dollars Average monthly rent in dollars Proportion of housing owner occupied Average value of housing by stratum Average value of housing by stratum (in percent) Average monthly rent by stratum Average monthly rent by stratum (in percent) Proportion of housing owner occupied by stratum Proportion of housing owner occupied by stratum (in percent) A residential vacancy chain Unweighted and weighted chain length distributions, 1969-1970 Unweighted and weighted chain length dis— tributions, 1964-1965 Multipliers and housing system populations viii 19 39 44 46 49 53 54 55 56 57 58 63 70 80 143 - :4 4 L in . “‘4'“; Leasing 'jfl‘il’ ‘ ‘ g . fa mm: m“"” LIST OF MAPS 213 214 1. RESIDENTIAL MOBILITY 1.1: Characteristics of Movers Traditionally the analysis of residential mobility has been approached from two different perspectives: the analysis of migration streams, which is concerned with the volume and direction of the flow of people who move more or less permanently, between different places (see e.g. Stauffer, 1940; Zipf, 1946; McGinnis and White, 1967); and the analysis of differential migration which seeks to account for migration by differences in the attributes and characteristics of migrants and non-migrants, such as age, sex, occupational status, etc. (Rossi, 1955; Arminger, 1966; Rogers, 1966; Simmons, 1968; Straits, 1968; Morrison, 1971). Residential mobility is seen as "the process by which families adjust their housing to the housing needs that are generated by shifts in family composition that accompany life—cycle changes (Rossi, 1955:9; see also Folger, 1957; Moore, 1966; Brown et a1., 1970; and White, 1970). Because the process of adjustment formulated is a negative—feedback process, mobility potential is highest when living conditions and a family's desires are most discrepant (Rossi, 1955:76ff.), and much research has been devoted to the identification of persons or families most likely to move. h 7. _>-{ The typical mover is a young person (or family) with a comparatively low income, who is currently renting an apartment. If this renter expects a rise in salary, or if he wants his own home, or if young children are part of the house— hold — or all three — the mobility potential is increased (Abu—Lughod and Foley, 1970:471). The emphasis on tenure status (the distinction between renters and owners) as the critical variable is widely supported (Butler et a1., 1969; Cave, 1969; Moore, 1969; Brown and Holmes, 1971; McAllister et a1., 1971), but there are several other variables that are associated with mobility. Those who are most likely to move tend to be: young adults; males; professionals; unemployed (Rogers, 1966:452); recently married; wage earners (Morrison, 1971:172); persons not rooted in the community (Arminger, 1966). Some authors have reported that Whites (Rogers, 1966:452) or Blacks (Lansing et a1., 1969:52) are more likely to move than the other group, but recent findings suggest that the critical variable is tenure status (McAllister et a1., 1971). The identification of people likely to move is important "a small proportion of frequent migrants accounts for a high because proportion of all migration" (Taeuber, 1961:118). It would appear that approximately 20 percent of the population change residence each year (Taeuber et a1., 1961:862n), but "measuring migration on the basis of the number of moves recorded ... overstates the number of migrants by about 80 percent" (Goldstein, 1964:1131). gnu-T's: The findings reveal a substantial degree of chronicity ... a tendency for observed mobility rates to be the product of repeated and frequent movement by the same individuals rather than single moves by the observed population at risk (Morrison, 1971:172). 1.2: The Migration Process The analysis of migration streams, the flow of movers, between different places has yielded a persistent finding: there is an inverse relationship between the size of the migrant flow between two places and the distance which separates them. The bulk of this research has focused on intercounty or interurban movement (see Taeuber et al., 1961; Goldstein, 1964; Rogers, 1966; and Morrison, 1967). The use of the county or the metropolitan area as the smallest area amenable to analysis has been in large part determined by the availability of the data (Taeuber, 1961), or rather, the lack of data concerning intra-country or intra- urban mobility, despite the evidence which indicates that the highest proportion of residential movement occurs within a single metropolitan area or a single county (Butler, et a1., 1969:2 et seq.; Simmons, 1968:622). Not only have research findings indicated this to be the case but this situation is an obvious conclusion of many of the theoretical and mathematical formulations which were put forward to account for interurban and intercounty movement. This relationship can be expressed in its most general form as a probability density function. The probability of movement ‘ a distance D apart a p (D) - Db (can8D ) biffifieteSS'to almost all the empirical data regarding migration . 7‘ A '_-V 4‘ -_.I~ Wore, 1966:19-20). -b p(D) - am A} fi‘such as the gravitational model developed by Zipf \. "i ‘k v: Other ratios, such as are formed by Simmons (1968:641) in—migration = a population (distance)b are but simple variations of the basic function.1 All of them yield similar positively skewed curves of population movement as a function of distance, as sketched in Figure 1.1, with the exact shape of the curve determined by the constants in the equation. Size Distance Figure 1.1. Size of migrant stream and distance separating the origin and destination of moves Clearly, if we examine the volume of migratory mobility as a function of the distance between the place of origin and the 1In fact, if we let population = P1 and a = P2 then Simmons' equation is identical with Zipf's. place of destination V (D) = a T U and then extrapolate to include movements within a political boundary as well as movements between political entities, the great volume of intra—urban residential mobility should surprise no one. For b y 0, V(D) increases at least as rapidly as D decreases (V(D) increases linearly at b = 0, and exponentially at b > 0). ' Moore points out, however, that within an urban area the Pareto function will not strictly hold if the opportunities for migration do not decrease monotonically, i.e., if the population is not symmetrically distributed (1966:21). But empirical estimates of intra-urban movement as a proportion of all residential mobility range from two-thirds (Simmons, 1968:622) to a high of 80 to 85 percent (Butler, et a1., 1969:2 et seq.). This is consistent with the type of model being discussed. The consistency reaches further when we note that within the metropolitan area 25 percent of all moves are found to terminate in the neighborhood of origin and 60 percent terminate within a five mile radius (Butler, et a1., 1969:9). 1.3: The Aim of the Dissertation This paper will attempt to analyze intra-urban residential mobility within the framework of a vacancy-chain model (White, 1970). JJ By focusing on the structure of an urban housing system, and the movement of vacancies through it we will attempt to examine the opportunity structure within which residential mobility takes place. The models of migration discussed above are largely "push" models (Butler, et a1., 1969; Brown and Moore, 1970; Brown, et a1., 1970). The impetus to move arises from some dissatisfaction with the existing dwelling, or from the emergence of certain needs which the existing dwelling does not fulfill. Only then is the decision made to move, and after that occurs the question "where to?" (Brown, et a1., 1970:176). Regardless of whether the "push" is couched simply in terms of dissatisfaction (Butler, et a1., 1969) or more rigourously in terms of maximizing place utility (Brown and Moore, 1970), in the language of the marketplace, theSe are analyses of housing "demand," i.e. "who is looking for new (different) housing? Why do people look for new housing? What type of housing are they looking for?" The questions of housing "supply" are not dealt with, regardless of the fact that "the selection of a new home depends not only on demand conditions, but also on supply constraints" (Simmons, 1968:637). ' Clearly the selection of a specific new dwelling can not be fully understood without knowledge of the existing available choices, and the vacancy-chain model is an attempt to examine those choice systems. Knowledge of the movement of vacancies through the housing system would allow for a fuller understanding of residential mobility, because the housing system is the system within which residential mobility occurs. The vacancy chain model, after testing, will then be applied to the examination of the filtering process in a housing system. Filtering can be seen as "the changing of occupancy as the housing that is occupied by one income group becomes available to the next lower income group" (Ratcliff, 1949:321-22), and is widely considered to be the major mechanism for the provision of housing to lower income groups (see e.g., Forrester, 1969). The extent of house filtering, and even its existence, has been repeatedly questioned (Lowry, 1960; Grigsby, 1963; White, 1971), so the application of the vacancy chain model to the question of filtering not only probes the utility of the model, but may provide useful information regarding this important question. 2. MODELS OF MOBILITY 2.1: Markov Processes "Mobility analysis is ... the study of families of temporal functions" (McGinnis, 1968:713), or "time dependent probability processes" (McGinnis, 1968:715) and the basic stochastic model used to describe and analyze residential and occupational mobility has been that of the population of movers as a Markov process. The Markov process describes an object moving according to some set of probabilities through a system of distinct and defined states. 0n the surface, a more appropriate model for the analysis of residential mobility could hardly be imagined. Consider a population of objects initially distributed in a set of states {1, 2, ... , k} and the matrix 3 = [pij] of transition probabilities,1 the Markov chain model can examine 1pij is the probability that an object which was in state i at time t will be in state j at time t+1. 10 Nxt = j|x0= i} c = 1, 2, the probability that an object, X, will be in state j at time t, given that it began (t = 0) in state i. Under certain conditions it can also be used to examine the probability that an object beginning in state i will ever get to state j; how many moves it would take to get there, and other questions about the movement of objects in the population. The tractability and usefulness of Markov chains as models of mobility processes cannot be denied, especially the tractability of first-order Markov chains. A first order Markov chain is one where P{x ,...,X=k,..X =1} n+t = JIX0 n ' n+t—1 = P{Xn+t = J X i}. n+t-l = This is the Markov property. It states that the probability of movement from any state i at time (n+t—1) depends only upon the state the object occupies at time (n+t—l). The transition probability pij is in no way affected by previous occupancies by X or by the route through the system whereby X came to be in state i at time (n+t—l). In other words, knowledge of X's history —- that it was in state 1 at time 0, state k at time n, and state i at time (n+t—1) yields the same Pij (n+t-1) as knowing only that X is in i at (n+t-l). There are, of course, higher order Markov chains, whose transition probabilities do depend on the history of the process, but the mathematics becomes so cumbersome as to make them impractical (see, for example, Hua, 1973). 11 It is the Markov property that makes first—order Markov chains so attractive. If the transition probabilities remain constant over time, then the initial population distribution and the set of trans- ition probabilities completely describe the system (see Anderson and Goodman, 1957:89ff). But for the Markov property to hold it must be specified that the objects moving within the system of states move independently of one another. That is to say, a uniform rate of movement from state i to state j at time t, rij(t) is applied to the entire population of i at time t, and no selection process exists. What any object in the system does is in no way influenced by the action of any other object in the system. This requires that the population be homogeneous, and in the specific case of residential mobility that people have no friends, no relatives, no social contacts. At the very least the assumption requires that they ignore these contacts when they move, even though the most effective means of becoming aware of and taking advantage of residence vacancies is personal contact (Rossi, 1955:151; Moore, 1966:29). The repeated use of first order Markov chains with constant transition probabilities by students of residential mobility attests to their attractiveness despite the stringent requirements imposed by the Markov property, and the assumptions regarding state classification, population homogeneity, and time stationarity. That the assumptions of the model are not met (or alternatively, that the madel is simply not an adequate representation of mobility processes in human populations) is attested to by the constant revision, restate— 12 ment, and refinement of such models. A great amount of energy and paper has been wasted attempting to "apply" various in- adequate models to data when the models' inade— quacy could more easily have been discovered — and perhaps remedied — by a careful theoretical analysis of the models' assumptions and/or their logical consequences (McFarland, 1970:472). 2.2: Refinements of the Simple Markov Model Because Markov theory is concerned with state changes by an individual, not the movements of an entire population, population homogeneity must be assumed to exist. Because it does not exist the most persistent problem with simple Markov chain formulations of movement between residences or movement between jobs has been the failure of the predicted nth — step transition matrix to coincide with the observed nth - step transition matrix. Blumen et a1. (1966), in a study of occupational mobility were the first to note that the observed transition matrix 2‘“) differed considerably from the predicted transition matrix 3? at the 8th step. In particular, they found that pii8 ) d < w k = 1 2, dPii ( d-kpii(t and lim dPii(t) = 1 d+m where d is the prior duration in state i (Myers, et a1., 1967:123)- It would appear that there are some theoretical as well as some practical problems in this line of enquiry. The questions being 15 explored are those of proper occupancy—state classifications and proper partitioning of the population into homogeneous sub—populations and associating with each sub—population the appropriate transition matrix. The question is the same as the one McFarland raises regarding the Markov models of social mobility: ... intergenerational social mobility is not a Markov Chain when states are defined thegway they defined them; the process might still be a Markov Chain if the states were defined differently (1970:464). The practical problem inherent in this approach is the increase in the number of transition matrices and the number of transition probabilities which must be estimated. If a system initially contains k states, then k(k—l) transition probabilities must be estimated-rl If we then classify our states into duration- specific states, with a maximum of h—l prior—elapsed time periods, and allow the transition from any state Si with occupancy—duration, d, to any state S with occupancy-duration 0, i.e., the transition 1 from S. to S. then we must estimate k(kr2) transitions from S. to d 1 .0 J d 1 on and k transitions from dSi to d+1Si' For h time periods then there are hk(k—1) = hk2 - hk transitions to estimate. The simple model requires the estimation of only k(k—l) transition probabilities. As the initial population is disaggregated into more homogeneous populations in an attempt to overcome the discrepancies of Markov Chain projections with heterogeneous populations, a second, 1He need only estimate k(k—l) instead of k2 transition probabilities k becausejglPij - l, for all i. l6 and perhaps more important, practical difficulty arises. Not only is there no guarantee that this particular disaggregation will fit the modelz, but the disaggregations are attained at great cost because of the information which must be gathered about the people in the system. As the subdivisions become more specific, more and more information must be gathered, at greater cost in time and money. Even if we consider only the two population "Mover-Stayer" model, we must wait a sufficient length of time for "Stayers" to reveal themselves (Morrison, 1971:177-178). It is impossible to inquire of people whether they are one or the other. If more information is required as.in the estimation of cohort—specific transition matrices (Rogers, 1966), or in the use of the exposure-residence concept (Taeuber et 81., 1961), then the difficulties increase. If we assume, for example, that the Axiom of Cumulative Inertia has meaning, then we require a reliable measure of duration—of—residence. To obtain it we must resort to "individual histories of movement" (Myers et al., 1967:125). Population partitions based on variables such as "rootedness in a community" (Arminger, 1966), or "satisfaction/dissatisfaction with current residence" (Butler, et a1., 1969) or "place utility" functions (Brown and Moore, 1970) would be even more tenuous than would partitions based on the commonly used "demographic" characteristics of migrants (age, sex, occupational status, etc.) 2Even in Spilerman's model, where virtually each individual in the system has his own transition matrix, the predicted diagonal entries pii(n) are still discrepant from the observed pii(n) (1972a:282n). 17 insofar as an analysis focusing on attitudinal questions about migratory behavior or intentions assumes that people understand their own complex behavior patterns —- an assumption which is probably unsound (Goldscheider, 1971:37). These refinements of state classification and population disaggregation are an attempt to meet the requirement that individual transitions be made independently. However, they do not rectify the basic source of non—independence in the process of residential mobi- lity. As stated above, the most effective means of discovering which dwellings are available for occupancy is personal contact (Rossi, 1955:161; Moore, 1966:29). As long as models of residential mobility are Markov Chain models of people moving through a system of occupancy— states, it does not appear that the independence requirement can be wt. .91? ‘V.V -‘fi‘ss. 3. THE VACANCY CHAIN MODEL 3.1: The Vacancy Chain Model White (1970) has recently developed an interesting and elegant model for the analysis of mobility within systems of positions and occupants, which manages to avoid many of the problems discussed above. Although the model was originally formulated to deal with systems of men in jobs, its potential application to the study of residential mobility was quickly recognized (White, 1970:320-321, 390; White, 1971; Bus, 1973). In fact there are earlier indications of the development of such a model within the housing field (Kristof, 1965; and Lansing et a1., 1969). The model proposed is an embedded Markov Chain of first— order, but the crucial distinction is that the population of interest is not the people who move through the housing system, but rather the vacancies which appear when people leave the system. These vacancies then move through the system occupying successively different dwellings until they finally leave the system. Consider the representation in Figure 3.1. Let A, B, and C represent addresses or dwelling units and a, b, c, and d represent people. The dashed line represents the boundary of the 18 19 d o 2 a I :1 : 5 A E 5 b : = Cal 2 E : : c 5 E c E d E Figure 3.1. The vacancy chain model housing system under scrutiny. Person a moves out of the system from position A, and b moves from B to A. In turn, person c moves from position C to B, and finally d fills C from outside the system. This sequence of linked moves by people can be regarded as a chain of vacancy movement. We cangconsider that a vacancy has moved in from outside the system to occupy A, and then moves from A to B to C and finally leaves the system. 20 In the housing system under consideration a vacancy can enter the system in several different ways. As in the above example, a vacancy is said to enter the system when the occupant of a dwelling leaves the system, either by moving away or dying, and leaves the dwelling vacant. If one spouse of a marriage were to die leaving the other in their previously joint residence, this would not create a vacancy. New housing may be built, or an existing house (1 residence) might be subdivided into r apartments, creating (r-l) new vacancies. The marriage or cohabitation of two or more persons each previously occupying his own dwelling would also create (n-l) new vacancies, where n is the number of residences previously occupied. Vacancies are said to leave the system when an existing vacancy is filled by a newcomer to the housing system, usually a migrant. The formation of new households —— the marriage and establishment of a separate household by two people each previously living with his parents, or the separation of a married or otherwise cohabiting couple, causing one to seek a new separate residence -— also cause vacancies to leave the system. Vacancies also leave the housing system when the dwellings they occupy are destroyed or are converted to some non-residential use. 3.2: The Mathematics of Vacancy Chains Mathematically, the model is straightforward, and can be presented very briefly: 21 Given a set of occupancy states, i = {1, 2, ...,s} where all states outside the system are denoted by i =0, let the probability of a vacancy in stratum i moving to stratum k be qik' Then 1 i = 1,2,...,s k = 0,1,2,...,s kéo qik = and E ksl qu = 1. If we let p_= [inJ’ a column vector and g= [qik] i,k = 1,2,...,s it can be shown that the probability of a chain of length j beginning in stratum j , Ej = Qi'lz: (3.1) .Ill E‘l‘I‘ lll‘l'llllllillvl.‘ 22 To explicate the model by way of a simple example, let us consider the system with only 1 state. The probability of remaining in the system is equal to q, and the probability of leaving in any time period is equal to p, and p+q = 1. The probability of remaining in the system for exactly one time period P(1) - p. The probability of remaining for exactly 2 time periods P(Z) = qp. It follows that P(3) = qu and in general, P(1) = q(j_1)p. (3.1a) To determine the mean length of time a vacancy will spend in the system we simply compute the vector of mean chain lengths by stratum of arrival _ w —l A =j=ljzj= (1 - 9) l (3.2) where I_is the identity matrix and l_is conformable column vector of 1's. If ffit) is the row vector of proportions of vacancy arrivals in year t by stratum, then the overall distribution of chain lengths may be computed by §(t)P , for all 21' The overall mean length of a 23 cohort of vacancy chains, j(t) becomes simply -l j(t) E fiflt)A_E fiflt) (L -,Q) .1 (3.3) In the one-state model, the mean length of time spent in the system by a vacancy ” =jzo jP(j) = 1 = 1 P l-q (3.2a) The vector M(t) = Eni(t)J of the total number of moves ever made by vacancies entering the system in stratum i can be expressed as M =°z° h __(t) h=Q F.(t)g where 2(t) is the vector of vacancy arrivals by stratum. This summation yields yr) = :(c) (_I_-g)‘l. (3.4) The total number of moves made by the r vacancies which enter our one-state system is simply the product of the number of Vacancies entering the system and the mean length of time (number of '\ 24 moves) each vacancy spends in the system. m = r (g) = r (iéa) . (3.4a) Although the above equations are sufficient to describe, verify, and analyze the properties of the vacancy chain model, some additional comments are necessary. The matrix (I:Q)_l is of great interest and importance in the study of housing vacancy chains. Just as in the simple model where is the mean number of moves a 1 1-q vacancy makes within the system, if we let —1 (29.) =[“1j] i.j = 1,2,...,s then 1113. is the mean number of times that an object which began the process in state i will appear in state j before reaching an absorbing state (in this case, before leaving the system). For this reason (179)-1 is called the multiplier matrix (White, 1970; White, 1971; Hua, 1972). The model presented is an embedded Markov chain of first— order, although there are major differences between it and the "standard" Markov models of residential movers. Conceptually, the population of objects moving through the system is a population of vacancies, not people, and different questions are posed by the two models. We are concerned with chains of vacancies and their 25 properties, such as length and persistence, and speed of movement from stratum to stratum. At the system level we are examining sequences of independent mobility acts which provide the framework within which residential mobility takes place. Another important distinction is that the qii of the vacancy-chain model refer only to EQZEE of vacancies within state i. The aggregation of address changes within state i and "no—moves" within i, which occurs in the pii of the "people" processes cannot occur here. The model is concerned only with EQXEE of vacancies, because there is no vacancy without a move. Insofar as Butler, et a1., show that one-quarter of all residential moves are to different places in the same neighborhood (l969:9) this distinction should prove quite useful. Furthermore, the consideration of moves only should alleviate the previously-mentioned persistent problem of the predicted nth step 311 underestimating the actual nth step Pii' Finally, because the model is one of vacancies moving through a system of housing, vacancies are the entities which are assumed to move independently. This represents an attempt to reconcile the mathematical theory with physical and social reality, and this assumption is much more plausible than one requiring that people move independently. 26 3.3: Mean First Passage Times One additional aspect of the vacancy chain model can be profitably studied to gain information about the structure of our housing system. The measure has been referred to as a measure of social distance (Beshers and Laumann, 1967) and functional distance (Brown and Morton, 1970), and reflects the degrees of connectedness which hold between the different housing strata in our housing system. We will examine the matrix of mean first passage times, i.e., the mean number of steps that will elapse before a vacancy starting its career in state i will arrive for the first time in state j. Mean first passage times provide a measure of a particular kind of contiguity -- one based on interchange probabilities rather than distance. Thus they may be viewed as indices of aspatial ... (interstrata) ... distance (Rogers, 1966: 454). 27 As before, let Q_= q11 q12 ........... q1k q21 q22 .......... q2k qkl qk2 .......: qkk R = €110 (120' ko and hm = (‘101 (102 qu) Construct a matrix 28 qll (112 00.0.0000. Qlk qZI q22. ......... . q2k qkl qkz 0.0.0.0... qkk WV j ww qu qoz .......... qu Using the terminology and notation of Kemeny and Snell (1960: Ch.4), let the fundamental matrix of a regular Markov chain be z_--- (1 - <3 - M‘l where 5 = [aij] _ (n) _ n 5 ‘ lim 3 ' lim 3— n+oo n+0!) (3.5) Then, the matrix of mean first passage times where [1]. lm u (3.6) a square matrix with each element equal to 1; .gdg is a diagonal 29 matrix formed by setting the off-diagonal elements of §_equal to 0 and 2 = [an] is a diagonal matrix formed by setting the off-diagonal elements of.Q d1j=0 ia‘j and setting the diagonal elements d, = . JJ 8 In the case of an independent trials process, a process at equilibrium, M_is simply 1 E: If the process is at equilibrium then P=A_ 1It should be noted that A has the form 30 and §.= (l- (t-A)>"l=f1=i; £1= (rats-spa reduces to E b @dgm = £1.) ; D becomes lo u a :l... H u I ... and M = 1 . Again, to explicate by means of a simple example. Consider the closed system with two states. Assume also that the system is at equilibrium so that 31 The probability then of an object from state 1 going to state 2 in one step is equal to p. (1) _ p12 ‘ P ‘ The probability (2) _ p12 qp is the probability of an object's going from state 1 to state 2 in 2 steps, i.e., staying in state 1 for 1 step and then moving to state 2. In general, the probability of an object's staying in state 1 for (n—l) steps and then moving to state 2 for the first time in the nth step is n n-l P12( ) = q P ' The mean of n, the mean first passage time m = 2 nqn p = . (3.63) The use of first mean passage times provides us with a measure of structural distance that allows us to consider all possible vacancy flows through the housing system from state i to state 3, and also allows us to account for asymmetrical interstrata 32 distances, i.e., m.. # m... i] 31 3.4: Testing the Model Even though it is not clear that there are any statistical tests appropriate to processes of absorbing Markov chains such as are represented by Q_(White, 1970:31n), the structure of the model lends itself to a relatively simple and straightforward examination of the fit of the model to the data. Once the qik are estimated, equation (3.1) yields a probability distribution of chain lengths which can be compared with the observed distribution. Other derivative statistics, such as j(t), A_and M(t) can also be compared with their observed counterparts. That the single pool of data provides a valid test of the fit of the model is clear. The same sample of chains can yield both the observed length distribution and after decomoposition into constituent moves, the transition probability estimates (White, 1970: 33). Because the predicted chain lengths are derived from the transition 33 matrix Q_= [higj’ and there is no way for the qik to be inferred from the observed distribution of chain lengths, the test is a valid one. 4. ESTIMATION OF THE PARAMETERS OF THE VACANCY CHAIN MODEL 4.1: The Study Area The contiguous cities of Lansing and East Lansing, Michigan were selected as the housing system for which a vacancy chain model of residential mobility was to be constructed. The vacancy moves which yielded the estimators of the qij of the transition matrix were derived (in the manner discussed below) from the LansinggCity Directory published by R. L. Polk and Company. Two Q_matrices were estimated, one for the period centered on 1969-1970, and the second based on vacancy moves of 1964-1965. City directories spanning the period 1961 to 1972 were required for the estimation of these parameters. The occupancy states through which these vacancies move were classified according to selected housing characteristics reported in 0.8. Bureau of the Census, Census of Housing:» 1970 Block Statistics Final Report HC(3) - 125 Lansing, Mich. Urbanized Area, (U-S- Government Printing Office: Washington, D.C., 1971). Only the cities of Lansing and East Lansing, Michigan, were included in the housing system under consideration, even though the Lansing urbanized area contains several smaller towns (Dimondale, Haslett, Holt, and Okemos, to name a few) as well as other large, lightly-populated areas. However, the residents of these places have 34 35 not been included in the Lansing City Directogy until 1971, with the publication of the Lansing Suburban Directory_(R. L. Polk and Co.). Consequently, because information regarding the mobility of residents to, from, and within these areas was not available, the areas were not classified into occupancy states. The area comprising the housing system umder consideration is shown in Map l. 4.2: Classification of Occupancy States Attempts to define and classify sub-areas of the city were the nineteenth-century precursors of one line of research in the field of Human Ecology (Levin and Lindesmith, 1961). The process is not a new one, but little agreement exists as to what criteria are necessary or even adequate for this process, though sophisticated techniques exist for manipulating, examining and measuring the vari- ables that are selected (see, for example, Berry and Marble (eds.), 1968). Since Burgess' concentric-ring model of the city, several different sets of criteria have been posited which would allow one to classify the urban area into some number of meaningful sub-areas, i.e. a set of sub-areas which is indicative of social structure, in that the particular classification selected has behavioral con- sequences (Beshers, 1962:88). The accepted criteria of classification range from a broad set of indices of social rank, urbanization, and segregation (Shevky and Bell, 1955) to support of some cash rent or price measure as the sole criterion of housing level (Hua, 1972:122). 36 In attempting to apply the Shevky-Bell Social Area Analysis to some Australian data, Jones found that the three dimen- sions social rank, urbanization (type of housing and household com- position), and segregation were not necessary. Almost as much predictive accuracy could be obtained with only two components -- a combined measure of socioeconomic status and ethnicity, and a measure of household composition (1968:438). In fact, in the housing field, where "housing conditions tend with few exceptions to correlate highly with all indices of socioeconomic status" (Michelson, 1970:18), one would expect the interchangeability of indices to hold: If we have a reasonable collection of indicator items then for most purposes it does not matter which subset we use to form our index (classificatory instrument). (Lazarsfeld, 1959:60). The universe of items from which our classification scheme was chosen was determined by published census block data (U.S. Bureau of the Census, 1971), and, as Beshers has stated: We can only study the distributions of those characteristics that the census chose to gather information on and tabulate; we must rely on the census definitions for the characteristics.... (1962:90). Consequently, the criteria finally chosen to classify urban sub-areas which could adequately stand for the occupancy states 1 of a Markov process were two: a measure combining the average value 1That this problem is not limited to the question of residential mobility is seen in White (1970:132ff). The reader will also find a good general discussion of the difficulties involved in state classification. 37 of housing and the average contract rent, and a measure of tenure status -- the proportion of housing owner-occupied. Average value of housing is the arithmetic mean of: the respondents' estimate of how much the property (house and lot) would sell for if it were for sale. Value data are limited to owner occupied one-family houses on less than ten acres (U.S. Bureau of the Census, l97l:viii). Average contract rent is the arithmetic mean of: the monthly rental agreed to, or contracted for, regardless of any furnishings, utilities, or services that may be included. Contract rent data exclude one-family homes on a place of ten acres or more (U.S. Bureau of the Census, 1971:viii). A housing unit is "owner occupied" if the owner or co-owner lives in the unit, even if it is mortgaged or not fully paid for. A co-operative or condominium unit is "owner occupied only if the owner or co-owner lives in it. All other occupied units are classified as "renter occupied" including units rented for cash rent and those occupied without payment of cash rent (U.S. Bureau of the Census, 1971:viii). Both of these measures are accepted as standard in defining housing sub-areas. The most relevant classificatory variables are price, tenure, size, and location. Any one or any combination of these variables defines the housing sector(s) of a housing system (Hua, 1973:4). Additionally, tenure status is seen as strongly influencing mobility. Renters are almost universally found to be more mobile than homeowners (Cave, 1969; see also, Moore, 1969; Brown and Holmes, 1971; McAllister, et a1., 1971; Hua, 1972; Pickvance, 1973). This 38 relationship cannot be attributed solely to the monetary investment in an owned home, for it persists today when long-term (20 to 30 year) amortization mortgages have all but eliminated the financial distinction between owner and renter. The reduced mobility of owners seems to involve social and psychological factors as well as the legal and financial impediments of home ownership. Moore (1969:23-24) finds the strongest correlation (r = -.72) between housing turnover rate and any other variable is the Private Home Index, the proportion of dwellings which are in single private units. In Lansing and East Lansing, we find the correlation between proportion of dwelling units owner occupied and proportion of single-family dwellings to be quite high (r = .944), so we would expect the relationship between tenure status and mobility in our sample to be quite high. A non-metric measure of association, a Guttman-Lingoes smallest space analysis (Lingoes, 1973) based on zero-order correlation coefficients, shows that tenure status is associated with the variety of life-cycle and life-style variables considered important to mobility (Rogers, 1966: Moore, 1969; Pickvance, 1973) as well as to the determination of social areas within the city (Jones, 1968:41, see also Shevky and Bell, 1955; Beshers, 1962). Selected aspects of the smallest space analysis are presented in Figure 4.1. The distinction between the two variables used is indicated by the distance separating rent and value from the cluster of items surrounding the tenure variable. 39 A Percent of Units in Percent of , Black population one—unit population over 62 structures under 18 owner occupied One-person Families units households ith female Black 0 heads renters 0 Owner occupied Black units . Units with population 1.01 or more Total housing units persons per room \ Renter occupied Units with roomers, lodgers units Total population Units in structures of 10 or more units A Vector 2 I Part of the 3 dimensional solution. Coefficient of Alienation - .091. Percent of population in group quarter;\\. Average rent ' 3‘ Average value \ Vector 1 Figure 4.1. Smallest space analysis: characteristics of housing units and population 40 The correlation between average value and the proportion of owner occupied dwellings was .295, while the correlation between average rent and proportion owner occupied was -.052. Some recent empirical findings and a theoretical con- sideration entered into the decision not to use race -- proportion of the population Black -- as a criterion of state classification. Although Blacks may appear to be more mobile than Whites, "the slightly greater mobility of Blacks is a result of their tenure status, rather than of racial, demographic, socio-economic, or attitudinal differences" (McAllister, et a1., 1971:452). As Table 4.1 shows, Black home owners are only slightly more mobile than White home owners and Black renters are actually less mobile than White renters.. Table 4.1 Mobility by Race and Tenure (McAllister, et alg, 1971:451) Moving Behavior Owners Renters Black White Black White Stayed (Z) 76.8 79.8 35.9 26.6 Moved (Z) 23.2 20.2 64.1 73.4 Total 100.0 100.0 100.0 100.0 n 82 738 181 488 Further evidence for this view is provided by a dummy variable regression analysis of mobility behavior showing a B-weight 41 of -.014 associated with the variable: "Race: non-White" (Morrison, 1971:175). These findings, and the fact that in areas where the proportion of non-Whites is increasing, Black in—migrants tend to be of the same SES level as the Whites who are moving out, pointed to the conclusion that race would not be particularly useful in the determination of urban sub-areas as they would affect mobility behavior. On theoretical grounds it was felt that the character- istic "race of occupant" was much less appropriate to the entity "vacancy" than were the characteristics "value of dwelling" (occupied by vacancy) and tenure status of dwelling. Consequently, "race" was not included as a variable of ocdupancy-state classification. 4.3: Sub-Areas of LansingrEast Lansing The variables "average value and rent" and "tenure status" resulted in the establishment of four sub-areas of Lansing-East Lansing. These four occupancy states are shown in Map 2, and the states are labelled simply 1) Low 2) Lower Middle 3) Upper Middle 4) High. While the establishment of four housing sub-areas is somewhat arbitrary, there are no rigid procedures for the establishment of such states (see again, White, 1970:132ff). Some have used simply 42 deciles of housing value (Hua, 1972), but it was felt in this case that the 100 transition probabilities generated by that procedure would be far too great a number for stable parameter estimation. Furthermore, the shapes of the distributions of average value and average rent in Figures 4.2 and 4.3 respectively, render the use of a measure such as deciles, quartiles, or even stanines inappropriate. 43 Table 4.2 Average Value of Housing Average Value in 1,000's of Number of Relative Frequency Cumulative Frequency Dollars Blocks (adjusted percent) (adjusted percent) 7 l 0.1 0.1 8 6 0.3 0.4 9 9 0.5 0.8 10 46 2.4 3.3 11 82 4.3 7.6 12 121 6.4 14.0 13 172 9.1 23.1 14 163 8.6 31.7 15 132 7.0 38.6 16 135 7.1 45.8 17 109 5.8 51.5 18 84 4.4 55.9 19 71 3.7 59.7 20 74 3.9 63.6 21 85 4.5 68.1 22 51 2.7 70.8 23 50 2.6 73.4 24 49 2.6 76.0 25 39 2.1 78.0 26 28 1.5 79.5 27 23 1.2 80.7 28 40 2.1 82.8 29 36 1.9 84.7 30 33 1.7 86.5 31 29 1.5 88.0 32 28 1.5 89.5 33 26 1.4 90.9 34 21 1.1 92.0 35 15 0.8 92.8 36 12 0.6 93.4 37 12 0.6 94.0 38 10 0.5 94.6 39 11 0.6 95.1 40 9 0.5 95.6 40+ 83 4.4 100.0 Total 1895 100.0 100.0 No Value Given 413 Total 2308 1N0 value is given for blocks which contain no owner occupied houses or for blocks which contain so few that to release average value information would actually be releasing specific information, in vio- lation of the confidentiality guaranteed by the census. 44 Number of Blocks ”160 -140 "120 ’ -100 n. 80 ‘ -’60 ~40 Figure 4.2. Average value of housing in thousands of dollars (Source: U.S. Bureau of the Census,l97l) 45 Table 4.3 Average Mbnthly Rent Average Rent Number of Relative Frequency Cumulative Frequency (Dollars) Blocks (adjusted percent) (adjusted percent) 50 3 0.3 0.3 60 4 0.4 0.7 70 12 1.2 2.0 80 35 3.6 5.5 90 74 7.6 13.1 100 157 16.1 29.3 110 194 19.9 49.2 120 144 14.8 64.0 130 94 9.7 73.6 140 52 5.3 79.0 150 44 4.5 83.5 160 50 5.1 88.6 170 34 3.5 92.1 180 20 2.1 94.1 190 21 2.2 96.3 200 11 1.1 97.4 210 9 0.9 98.4 220 3 0.3 98.7 230 5 0.5 99.2 240 2 0.2 99.4 250 2 0.2 99.6 260 1 0.1 99.7 270 2 0.2 99.9 280 0 0.0 99.9 290 0 0.0 99.9 300 0 0.0 99.9 310 l 0.1 100.0 Total 974 100.0 100.0 No Value Given1 1334 Total 2308 1; 1No value is given for blocks which contain no renter occupied dwellings or for blocks which contain so few that to release average rental information would actually be releasing specific information, in violation of the confidentiality guaranteed by the census. 46 Number of blocks. r-180 _.160 ..140 n 120 ..100 ..80 -'60 n.40 F'ZO Rent 50 100 150 200 250 300 Figure 4.3. Average monthly rent in dollars (Source: U.S. Bureau of the Census, 1971) 47 Five levels of "average value" were determined, which were combined with four levels of "average contract rent," to yield the combined measure of rent and value. rent and value level = average value level + average rent level 2 Table 4.4 Average Value Levels Number of Relative Cumulative Level Blocks Frequency Frequency 1. $ 6,500-14,499 600 31.7 31.7 2. 14,500-19,499 531 28.0 59.7 3. 19,500-27.499 399 21.0 80.7 4. 27,500-40,499 282 14.9 95.6 5. Over 40,500 83 4.4 100.0 Total 1895 100.0 100.0 No value given 413 Total 2308 100.0 100.0 Table 4.5 Average Rent Levels Number of Relative Cumulative Level Blocks Frequency Frequency 1. $ 45- 94.99 128 13.1 13.1 2. 95-124.99 495 50.8 64.9 3. 125-149.99 190 19.5 84.4 4. Over 155 161 16.5 . 100.0 Total 974 100.0 100.0 No value given 1334 Total 2308 100.0 100.0 48 The distribution of the tenure status indicator "proportion of housing owner occupied," was no more amenable to equal-sized divisions, as can be seen in Table 4.6 and Figure 4.4. Table 4.6 Proportion of Housing Units Owner Occupied Percentage Number of Relative Cumulative Owner Occupied Blocks Frequency Frequency 0- 4.9 82 3.8 3.8 5- 9.9 42 1.9 5.7 10-14.9 42 1.9 7.6 15-19.9 51 2.3 9.9 20-24.9 49 2.3 12.2 25-29.9 43 2.0 14.2 30-34.9 43 2.0 16.2 35-39.9 52 2.4 18.6 40-44.9 38 1.7 20.3 45-49.9 69 3.2 23.5 50-54.9 65 3.0 26.5 55-59.9 89 4.1 30.6 60-64.9 92 4.2 34.8 65-69.9 136 6.3 41.1 70—74.9 119 5.5 46.6 75-79.9 161 7.5 54.1 80-84.9 208 9.6 63.7 85-89.9 222 10.2 73.9 90-94.9 264 12.2 86.1 95-99.9 75' 3.5 89.6 100.0 226 10.4 100.0 Total 2173 100.0 100.0 No value given1 135 Total 2308 1N0 value is given for blocks where the release of general occupancy information would actually be releasing specific information, in violation of the confidentiality guaranteed by the census. Number of blocks ..300 p— 250 .— 200 -150 _100 49 Percentage of housing owner occupied Figure 4.4. Proportion of housing owner occupied (Source: Bureau of the Census, 1971) U. S. 50 Table 4.7 Proportion Owner Occupied Levels Number of Relative Cumulative Level Blocks Frequency Frequency 1. 0-39.9Z 409 18.8 18.8 2. 40-64.9 353 16.2 35.0 3. 65-84.9 624 28.7 63.7 4. 85-99.9 561 25.8 89.5 5. 100.0 226 10.4 100.0 Total 2173 100.0 100.0 No value given 135 Total 2308 100.0 100.0 Five levels of "proportion of dwelling units owner occupied" were determined (Table 4.7) and combined with the above four levels of average rent and value to yield four housing sub-areas. sub-area = rent and value level + proportion owner occupied level 2 This final combination resulted in the delimitation of the four housing sub-areas shown in Map 2. Certain "smoothing" pro— cedures were followed in assigning all the blocks to a sub-area: 1) On the assumption that these characteristics are not randomly distributed in space, but rather that "sub-areas near one another have similar characteristics" (Hawkes, 1972:1219), blocks with 51 no information available were assigned to the stratum of the majority of their contiguous neighbors.1 2) Blocks for which there was information re- garding only one or two of the criterion variables were assigned to strata according to the information available. 3) "Small islands" were not permitted. Groups of less than four continguous blocks of any stratum i, surrounded by stratum j, or strata j's, were converted to the appropriate stratum j by means of rule 1 above, applied recursively. Characteristics of the sub—areas so defined are shown in Table 4.8 and graphically represented in Figures 4.5 through 4.10. As can be seen in Table 4.8, the rank-ordering of each of the variables used in assigning city blocks to sub-areas is preserved, but it appears that some distinctions were based more on one variable than on the others. Sub-areas 1 and 2 differ not so much in terms of average value of housing ($14,400 and $15,300 respec- tively) or average rent ($110 and $121), but mostly in terms of 1Blocks which shared a common border were considered contiguous. Blocks which shared a common point were not. 52 Table 4.8 Characteristics of Housing Sub-Areas Blocks Housing Units Population Area Number Z Number % Number % 1. Low 537 26.5 18,362 31.1 45,139 26.1 2. Lower Middle 864 42.6 24,535 41.6 73,783 42.7 3. Upper Middle 361 17.8 10,125 17.2 33,205 19.2 4. High __g§1. 13.2 5,990 10.2 20,695 12.0 Total 2,029 100.1 59,012 100.1 172,822 100.0 Mean Value Mean Proportion Area of Housing, Mean Rent of Housing Owned 1. Low $14,400 $110 .40 2. Lower Middle 15,300 121 .75 3. Upper Middle 23,700 169 .83 4. High 34,500 _l§4 .i§§ Average $19,100 $121 .69 53 Number of blocks Strata 1. ---- 2 3. 4. Value ask-cu .I 60 55 50 45 Average value of housing by stratum Figure 4.5. 54 Percent -20 Strata l. 2. 30 4 o O ...... n .0 .0 O O. A I I0 sun“ ”at x on .s a 55 50 45 40 35 30 25 20 15 10 Average value of housing by stratum (in percent) Figure 4.6. 55 Rent ‘ o I o I I a ‘ o I . t o a u a . . u u I o I C t n o I I S n o 00; 1 2 3 4 o n S; I on on on on .- I I. o n x.— n. O I... .- oo - o n o n .. s \ I so. a I noon-o... ---§ u 000000 0-- a .----‘0 I Number of blocks 300 250 200 150 100 50 Average monthly rent by stratum Figure 4.7. 56 Percent -»25 Strata -'20 1. 2. 3. 4 Rent ..A 300 250 200 150 100 50 Average monthly rent by stratum (in percent) Figure 4.8. 57 blocks Number of Percentage of housing owner occupied 0 m ll.....""""""”””””””” 00000 “ s....O...‘ IIIIIIIIIIOOOIOI ‘ O .. . ' .. I ......O.. ‘ 9 . ....... . C. ......I. I o 0 .......... ‘.” ' 8 o I ...... -- - l. 1 m oooooooo -9 a. 000.000.000.003. 0 ... ‘--- .... . ' 0 9 0 fl 6 0" ‘ so. 0. ‘0 "" 00000 00" so 0 --- ... A. ". .. “ 5 ‘k‘ 000 o ‘ "‘ ... O "' .. ~ . Ito ‘ O I ‘ O O A. I I O o I I I I a . u u ,1 o . a . . u I . AU 0 I I I 3 u ‘ u I 9‘ 00 s I S I I I o “ u o 1 2 3 4 o.‘ u 2 l . I ’0 W 0 I o 1.. ~ A& ‘ p . “III I- . D b 0 q 0 0 m w 8 6 4 2. l 1 Proportion of housing owner occupied by stratum Figure 4.9. 58 Percent -.50 , 3 -'£ Strata 55 -4O 1..---. g; 2.Iooooooo SE 3,— 5 E 4....IIIIIII 5 E - 30 : 1 +20 Percentage of housing )4_ owner occupied ' 10 20 30 40 50 6O 70 80 90 100 Figure 4.10. Proportion of housing owner occupied by stratum (in percent) 59 owner occupancy. Almost twice as great a proportion of homes per block are owned in sub-area 2, as are owned in sub-area 1 (75 percent compared to 40 percent). The differences in average value and rents are due, not surprisingly, to a greater proportion of homes in area 2 being valued between $16,000 and $20,000, and a greater proportion of rents between $130 and $160 per month. There is a sizeable difference in average values and rents between sub-areas 3 and 2, however. On the average, homes in area 3 are valued at 55 percent more than homes in area 2 and rents are 40 percent more. Homes in sub-area 4 are valued at 45 percent more than homes in area 3, while rents in area 4 are less than 10 percent higher. Owner occupancy, on the average, is only 5 per- cent greater in area 4 than in area 3 and 13 percent greater than in area 2. So it would appear than area 1 can be distinguished largely in terms of its lower proportion of homes owned -- almost 70 percent of all blocks have less than 50 percent of dwellings owner occupied, while the distinction between areas 2,3 and 4 is based largely on economic grounds. To a large extent this is true, but we cannot ignore the facts shown vividly in Figure 4.10: areas 3 and 4 both have more than 50 percent of their blocks with 90 per- cent owner occupancy -4 in fact over 50 percent of blocks in area 4 are 95 percent owner occupied -- while area 2 has only 20 percent of blocks with 90 percent owner occupancy. The high average for area 2 60 is obtained with a high proportion of blocks of over 70 percent owner occupancy, and a very small proportion of blocks with less than 50 percent owner occupancy. The differences in these distri- butions should not be overlooked because of the similarities in their averages. 4.4: Estimation of the Transition Probabilities The transition probabilities, the qij were estimated by the use of changes in the occupancy of dwelling units as reported in the Lansing City Directory (R.L. Polk and Co.). Sampling techniques were used which are felt to yield an unbiased sample of vacancy chains, even though the city directory is not a listing of vacancy chains, but addresses, and the population of vacancy chains from.which our sample was drawn is "hidden." Occupancy as reported in the city directory is generally considered to be an accurate "complete enumeration of the entire adult population of the community" (Goldstein, 1954:170). The data collection methods are quite thorough, involving, when necessary, two or more house calls, return postcards, telephone calls, and telephone calls to neighbors and reported places of work in order to identify occupants.l The accuracy of city directories is quite high, and there is substantial agreement among users that they 1Personal communication with R. L. Polk Detroit Production Manager, Mr. Head, who estimates the accuracy of the City Directory to be about 952 at the time of publication. 61 are reliable and useful sources of data (Albig, 1936; Goldstein, 1954; Ianni, 1957; Brown and Holmes, 1971). Comparisons of city directory and census counts of the adult male population of Norristown, Pennsylvania, show that "from 1930 on there is virtually 100 percent coverage by the directories" (Goldstein, 1954:172). In no year was the discrepancy more than 2.3 percent (Goldstein, 1954: 174). In the present case, an estimate of housing units in Lansing and East Lansing derived from the 1970 City Directogy yields a total of 56,160 addresses. The census count of year-round housing units for 1970 is 56,494 (U.S. Bureau of the Census, 1971:l). The dif- ference is slightly greater than one-half percent. The directory lists by street address the names of household members over the age of 18 and indicates their occupation, place of work, marital status and tenure status. In addition, an alphabetical list of residents, with addresses and the above- mentioned other data, is provided. In essence, the city directory is two directories, and it is this dual listing which permits us to infer vacancies and vacancy changes from changes in successive occupancies. The technique is as follows: Consider that our sample of addresses consists of every nth address in the 1970 Lansing City Directory, and that the knth address is 123 First Street. The occupant of 123 First Street in 1970 is given as John Jones. This is then compared with the 62 information reported in the 1969 directory. If we find that the occupant in 1969 is also John Jones, then no change and, consequently, no vacancy movement is said to have occurred. However, if we find that the 1969 occupant of 123 First Street is someone other than John Jones, say, Peter Smith, then we conclude that a change of occupancy has occurred and that a vacancy must have passed through 123 First Street, and we proceed to trace out the complete vacancy chain. First we find the 1970 address for Peter Smith, and see that it is 456 Second Street. The 1969 occupant of 456 Second Street is given as Jane Johnston. Jane Johnston, however, is no longer listed in the 1970 directory and we infer that she left the housing system, and further, that this particular vacancy entered the system by her departure. We then must complete the chain by tracing it out the other way, by finding John Jones' 1969 address. Let it be 789 First Street. The 1970 occupant of 789 First Street is Jack WOng, who is not listed in the 1969 directory, so we infer that he has just entered the housing system and it is by his entry that the vacancy leaves the system. In this example, we have a vacancy entering the system at 456 Secdnd Street (when Jane Johnston leaves the system), and moving then to 123 First Street, then to 789 First Street, and finally leaving the system from 789 First Street when Wong moves in. The process is represented pictorially (and perhaps more clearly) in Figure 4.11. .63 A Johnston 456 Second St. Smith 123 First St. 789 First St. D Wong Figure 4.11. A residential vacancy chain Each vacancy move (A, B, C, and D in Figure 4.11) is then assigned to a stratum of origin and destination according to its addresses of origin and destination, with the "outside" labelled stratum 0. Information was also collected concerning the type of dwelling: apartment, house, townhouse; the marital status of the occupant: married, single, widow; tenure: owned, rented; and 64 occupational status of movers. Occupational status was classified according to the Rice "modified white-collar, blue-collar code" (Robinson et a1., 1969:342ff): 1) High status white-collar 2) Low status white-collar 3) High status blue-collar 4) Low status blue-collar 5) Farm occupations with the addition of the codes 6) Student 7) Retired 8) Military. The selection process described above insures that the address which is the beginning of a chain need not be the address initially sampled for the chain to be included in our sample of vacancy chains. If we find any address in a vacancy chain we must find the entire chain. A 1/7 systematic sample of the addresses listed in the 1970 Lansigg City Directogy was selected and the same propOrtion of addresses was selected from the 1965 City Directogy. If we can reasonably expect the preportion of addresses involved in moves, r, to be .2 s r s .3 (Taeuber, et a1., 1961:826n) this would yield an estimated 1600 to 2400 vacancies in 1969-1970 and 1400 to 2100 65 vacancies in 1964-1965. Both sample sizes are large enough to permit the accurate estimation of overall vacancy rates (r i .01) as well as the q1k (Cochran, 1963: Ch. 3, 4, 5, 5A). The fact that our sample of vacancy chains was not derived from a sampling frame of vacancy chains, but from a sampling frame of vacancies, complicates only slightly the estimation of the transition probabilities. If the sample were drawn from a population of vacancy chains, such that all chains had an equal probability of being selected then the q1k could be estimated by aik(t) q1k = s (4.4.1a) a (t) Where aik(t) is the number of observed vacancy moves in a cohort from state i to state k. tack(‘) f a (4.4.2a) 1 a (t) i=1 03 could then be used as the estimator of the fk(t),the proportion of vacancy creations-in stratum k. However, because we are initially sampling vacancies, the chains do not have an equal probability of being selected. In fact, the probability of a chain of length 7 being selected is 66 exactly 7 times as great as the probability of a chain of length 1, since there are 7 times as many addresses in the chain of length y. To compensate for this bias in the estimators of transition proba- bilities (4.4.la and 4.4.23), each vacancy move is assigned a weight equal to -%-, where y is the length of the chain the move appears in, so that the contribution of any particular move to both the numerator and the denominator of (4.4.1a) or (4.4.2a) is now -$- . Because of a small amount of non-response, the weighting factor -%%-tends to underestimate the contribution of longer chains. Of the 8022 addresses sampled in 1970, 505 were listed as having submitted no return. This yields a response rate _ 505 a 8022 distributed then the probability of a chain of length 7 being of 1 .937. If "no-returns" are assumed to be independently completed is .937Y. Thus, longer chains are more likely to be lost from the sample due to a failure to-respond at any one of y addresses, and a second weighting factor was introduced. Each vacancy move was therefore assigned a second weight, --fL17- , where y is the length 1937 of the chain in which the move is found. The weighted contribution of any move from state i to state j was then 1 c = --—-—-- 13 y(.937Y) If we let w1k(t) be the sum of the observed weighted contributions of vacancy moves from state i to state k at time t, 67 the resulting estimator of qik is wik(t) dik(t) - S (4.4.lb) w (t) hZO 1“ and the estimator of the proportion of vacancy chain creations is . w (t) £k(c) = 0k . (4.4.2b) i . woj(t) j 1 Using these estimators, two vacancy models of residential mobility in Lansing - East Lansing were established, one based on qik for 1969-1970, and one based on the estimators for 1964-1965. These two models are discussed in Chapter 5. 5. VACANCY CHAINS IN LANSING-EAST LANSING 5.1: The General Model, 1969-1970 The sampling procedures described in Chapter 4 yielded a sample of 1397 complete vacancy chains distributed as shown in Table 5.1. The average length of chains was 1.378 moves with a stan- dard deviation of .655 moves. The median length was 1.211 moves and the longest chains were 5 moves. Table 5.1 Distribution of Unweighted Vacancy Chains by Leggth, 1969-1970 Length Number of Proportion Cumulative Chaigg, . Proportion l 983 .704 .704 2 318 .228 .931 3 81 .058 .989 4 12 ' .009 .998 5 3 .002 1.000 Total 1397 1.001 Of these 1397 vacancy chains, 439 entered the system in stratum 1, while 560 arrived in stratum 2 and 289 and 109 chains made their entries in strata 3 and 4 respectively. The vector of vacancy arrivals by stratum, then V(t) - (439 560 289 109) while the vector of vacancy departures, Qfit) - (503 550 260 84). As the two vectors show, slightly more vacancies left the system via stratum 1 than arrived there. 68 69 - Table 5.2 Unweighted Vacancy Chain Arrivals and Departures by Stratum,,l969-1970 Stratum Number of Proportion Number of Proportion of Proportion Arrivals of Arrivals Departures Departures of Total Housing 1. Low 439 .314 503 .360 .31 2. Lower 560 .401 550 .394 .42 Middle 3. Upper 289 .207 260 .186 .17 Middle 4. High 109 .078 84 ‘ .060 .10 Total 1397 1.000 1397 1.000 1.00 It should be noted that the preportion of vacancy arrivals and de- partures by stratum coincides very closely with the distribution of dwelling units within the strata. No stratum is undergoing a dis- prOportionate inflow or outflow of vacancies, although stratum 4 appears slightly under-active. - Before these raw data can profitably be interpreted, the system of sampling weights must be taken into account. Because the sampling frame was not a frame of vacancy chains, chains of length y were 7 times as likely to be included in the sample as were chains of length l. A second weighting factor was required to compensate for the loss of chains in the sample due to non-response. If the probability of collecting occupant information at any address is .937, then the proba- bility of completing a chain of length 7 is .937Y. Consequently, a weighting factor, a function of chain length V(Y) - 1 y(.937)Y was assigned to each vacancy move. Applying this weighting factor to our observed distribution of chain lengths yields the distribution of 70 weighted chain lengths shown in Table 5.3. The frequency distribu— tions of weighted and unweighted chain lengths are shown in Figure 5.1 below. Percent Percent Unweighted " Weighted Figure 5.1. Unweighted and weighted chain length distributions, 1969-1970 71 Table 5.3 Distribution of weighted Vacancy Chains by Length, 1969-1970 Length Observed Weighting Weighted Proportion Cumulative Number of Factor Number of Proportion Chains Chains (A) (B) (A x B) 1 983 -§%7—-=1.O672 1049.093 .829 .829 2 318 1 2= .5695 181.104 .142 .971 2(.937) 3 81 1 3= .4052 32.820 .024 .995 3(.937) 4 12 -—l——4= .3242 3.892 .004 .999 4(.937) 5 3 1 5= .2769 .831 .001 1.000 5(.937) Total 1397 1267.739 1.000 The mean length of weighted chains is 1.206 moves with a standard deviation of .490 moves. The median length is 1.104. The pattern of weighted vacancy arrivals by stratum is similar to that of the raw chains, 9 gm - (411.488 512.064 258.704 85.424), as is the pattern of vacancy departures: Qfit) - (441.564 508.141 244.896 73.072). Though.the patterns are similar, Table 5.4 shows slightly smaller differences between the number of arrivals and departures 72 Table 5.4 weighted Vacanc14Chain Arrivals and Departures by Stratum, 1969-1970 Stratum. Number of Proportion of Number of Proportion of Arrivals Arrivals Departures Departures 1. Low 411.488 .325 441.564 .348 2. Lower 512.064 .404 508.141 .401 Middle 3. Upper 258.704 .204 244.896 .193 Middle 4. High 85.424 .067 73.072 .058 Total 1267.6801 1.000 1267.6801 1.000 1Discrepancy with Table 5.3 due to raunding. within strata for the weighted chains, and also that area 4 is under- represented both in vacancy arrivals and departures. The implications of this latter fact will be discussed below. Disaggregation of the 1397 raw vacancy chains into their 1925 constituent moves and weighting them as described in section 4.4 yielded the estimators of the matrix 9. - [91k] and the vector 2. ' [910] - The 1397 entrance moves were weighted and used to estimate the prepor- tion of vacancy arrivals by stratum, i (t). The vectors and matrices describing the 1969-1970 model are presented below: .§(t) - (411.488 512.064 258.704 85.424) 51:) - (.325 .404 .204 .067) 73 g_ = .095 .032 .012 .004 .057 .084 .019 .006 .050 .062 .050 .020 .049 .098 .080 .083 g_ a .857 .835 .817 .690 (1:9)‘1. 1.109 .040 .015 .005 .070 1.096 .023 .008 .065 .077 1.057 .024 .073 .126 .095 1.093 = .857 .121 .019 .003 0 .835 .138 .023 .004 .001 .817 .150 .027 .005 .001 .690 .246 .052 .010 .002 1.1:“ The main tests of the fit of the vacancy model are the predictions of chain length distributions for the model taken as a whole, and predicted distributions across strata (White, 1970:33ff.). The prediction of vacancy chain lengths for the entire model is given by yogi and the vector of mean chain lengths by stratum of arrival is given by fl -1 A I 2 P - I- 1 o 30 2 _ P111 (_ 9.) _ < > The average length of vacancy chains for the entire model is given simply by j(t) - _f_(t) A . (3.3) we can also compare the expected number of predicted vacancy moves ever made from a stratum in our system with the observed 74 total moves made from strata by examining gm - :(t) (1-99’1 (3.4) and Qfit), the vector of observed vacancy moves by stratum. We can also compare the total number of moves predicted, M(t), with the total number of observed moves, 0=(t). The first test, the comparison of actual and predicted chain length distributions for the model is presented in Table 5.5. Although there are no statistical tests which can preperly be applied to this data, an index of dissimilarity was computed to facilitate the comparison of the distributions. The index of dissimilarity is l A” '2' 2 “P11 " 1’21)| where P11 and P21 are the proportions of cases found in state i in distributions 1 and 2 respectively. "Put as simply as possible, the Index of Dissimilarity indicates the minimum proportion in one or the other population which would have to change categories in order for the two distributions to be identical" (Matras, 1973: 157). The particular utility of the index of dissimilarity is that it allows the comparison of entire distributions and not simply central tendencies. The congruence between observed and predicted chain length distributions seen in Table 5.5 is exceptional! There are virtually no differences between the two, and in fact, the predicted mean chain length, j(t), is identical with the observed mean chain length, i'- 1.206. The index of dissimilarity, A = .003. 75‘ Table 5.5 Comparison of Observed and Predicted Chain Length Distributions for the Complete Model, 1969-1970 Chain Observed Predicted Observed Predicted Length Frequency Frequency Proportion Proportion F t P f P (_( )1) (_(t)_i) 1 1049.093 1050.522 .828 .829 2 181.104 180.275 .143 .142 3 32.820 31.023 .026 .024 4 3.892 5.430 .003 .004 5 .831 .171 .001 .000 Total 1267.739 1267.421 1.001 .999 3? = 1.206 j(t) - 1.206 A - .003 This extreme goodness-of-fit is reflected in the other tests of the model. A comparison of_L, the predicted average length of chain by stratum of origin, and L, the observed average length of chain by stratum, is shown in Table 5.6. In only one of the strata, stratum 4, does the discrepancy between predicted and observed values exceed one-half-percent, and this occurs in the stratum with the fewest vacancy creations (85.423). Table 5.6 Comparison of Observed (L) and Predicted (L) Mean Chain Lengths by;Stratum of Origin, 1969-1970 Stratum Mean Chain Length Difference as a Observed Predicted Pro ortion of L (a) (A) ( LEM/L1) 1. Low 1.165 1.169 .003 2. Lower Middle 1.203 1.198 .004 3. Upper Middle 1.228 1.223 .004 4. High 1.447 1.388 .006 76 The last test, a comparison of the predicted total number of moves ever made from a stratum 14m - yr) <1-9>‘1 with the observed total number of moves made from strata, Qflt), also indicates an excellent fit. Mk) 8 (411.488 512.064 258.704 85.424) 1.109 .040 .015 .005 .070 1.096 .023 .008 .065 .077 1.057 .024 .073 .126 .095 1.093 _th) = (515.421 608.635 299.515 105.731 and the observed number of moves by stratum of origin Qfit) - (515.135 608.573 299.813 105.891). Obviously, the total numbers of observed and predicted vacancy moves coincide. M(t) - 1529.438 and _Q_(t) = 1529.412 |g(c) - mm = ‘|1529.412 - 1529.438l = .00002 gm 1529.412 In addition to the excellent fit of the model, one other finding should be noted here - the extremely high proportion of vacancy chains that leave the Lansing-East Lansing area in their first move. This preportion can be computed by fjt) p_= (.325 .404 .204 .067) .857 .835 .817 .690 _f_(t) p - .829 Over 80 percent of vacancies leave the system on their first move, so that very little vacancy movement is generated by the entrance of vacancies into the system. This low level is indicated by the value 77 of j(t) = 1.206 -- each vacancy entrance generates, on the average, 1.206 vacancy moves. This is a result of the extreme shortness of vacancy chains originating in the lower strata, and the small numbers of chains originating in the strata with the highest average lengths or multipliers. Sub-area 4, has the highest multiplier, 14 = 1.388, but only 6.7 percent of vacancies enter the system at this point. 5.2: The General Model, 1964-1965 The frequency distribution of unweighted vacancy chains sampled in 1964-1965 is shown in Table 5.7. The 805 chains had an average length of 1.256 moves, with a standard deviation of .805 moves. The median length was 1.257 moves and the longest chains were of length 6. The arrival and departure distributions of the unweighted chains are shown in Table 5.8. , Table 5.7 Distribution of Unweighted Vacancy Chains by Length, 1964-1965 Length : Number of Proportion Cumulative ' Chains Proportion 1 532 .661 .661 2 189 .235 .896 3 59 ..073 .969 4 19 .024 .993 5 4 .005 .998 6 2 .002 1.000 Total 805 1.000 78 Table 5.8 unweighted Vacancy Chain Arrivals and Departures by Stratum, 1964-1965 Stratum Number of Proportion Number of Preportion Proportion Arrivals of Arrivals Departures of of Total Departures Housing 1. Low 264 .328 310 .385 .31 2. Lower 328 .407 322 .400 .42 Middle 3. Upper 147 .183 129 .160 .17 Middle 4. High 66 .082 44 .055 .10 Total 805 1.000 805 1.000 1.00 It should be noted that, just as in 1969-1970, the prOpor- tions of vacancy arrivals by strata follow very closely the interstratum distribution of housing units. The higher level of vacancy departures from stratum 1 in 1964-1965 seems to indicate a higher level of vacancy activity in this time period than in 1969-1970. Weighting procedures identical to those used in 1969-1970 were applied to the observed distribution of chain lengths for 1964— 1965 to yield the distribution of weighted chain lengths presented in Table 5.9. The mean length of weighted chains is 1.256 with a standard deviation of .60 moves. The median length is 1.123 moves. A compari- son of the distributions of unweighted and weighted chains is shown graphically in Figure 5.2. The relative distributions of the unweighted and weighted chains are not unlike their counterparts for 1969-1970, although the difference between them is somewhat greater for 1964-1965. This is due to the smaller porportion of unweighted chains of length l in 1964-1965. 79 Table 5.9 Distribution of weighted Vacangy Chains by Length,il964-l965 Length Observed Weighting Weighted Pr0portion Cumulative Number of Factor Number of Proportion Chains Chains (A) (B) (A x B) 1 532 -§%7-=1.0672 567.769 .803 .803 2 189 1 2- .5695 107.637 .152 .955 ' 2(.937) ‘ 3 59 1 3= .4052 23.906 .034 .989 3(.937) 4 19 1 4= .3242 6.162 .009 .998 4(.937) 5 4 1 5: .2769 1.108 .002 .999 5(.937) 6 2 1 6- .2463 .493 .001 1.000 6(.937) Total 805 707.075 1.001 The distribution of weighted vacancy chain arrivals and departures is much the same as in the unweighted case, except that the discrepancy between rates of arrival and departure within strata has been reduced, though more vacancies still exit the system through stratum.l than enter it there, and stratum 4 is still under-represented both.in vacancy arrivals and departures. 100 1 90- 80- 80 PerCent 100 1 90- 80 ..... ...... 70.1 70 "' ...... ............. ........... ......... '. CI II‘.."'.ID ... ...“n .‘ I. . ... " 0" .. . . . II . _ I. ..,_,I. -. .- ...... .'-".,.. 4....- . ...-h ”...... I I o '1‘. .. ' III I.' o. . ‘ .u."" . ...'.'-'.I .III... I . .... . “up ... I .‘l. l I ~... ‘- I. .. .... ... ..- . . . . ‘ . e ' . a ... . | 3 a... 0. .... .... ' e N...- . .0 '0 p 3. p Unweighted weighted Figure 5.2.. Unweighted and weighted chain length distributions, 1964-1965 81 Table 5.10 WeLghted Vacancy_Chain Arrivals and Departures by Stratum; 1964-1965 Stratum Number of Proportion Number of Proportion Arrivals of Arrivals Departures of Departures 1. Low 236.625 .335 259.159 .367 2. Lower- 290.537 .411 286.725 .406 Middle 3. Upper- 127.230 .180 118.366 .167 Middle 4. High 52.672 .074 42.812 .061 Total 707.0641 1.000 707.0621 1.001 1Discrepancy with Table 5.9 due to rounding. The 805 chains were disaggregated and the 1195 vacancy moves and the 805 entrance moves were weighted and used to estimate the transition probabilities of the 1964-1965 vacancy chain model. The model has the following characteristics: (236.625 290.537 127.230 52.672) _1_"_(t) (.335 .411 .180 .075) £(t) g_= .138 .039 .011 0 .073 .108 .019 .003 .064 .094 .050 .009 .041 .064 .107 .052 .813 .797 .783 .737 (;:9)'1 '1.165 .053 .141 0 .098 1.128 .024 .004 .089 .116 1.057 .010 .066 .092 .121 1.056 = (.813 .151 .029 .006 .001 4.1:“ .797 .162 .032 .006 .001 .783 .172 .035 .007 .001 .737 .206 .046 .009 .002 0000 82 The comparison of predicted chain length distributions with observed is shown in Table 5.11. Table 5.11 Comparison of Observed and Predicted Chain Length Distributions for the Complete Mode1,,l964-l965 Chain Observed Predicted Observed Predicted Length Frequency Frequency Proportion Proportion F t P f t P (_( )_j) (_( )1) 1 567.769 562.374 .803 .796 2 107.637 115.531 .152 .164 3 23.906 23.035 .034 .033 4 6.162 4.528 .009 .006 5 1.108 .760 .002 .001 6 .493 O .001 0 Total 707.075 706.228 1.001 1.000 3? - 1.256 j(t) = 1.256 A = .011 The goodness of fit indicated in Table 5.11 above is sup- ported by the comparison of L_and L shown in Table 5.12. The largest discrepancy between observed and predicted chain lengths by stratum of arrival is only .016, and this discrepancy occurs in stratum 4, where the number of vacancy creations is smallest -- only 52.722 vacancy chains, 7.4 percent of the total, began their careers in stratum 4. Table 5.12 Comparison of Observed (L) and Predicted (A) Mean Chain Lengths by Stratum of Origin, 1964—1965 Stratum Mean Chain Length. Difference as a Proportion Observed Predicted of L (ILi-Ail /Li) 1. Low 1.232 1.215 .014 2. Lower 1.253 1.254 .001 Middle 3. Upper 1.272 1.276 .003 Middle 4. High 1.335 1.377 .016 83 Finally, the predicted total number of moves ever made from a stratum, M(t) was compared with the observed distribution of moves made from strata, Qfit). -l 110:) = yr) (1:3) M(t) = (318.954 359.725 151.096 58.098) While git) = (318.939 359.660 151.100 58.118). The predicted total number of moves M(t) is simply the sum of the M1(t), so that M(t) 887.873 and 0(t) 887.818, This represents a discrepancy of 887.873-887.818 = .0001. 887.818 The fit of the 1964-1965 model to the data is quite good. In general terms, the model is very similar to the 1970 model. The proportion of vacancy chains leaving the system in their first move is very high: fjt) p_= .796 and the number of moves generated by entering vacancies is corres- pondingly low. The multiplier j(t) - 1.256. Although the multiplier varies among strata with a low of_1 - 1.232 and a high value of l 14 - 1.335, the stratum with the largest multiplier has the lowest pro- portion of vacancy arrivals (f4 - .074), so that the overall multip- lier is only minimally influenced by it. 84 5.3: The Basic Models -- Discussion In the most general terms, the two models described above, the complete vacancy chain models for 1969—1970 (to be called simply the 1970 model) and 1964-1965 (the 1965 model) are very similar. If one compares the two predicted chain length distributions, as in Table 5.13, there is little to choose between them. In fact the index of dissimilarity is equal to .026. Table 5.13 Predicted Chain Length Distributions, 1969-1970 and 1964-1965 1969-1970 1964-1965 Length Number Proportion Number Proportion 1 1050.522 .829 567.769 .803 2 180.275 .142 107.637 .152 3 31.023 .024 23.906 .034 4 5.430 .004 6.162 .009 5 .170 .001 ' 1.108 .002 6 O O .493 .001 Total 1267.420 1.000 707.075 1.001 j(t) = 1.206 j(t) = 1.256 A - .026 The mean chain length for 1964-1965 is marginally longer, but only two and one-half percent of the population would have to change categories for the two distributions to be identical. Both models are characterized by an extremely high degree of fit, indicated by all tests. The comparisons of predicted and observed chain length distributions, mean chain lengths by stratum, and moves made from strata are all very close. In none of these 85 comparisons does the discrepancy between observed and predicted values exceed 3 percent. These findings lead one to conclude that the housing vacancy transfers occurring here can be modelled adequately by a first-order Markov chain. The problems of state classification and non-independence of transitions which seem to confound models of people as the population of movers (cf. Chapter 2) seem not to have arisen. In fact the fit is so good that one begins to suspect that state classification plays very little part in the determination of the model other than in terms of housing policy and that the more salient criterion for goodness-of-fit in the first-order Markov chain is the independence of the vacancy moves. If the moves are being made independently, then, the state classification is important only in terms of substantive theory. Mathematically, the classification is arbitrary. I That the vacancy moves are independent can perhaps be supported by the following oversimplification of the 1970 model: Let the housing system consist of only one state. The model can then be characterized as a series of Bernoulli trials with the probability of leaving the system on any trial p = 1 = 1 = .829. j(t) 1.206 The probability of remaining in the system on any trial is then simply q 3 l-p = .1710 The distribution of chain lengths is then given by the function P(J) = 91719 - 8' t: ('7! 7. ‘II..‘.II."1 86 The number of chains expected at each length -1 N(3) = N91 9 where N is the total number of vacancy arrivals, 1267.739. Table 5.14 Comparison of Observed and Expected Chain Length Distributions, Using N(j) = gi-lp, 1970 Predicted Observed Chain Length Num 3 Pro oition Number Preportion (Nq p) (q -p) 1 1050.951 .829 1049.093 .828 2 179.713 .142 181.104 .143 3 30.731 .024 32.820 .026 4 5.255 .004 3.892 .003 5 .899 .001 .831 .001 6 .154 ** O O 7 or more .036 ** 0 0 1267.739 1.000 1267.739 1.001 A = .003 ** Less than .001 The findings for the 1965 model are identical. With ...11. - __le__ 9 j(t) 1.256 - .796 and N - 707.075, the Bernoulli trials model yields a predicted distri- bution of chain lengths which fits the data extremely closely. The excellent fit of the simplified model serves not only to lend credence to the assumption of independent mobility of vacancies, but also to illuminate the extreme goodness—of—fit of the articulated model. To a certain extent the fit is an artifact of the low average chain length, or the high exit probabilities of vacancy chains in Lansing-East Lansing. The model is not tautological, to be sure, but 87 Table 5.15 Comparison of Observed and Expected Chain Length Distributions, Usigg:N(j) = N j-lp, 1965 Predicted Observed Chain Length Number Proportion Number Proportion -l -1 (N91 10 (<11 2) 1 562.832 .796 567.769 .803 2 114.818 .162 107.637 .152 3 23.423 .033 23.906 .034 4 4.778 .007 6.162 .009 5 .975 .001 1.108 .002 6 .199 ** .493 .001 7 or more .050 ** O 0 Total 707.075 .999 707.075 1.001 A = .011 **Less than .001. an average chain length of 1.206 moves requires the vast majority of chains to be of length one. Since there is no such thing as a chain of length zero, there is no other way for this low average to be achieved. This, then, severely constrains the number of chains avail- able for assignment, if you will, to the other four or five chain length categories. The result is that reasonable estimation of only the initial exit probabilities ensures that proportional discrepancies between observed and predicted numbers of chains of length 2 or more must be small. This domination of the model by the exit probabilities is seen in the accuracy of the predictions made assuming simply a Bernoulli process of vacancy movement. The fit of the model does not guarantee that vacancies do move independently of each other, but it certainly makes that assumption 88 much more attractive in this formulation than in traditional models of people moving through housing systems. What we are modelling is a housing system. The questions that we can ask and the answers that we get are different than the questions and answers we confront when we model flows of people through the system. The g_matrix, the matrix of vacancy transitions between states of the housing system most closely resembles elements found in models of people. However, even 2 differs from the more familiar transition matrix of movers. For example, in 1970 .095 .032 .012 .004 970 = .057 .084 .019 .006 .050 .062 .050 .020 .049 .098 .080 .083 The qii represent vacancy moves within the strata, and not the sum of vacancy moves within a stratum and the vacancy "not—moves" within a stratum. The diagonal entries in the simple models of people would represent both intrajstratum.moves and "not moves." In fact, there is no such thing as a vacancy which doesn't move. The model is only concerned with entities that move. In__Q7O above, for example, q11 - .095 is interpreted as .095 of vacancies arriving in stratum 1 move to another address in stratum 1 in the next time period; and .084 of the vacancies arriving in stratum 2 move to another address in stratum 2. .138 .039 .011 0 .965 .073 .108 .019 .003 .064 .094 .050 .009 .040 .064 .107 .052 -g65 is very similar to Q70. The proportion of vacancies which 89 remain within the system are low in both of them. The row sums of the matrices are on the order of .2. In addition, the flow of vacancy transfers within the system follows the same pattern: the proportion of intra-stratum moves is inversely proportional to the stratum of origin. In both time periods, in strata 1 and 2 over 50 percent of vacancy moves which remain within the housing system end in the stratum of origin. In fact in 1965 accounts for over 90 percent 9 qll of the within-system moves originating in stratum 1. In strata 3 and 4, however, infra-stratum moves account for less than 30 percent of the within-system moves originating in these strata. This indicates a flow, or rather, a trickle, of housing vacancies from areas of higher to lower housing level. The reverse flow of housing, not surprisingly, doesn't exist. But this downflow of housing from strata 3 and 4 to strata 1 and 2, mirroring the improving accomodation of previous residents of the lower housing strata, is almost overwhelmed by the flood of vacancies leaving the Lansing-East Lansing area at all strata. This is perhaps more clearly seen in an examination of the (L:Q)-l matrix, the fundamental (Kemeny and Snell, 1960) or the multiplier matrix (Kristof, 1965; White, 1970; 1971; Hua, 1973) of our vacancy chain model. (11g);3 = 1.109 .040 .015 .005 .070 1.096 .023 .008 .065 .077 1.057 .024 .073 .126 .095 1.093. 90 (1:9)31 - 1.165 .053 .014 0 .098 1.128 .024 .004 .089 .116 1.057 .010 .066 .092 .121 1.056 (L:Q)_l is called the multiplier matrix because if (L:Q)-1 = [nij1 , then nij is the total number of moves generated in state 1 by a vacancy chain which began its career in state i. In other words, the vector of the row sums of (L:Q)-1 is equal to A, the vector of mean chain lengths by stratum of origin. Inspection of the (I-Q);1 immediately reveals the low level of vacancy movement within the system. Inasmuch as the average chain lengths by strata are on the order of 1.2 to 1.25 and at least 1 move is accounted for by the stratum of origin (the minimum being 1.056 moves for n44 in 1965, with the maximum being 1.165 in n11 in 1965), there is very little room for moves to be generated elsewhere in the system. What little internal movement there is seems to be in a downward direction. If it is not stretching a point, the matrices may be said to be lower triangular. Many more vacancy moves are generated in strata below than above the stratum of origin and, in that sense, vacancies seem to flow downward. The highest stratum appears to generate the greatest prOportional level of vacancy movement and stimulates it in all 3 lower strata. Stratum 3 generates vacancy movements in both stratum l and stratum 2. It should be noted that in 1970, stratum 2 receives the greatest benefit of vacancies entering the system in stratum 4. 91 But the most striking finding has been the extremely high rate at which vacancy chains leave the cities of Lansing and East Lansing. In part this is due to the definition of the housing system used in this study. Because the urban area has been specified to include only the cities of Lansing and East Lansing, there is no way to estimate what proportion of chains leaving the system arrive in places such as Haslett, DeWitt, Okemos, or Holt, or other places comprising the Greater Lansing area. This omission is unfortunate in some senses, but the figures arrived at are important in terms of municipal housing policy. If the cities of Lansing and East Lansing are involved in housing programs, the benefits of these programs must be considered as they affect residents and taxpayers of the two cities. How much benefit do they derive if each unit created only generates 1.2 vacancy moves? How many households improve their situa- tion as a result of such programs, and at what cost; or do the programs mainly allow outsiders to come into the city, with little housing relief for previous residents? In other words, what is the rate of new household formation from within the cities, compared to rates of house— hold in-migration. Unless the former rates are high, the major bene- ficiaries of such housing programs, at least in the short run, would be non-residents of Lansing-East Lansing. It was also considered that the shortness of chains might be due to the large number of students in the Lansing-East Lansing popula- tion-— well over 10 percent of the population was comprised of Michigan 92 State University students. To examine this possibility, an analysis of the vacancy flows in chains containing no students was conducted and the results are presented below. For 1970: Efit) - (330.409 439.014 181.157 63.331) 9_ = .098 .026 .006 .001 .056 .089 .015 .004 .052 .081 .037 .025 .061 .098 .096 .061 .867 2 = 0835 .804 .638 (1;g)’1 = 1.111 .033 .007 .002 .070 1.102 .018 .005 .068 .098 1.044 .029 .087 .128 .110 1.069 Ej = .869 .112 .016 .002 0 .835 .138 ..022 .004 .001 .804 .160 .030 .005 .001 .683 .254 .051 .009 .002 .5 = (1.153 1.196 1.239 1.392) j(t) = 1.202 No students were found in 80 percent of vacancy chains occurring in 1969-1970. The average length of the 1014 chains was 1.202 moves - slightly shorter, in fact, than the average of 1.206 for all chains -- and L_for the model containing no students is (1.153 1.196 1.239 1.392) compared to _L = (1.169 1.197 1.223 1.387) for the full model. Clearly, then, in 1970, the presence of students in vacancy 93 chains does not shorten the chains. The students' higher level of mobility is shown by their overrepresentation in chains, and while the I9_matrices differ between the populations with and without students, the differences appearing in the values of j(t) and L_must be considered negligible. The findings for 1965 are similar: §(t) = (209.722 268.014 114.761 49.320) 9_ = .139 .044 .121 0 .067 .112 .018 .002 .068 .100 .053 .009 .038 .053 .115 .050 p_ = .805 .801 .770 .745 (;:Q)'1 = 1.167 .060 .016 0 .090 1.133 .023 .002 .094 .125 1.060 .011 .063 .080 .131 1.054 gj = .805 .156 .031 .006 .001 0 .801 .159 .032 .006 .001 0 .770 .182 .038 .007 .002 0 .745 .198 .045 .009 .002 0 There are 641.831 chains containing no students, and this comprises 90 percent of all chains. Chains without students have an average length, j(t)= 1.260 moves while the full model has an average of j(t)= 1.256. For chains containing no students L_= (1.243 1.248 1.290 1.327) while, overall L_= (1.232 1.253 1.272 1.335). 94 Although the Q_and (270)-1 matrices show small differences between the two models, the shortness of chains cannot be accounted for by the presence of students in the population. We must consider,however, the fact that the sample was drawn from city directories, from data collected at one-year intervals. The number of chains being discussed actually constitutes the minimum number of chains which could represent the data. Because we examine an address at only 2 points in time a year apart, we may miss several occupants of an addreSS, i.e., several chains passing through an address. For example, if we sample address 1 at timel and find person A there, and then at time we find person B at addressl, we infer 2 simply that A moved out and B moved in. We do not consider the fact that persons C,D,E,F, and C may have successively occupied addressl in the time interval between A's departure and B's arrival. Despite these factors which may tend to shorten the chain lengths in our sample, the average length of chains appears to fit in with a general pattern. It seems axiomatic that "large cities provide more migration opportunities than small cities, and a large observa- tion unit allows people to move farther without crossing a boundary (leaving the system)" (Simmons, 1968:627).1 An examination of multi- pliers for various areas in Table 5.16 seems to support this reasoning. A regression equation of the form j(t) - a + b log P to predict the size of the multiplier yields the solution 1Parentheses mine. 95 3 (t) = —1.300 + .577 log P with r2 = .36. However, if we delete the data for Cleveland, 1938-40, from the analysis, because of the great temporal and economic dis- crepancies involved, our predictor becomes 3 (t) = -2.992 + .773 163 P 2 = .99. Our computed value of 3 (t) = 1.208 fits in quite well and r with this model. Bearing in mind the differences in sample size and data collection methods in the 4 studies, this finding indicates a remarkable consistency in the size of multipliers as a function of population size. Table 5.16 Multipliers and Housing Area Populations Area Multiplier Population log(Population) Source U.S.A., 1966 3.5 195,857,000 8.292 Lansing 2L El, 1969 New York, 1960 2.4 10,695,000 7.029 Kristof, 1969 Clydeside, 1970 1.7 2,000,000 6.301 Watson, 1974 Cleveland, 1939 3 7 1,195,000 6.077 Hua, 1972 Lansing, 1970 1.2 179,000 5.253 Though the methods used may underestimate the number of vacancy chains in the system, there are advantages to the method which may counterbalance these disadvantages, especially in light of the accuracy of the model and its ability to describe the housing system. The advantages lie primarily in the realm of data collection, and pro- vide some relief in terms of time and money costs. The data are readily available -- city directories are public information, as is the census. They are both relatively non-reactive data sources, and are largely free of contamination by researchers, i.e. the data are not 96 altered by their use. Additionally, census data and city directory data is available for hundreds of cities in North America over a time- span of several decades, allowing comparative research to be conducted. The use of city directory data also forces one to deal with human behavior. we only discuss moves as indicated by a change of occupancy at an address. Subjects are not required to respond verbally regarding their mobility behavior or their attitudes towards mobility. The study is one of mobility behavior not verbal behavior. The use of the vacancy chain model forces one to focus on system properties -- the system of housing Opportunities within which we move -- and not indi- viduals moving within a system considered as a given. 5.4: The Housingjgub-Systems, 1969-1970 Further exploration of the housing system requires that the system be decomposed into several sub-systems. In this way, we can examine the differences between vacancy chains beginning in houses and chains beginning in apartments; chains starting in newly constructed units and chains starting in existing units, etc. The 1397 unweighted chains can be classified according to the two dimensions of type of dwelling unit and age of unit. Age was considered to consist only of the two categories "new unit" and "existing unit." This classification is presented in Table 5.17. We can also consider a second type of submarket in light of the effects of tenure and housing type on mobility. This is the class of pure chains -- chains which contain only houses or only apart- ments. It should be stated immediately that in 1969-1970 pure chains 97 Comprise 91.8 percent of all unweighted vacancy chains. Their distri- bution is seen in Table 5.18 Table 5.17 Distribution of Unweighted Chains by First Unit in Chain, 1969-1970 3&8? Type of Dwelliog; House , Apartment Other , Total # of (Z of 7 Chains Total) 149 (10.7» 86 ( 6.2) 10 ( .1) 245 (17.5) NEW (Z of (X of (17.1) (60.8) (16.9) (35.1) (58.8) (4.1) (17.5)(100.0) Column) Row) 721 (51.6) 424 (30.4) 7 ( .1) 1152 (82.5) EXISTING (82.9) (62.6) (83.1) (36.8) (41.2) ( .1) (82.5)(100.0) 870 (62.3) 510 (36.5) 17* ( .1) 1397 (100.0) TOTAL (100.0) (62.3)F100.0) (36.5) (100.0) ( .1) (100.0)(100.0) The 17 "Other" dwellings consist of 5 old trailers and 10 new and 2 old townhouse units. . Table 5.18 Distribution of Unweighted Pure Chains by First Unit in Chain,gl969-197O Age Type of Dwelligg ,House ,Apartment. Total # of (Z of Chains Total) 135 (10.5) 77 ' ( 6.0) 212 (16.5) NEW (2 of (2 of (16.6) (63.7) (16.4) (36.3) (16.5)(100.0) clump) Row), - p 678 (52.8) 393 (30.6) 1071 (83.5) EXISTING (83.4) (63.3) (83.6) (36.7) . (83.5)(100.0) TOTAL 813 (63.4) 470 (36.6) 1283 (100.0) ,1 (100.0) (63.4) (100.0) (36.6) (100.0)(100.0) Applying the weighting procedures described previously to correct for sampling biases, the distribution of weighted chains in 98 Table 5.19 is generated. Table 5.19 shows that of the approximately 1268 chains occurring in our sample in 1969-1970, about one-sixth ori- ginate in new units and the remainder in existing housing. Chains began in houses 60 percent of the time (single unit structures accounted for 57 percent of housing units in Lansing-East Lansing in 1970 (U.S. Bureau of the Census, l970:1)). The one-sixth of chains beginning in new housing was proportionately distributed in apartments and houses. Table 5.19 Distributiog_of Weighted Vacancy Chains by First Unit in Chain, 1969-1970 Ago Type of Dwelling House Apartment Other Total I of (2 of Chains Total) 123.139 (9.7) 78.361 (6.2) 9.432 (.7) 210.932 (16.6) NEW (2 of (2 of . Column) Row) (16.0) (58.4) (16.3) (37.1) (59.9) (4.5) (16.6) (100.0) 646.742 (51.0) 403.755 (31.8) 6.311 (.5) 1056.808 (83.4) EXISTING . ' (84.0) (61.2) (83.7) (38.2) (40.1) (.6) (83.4) (100.0) 769.882 (60.7) 482.115 (38.0) 15.742*(1.2) 1267.739 (100.0) TOTAL (100.0) (60.7) (100.0) (38.0) (100.0) (1.2) (100.0) (100.0) fir *Conoiots of approximately 5 old trailers, and 9 new and 2 old townhouses. The distribution of pure chains in Table 5.20 is virtually identical, with pure chains accounting for 952 of all chains! Of chains beginning in houses, 93.1 percent are pure, as are 94.6 percent of chains beginning in new houses. Pure chains also account for 95.8 percent of chains beginning in an apartment and 94.7 percent of chains beginning in new apartments. The implications of this high degree of separation between house and apartment sub—systems will be discussed in section 5.6 and 99 Chapter 6 below, but it should be remembered that this separation is a function of the shortness of chains. If 80 percent of chains leave the system in their first move, then only 20 percent of them can possi- bly contain both types of dwellings. Table 5.20 Distribution of Weighted Pure Chains by First Unit in Chain, 1969-1970 Age Type of Dwelling House .Apartment Total # of '(Z of Chains Total) 116.478 (9.7) 74.219 (6.2) 190.697 (15.8) NEW (2 of (z of (15.7) (61.1) (16.1) (38.9) (15.8) (100) Column) Row) EXISTING 625.794 (52.0) 387.414 (32.2) 1013.208(84.2) (84.3) (61.8) (83.9) (38.2) (84.2) (100) TOTAL 742.272 (61.7) 1203.905(100) 461.633 (38.3) (100) (61.7) (100) (38.3) (100) (100) Perhaps the easiest way to inspect the sub-system.models is to present a summary table of the eight mixed sub-system models and then another summary table of the six pure sub-system models. These tables show observed and predicted chain length distributions as well as i; the observed mean chain length, j(t), the predicted mean chain length, and A, the index of dissimilarity. This information for the mixed sub- systems is presented in Table 5.21. Pure systems are described in Table 5.22, where the same data for the complete model is also shown for comparative purposes. Clearly the sub-system models fit the data. In no case does A exceed .015 and in every case the predicted mean chain length matches the observed mean length exactly. 100 Table 5.21 Observed and Predicted Chain Length Distributions, Mixed Sub-Systems,41262-l970 Z of ‘_ Chain Length All X All Chains l“ 2 3 4 5 Total Chains,](€) A Observed 1049.093 181.104 32.820 3.892 .83] 1267.739 100 1.206 Predicted 1050.522 180.275 31.023 5.430 .171 1267.421 1.206 .003 Type of First Unit: New 164.354 35.870 8.104 2.595 0 210.932 1.284 163.751 37.038 7.953 1.680 .235 210.658 1.284 .007 Existing 884.739 145.225 24.716 1.297 .831 1056.808 1.191 887.265 142.234 22.579 4.001 .684 1056.764 1.191 .005 1267.740 100.0 House 615.795 126.431 23.906 2.919 .831 769.882 1.242 619.386 121.756 23.033 4.555 .976 769.706 1.242 .007 Apartment 419.424 53.534 8.509 .649 0 482.115 1.150 419.433 54.476 7.280 1.030 .033 482.252 1.150 .003 - 125l.997* 98.8* New House 89.648 26.197 5.673 1.622 0 123.139 9.7 1.344 90.836 24.530 5.990 1.359 .320 123.035 1.344 .015 New 66.169 9.112 2.431 .649 0 78.361 6.2 1.203 Apartment 65.487 10.411 1.910 .401 .090 78.300 1.203 201.5** 15.9** Existing 526.147 100.233 18.233 1.297 .831 646.742 51.0 1.223 House 528.739 96.992 17.571 3.058 .670 647.030 1.223 .006 Existing 353.255 44.422 6.078 “0 .0 403.755 31.8 1.140 Apartment 354.343 42.926 5.529 .728 .024 403.551 1.140 .005 1050.497**§2.9*** Total 1251.997* 98.8* * 15.742 chains start in units other than a house or apartment. ** 9.432 chains start in units other than a house or apartment. *** 6.311 chains start in units other than a house or apartment. 101 Table 5.22 Observed and Predicted Chain Length Distributions, Pure Chains, 1969-1970 2 of ~ All __ Chain Length Chains .X Houses only 1 2 3 4 5 Total A(1267.739)j(t) A Observed 615.795 108.776 15.802 1.622 .277 742.272 58.6 1.197"’E Predicted 618.986 103.459 16.446 2.579 .189 741.659 1.197 .007 Apartmenta- 419.424 38.157 4.052 0 0 461.633 36.4 1.100 Only . 419.834 37.916 3.532 .493 0 461.775 1.100 .002 1203.905 95.0 New First Unit: Houses‘ 89.648 22.211 3.647‘ .973 '0 116.478 9.2 1.278 90.415 20.861 4.261 .781 .137 116.456 1.278 .013 Apartments 66.169 6.834 5.267 O 0 74.219 5.9 1.125 190.697 15.0 Existing First Unit: Houses 526.147 86.565 12.156 .649 .277 652.794 49.4 1.182 528.964 82.288 12.416 1.861 .100 625.629 1.182 .007 Apartments 353.255 31.323 2.836 ~0 0 387.414 30.6 1.095 353.822 30.609 2.806 .323 0 387.561 1.095 .002 1013.208 79.9 Total 1203.905 95.0 The agreement between the model and the data is also seen in the comparison of A, the vector of predicted mean chain length by stratum*with L, the vector of observed mean chain lengths by stratum, shown in Table 5.23. 102 Table 5.23 L and L_, Mixed Sub-Systems, 1969-1970 Type of First Unit Stratum of Origin 1 2 3 4 New Unit L_ 1.154 1.174 1.349 1.528 A 1.222 1.201 1.340 1.491 Existing Unit L_ 1.150 1.210 1.200 1.367 A 1.162 1.197 1.200 1.320 House L 1.179 1.228 1.358 1.560 E; 14182 1.222 1.332 1.515 Apartment L 1.129 1.138 1.140 1.212 5; 1.142 1.155 1.136 1.225 New House L 1.185 1.170 1.591 1.666 'Z_ 1.168 1.232 1.472 1.614 New Apartment L 1.250 1.184 1.150 1.125 EL 1.320 1.183 1.173 1.205 Existing House L 1.178 1.227 1.130 1.435 7: ‘ 1.183 1.219 1.298 1.409 Existing Apartment L 1.117 1.165 1.131 1.250 7; 1.123 1.148 1.132 1.238 The predicted chain lengths agree very well with the observed distribution. The largest discrepancy occurs in chains begin- L1'11 . .056. L1 sider that only 11.8 percent Of chains originate in this cell. ning in new apartments where However, we must con- Substantively, it should be noted in Table 5.21 that j(t) ranges from a low of 1.14 for chains beginning in existing apartments to a high of 1.344 for chains beginning in new houses. Chains beginning in houses are longer than those beginning in apartments and chains beginning 103 in new units are longer than chains beginning in old ones. Not surprisingly this ordering of chain lengths is preserved for pure chains as well. Although the range is still narrow, the longest chains (j(t) - 1.278) in new houses and the shortest (j(t) = 1.095) originate in existing apartments. 5.5: The Housing Sub-Systems, 1964-1965 The distribution of unweighted vacancy chains for 1964-1965 according to the age and type of first unit is shown in Table 5.24 and the distribution of pure chains in Table 5.25. The marked separation of housing subsystems in 1965 is immediately evident. Over 95 percent of all chains are pure chains -- 86.8 percent of chains beginning in apartments and 97.7 percent of chains beginning in houses. Table 5.24 Distribution of Unweighted Chains by First Unit in Chain. 1964-1965 Ago Type of Dwelling House Apartment Other Total I of (I of Chains Total) 183 (22.7) 35 (4.3) 4 (0.5) 222 (27.6) ugw (Z of (Z a: (26.3) (82.4) (33.0) (15.8) (100.0) (1.8) (27.6)(100.0) \ 512 (63.6) 71 (8.8) O (0) 583 (72.4) EXISTING (73.7) (87.8) (67.0) (12.2) (0) (0) (72.4)(100.0) 695 (86.3) 106 (13.2) 4* (0.5) 805 (100.0) TOTAL (100.0) (86.3) (100.0) (13.2) (100.0) (0.5) (100.0)(100.0) *4 new townhouses. Weighting the chains results in the distribution of chains shown in Table 5.26: 85 percent of chains begin in houses and almost 30 percent of chains begin in new units - almost twice as many as in 1970. 104 Table 5.25 Distribution of Unweighted Pure Chains by First Unit in Chain, 1964—1965 Age Type of Dwelling, House Apartment Total # of (z of 179 (23.2) 31 (4.0) 210 (27.3) Chains Total) NEW (2 of (Z of (26.4) (35.2) (34.1)(14.3) (27.3)(100.0) Column) Row) 500 (64.9) 60 (7.8) 560 (72.7) EXISTING (73.6) (89.3) (65.9)(10.7) (72.7)(100.0) 679 (88.2) 91 (11.8) 770 (100.0) 'TOTmfl. (100.0) (88.2) (lO0.0)(ll.8) (100.0)(100.0) Table 5.26 Distribution of weighted Vacancy Chains by First Unit in Chain, 1964-1965 15521 Type of Dwelling House Apartment Other Total 7 of‘ (I of Chains Total) 158.399 (22.4) 33.957 (4.8) 4.269 (0.6) 196.626 (27.8) NEW (1 of (I of (26.4) (80.6) (33.3) (17.3) (100.0) (2.2) (27.8) (100.0) ..Qslassl. Row) 442.387 (62.6) 68.063 '(9.6) 0 (0) 510.449 (72.2) EXISTING (73.6) (86.7) (66.7) (13.3) (0) (0) (72.2) (100.0) 600.786 (85.0) 102.020 (14.4) 4.629* (0.6) 707.075 (100.0) TOTAL , (100.0) (85.0) (100.0) (14.4) (100.0) (0.6) (100.0) (100.0) *4.269 new townhouses. 105 New chain starts are not evenly distributed in 1964-1965 as they were in 1970, either. New units account for one—quarter of the chains originating in houses, and for onenthird of the chains beginning in apartments. The pure chains in Table 5.27 account for 97.2 percent of weighted chains -- 92.1 percent of chains beginning in apartments and 98.8 percent of chains beginning in houses. Table 5.27 Distribution of Weighted Pure Chains by First Unit in Chain, 1964-1965 Age Type of Dwelling House Apartment Total # of (Z of ' Chain, Total) 156-612 (22.8) 31.925 (4.6) 188.537 (27.4) NEW (2 of (Z of Column) Row) (26.4) (83.1) (34.0) (16.9) (27.4) (100.0) 436.945 (63.6) 62.043 (9.0) 498.988 (72.6) EXISTING (73.6) (87.6) (66.0) (12.4) (72.6) (100.0) 593.557 (86.3) 93.960 (13.7) 687.525 (100.0) TOTAL (100.0) (86.3) (100.0) (13.7) (100.0) (100.0 The distributions of observed and predicted chain lengths by sub-system along with the models' associated measures, i; j(t), and A, are presented in Table 5.28. The longest chains are found beginning in houses of both ages, and the shortest beginning in new apartments. The age of the first unit appears to have little bearing in 1965 on the average length of chains when compared to type of dwelling. bution of pure chains is seen in Table 5.29, where the patterns dis- cussed above are preserved. 106 Table 5.28 The distri- Observed and Predicted Chain Length Distributions, Mixed Sub-Systems, 1964-1965 Z of Chain Length A1 1 '1? All Chains 1 2 3 4 5 6 Total Chains j(t) A Observed 567.769 107.637 23.906 6.162 1.108 .493 707.075 100 1.256 Predicted 562.374 115.531 23.035 4.528 .760 0 706.227 1.256 .011 Type of First Unit: New 162.220 23.350 8.509 2.270 .277 0 196.626 27.8 1.246 157.662 31.368 6.153 1.144 .224 .0 196.551 1.246 .041 Existing 405.550 84.287 15.397 3.892 .831 .493 510.449 72.2 1.260 404.733 83.560 17.200 3.572 .727 0 509.791 1.260 .004 707.075 100.0 House 472.785 97.386 23.501 5.514 1.108 .493 600.786 85.0 1.279 468.476 104.249 21.984 4.725 1.045 .044 600.523 1.279 .011 Apartment 90.715 10.251 .405 .649 O 0 102.020 14.4 1.127 90.840 9.684 1.318 .329 .036 0 102.207 1.127 .009 702.806* 99.4* New House 127.001 21.072 8.104 1.946 ‘.277 0 158.399 22.4 1.279 123.342 27.828 5.834 1.166 .246 .025 158.441 1.279 .043 New 30.950 2.278 .405 .324 .0 0 33.957 4.8 1.120 Apartment 30.404 3.108 .396 .049 0 .0 33.957 1.120 .024 192.356* 27.2* Existing 345.784 76.314 15.397 3.568 .831 .493 442.387 62.6 1.279 House 345.198 76.444 16.328 3.512 .722 .019 442.233 1.279 .002 Existing 59.765 7.973 0 .324 .0 '0 68.063 9.6 1.131 Apartment' 60.526 6.632 .964 .150 .030 ‘0 68.302 1.131 .023 ' 510.450 72.2 Total *4.269 chains start in units other than house or apartment. 702.806* 99.4* Observed end Predicted Chain Length Distributions, Pure Sub-Systems, 1964—1965 107 Table 5.29 2 of All ._ Chains X A Houses Only 1 2 3 5 6 Total (707.075)gj(t) Observed 472.785 93.399 21.880 3.892 1.108 .493 593.557 83.9 1.262 Predicted 469.106 99.483 20.000 4.028 .735 0 593.352 1.262 .011 Apartment 90.715 2.848 .405 .0 0 ‘0 93.968 13.3 1.039 ” 90.475 3.344 .149 o o 0 93.968 1.039 .005 687.525 97.2 New First Unit: House 127.001 19.933 8.104 1.297 .277 ”0 156.612 22.1 1.263 123.518 26.545 5.315 1.027 .181 0 156.586 1.263 .042 Apartment 30.950 .570 .405 0 0 0 31.925 4.5 1.043 30.588 1.267 .039 0 0 -0 31.894 1.043 .022 “ 188.537 26.7 Existing First Unit: House 345.784 73.467 13.776 2.595 .831 .493 436.945 61.8 1.262 345.751 72.595 14.883 2.996 .695 0 436.921 1.262 .003 Apartment 59.765 2.278 0 0 0 .0 62.043 8.8 1.037 59.883 2.006 .115 0 .0 0 62.004 1.037 .004 - 498.988 70.6 Total 687.525 97.2 The fit of the model is extraordinarily good, especially when one considers that the occupancy states were classified according to 1970 data. The index of dissimilarity ranges from a minimum of .002 to a maximum of .043 in the sub-system beginning in new houses. This discrepancy is carried through the new unit sub-system (A=.O41) because houses comprise over 80 percent of all new units. As well, there is a discrepancy of A=.024 in the new apartment sub-system which also contri- butes to the 4.1 percent difference in the observed and predicted dis— tributions of chains beginning in new units. 108 Table 5.30 §_and A“, Mixed Sub-Systems, 1964-1965 Type of First Unit Stratum of Origin 1 2 3 4 New Unit k_ 1.120 1.144 1.277 1.500 __ 1.223 1.192 1.292 1.378 Existing Unit L 1.223 1.290 1.222 1.320 i' 1.235 1.284 1.258 1.293 House £_ 1.209 1.278 1.350 1.477 A_ 1.233 1.276 1.322 1.410 Apartment L 1.278 1.065 1.037 1.000 X' 1.231 1.086 1.079 1.000 New House 2_ 1.200 1.169 1.361 1.600 A_ 1.219 1.221 1.334 1.424 New Apartment 1, 1.112 1.063 1.200 1.000 A_ 1.238 1.081 1.141 1.000 Existing House L 1.204 1.275 1.297 1.421 X' 1.236 1.300 1.312 1.395 Existing Apart- L 1.167 1.067 1.059 1.000 ment i’ 1.231 1.095 1.034 1.000 A comparison of E and 5, observed and predicted mean chain lengths by stratum is given in Table 5.30. Though there are some minor internal discrepancies originating in sub-systems with few chains, the general fit is quite good. 5.6: The Housing;§ubs§ystems -- Discussion The decomposition of the complete models in order to examine selected aspects of the housing system structure, i.e., the 109 nature of the housing submarkets, has yielded some interesting and useful information about both the housing system and the model. In general, the vacancy chain model fit the data very well in 1969-70 and only slightly worse in 1964-65. Perhaps this was due to the classification of occupancy states on the basis of 1970 data, and their inaccuracy when applied to the city of 1965. In any event, the largest index of dissimilarity for chain length distributions in 1964-65 was less than .05. Overall predictions tended to be more accurate than were predictions of more specific values. Chain length distributions, mean lengths, j(t), and total numbers of moves were predicted more accurately than were values such as the Ai' Values for the complete model were predicted more accurately than values for specific sub- systems. This pattern is due, in part,to the smaller cell frequencies that the more specific predictions are based on, and, in part, to the fact that in the more general predictions the smaller discrepancies of their more specific components tend to cancel each other out. The most interesting discovery concerning the housing sub-system structure of Lansing-East Lansing is the extreme separation of the house and apartment sub-systems. Only 5 percent of all vacancy chains in 1970 contain both houses and apartments. Only 3 percent do so in 1965. Much of this phenomenon is attributable to the extreme shortness of chains found in this sample. With exit probabilities of .829 in 1970 and .796 in 1965, 83 percent and 80 percent of chains 110 respectively are only one move long, and consequently contain only one dwelling type. To control for the confounding influence of chains of length one on the proportion of pure chains in the sample, a second measure was calculated: pure chains as a prOportion of all chains of length 2 or more. This information for 1970 is presented in Table 5.31 and for 1965 in Table 5.32. Table 5.31 Pure_§hatgs as 3 Proportion of All Chains of Length 2 or More, 1969-1979 Type of First Unit Pure Chains All Chains Proportion .._____... ._ (A) (13} (M B) House 126.477 154.087 .821 Apartment 42.209 62.291 .673 New House 26.830 33.491 .801 Existing House 99.647 120.595 .826 New Apartment 48.050 12.192 .660 Existing Apartment 34.159 50.500 .676 Table 5.32 Pure Chains as a Froportion of A11 Chains of Length 2 or More,,l964-196? Type of First Jni; Pure Chains A11 Chains Proportior - ...... .. 5 (A) (3) (A13) House 120.772 128.001 .944 Apartment 3.253 11.305 .288 New House 29.611 31.398 .943 Existing House 91.161 96.603 .944 New Apartment .975 3.007 324 Existing Apartments 2.278 8.298 .275 w .-— “ cu. — While the elimination of l-move chains from the analvsis improves the situation somewhat in 1970. over four~£ifths of chain. beginning in houses are pure as are two-thirds o‘ the rhains begin .32 111 in apartments. This does not indiusts a substantial flow at vssenuies from houses to apartments. The relative decrease in the proportion of pure chains for 1965 is much greater for apartments - less than 30 percent of chains beginning in apartments are pure, but this is based on a sample of only 11.305 chains. As in 1970, the ratio of pure chains beginning in houses is extremely high. This is surprising because previous research has shown substantial shifts in the tenure of occupants from renters to owners. This flow should have revealed itself in this study in a flow-of vacancies from houses to apartments. A national sample indicates that one-third to one-half of renters in the early positions of a vacancy chain become owners (Lansing, et a1., 1969:30), and Butler also shows that about 50 percent of previous renters have become owners in the last move (Butler, et a1., 1969:10). Perhaps this shift was not found here because of the shortness of chains induced by the boundaries of our housing system. That is to say, that possibly a substantial pro- portion of vacancies beginning in houses which leave the system immediately move to apartments which are within the Lansing Metro- politan Area, but outside the Lansing-East Lansing municipal boundaries. Also in contrast to the reported national situation is the proportion of vacancy chains created by new construction. New units accounted for only one-sixth.of all chains in 1970 and one- quarter of the chains in 1965 while "sequences of new construction 112 account for about half the sequences of moves in the nation" (Lansing et a1., 1969:62). This discrepancy is either a reflection of sampling error or local differences in housing construction, or both. The examination of the sub-systems yields some useful information regarding the vacancy multipliers -- the j(t), which describe the number of vacancy moves generated by each entering vacancy. While all the j(t) in our sample fall into a narrow range, 1.095 to 1.344 in 1970, and 1.037 to 1.279 in 1965, some patterns seem to emerge. Vacancy chains beginning in new units and houses tend to be longer, with chains beginning in new houses the longest of all. A rank-order correlation coefficient, Spearman's p was computed to compare the order of mean chain lengths by sub-system for the two time periods and p - .633 (p<.10). Within sub—systems, chains beginning in higher strata tended to be longer. Tables 5.23 and 5.30 show 64 values of 11 by stratum. Of these 64 values, 46 fit the descending pattern described above. Of the 18 non-conforming values of A 9 are caused by a 1! single value -- in 1965, for chains beginning in apartments, 14 - 1.000. This pattern of chain length being prOportional to the value of the original vacancy has been noted previously (Lansing et a1., 1969: l7ff.) and seems a natural consequence of housing market structure. In fact, this process forms the subject matter of Chapter 6. 113 5.7: Summary of Findings A sample of 1397 vacancy chains drawn from the Lansing City Directory (1970) and 805 chains drawn from the Lansing City Directory (1965) formed the basis for the construction of two vacancy chain models. The chains were disaggregated and the individual vacancy moves were weighted according to their relative sampling probabilities and models were developed based on 1528.039 weighted vacancy moves for 1970 and 888.086 weighted moves for 1965. The most striking substantive finding was the ex- treme shortness of vacancy chains beginning in Lansing-East Lansing. The average length of chains was found to be 1.206 moves in 1970, and 1.256 moves in 1965. Much of the cause for this shortness can probably be found in the size-of the housing system and the boundaries of this specific system. It is clear that the larger a housing system is, the more opportunities for movement within the system exist, and vacancy chains will be longer (see Simmons, 1968; Hua, 1972: 222n.). Defining the Lansing-East Lansing housing system in terms of the municipal boundaries of the two cities established a system with fewer than 60,000 addresses. Furthermore, such a definition causes vacancy moves to parts of the Lansing Metropolitan Area which are outside of Lansing and East Lansing to be counted as moves outside of the system. Both these factors tend to shorten chain lengths. 114 It should also be noted that the models being considered are minimal models, in that they represent the minimum numbers of chains and mobility rates which could fit the data. Insofar as vacancies are inferred from changes in occupancy reported at one-year 1ntervals, the estimates of the numbers of chains are minimum esti- mates. If n people occupy one address during the course of a year, there would be n-l moves through that address. Our sampling method would only find the first and last occupants and we infer only a single move through that address. Vacancy arrivals and departures by stratum were found to match the distribution of housing units by stratum very closely, although slightly more vacancies left the system via stratum 1 than arrived there. This difference was only 2 percent in 1970 and 3 percent in 1965 and indicates a slight downward flow of vacancies through the system. The multiplier matrices. (I:Q)-1. show this trickle slightly more clearly and also show that the upward flow of vacancies is almost nil. Examination of the housing sub-systems shows that, in general, vacancies created by new construction generated more vacancy moves than did vacancies entering via existing units; house entrances generated more moves than vacancies entering in apartments, and vacancies entering in higher strata generated more moves than vacancies entering the system at lower strata. 115 The sub—system analysis also shows that very few vacancy chains in our sample cross dwelling types. This is not merely a necessary conclusion of the fact that four-fifths of the chains in the sample are only one move long. In 1970, 78 percent of chains longer than 1 move did not cross dwelling type and 89 percent of chains longer than 1 move in 1965 did not. The implications for housing policy of the very high exit provabilities, the slight downward flow of vacancies, the order of the multipliers by sub-system, and the separation of the house and apartment sub—systems form the substance of Chapter 6, below. The major theoretical finding is the extraordinarily good fit which the vacancy chain model achieves with the data. Indices of dissimilarity based on the differences between observed and predicted chain length distributions for the complete models show A7O =.OO3 and A65 = .011. The model also predicts chain length distributions for the sub-systems extremely well. In no case does A exceed .05. This goodness—of—fit is partially a function of the very high exit probabilities found in this sample. Exit probabilities on the order of .8 imply that if this one parameter is estimated with reasonable accuracy, then the predicted distribution must match the observed with considerable accuracy. The domination of the model by the exit probabilities is confirmed by the use of a simple Bernoulli trials model to represent the data. In this model, the probability 116 that a chain will be of length j is P(j) = qj-lp Where p is the probability of leaving the system on any turn, q = l - p, and the trials are independent. This model predicts a chain length distribution for 1970 which, when compared with the observed distribution yields an index of dissimilarity, A = .003. At the same time as it shows the domination of the model by the exit probabilities, the Bernoulli trials model lends support to the assumption of the independence of vacancy movement. What we have is a first-order Markov chain model of residential mobility where the independence assumption is met. The model is not a model of people moving through a housing system, but it is a model of that system itself, and of housing vacancies moving through it. It appears that the problems of independence when people are the population which moves have not arisen. It also appears that when the independence assumption is met, the problems of state classification are minimized. This seems only logical, because state classification is critical only when the independence assumption is not met. Then the states must be classified so that the population within each state is homogeneous. It is this homogeneity that allows such Markov models of populations to assume the independence of transitions. If, however, the transi- tions are independent prior to state classification, then the classi- fication of states can become a substantive theoretical, rather than a mathematical theoretical problem. 117 In addition to the advantages presented by the first- order Markov chain model in mathematical and analytical terms. the model discussed here has certain methodological advantages which should not be ignored. The model permits an analysis of mobility behavior, and need not resort to questionnaire data. This eliminates the specific problems of questionnaires about mobility (see, e.g., Goldscheider, 1971:37) and the more general problems of the discre- pancies between verbal and actual behavior which have been documented since LaPiere's "Attitudes vs. Actions" (1934). The data sources used are available today for hundreds of cities in North America, and around the world. In North America, they are available more or less continuously, for at least 50 years. The data is public, non-reactive and relatively uncontaminated and uncontaminable. To overcome the problem of year-long intervals between occupancy checks, more complete records might be used -- perhaps records of telephone connections or power connections, disconnections and reconnections. Such lists would certainly yield a more complete picture of vacancy movement. To further examine the fit of the model, to determine the extent to which the shortness of chains affects the predictive accu- racy of the model, larger housing systems should be studied. It would seem that the analysis of metropolitan areas with at least 150,000 housing units and preferably 250,000 units would allow the construction of models with a greater chain length. This longer average chain 118 length would possess a larger maximum possible number of chain length distributions, and therefore allow a better evaluation of the fit of the model. It would also allow the further analysis of multipliers as a function of population size. 6. VACANCY CHAINS AND THE FILTERING OF HOUSING 6.1: The Filtering Process Filtering, as a process which indirectly provides housing for lower-income groups when new, high-quality housing is built for higher income groups, has been a "well-recognized phenomenon"(Ratcliff, 1949:321) in the housing field since Hoyt first formulated it in his classic (1939) study of residential uses within a city (Smith, 1970:64). In the most general terms higher-income people moving into new high- quality homes leave their previous residences free for occupancy by lower-income people, who free their previous residences for occupancy by families of even lower incomes. The process continues, ideally, until this process has provided at least some new housing opportunities at all income levels in the community and consequently has resulted in the improvement of living standards at all economic levels. The problems with the filtering process arise from a theoretical confusion regarding which of the several related processes mentioned above do we mean by filtering: the change in occupancy, the change in housing value, or the change in housing standards? The problem is further compounded by the fact that "filtering" is assumed to exist naturally in the housing market, and is often used as a tool of housing policy (see, e.g., Forrester, 1969). As a result, the questions regarding the existence and nature of some filtering process 119 120 must often be dealt with at the same time as "value" questions re— garding the filtering process: is it legitimate to tamper with the filtering process as it exists in the housing market? Because of its widespread currency, the small number of attempts to rigourously define and explicate the concept of filtering is surprising. The first attempt at a more formal definition of the filtering process was made by Ratcliff in 1949 who said that filtering;down is "the changing of occupancy as the housing that is occupied by one income group becomes available to the next lower income group as a result of decline in market price" (1949:321-22). This definition, however, contains two elements: the change in occupancy and the change in value (Fisher and Winnick, 1951:49; Grigsby, 1963:85). In an attempt to develOp a better measure of filtering (not just filtering-down), Fisher and Winnick eliminate what some consider to be the key element in Ratcliff's definition: the change of occupancy (Grigsby, 1963:88), and focus only on the change in relative housing value. "Filtering is defined as a change over time in the position of a given dwelling unit or group of dwelling units within the distribution of housing rents and prices in the community as a whole" (Fisher and Winnick, 1951:52). This is a much more measurable quantity than Ratcliff's earlier definition yields and certainly it would correlate "filtering- down" with successive occupation of a housing unit by relatively lower classes (Fisher and Winnick, 1951:54). But it is clear that the definition "is not intended to answer the question of whether the filtering process is bringing dwellings within the range of low 121 income groups" (Grigsby, 1963:90). Because the definition rests on changes in relative costs between housing units, substantial filter- ing could be indicated by this measure without it being available to lower-income groups. Grigsby (1963) attempts to resolbe this problem by re- injecting occupancy into the definition of filtering, by speaking of an improvement in housing standards, a concept implicit in all dis- cussions of filtering. "Filtering occurs only when value declines more rapidly than quality, so that families can obtain either higher quality and more space at the same price, or the same quality and space at a lower price than formerly" (Grigsby, 1963:97). Following Lowry's (1960) lead, Grigsby is talking in terms of real dollars, so that the question becomes: are incomes rising faster than housing costs (Grigsby, 1963:97). As the above description illustrates, there is no con- census as to what constitutes housing filtering, even though.we seem to have little trouble understanding and using the term inter alia (Grigsby, 1963:85-86). But the problem extends beyond even the resolution of the question of which process in the market constitutes filtering. There are other theoretical and empirical problems which must be overcome. In assessing the efficacy of filtering as a mechanism for the provision of lower-income housing it is not enough to say that ”the shortcomings of the filtering pro ess ..(can be attributed to) ...the failure of the relatively well—to—d: to place good quality existing housing on the market in such volume as to produce a significant reduction in its relative prices" (Winnick, 1960: 18). Given the structure of American society and its corresponding pyramid- shaped housing systems (a high proportion of housing units at the bottom and a very low proportion of units at the highest levels), it is ridiculous to assume that the 10 percent of population in the highest levels could ever place sufficient housing units on the market to accomodate the 50 percent of population in the lowest levels. The only way this could come about would be if the housing market took on the form of other consumer-durables markets (automobiles, heavy appliances, etc.) which would result in the high-income segments of satiety changing homes every 2 or 3 years. In the arguments by pro- ponents of filtering models, "it is not altogether clear whether the argument is that the housing market i§_1ike the automobile market or that it could become like the automobile market" (Lowry, 1960: 364). Nor is the problem simply one of untested assumptions in the filtering model: Do people only move to better housing? Do rapid rates of depreciation really have no adverse effects on housing quality (Grigsby, 1963: 96-97)? Does the required surplus of housing exist at all housing levels? At any housing levels (Ratcliff, 1949:323)? The problems are such that the process called filtering might not even exist. Lowry's (1960) major contribution to the discussion 123. was not his introduction of constant or real dollars to the value debate, but rather his analysis of the process adduced to result in filtering. The mechanisms which lead to price-filtering in Fisher's and Winnick's terms, could just as easily and as logically result in a deliberate program of housing under-maintenance and disinvestment resulting in an extremely rapid decline in the quality of housing. Such a decline in quality would provide no housing at the lower end of the value and income scale. Lowry also questions the validity of attributing im- provements in housing standards to the filtering process, as a part of the housing system. Analysis of variables endogenous and exogen- ous to the housing market shows that the causes of relative price declines of housing are exogenous ones. "If, for example, rising incomes cause filtering and result in an improved living environ- ment, it is the increment to earning power, not the intermediate market consequence, which should be given credit" (Grigsby, 1963:94). If these kinds of events are occurring, does it seem reasonable to talk of filtering bringing about improvements in housing standards (Lowry, 1960: 366ff.). This argument is carried one step further by White (1971). Not only must the flow of housing be considered as an exogenous variable, but the housing situation cannot even be considered a market in classical terms, because the mobility actions of each family change 124 the context in which other households must act. What is required is a "model of continuing realignment between existing stocks of housing and families" (White, 1971:88-89). 6.2. Vacancy Chains and FilteringgEffects Regardless of the definition of filtering used —- changes in occupancy, changes in value, or changes in housing standards -- there seems to be agreement on the social effects of this process. When filtering is not directly cast in terms of changes in occupancy, the effect of the filtering process is "the succession of occupancy by lower-income classes" (Fisher and Winnick, 1951:49); see also Grigsby, 1963; Kristof, 1965; White, 1971). In such cases~the decline in housing value is often seen as the mechanism which allows such a change of occupancy to take place. The simplest way to assess this type of filtering with the vacancy chain model is to examine multipliers, the number of moves generated by each vacancy entrance and to examine the attributes of households at different positions in the sequences of moves (Kristof, 1965; Lansing et a1., 1969; watson, 1974). Kristof finds, for example, in New York that "at each successive link in the chain families with generally lower incomes than their predecessors moved into turnover units (1965:241) and in a national sample of housing "there is a strong tendency for monthly rents to decline from one position to the next" (Lansing et a1., l969:7). . If the vacancy chain model is more fully explicated, there are several structural aspects of the system which can be examined 125 in order to study the flows of vacancies between strata. Ideally, filtering—down would be indicated by a lower-triangular multiplier matrix (Hua, 1972: 87), i.e. the multiplier matrix -1 (1:9,) - [n11] would have the appearance: -1 (179) = nll O O “21 “22 0 “31 “32 “33 “41 “42 “43 “44 Because we cannot expect the evidence to be so clear cut we will examine not only the multiplier matrices of the complete models for the two time periods, but we will also examine selected sub-system matrices. We will also examine the matrices of transition probabilities, the Q_matrices, in order to gain a fuller understanding of the process, and M, the matrix of mean first passage times, to gain another perspective on the structure of the housing system and flows within it. As has been stated previously, mean first passage times provide a measure of a particular kind of contiguity -- one based on inter- change probabilities rather than distance. Thus, they may be viewed as measures of aspatial ... (interstrata) ... distance (Rogers, 1966: 454). 6.3: Findings A cursory examination of the complete models for 1970 and 1965 indicates low levels of filtering (up or down) occurring 126 in Lansing-East Lansing. With multipliers, j(t) = 1.206 in 1970 and j(t) - 1.256 in 1965, we know there is only minimal vacancy movement within the city. But to determine the nature and direction of these flows we must study the Q_matrices of vacancy transitions, and the (I:Q)-1, multiplier, matrices. .095 .032 .012 .004 1.109 .040 .015 .005 _ .057 .084 .019 .006 (I19) -1 = .070 1.096 .023 .008 970 .050 .062 .050 .020 —- 70 .065 .077 1.057 .024 .049 .098 .080 .083 .073 .126 .095 1.093 .138 .039 .011 0 1.165 .053 .141 0 .073 .108 .019 .003 -l .098 1.128 .024 .004 965 ' .064 .094 .050 .009 (l:9)65 “ .089 .116 1.057 .010 .041 .064 .107 .052 .066 .092 .121 1.056 Inspection.of the multiplier matrices shows that some filtering-down of housing seems to be occuring in the city. The matrices might charitably be said to be somewhat lower-triangular. In both time periods, all elements above the diagonal are less than .05. In 1965, the elements below the diagonal approach or exceed .1 moves, while in 1970, the lower elements all exceed .05 and nl‘2 - .126 moves. Clearly, more vacancy moves are generated down- wards, than upwards. A more precise measure of filtering-down might be computed by calculating the "filtering-down" ratio -- the ratio of the sum of vacancy moves below the diagonal in (139)-1 to the sum of vacancy moves above the diagonal. 2n ...—igu— 1., ji I In 1970, this ratio was fd = .506 .115 and in 1965, it was fd = .582 .232 In other words 4.4 127 4.40 2.51. times as many vacancy moves are generated downwards in 1970 and 2.5 times as many moves in 1965 are to lower strata. Obviously, fd < 1 indicates "filtering-up." To compute M, the matrix of mean first passage times, the vectors p_and f were concatenated to g_in the following fashion to create It 0 9'2. £ The matrices of described in section 3.3. 5.429 5.667 5.722 5.844 4.851 I mean 4.823 4.596 4.710 4.645 3.840 9.668 9.623 9.328 9.139 8.641 28.535 28.471 28.076 26.411 27.493 first passage times were computed as 1.169 1.198 1.223 1.387 2.206 4.999. 5.357 5.421 5.597 4.593 4.745 4.433 4.504 4.675 3.747 10.966 10.887 10.553 9.938 9.883 28.860 28.781' 28.635 27.436 27.637 1.232 1.253 1.271 1.335 2.255 _65 What is immediately evident is the complete domination of the M matrices by the exit probabilities. The first indication of this is seen in the relatively low values of the entries in the fifth 128 columns of both matrices. Vacancies move much more quickly to the outside than to any other stratum in the system. The result of this is that the rows of the matrices become very similar. because arrival times are determined mostly by the exit and entrance prob- abilities-—-the p and f_vectors-——and not by internal vacancy transi- tions. The ratio of any element m to any other element, m ik’ of to fj -the inverse 13 the same row is very close to the ratio of fk ratio of their entrance probabilities. For example, in 1965 “113 = 10.966 = .380 [1114 28.860 and f4 = .074 = .411 £3 .180 The construction of 2_with the assumption that vacancies leaving the system would be reflected back into it has rendered £1 virtually useless for the analysis of internal system structure, given the high exit probabilities found in this sample. Consequently, a second matrix of mean first passage times, §* , based only upon 1 intra.system moves, was derived. To compute y} , a matrix, 1}:- [r13], where “13 " “11 4 X <11k k-l' was defined. Substituting R for 21in equation 3.5 (section 3.3) and bland 11* for all sub-systems are presented in Appendix A. 129 a following the identical procedure then yields the 1!: the matrices of mean first passage times based on intra-city moves only. We now have our measure of intra-system "distances". In 1970, for example, .666 .342 .274 .158 .225 .507 .340 .317 The mean first 2.088 .085 .113 .275 .258 4.132 9.982 2.939 9.548 3.472 7.828 3.478 7.648 4.454 14.320 2.961 13.443 3.203 11.386 3.714 8.070 .025 .038 .111 .267 passage times are presented below: 25.601 25.003 22.784 18.848 116.391 113.352 109.968 91.190 - These two matrices clearly show that some downward flow of housing vacancies is taking place. The smaller values of m * 13 when j the sub-systems are similar to those of the complete model, with the previously noted ex- ception of chains beginning in apartments. Not only is the filtering- down ratio smaller, but the entire system seems more compact. Vacancies beginning in apartments seem to move moresquickly to other strata above 133 and below the stratum of origin. It only takes 3.5 times as long for a vacancy to move from stratum l to stratum 4 as it does for a vacancy to move from state 4 to state 1. In less extreme cases, it only takes 2 to 2.5 times as long for a vacancy to move from stratum i to stratum j above it, compared to the move from j to i. In the other sub-systems, with the exception of moves between strata l and 2, moves upward take anywhere from 3 to 10 times as long as the corresponding vacancy moves downward in strata. It should also be noted that vacancies originating in stratum 4 move more quickly to stratum 2 than to stratum l, and much more quickly than to stratum 3. This phenomenon is much less pro- nounced in the set of chains beginning in apartments. This pattern of vacancy movement is found in 1965 in the three sets of chains which do not begin in apartments. Moves are made at more nearly equal speed between strata l and 2, than between any other pair of strata in the 3 models. With the exception of moves between strata l and 3 and strata 2 and 3 for chains beginning in new units (and strata 1 and 2 in all cases) moves upward through strata take 5 to 50 times longer than the corresponding moves downward. The "skip" phenomenon is more pronounced in 1965. Vacancies originating in stratum 4 move to stratum 1 or 2 rather than stratum 3 relatively more quickly than in 1970; furthermore, in 1965 they move to stratum 1 just as quickly as to stratum 2. The mean first passage times for the sub-systems in 1965 are presented below: 134 Chains beginning in new units: 2.731 3.252 4.843 53.136 3.206 2.497 5.394 53.687 - 3.622 2.761 4.811 48.294 3.992 3.895 3.064 39.361 Chains beginning in existing units: 1.644 4.778 21.668 148.924 2.758 3.079 20.176 144.146 - 2.839 3.419 16.995 147.565 3.617 2.834 15.098 125.992 Chains beginning in houses: 1.802 4.005 15.811 133.990 2.750 2.786 15.049 130.765 - 2.974 2.905 13.048 129.351 3.582 3.478 8.840 105.978 .M4 for chains beginning in apartments could not be com- puted. Because one row of.9 contained nothing but zeros (q41 - q,‘2 - q43 - q44 = 0), the matrix §_became the transition matrix of a Markov chain with an absorbing state.1 Consequently, gr, the matrix of mean first passage times, could not be computed. 6.4: Filtering;Effects in Lansing:East Lansigg;-- Discussion The consideration of the filtering process in Lansing-East Lansing as a mechanism for the provision of housing at all levels of the housing system, and the effects of such a process must proceed at two levels. On a macrosCOpic scale, at the system level, we can only conclude that vacancies entering the cities of Lansing-East Lansing do not generate large numbers of vacancy moves within the city. With multipliers in both time periods not exceeding 1.3 moves, the system might be said to be relatively insensitive to vacancy creations. 1 See Appendix A, p. 188. 135 This low level of vacancy transfer within the system is further aggravated by the high degree of separateness of the housing sub- markets. Although not surprising (see White, 1971: 90) the limited exchange of vacancies between house and apartment sub-systems can only decrease the number of vacancy transfers in the system. The vacancy chain model -the (1:9):1 multiplier matrix, and HF, mean first passage time matrix in particular--permits us to examine in detail the flows of vacancies through the system. These two matrices indicate that, in general, vacancies flow from higher strata to lower strata through the system. This downward flow of vacancies corresponds to an upward flow of people from lower to higher standards of housing. In 1970, more than 4 times as many vacancies travelled downward, and in 1965 more than 2.5 times as many vacancies travelled downward as travelled up. Filtering-down ratios of this magnitude were found to exist in all the subsystems studied, except in the sequences of vacancy chains beginning in apartments (see Table 6.1). In this case, in 1970, only twice as many vacancy moves were generated downward as were generated upward, and in 1965 more vacancy moves were actually genera- ted to strata above the stratum of origin. This extreme discrepancy in 1965 is largely due to the small number of chains entering the system via apartments in strata 3 and 4 (only 35.3 chains do so-—-and all but 2.9 exit the system immediately), but there definitely seems to be a difference between the house and apartment sub-systems. 136 2.474 5.169 7.252 16.456 (yr 4.259 3.644 6.975 14.249 7°(Apt‘) 4.544 4.880 4.753 14.369 4.655 5.089 7.077 9.007 To use 1970 as an example, we can see that the sub—system is more compact than other sub-systems. The distances are relatively are of the same 13 There is relatively little difference between m small and, with the exception of the mi4*, the m .* and m order. 13 *0 ji Filtering-down in this model is indicated mainly in a negative way, by the length of time it takes vacancies to reach the highest stratum, and not by the Speed at which vacancies flow downward- Even this span, as indicated by the m14* is not large when it is compared to other systems. Several factors could account for the compactness of this system, the relative ease with which vacancies move both up and down between strata. The largely rental market of apartment units provides much greater potential for forced moves on short notice than does the house market, and this might account for some of the upwards vacancy movement. Perhaps, apartments have higher substitutability than houses, or perhaps apartment dwellers are more downwardly mobile than house dwellers. A more reasonable explanation might be that apartment buildings, especially newer ones, might be non-conforming in our scheme of state classification. It is not uncommon for new apartments of high rent, status and dwelling standards to be built in the urban core or other urban redevelopment areas, resulting in an apartment building 137 of-much higher standard than its surrounding area. If this apart- ment occupied one, or at most, two blocks, the area would be too small to be classified as a high-level area, and the apartment would be classified as being, most likely, in stratum 1 or 2. Occupant moves to this apartment building from stratum 2 or 3 which are in- stances of upward housing mobility (i.e. vacancies filtering down, from the point of reference of the apartment building) would appear, because of state classification, to be instances of vacancies filtering up. The extent to which this occurs is an empirical question. A second notable finding was the fact that vacancies originating in stratum 4 seem to skip stratum 3 in their downward move- ment through the system. In both time periods, in all sub-systems, vacancies beginning their careers in stratum 4 moved to stratum 2 and stratum 1 much sooner than they did to stratum 3. As Table 6.2 (excerpted from Table 4.8) shows, strata 3 and 4 contain housing of a much higher standard than strata l and 2. They also contain a much higher proportion of owner occupied homes. Selected Characteristics of Housing Sub-Areas Mean Value Mean Proportion Sub-Area of Housing Mean Rent of Housing Owned 1. Low $14,400 $110 .40 2. Lower Middle 15,300 121 .75 3. Upper Middle 23,700 169 .83 4. High 34,500 184 .88 138 If, "most moves are undertaken voluntarily and are motiva- ted by the changes in family size which rendered the old dwelling's space inadequate to its requirements" (Rossi, 1955:175), then it may be that the "skip" phenomenon can be accounted for in terms of house- hold life-cycle. It is not unreasonable that occupancy moves from strata l and 2 to stratum 4 represent moves made by families entering the "child-bearing" or expansion stage of the life-cycle. Residents of stratum 3 would in many cases already have made such a move to accomodate their increased need for space, and would be less likely to move to a new, larger home in stratum 4. Moves into strata 3 and 4 could well represent the same type of be- havior'— households entering their child-rearing periods- and result in the pattern of vacancy movement described. Bearing in mind the dangers of ecological correlation, such a pattern of behavior with the lower levels of owner-occupancy in strata 1 and 2, could also indicate a shift of previous renters becoming home-owners in Lansing-East Lansing. Such tenure changes are very common (Lansing, et al., 1969: 30; Butler, et a1., 1969: 10) and their absence would be curious. - We should also note that the lower ratios of reciprocal mean first passage times between strata 1 and 2 when compared with other adjacent strata, and the "skip" phenomenon differentiating them only slightly lends some support to the notion of cash rent or price as the 139 sole criterion of housing levels (Hua, 1972: 122). As Table 6.2 shows, the major difference between the two strata is not the value of housing or rent, but the proportion of housing owner-occupied. The differences in rents and values are minimal when compared to the differences between strata 2 and 3 and strata 3 and 4. The vacancy chain model has enabled us to examine closely the internal filtering processes of Lansing-East Lansing. But we should not let this micro-analysis blur the major finding: a multiplier of 1.3 moves per vacancy arrival does not yield many opportunities for occupants to change their housing standards, even if 70 percent of those who change, improve them. 7. CONCLUSIONS The purpose of this study has been two-fold. The first aim was the development of a vacancy chain model of intra-urban residential mobility and the assessment of the adequacy of this model to represent the transfer of housing vacancies in the urban area. The second aim*was to apply the model, if it proved adequate, to the analysis of the filtering process in the same urban housing system -- Lansing-East Lansing, Michigan. Does the model fit the data? The model more than ade- quately represents the process of housing vacancy transfer in Lansing- East Lansing. The fit of the model to the data is exceptional. Comparisons of predicted and observed values for the complete model and numerous sub-system models in each of two time periods never yields a discrepancy of more than 5 percent, and only rarely do these -..” ... -....— discrepanciee exceed 2 percent. The transitions of individual vacancies appear to be independent of one another.' As a result, we can construct a first— order Markov chain that represents vacancy movements extremely well. The problem of non-independence of transitions which arises in models of people as the population of movers does not arise when vacancies are the population of movers. This provides two distinct benefits: 140 141 the first is the use of first-order Markov chains in the analysis of residential mobility, and the second is to allow us to focus on state classification from a substantive or a policy perspective. If the individual transitions are not made independently, as is the case with peOple changing addresses (Rossi, 1955; Moore, 1969), then one of the functions of state-classification is to ensure popu- lation homogeneity within occupancy states. This homogeneity allows us to assume the independence of transitions for practical purposes. If, however, the transitions really are made independently, then state classification forms an arbitrary partition of the occupancy space. Different partitions will obviously result in different transition parameters but will not affect the fit of the model. This allows states to be classified on the basis of substantive housing policy or theo- retical concerns, free of any mathematical-theoretical constraints. In our case, the model can be just as well represented by a single-state model of the residential system. This model predicts vacancy chain length distributions just as accurately as the 4-state model. While this lends support to our belief in the assumption of independent vacancy transitions (the l-etate model reduces to a series of Bernoulli trials), this fit also serves to explain the extraordinarily good fit of the complete model. Because the average length of chains is so low the vast majority of chains must be only 1 move long. This sharply limits the number of chains which could possibly be distrihutea at lsnaths other than 1 move. Consequently. '1"- 142 reasonably accurate estimation of the exit probabilities guarantees that overall differences between the model and the data will be minimal. That is to say, the high exit probabilities in this sample imply that the good fit of the model rests not on the estimation of 5x4 = 20 transition probabilities, but only on the estimation of the 4 exit probabilities -- especially when these are all so similar. The low average chain length determined by these exit probabilities is a function of the size of the housing system under study. Lansing-East Lansing contains fewer than 200,000 people and the multiplier lengths of 1.2 to 1.3 fit quite well with other recent estimates of multipliers in areas of different sizes (Kristof, 1965; Lansing et a1, 1969; Watson, 1974). This pattern is shown in Figure 7.1. Chain lengths varied only slightly among the several sub— systems studied in the two time periods. Generally, chains beginning in houses, new units, and higher strata tended to be longer. The shortest chains were pure chains beginning in apartments in 1965 (i'- 1.04) and the longest began in new houses in 1970 (i'= 1.34). The sub-systems also seem to be highly separated in that 78 percent of chains longer than 1 move in 1970 and 89 percent of such chains in 1965 contained only houses or apartments. There was a mini— mal flow of vacancies between these two dwelling types. The vacancy chain model is an exceptional tool for the study of the filtering process. The use of mean chain lengths as an indicator 143 Multiplier “'1 e (Lansing g£_al,,l969) 3.- '(Kristof,l965) 2J e (Watson,l974) e(Current study) 1- log(pOpulation) Figure 7.1. Multipliers and housing system populations of the amount of filtering is not new (Kristof, 1965; Lansing et a1., 1969; Watson, 1974). But the complete model yields a wealth of infor- mation regarding internal vacancy movement as well. In fact, the volume and extent of such micro-system data provided is such that one might ignore or forget the macroscopic findings with their system- level implications. . With the exception of chains beginning in apartments, some vacancy filtering was found to occur in all sub-systems. Overall, 144 4.4 times as many vacancy moves were generated downward in 1970 as were generated upward. In 1965, the ratio of moves downward in strata to moves upward was 2.5 : l. Chains beginning in apartments, however, had the lowest filtering-down ratio in each time period. In fact, in 1965, filtering-up from apartments was indicated. How much of this apparent vacancy filtering-up is due to the construction of high- standard apartments in areas of low housing standards as parts of urban renewal programs must be determined. Vacancies were also found to move more quickly from the highest housing level to the two lowest levels, skipping one stratum in the process, most likely reflecting life-cycle differences between the populations of strata 1 and 2 and stratum 3. Reciprocal vacancy movement between the 2 lowest housing levels was the most nearly equal of any pair of housing strata. Insofar as these strata are also the most similar in terms of housing values and rents, we must consider again the notion of using only a measure of cash rent or price as the sole criterion of housing levels (Hua, 1972: 122). The internal analysis of vacancy filtering points to the previously stated conclusion that anyone who moves benefits from such a ‘move (Lansing et a1., 1969: 65). The problem with the filtering pro- cess as a mechanism for the provision of new housing units throughout the housing system is that not many people move! Only 1.2 or 1.3 moves occur per vacancy creation. This pattern would surely change if we consider filtering on a regional or even national basis, for surely 145 filtering need not only meet the needs of local residents (Watson, 1974: 349), especially in the light of recent flows of migrants from rural to urban areas. It is clear, however, that the major bene- ficiaries of the filtering process are not the residents of Lansing- East Lansing. It would seem then that further study is required, both of the model and method employed, and of the Lansing-East Lanaing housing system. Inclusion of the other portions of the Lansing Metro- politan Area would permit greater understanding of mobility processes in Lansing. Greater specificity of housing level classification, especially in the case of apartment buildings might allow the recon- ciliation of vacancy movements from apartments with other movements. The model could be more fully explored by the selection of a larger study area. A larger system can be expected to have a longer average chain length. A longer average chain length reduces the extent to which the fit of the model is determined by the exit probabilities. This would allow a sounder evaluation of the fit of the model. The use of directory data facilitates the study of larger metropolitan areas. The use of directory data in combination with questionnaires to trace the places of origin of in-migrants and the extension of our model to include all such places of origin, say, within a state, would increase costs, but would permit the analysis of filtering as a regional process. 146 The vacancy chain model is promising. It is a model of the housing system within which residential mobility occurs. It can be based on data which is now available for a large number of cities over a considerable time span. These factors together with the fit of the model within early studies, and the congruences between early studies call for its further exploration, and its application in the analysis of residential mobility. APPENDICES APPENDIX A DATA MATRICES This appendix presents the raw data from which the vacancy chain models of the complete housing system, and all the housing sub=systems, for both time periods were constructed. The unweighted and weighted vacancy moves are presented in matrix form. If !_= [Vij]’ the matrix of (weighted or unweighted) vacancy moves, then vij is the number of moves from stratum i to stratum j. In addition to the raw data, the f, p, and Q_matrices -- vacancy arrival, departure, and transition probabilities -- are presented, as are the derivative matrices and parameters of the model: (I:Q)-l - the multiplier matrix; A_ - the vector of mean chain lengths by stratum of origin; j(t) - the mean chain length; P - the probability distribution of -j expected chain lengths by stratum of origin; yr - the matrix of intra-system mean first . passage times; 11 - the matrix of mean first passage times including moves outside the system. The appendix also presents a comparison of the observed and predicted chain length distributions. 147 _F_‘(t) _f_(t) {1,-3Y1 I» J(t) DWNH 148 ALL CHAINS, 1969 - 1970 439 98 72 31 10 441.488 49.042 34.380 15.031 5.202 .325 .095 .056 .050 .049 1.109 .070 .065 .072 1.169 1.206 Unweighted Vacancy Moves 2 3 4 560 289 109 32 13 4 101 22 7 37 31 12 22 16 20 Weighted Vacancy Moves 512.064 258.704 85.424 16.536 6.208 1.785 50.895 11.334 3.822 18.681 15.109 6.096 10.396 8.457 8.764 Transition Probabilities .404 .204 .607 .032 .012 .003 .083 .019 .006 .062 .050 .020 .098 .080 .083 Multiplier Matrix .040 .015 .004 1.096 .023 .008 .077 1.057 .024 .126 .095 1.093 Mean Chain Length by Stratum 1.198 1.223 1.388 Mean Chain Length 503 550 260 84 441.564 508.142 244.896 73.072 .857 .834 .817 .690 J-‘UJNV— ODWNH Probability Distribution 1 .857 .835 .817 .690 938 1049.093 2.088 3.117 3.443 3.923 Mean 5.429 5.667 5.722 5.844 4.851 149 2 .121 .138 .150 .246 318 3 .019 .023 .027 .052 81 32.820 4 5 .003 .000 .004 .001 .005 .001 .010 .002 Observed Chain Length Distribution 12 3 Weighted Chain Length Distribution 181.104 3.892 .831 Mean First Passage Times 2 4.032 2.939 3.472 3.478 2 4.823 4.596 4.710 4.647 3.840 3 9.982 9.548 7.828 7.648 3 9.668 9.633 9.328 9.139 8.641 4 25.601 25.003 22.784 18.848 4 28.535 28.471 28.076 26.411 27.493 of Chain Lengths First Passage Times (Including Outside) 0 1.169 1.198 1.223 1.287 2.206 150 CHAINS STARTING IN NEW UNITS, 1969 - 1970 Unweighted Vacancy Moves l 2 3 4 0 0 43 100 53 49 1 16 4 1 1 65 2 13 22 2 2 108 3 7 8 8 7 41 4 8 13 7 8 31 Weighted Vacancy Moves Eflt) = 38.758 92.218 43.822 36.472 7.474 2.278 .324 .569 48.418 6.092 10.888 .894 .975 96.364 2.513 3.737 3.489 3.577 38.370 4.227 6.092 3.496 3.244 27.778 Transition Probabilities £(c) = .184 .437 .206 .173 p_ .127 .039 .005 .010 .820 Q_ = .053 .095 .008 .008 .836 .049 .072 .068 .069 .742 .094 .136 .078 .072 .620 _1 Multiplier Matrix (Ifg) = 1.150 .052 .008 .013 .069 1.110 .011 .012 .075 .102 .081 .082 .133 .176 .093 1.088 Mean Chain Length by Stratum .1 - 1.223 1.201 1.340 1.491 Mean Chain Length J(c) - 1.284 1'" l3 J-‘wNH OmeI—I 151 Probability Distributions of Chain Lengths 1 .820 .836 .742 .620 154 164.354 2.013 2.259 3.929 3.666 Mean 8.157 8.795 8.884 8.560 8.157 2 .146 .133 .193 .294 63 35.87 Mean First Passage Times 2 4.370 2.774 3.754 3.522 2 4.447 4.182 4.354 4.194 3.440 3 .028 .024 .050 .068 20 8.104 3 21.192 20.631 15.306 16.816 3 9.878 9.835 9.322 9.355 8.733 4 .005 .005 .011 .147 8 2.595 4 15.872 15.786 11.417 12.909 4 11.282 11.276 10.655 10.746 10.120 5 .001 .001 .002 .003 Observed Chain Length Distribution 0 Weighted Chain Length Distribution 0 First Passage Times (Including Outside) 0 1.223 1.201 1.340 1.491 2.284 git) ((L) (1:9) J(t) 152 CHAINS STARTING IN OLD UNITS, 1969 - 1970 bLaJNH 396 82 59 24 372.735 41.568 28.289 12.518 .975 .353 .091 .057 .050 .016 1.104 .071 .063 .030 1.162 1.191 Unweighted Vacancy Moves 2 3 460 236 28 12 79 20 29 23 9 9 4 60 NU'IU'IUO Weighted Vacancy Moves 419.857 215.222 14.258 5.884 40.007 10.440 14.944 11.620 4.304 4.961 48.952 1.216 2.847 2.519 5.520 Transition Probabilities .397 .204 .031 .013 .081 .021 .060 .047 .070 .081 Multiplier .039 .016 1.093 .026 .072 1.053 .092 .096 Mean Chain Length by Stratum 1.197 1.200 .046 .003 .006 .010 .090 Matrix .004 .007 .012 1.101 1.320 Mean Chain Length 438 442 219 53 393.154 411.790 206.527 45.294 .862 .835 .832 .742 II: n bump—- 0990191— Probability Distribution of Chain Lengths 153 1 2 3 4 5 .862 .117 .017 .003 .000 .835 .139 .022 .004 .001 .832 .140 .022 .004 .001 .742 .207 .042 .007 .001 Observed Chain Length Distribution 829 255 61 4 3 Weighted Chain Length Distribution 844.739 145.225 24.716 1.297 .831 Mean First Passage Times 1 2 3 4 2.099 4.081 8.904 33.307 3.099 2.974 8.422 32.521 3.307 3.391 7.063 31.490 4.443 3.571 5.926 21.859 Mean First Passage Times (Including Outside) 1 2 3 4 0 5.076 4.912 9.634 41.460 1.162 5.279 4.693 9.577 41.351 1.197 5.320 4.794 9.331 41.167 1.200 5.605 4.821 9.043 37.921 1.320 4.440 3.932 8.622 40.436 2.191 £(t) £(t) (L—gfl J(L) 154 CHAINS STARTING IN HOUSES, 1969 - 1970 L‘WNH Unweighted Vacancy Moves 1 2 3 4 289 384 128 69 69 25 7 3 60 79 15 2 21 28 12 8 7 19 14 10 Weighted Vacancy Moves 267.841 345.889 106.270 49.862 34.493 12.713 3.201 1.380 28.203 39.352 7.593 1.139 10.564 13.720 5.767 4.146 3.658 8.851 7.318 4.219 Transition Probabilities .348 ..449 .138 .065 .100 .037 .009 .004 .067 .094 .018 .003 .081 .105 .044 .032 .060 .146 .120 .069 Multiplier Matrix 1.116 .048 .012 .005 .085 1.110 .022 .004 .107 .133 1.054 .037 .099 .194 .140 1.080 Mean Chain Length by Stratum 1.182 1.222 1.332 1.551 Mean Chain Length 1.242 342 379 107 42 292.968 344.240 95.951 36.700 .850 .819 .737 .604 Mi: 1 3 J-‘LJNH ObUJNl-d 155 Probability Distribution of Chain Lengths 1 .850 .819 .737 .604 577 2 .124 .149 .207 .301 222 3 .021 .027 .045 .075 59 4 .004 .005 .009 .016 9 5 .001 .001 .002 .003 Observed Chain Length Distribution 3 Weighted Chain Length Distribution 615.795 1 1.977 2.811 3.085 3.606 126.431 Mean First Passage Times 2 3.804 2.757 3.061 3.041 23.906 3 12.312 11.732 10.568 8.720 2.919 4 32.850 33.201 29.426 27.177 .831 Mean First Passage Times (Including Outside) 1 5.006 5.202 5.199 5.423 4.406 2 4.319 4.104 4.120 4.052 3.334 3 13.668 13.577 13.260 13.299 12.652 4 30.208 30.277 29.461 28.408 29.179 0 1.181 1.222 1.322 1.515 2.242 yr) _f_(t) (39* JR) II 150 CHAINS STARTING IN APARTMENTS, 1969 - 1970 «DWNH 146 27 12 139.893 13.655 6.177 4.142 1.544 .290 .083 .034 .025 .037 1.092 .041 .030 .047 1.142 1.150 Unweighted Vacancy Moves 2 169 Weighted Vacancy Moves 158.724 150.473 33.022 3.822 2.683 .405 11.543 3.741 2.683 4.392 9.342 1.949 1.544 1.139 3.734 Transition Probabilities .329 .312 .068 .023 .016 .002 .064 .021 .015 .026 .056 .012 .037 .027 .089 Multiplier Matrix .028 .019 .004 1.071 .025 .018 .031 1.061 .014 .046 .034 1.099 Mean Chain Length by Stratum 1.155 3 158 1.136 4 37 m-L‘UIH 1.225 Mean Chain Length 157 163 151 39 144.845 155.880 147.553 33.832 .876 .866 .882 .809 DWNH ObWNH 157 Probability Distribution of Chain Lengths 1 .896 .866 .882 .809 393 419.424 2.474 4.259 4.544 4.665 Mean 6.267 6.594 6.645 6.630 5.699 2 5.995 5./58 5.969 5.975 5.012 First Passage Times 3 6.455 6.434 6.193 6.451 5.433 2 3 4 5 .109 .014 .002 0 .116 .016 .002 0 .103 .013 .002 0 .161 .025 .004 .001 Observed Chain Length Distribution 24 21 2 0 Weighted Chain Length Distribution 53.534 8.509 .649 0 Mean First Passage Times 2 3 4 5.169 7.252 16.456 3.644 6.975 14.249 4.880 4.753 14.369 5.089 7.077 9.007 (Including Outside) 4 27.095 26.753 26.828 24.802 26.043 0 1.142 1.155 1.136 1.225 2.150 158 CHAINS STARTING IN NEW HOUSES, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 f o o 29 52 29 39 1 6 3 o 1 45 2 11 15 o o 62 3 3 6 4 7 20 4 6 2 7 8 22 Weighted Vacancy Moves £(t) = 26.968 47.039 22.336 26.795 3.088 1.708 O .569 34.194 5.117 7.231 0 0 51.915 1.135 2.762 1.704 3.577 18.358 3.253 5.522 3.496 3.244 18.671 Transition Probabilities £(t) = .219 .382 .181 .218. B_ .078 .043 0 .014 .864 a .080 .113 0 0 .808 9 . .041 . 100 .062 .130 .667 .095 .162 .102 .094 .546 Multiplier Matrix 1.091 .057 .002 .018 (1.9)-1 0°97 1e132 0 e002 - .078 .155 1.083 .157 .141 .225 .123 1.125 Mean Chain Length by Stratum A_ 1.168 1.232 1.472 1.614 Mean Chain Length 1(1) 1.344 .1?“ "I «L‘WND—i O§wNH Probability Distribution of Chain Lengths 1 .864 .808 .667 .546 84 89.648 2.131 2.413 3.758 3.421 Mean 7.299 7.316 7.702 7.383 2 .110 .160 .229 .333 46 Mean First Passage Times 2 3:100 2.285 3.123 2.976 2 4.768 4.493 4.631 4.455 3.854 3 .020 .027 .078 .091 14 5.673 3 66.707 69.121 49.550 50.044 3 11.032 11.114 10.486 10.212 9.884 4 .004 .004 .020 .023 5 1.622 4 16.664 19.077 10.808 13.718 4 8.904 9.104 8.035 8.446 7.885 5 .001 .001 .005 .005 Observed Chain Length Distribution 0 Weighted Chain Length Distribution 26.197 0 First Passage Times (Including Outside) 1.168 1.232 1.473 1.614 2.344 160 CHAINS STARTING IN NEW APARTMENTS, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 0 0 14 41 22 9 1 9 1 0 0 20 2 2 7 2 2 38 3 3 1 4 0 20 4 2 1 0 0 8 Weighted Vacancy Moves Eflt) = 11.791 37.708 20.252 8.610 4.061 .569 0 0 14.224 .975 3.658 .894 .975 36.409 1.054 .405 1.785 0 19.687 .975 .569 0 0 8.040 Transition Probabilities ‘g1c) = .150 .481 .258 .110 2_ .215 .030 0 0 .754 = .023 .085 .021 .023 .848 -9- .046 .018 .078 0 .859 .102 .059 0 0 .839 Multiplier Matrix 1.276 .042 .001 .001 (1- )-1 = .036 1.096 .025 .025 «- 9 .064 .023 1.085 .001 .132 .069 .002 .002 Mean Chain Length by Stratum 5‘ - 1.320 1.183 1.173 1.205 Mean Chain Length 1“) - 1.203 .1" H! bum.— ObtoNv— Probability Distribution of Chain Lengths 1 .754 .848 .859 .839 62 66.169 1.521 4.233 3.397 2.561 Mean 1 9.157 11.213 10.948 10.360 10.364. 2 .188 .126 .116 .127 16 9.112 Mean First Passage Times 2 8.133 4.251 8.094 6.139 2 4.380 4.023 4.309 4.155 3.229 3 .044 .020 .020 .027 6 2.431 3 30.981 22.849 13.930 28.983 3 8.309 7.992 7.530 8.189 6.996 4 .010 .004 .004 .006 2 .649 4 30.351 22.219 30.313 28.353 4 18.142 17.573 18.003 18.015 16.839 5 .002 .001 .001 .001 Observed Chain Length Distribution 0 Weighted Chain Length Distribution 0 First Passage Times (Including Outside) 1.320 1.183 1.173 1.205 2.203 162 CHAINS STARTING IN OLD HOUSES, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 O O 260 332 99 3O 1 63 -22 7 2 297 2 49 64 15 2 317 3 18 22 8 1 87 4 1 7 7 2 20 Weighted Vacancy Moves £(t) 240.874 298.856 83.934 23.067 31.405 11.005 3.201 .810 258.778 23.087 32.122 7.593 1.139 292.331 9.429 10.958 4.063 .569 77.593 .405 3.329 3.822 ~.975 18.030 Transition Probabilities £(t) = .372 .462 .130 .036 p_ .102 .036 .010 .003 .848 = .064 .090 .021 .003 .821 9 .092 .107 .040 .006 .756 ' .015 .125 .144 .037 .679 Multiplier Matrix 1.120 .046 .014 .003 -1 .082 _ 1.106 .026 .004 (1‘9) " .117 , .128 .046 .007 .046’ .164 .160 1.040 Mean Chain Length by Stratum A_ - 1.183 1.219 1.298 1.409 4 Mean Chain Length J(t) - 1.223 163 Probability Distribution of Chain Lengths 1 2 3 4 5 .848 .127 .021 .003 .001 .821 .147 .027 .005 .001 .756 .199 .037 .007 .001 .679 .250 .058 .011 .002 Observed Chain Length Distribution 493 176 45 4 3 Weighted Chain Length Distribution "I book).— C§UJNH Mean First Passage Times (Including Outside) 1 4.708 4.918 4.837 5.281 4.088: 2 4.238 4.033 4.022 3.991 3.242 6.140 3 14.345 14.208 14.003 12.524 13.355 4 55.840 55.837 55.768 54.099 54.837 526.147 100.233 18.233 1.297 .831 Mean First Passage Times 1 2 3 4 1.922 3.888 10.789 54.856 2.804 2.813 10.059 54.829 2.761 3.021 9.612 54.558 3.760 2.866 49.445 1.183 1.218 1.298 1.409 2.222 164 CHAINS STARTING IN OLD APARTMENTS, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 O O 132 128 136 28 1 18 6 5 l 137 2 10 15 5 3 125 3 6 7 15 4 131 4 l 2 2 8 31 Weighted Vacancy Moves §(t) = f128.102 121.016 130.221 24.412- 9.594 3.253 2.683 .405 130.621 5.202 7.885 2.847 1.708 119.471 3.088 3.986 7.557 1.949 127.866 .569 .975 1.139 3.734 25.792 Transition Probabilities ffit) = .317 .300 .323 .060 p_ .065 .022 .018 .003 .891 9 .021 .028 .052 .013 .885 .018 .030 .035 .116 .801 Multiplier Matrix 1.072 .026 .021 .004 (I _ )-1 a .044 1.063 .025 .016 — 9» .026 .032 1.057 .017 ' .024 .038 .044 1.132 . Mean Chain Length by Stratum .1 - 1.123 1.148 1.132 1.238 Mean Chain Length' J(t) 1.140 Ltd 165 Probability Distribution of Chain Lengths 1 2 3 4 ’ 5 .891 .096 .011 .001 0 .871 .112 .014 .002 0 .885 .100 .013 .002 0 .801 .166 .030 .004 .001 Observed Chain Length Distribution 331 78 15 O O Weighted Chain Length Distribution H! bush).— OL‘WNH 5.896 6.084 6.175 6.292 5.196 - 2 6.513 6.302 6.483 6.551 5.553 3 6.187 6.191 5.982 6.169 5.192 4 30.157 - 29.874 29.826 26.828 29.142 353.255 44.422 6.078 0 0 Mean First Passage Times 1 2 3 4 2.908 4.829 5.974 14.879 4.375 3.698 6.013 13.493 5.095 4.655 4.246 12.890 6.142 5.390 5.842 6.662 Mean First Passage Times (Including Outside) 1.123 1.148 1.132 1.238 2.140 yr) 3(t) (_I_ - 9,)"1 [>- J(t) bump-I O 166 PURE CHAINS, HOUSES, 1969 - 1970 273 55 52 11 260.170 29.189 24.762 5.936 3.253 .351 .090 .062 .050 .064 1.102 .076 .069 .091 1.141 1.197 Unweighted Vacancy Moves 2 3 4 365 120 55 16 4 O 65 8 1 26 10 7 11 9 7 Weighted Vacancy Moves 336.349 102.043 43.694. ' 8.290 1.821 0 33.324 4.227 .569 12.874 5.038 3.577 5.526 4.880 3.084 Transition Probabilities .453 .137 .059 .026 .006 0 .084 .011 .001 .109 .043 .030 .109 .096 .061 Multiplier Matrix .032 .007 0 1.096 .013 .002 .131 1.050 .034 .142 .109 1.068 Mean Chain Length by Stratum 1.187 1.284 1.410 Mean Chain Length 322 357 97 37 284.005 333.481 90.585 34.182 .878 .841 .768 .671 er 167 Probability Distribution of Chain Lengths 1 2 3 4 5 .878 .105 .014 .002 0 .841 .135 .020 .003 0 .768 .189 .036 .006 .001 .671 .262 .055 .010 .002 Observed Chain Length Distribution 577 191 39 5 1 Weighted Chain Length Distribution H! #wND-I O-l-‘wNI—o Mean First Passage Times (Including Outside) 1 5.044 5.219 5.351 5.369 4.417' 2 4.331 4.114 4.066 4.146 3.321 3 14.268 14.232 13.818 13.126 13.223 4 33.926 33.914 32.987 32.020 32.794 615.795 108.776 15.802 1.622 .277 Mean First Passage Times 1 2 3 4 1.715 4.420 18.681 91.078 2.669 2.967 18.022 89.451 3.317 2.928 15.195 76.820 3.490 3.325 12.968 70.601 1.140 1.187 1.284 1.410 2.197 168 PURE CHAINS, APARTMENTS ONLY, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 0 0 142 145 149 34 1 21 4 5 0 143 2 5 11 2 2 149 3 3 7 14 2 145 4 2 2 1 6 33 Weighted Vacjacy Moves .§(c) = 137.779 145.794 146.414 31.642 11.302 2.278 2.683 0 138.184 2.519 5.607 1.139 1.139 148.236 1.708 3.822 7.316 .975 144.301 1.139 1.139 .569 2.924 30.908 Transition Probabilities _§(1) = .298 .316 .317 .069, 2_ .073 .015 .017 0 .895 3 .016 .035 .007 .007 .934 9- .011 .024 .046 .006 .913 .031 .031 .016 .080 .843 Multiplier Matrix 1.079 .017 .020 0 -1 _ .018 1.037 .008 .008 (179) ‘ .013 . .027 1.049 .007 .037 . .036 .019 1.087 Mean Chain Length by Stratum '1.117 1.072 1.096 1.179 Mean Chain Length 1» I J(t) - 1.100 169 Probability Distribution of Chain Lengths 1 2 3 4 5 .895 .095 .009 .001 0 P = .934 .060 .005 .001 0 -j .913 .080 .007 .001 O .843 .138 .017 .002 0 Observed Chain Length Distribution 393 67 10 0 O Weighted Chain Length Distribution 419.424 38.157 4.052 O 0 Mean First Passage Times 1 2 3 4 1 2.589 5.657 6.731 20.840 Me 2 4.702 3:419 7.536 17.322 " 3 5.643 4.384 4.463 17.761 4 5.034 5.165 7.736 10.289 Mean First Passage Times (Including Outside) 1 2 3 4 O 1 6.278 6.281 6.332 28.665 1.117 M 2 6.618 6.112 6.358 28.412 1.072 —' 3 6.675 6.201 6.132 28.460 1.096 4 6.605 6.227 6.402 26.434 1.179 O 5.268 6.337 27.556 2.100 5.660 ' :(t) gt) (1:9)’1 [v 1(t) 170 PURE CHAINS, STARTING IN NEW HOUSES, 1969 - 1970 waI— O 25 O‘HO-b 24.690 2.114 4.307 .405 3.253 .212 .061 .073 .016 .109 1.066 .086 .046 .144 1.086 1.278 Unweighted Vacancy Moves 2 3 4 50 27 33 1 0 0 12 0 O 6 2 6 8 5 6 Weighted Vacancy Moves 46.065 21.361 24.361 .569 0 0 5.932 0 0 2.762 .975 3.008 3.982 2.602 2.515 Transition Probabilities .395 .183 .209 .016 0 0 .100 0 0 .111 .039 .121 .133 .087 .084 Multiplier Matrix .019 0 0 1.113 O 0 .151 1.053 .139 .179 .100 1.105 Mean Chain Length by Stratum 1.199 1.389 1.527 Mean Chain Length 40 56 19 20 32.085 49.072 17.789 17.532 .923 .827 .713 .587 Probability Distribution of Chain Lengths 1 .923 .827 .713 .587 171 2 .070 .150 .205 .322 3 .007 .020 .065 .073 4 .001 .002 .014 .015 5 0 0 .002 .002 Observed Chain Length Distribution 84 39 9 3 0 Weighted Chain Length Distribution H! wav—I C-l-‘UJNH 89.648 22.211 3.647 .973 0 Mean First Passage Times 1 2 3 4 1.505 4.708 1.329 1.441 2.378 2.980 1.329 1.441 3.791 3.255 1.000 .739 3.221 3.673 .978 1.000 Mean First Passage Times (Including Outside) l 2 3 4 0 7.634 4.780 10.907 9.373 1.086 7.596 4.475 11.020 9.487 1.199 8.092 4.494 10.643 8.445 1.389 7.485 . 4.509 10.283 8.882 1.528 7.054_, 3.781 9.821 8.288 2.279 £(t) _f_(t) (1:9)" IV j(t) 172 PURE CHAINS STARTING IN NEW APARTMENTS, 1969 - 1970 L‘WNH O 13 HONO‘ 11.221 3.088 .975 0 .569 .151 .195 .025 .062 1.242 .035 .001 .079 ' 1.242 1.125 Unweighted Vacancy Moves 2 36 HHU‘IO 3 19 OI—IOO 4 OOHO \o Weighted Vacancy Moves 35.270 0 2.519 .405 .569 19.118 0 0 .569 O 8.610 0 .569 0 0 Transition Probabilities .475 O .065 .021 .062 .257 .116, Multiplier Matrix 0 1.070 .023 .066 0 0 1.030 0 0 .016 0 1.001 Mean Chain Length by Stratum 1.121 1.054 1.147 Mean Chain Length 16 35 18 12.765 34.700 18.712 8.040 .805 .895 .950 .876 Probability Distribution 1 .805 .895 .950 .876 1.1.", 62 66.169 1 1.000 3.683 6.086 2.841 bump—- Mean 1 9.949 11.890 12.160 11.473 11.114 OJ-‘LoNt—a * 10 Value exceeds 10 173 2 .157 .091 .046 .105 12 6.834 Mean First Passage Times 2 * * * 2 4.477 4.069 4.196 4.112 3.235 3 .031 .011 .003 .015 3 5.267 3 2.403 2.403 1.000 2.403 3 8.439 8.318 8.012 8.344 7.197 of Chain Lengths 4 .005 .002 O .003 0 0 :I-a-x-x- «‘5 4 17.296 16.904 17.101 17.183 16.053 5 .001 O 0 0 Observed Chain Length Distribution 0 Weighted Chain Length Distribution 0 First Passage Times (Including Outside) 1.242 1.121 1.054 1.147 2.125 £(t) gt) (gm-1 p! 1(t) PURE CHAINS STARTING IN OLD HOUSES, 1969 - 1970 waI— O 248 51 43 10 235.480 27.075 20.455 5.531 0 .376 .094 .061 .059 1.106 .074 .077 .014 1.147 1.182 174 Unweighted Vacancy Moves 2 3 4 315 93 22 15 4 0 53 8 1 20 8 l 3 4 1 Weighted Vacancy Moves 290.290 80.681 19.333 7.721 1.821 0 27.393 4.227 .569 10.111 4.063 .569 1.544 2.278 .569 Transition Probabilities .464 .129 .031 .027 .006 0 .081 .013 .002 .109 .044 .006 .073 .108 .027 Multiplier Matrix .033 .008 O 1.093 .015 .002 .127 1.049 .007 .118 1.029 .097 Mean Chain Length by Stratum 1.184 1.259 1.257 Mean Chain Length 282 301 78 17 251.924 284.415 72.796 16.650 .873 .844 .782 .791 Ltd H! wan—o C§L~NH 175 Probability Distribution of Chain Lengths 1 .873 .844 .782 .791 493 526.147 1.721 2.691 3.072 4.067 Mean 4.731 4.920 4.981 5.277 4.087 ' 2 .109 .133 .183 .168 152 3 .015 .020 .030 .034 30 12.156 4 5 .002 0 .003 O .005 .001 .006 .001 Observed Chain Length Distribution 2 1 Weighted Chain Lengtthistribution 86.565 .649 .277 Mean First Passage Times 2 4.375 2.959 2.844 2.844 2 4.255 4.050 3.987 4.108 3.242 3 16.460 15.594 13.474 7.449 First Passage Times 3 15.156 [15.085 14.669 13.651 14.122 4 177.933 174.825 171.011 151.157 (Including Outside) 4 66.630 66.544 66.307 64.883 65.490 1.147 1.184 1.259 1.257 2.182 gm £(t) (I—gf [y 1(t) PURE CHAINS STARTING IN OLD APARTMENTS, 1969 - 1970 waH O 129 15 F‘h’u' 126.558 8.214 1.544 1.708 .569 .327 .059 .013 .012 .021 1.064 .014 .014 .025 1.104 1.095 176 Unweighted Vacancy Moves C 3 4 109 130 25 4 5 0 6 2 1 6 13 2 1 1 6 Weighted Vacancy Moves 110.524 127.297 23.032 2.278 2.683 0 3.088 1.139 .569 3.417 6.746 .975 .569 .569 2.924 Transition Probabilities .285 .329 .059 .016 .019 0 .026 .010 .005 .025 .049 .007 .021 .021 .106 Multiplier Matrix .019 .022 0 1.027 .011 .006 .027 1.052 .008 .025 , .025 1.119 Mean Chain Length by Stratum 1.058 1.102 1.195 Mean Chain Length 127 114 127 25 125.419 113.536 125.588 22.868 .904 .947 .907 .832 .1?" 177 Probability Distribution of Chain Lengths 1 2 3 4 5 .905 .087 .008 .001 ' 0 .947 .049 .004 0 O .907 .085 .007 .001 0 .832 .146 .020 .003 0 Observed Chain Length Distribution 331 55 7 O O Weighted Chain Length Distribution HZ baron- Owar—s Mean First Passage Times (Including Outside) 1 5.857 6.099 6.143 6.172 5.125," 2 6.876 6.772 6.816 6.924 5.898 3 6.043 6.062 5.864 6.114 5.067 4 32.943 32.740 32.699 29.519 31.846 353.255 31.323 2.836 0 0 Mean First Passage Times 1 2 3 4 3.134 5,075 5.183 20.716 5.206 3.569 5.514 18.076 6.058 4.463 3.631 18.045 6.466 5.891 6.278 7.987 1.104 1.058 1.102 1.195 2.095 178 CHAINS WITHOUT STUDENTS, 1969 - 1970 Unweighted Vacancy Moves 1 2 3 4 0 O 349 479 205 81 l 80 20 5 1 414 2 6O 92 15 4 469 3 22 33 16 10 174 4 9 16 14 11 57 Weighted Vacancy Moves §(t) = 330.409 439.014 181.157 63.331 40.560 10.897 2.438 .569 361.104 29.246 46.555 7.840 2.114 434.881 10.727 16.696 7.717 5.285 166.046 4.633 7.471 7.318 ' 4.624 51.878 Transition Probabilities .§(t) = .326 .433 .179 .062 p_ .098 .026 .006 .001 .869 = .056 .089 .015 .004 .835 9- .052 .081 .037 .026 .804 .061 .098 .096 .061 .683 Multiplier Matrix 1.111 .032 .007 .002 -1 _ .070 1.102 .018 .005 (179) ‘ .068 .098 1.044 .029 .087 _ .128 .110 1.069 - Mean Chain Length by Stratum §_ - 1.153 1.196 1.239 1.392 Mean Chain Length 1(t) - 1.202 Probability Distribution of Chain Lengths 1 .869 .835 .804 .683 2 .112 .138 .160 .254 179 3 .016 .022 .030 .051 4 .002 .004 .005 .009 5 0 .001 .001 .002 Observed Chain Length Distribution 787 259 56 11 1 Weighted Chain Length Distribution 500191-— OkwNv—o 5.371 5.633 5.686 5.741 4.813 2 4.543 4.287 4.350 4.375 3.531 3 11.117 10.044 10.811 10.253 10.045 4 31.117 31.060 31.409 29.400 30.022 839.915 147.503 22.690 3.568 .277 Mean First Passage Times 1 2 3 4 1.806 4.653 15.603 42.043 3.068 2.983 14.386 40.692 3.399 3.345 12.454 35.799 3.699 3.608 10.493 32.638 Mean First Passage Times (Including Outside) 1.153 1.196 1.238 1.392 2.202 £(t) _f_(t) (1:9)“ I» 10:) buNo—I O 180 ALL CHAINS, 1964 - 1965 264 98 56 20 236.625 43.907 26.382 9.674 2.353 .335 .138 .073 .064 .040 1.165 .098 .089 .066 1.232 1.256 Unweighted Vacancy Moves 2 3 4 328 147 66 28 8 O 81 14 2 3O 17 3 8 13 6 Weighted Vacancy Moves 290.537 127.230 52.672 12.464 3.408 0 38.78} 6.697 1.139 14.205 7.559 1.299 3.737 6.209 3.008 fransition Probabilities .411 .180 .074 .039 .011 0 .108 .019 .003 .094 .050 .009 .064 .107 .052 Multiplier Matrix .052 .014 0 1.128 .024 .004 .116 1.057 .010 .092 .121 1.056 Mean Chain Length by Stratum 1.253 1.272 1.335 Mean Chain Length 310 322 129 44 259.160 286.725 118.366 42.812 .813 .797 .783 .737 #93531— Owar— 181 Probability Distribution of Chain Lengths 1 .813 .797 .783 .737 532 567.769 1.775 2.866 3.107 3.684 Mean 4.999 5.357 5.421 5.597 4.593 2 .151 .162 .172 .206 189 107.637 3 .029 .032 .035 .046 59 23.906 Observed Chain Length Distribution Weighted Chain Length Distribution 4 5 6 .006 .001 0 .006 .001 O .007 .001 0 .009 .002 O 19 4 2 6.162 1.108 .493 Mean First Passage Times 2 4.454 2.961 3.203 3.714 2 4.745 4.433 4.504 4.675 3.747 3 14.320 13.443 11.386 8.070 First Passage Times 3 10.966 10.887 10.553 9.938 9.883 4 116.391 113.352 109.968 91.190 (Including Outside) 4 28.860 28.781 28.635 .27.436 27.637 1.232 1.253 1.271 1.335 2.255 Doctor—4 I: ObOJNr—I 181 Probability Distribution of Chain Lengths l 2 3 4 5 6 .813 .151 .029 .006 .001 O .797 .162 .032 .006 .001 0 .783 .172 .035 .007 .001 0 .737 .206 .046 .009 .002 0 Observed Chain Length Distribution 532 189 59 19 4 2 Weighted Chain Length Distribution 567.769 107.637 23.906 6.162 1.108 .493 Mean First Passage Times 1 2 3 4 1.775 4.454 14.320 116.391 2.866 2.961 13.443 113.352 3.107 3.203 11.386 109.968 3.684 3.714 8.070 91.190 Mean First Passage Times (Including Outside) 1 2 3 4 O 4.999 4.745 10.966 28.860 1.232 5.357 4.433 10.887 28.781 1.253 5.421 4.504 10.553 28.635 1.271 5.597 4.675 9.938 ‘27.436 1.335 4.593 3.747 9.883 27.637 2.255 git) git) (I—Q)’ [y j(t) war— 0 25.163 .521 .044 .248 .704 V-‘LJO‘w .128 .089 .052 .055 .055 1.104 .065 .074 .078 1.223 1.246 182 CHAINS STARTING IN NEW UNITS, 1964 - 1965 Unweighted Vacancy Moves 2 3 4 103 55 36 4 4 0 21 6 0 14 7 3 2 10 4 Weighted Vacancy Moves 97.410 46.524 27.528 1.949 1.704 0 9.419 2.762 0 6.123 3.168 1.299 .729 4.500 2.114 Transition Probabilities .495 .237 .140 .049 .043 O .081 .024 O .104 .054 .022 .024 .145 .068 Multiplier Matrix .065 .051 .001 1.096 .031 .001 .126 1.068 .025 .051 .170 . 1.077 Mean Chain Length by Stratum 1.192 1.292 1.378 Mean Chain Length 44 104 51 23 32.506 97.406 44.820 21.893 .819 .842 .764 .708 bush).— OwaI-I Probability Distribution of Chain Lengths 1 .819 .842 .764 .708 152 162.220 2.731 3.206 3.622 3.992 Mean 1 11.122 11.539 11.541 11.573 11.068 2 .147 .130 .190 .224 41 23.35 Mean First Passage Times 2 3.252 2.497 2.761 3.895 2 3.967 3.819 3.804 4.174 2.992 183 3 .028 .023 .037 .054 21 8.509 3 4.843 5.394 4.811 3.064 3 7.576 7.705 7.529 6.838 6.744 4 .005 .004 .007 .011 7 2.27 4 53.136 53.687 48.294 39.361 4 15.204 15.181 14.930 14.273 '13.999 5 .001 .001 .001 .002 Observed Chain Length Distribution 1 Weighted Chain Length Distribution .277 First Passage Times (Including Outside) 0000 1.223 1.192 1.293 1.378 2.246 Eflt) _f_(t) (1:9,)- I? J .btoNo— O 236 89 43 13 211.462 40.386 20.337 6.425 .649 e414 .145 .083 .070 .024 1.175 .113 .096 .048 1.235 1.260 184 CHAINS STARTING IN OLD UNITS, 1964 - 1965 Unweighted Vacancy Moves 2 3 4 225 92 3O 24 4 0 60 8 2 16 10 0 6 3 2 Weighted Vacancy Moves 193.132 80.706 25.144 10.515 1.704 0 29.368 3.935 1.138 8.081 4.391 0 3.008 1.708 .894 Transition Probabilities .378 .158 .049 .038 .006 0 .120 .016 .005 .087 .047 0 .111 .063 .033 Multiplier Matrix .051 .008 0 1.144 .020 .005 .109 1.052 .001 .139 .071 1.035 Mean Chain Length by Stratum 1.284 1.258 1.293 Mean Chain Length 226. 189. 73. 20 266 218 78 21 655 324 547 .919 .812 .776 .796 .770 185 Probability Piri— 4.964 5.528 5.310 5.526 4.527. 2 4.541 4.244 4.261 4.481 3.561 3 11.732 11.668 11.278 10.413 10.646 4 29.613 29.541 29.505 27.934 28.389 472.785 97.386 23.501 5.514 1.108 .493 Mean First Passage Times 1 2 3 4 1.802 4.005 15.811 133.990 2.750 2.786 15.049 130.765 2.974 2.905 13.048 129.351 3.582 3.478 8.840 105.978 Mean First Passage Times (Including Outside) 1.233 1.276, 1.321 1.410 1.279 Fit) git) (I-Sf'l = [V 1(t) bLANI-d 188 CHAINS STARTING IN APARTMENTS, 1964-65 35.636 7.158 .324 .349 .166 .011 1.199 .014 1.231 1.127 Unweighted Vacancy Moves 2 31 OOUJH 3 26 OWNN oo Ob—‘OO Weighted Vacancy Moves 31.093 .324 1.463 0 0 26.753 .729 1.139 1.299 0 8.538 0 0 .569 0 Transition Probabilities .305 .008 .045 0 0 Multiplier Matrix .009 1.047 0 0 Mean Chain Length by Stratum 1.080 .262 .017 .035 .043 O .022 .038 1.046 0 1.079 .084 0 O .019 O O .001 .020 1.000 1.000 39 30 28 34.906 30.279 27.727 9.107 I’d .810 .921 .927 1.000 HZ DWNH OJ-‘UJNH 5.034 5.890 5.818 5.807 4.807 189 Probability Distribution of Chain Lengths 1 2 3 4 5 6 .810 .157 .028 .005 .001 0 .921 .073 .006 0 0 0 .927 .068 .005 .001 0 0 1.000 O 0 0 0 0 Observed Chain Length Distribution 85 18 1 2 0 O Weighted Chain Length Distribution 90.715 10.251 .405 .649 O 0 Mean First Passage Times 1 2 3 4 M? is undefined Mean First Passage Times (Including Outside) 2 3 4 0 6.992 7.581 24.056 1.231 6.602 7.318 23.904 1.086 6.902 7.255 23.440 .1.079 6.824 7.507 23.835 1-000 5.824 6.507 22.835 2.128 190 CHAINS STARTING IN NEW HOUSES, 1964 - 1965 Unweighted Vacancy Moves 1 2 3 4 l 1 O 21 83 45 34 1 8 4 2 O 38 2 13 21 4 O Bh 3 6 14 5 2 39 4 4 2 10 4 20 Weighted Vacancy Moves 5(t) = 19.595 77.061 36.349 25.394 2.952 1.949 .975 0 27.141 6.044 9.419 1.623 0 /8.l9h 2.924 6.123 2.438 .730 13.670 1.704 .730 4.500 2.114 !9.189 Transition Probabilities [(L) = .124 ..487 .229 .160 p g = .089 .058 .029 O 835 .063 .099 ,,017 0 V" .064 .133 .053 .016 71¢ .060 .026 .159 .075 .680 Multiplier Matrix _1 1.105 .077 .036 .001 (I-Q) = .079 1.118 .023 O .087 .164 1.065 .018 .089 .065 .186 1.084 Mean Chain Length by Stratum - 1.219 1.221 1.334 1.424 Mean Chain Length 1(t) = 1.279 ..lf" bWNH ObWNl-d Probability Distribution of Chain Lengths .823 .821 .734 .680 119 127.001 1 2.537 2.945 3.378 3.807 Mean First Passage Times (Including Outside) 1 10.870 11.154 11.184 11.250 10.796 2 .143 .146 .212 .239 37 21.072 Mean First Passage Times 2 2.792 2.210 2.347 3.515 2 3.943 3.790 3.730 4.197 3.017 191 3 .027 .027 .044 .064 20 8.104 3 7.380 8.050 7.039 3.966 3 7.983 8.088 7.869 7.003 7.045 4 .005 .005 .009 .014 6 1.946 4 121.222 121.891 113.842 90.276 4 13.650 13.654 13.539 12.787 12.438 5 .001 .001 .002 .003 Observed Chain Length Distribution 1 Weighted Chain Length Distribution .277 0 1.219 1.220 1.334 1.425 1.280 O .001 £(t) gm (1:951 |>v 1(t) bLaJNl-d \l CHOP 5.568 .569 .324 .164 .088 .025 1.100 .002 .030 1.238 1.120 192 Unweighted Vacancy Moves 2 16 COCO 10 ONNN N Of—‘OO Weighted Vacancy Moves 16.080 0000 Transition Probabilities .474 0000 10.175 .730 .139 .730 O .300 .113 .071 .057 O Multiplier Matrix 0 1.000 O 0 .132 .075 1.064 0 CHAINS STARTING IN NEW APARTMENTS, 1964-1965 2.134 .569 .006 .003 .047 1.000 Mean Chain Length by Stratum 1.081 1.141 Mean Chain Length 1.000 .799 .929 .873 1.000 ..lf" 193 Probability Distribution of Chain Lengths I3 1 2 3 4 5 6 .799 .169 .028 .004 0 0 .929 .062 .008 .001 0 0 .873 .115 .011 .001 O O 1.000 0 0 O 0 0 Observed Chain Length Distribution 29 4 1 1 O O Weighted Chain Length Distribution 30.950 2.278 .405 .324 0 0 Mean First Passage Times 1 2 3 4 l 2 ‘MS is undefined 3 4 Mean First Passage Times (Including Outside) l 2 3 4 O 1 11.139 4.633 5.351 26.700 1.238 2 12.076 4.476 5.511 26.610 1.081 3 11.829 4.536 5.635 25.497 1.141 4 12.018 4.395 5.855 26.619 1.000 0 11.018 3.395 4.855 25.619 2.120 git) £(t) (I-gfl l)’ 1(t) 194 CHAINS STARTING IN OLD HOUSES, 1964-1965 bWNH 202 77 43 13 181.394 33.798 20.337 6.426 .649 .410 .139 .089 .085 .031 1.169 .122 .119 .067 1.236 1.279 Unweighted Vacancy Moves 2 3 4 210 76 24 23 4 0 57 8 2 16 9 0 6 3 2 Weighted Vacancy Moves 178.119 64.128 18.741 10.190 1.704 0 27.905 3.935 1.139 8.081 3.822 0 3.008 1.708 .894 Transition Probabilities .403 .145 .042 .042 .007 0 .123 .017 .005 .107 .051 0 .145 .082 .043 Multiplier Matrix .057 .010 O 1.150 .022 .006 .135 1.057 .001, .187 .095 1.046 Mean Chain Length by Stratum 1.300 1.312 1.395 196. 173. 56. 14. 233 202 62 15 912 987 969 515 .812 .765 .757 .699 1.1.1, 195 Probability Distribution of Chain Lengths 1 2 3 4 5 6 .812 .151 .030 .006 .001 O .765 .183 .040 .009 .002 O .757 .190 .042 .009 .002 O .699 .228 .057 .013 .003 .001 Observed Chain Length Distribution 324 134 38 11 3 2 Weighted Chain Length Distribution bUJNl-J ObWNH 1 4.156 4.414 4.439 4.738 3.621 2 4.781 4.435 4.511 4.362 3.799 3 13.945 13.839 13.389 12.968 12.839 4 50.587 50.374 50.643 48.531 49.366 Mean First Passage Times (Including Outside) 0 1.236 1.300 1.311 1.394 2.279 345.784 76.314 15.397 3.568 .831 .493 Mean First Passage Times 1 2 3 4 1.701 4.296 19.676 139.067 2.684 2.930 18.472 134.771 2.759 3.166 16.039 137.937 3.551 2.694 13.904 117.840 git) {(1) (1-951 [>8 j(t) «DWNH 30.068 6.589 0 0 0 .442 .179 1.231 1.131 196 CHAINS STARTING IN OLD APARTMENTS, 1964 - 1965 Unweighted Vacancy Moves 2 3 4 15 16 6 1 0 0 3 0 0 0 l 0 0 O 0 Weighted Vacancy Moves 15.013 16.578 6.403 .324 0 O 1.463 0 O 0 .569 O O 0 0 Transition Probabilities .221 .244 .094 .009 0 0 .087 0 0 0 .033 0 0 0 0 Multiplier Matrix .012 0 0 1.095 0 0 0 1.034 O 0 0 1.000 Mean Chain Length by Stratum 1.095 1.034 1.000 Mean Chain Length 33 16 16 29.744 15.337 16.578 6.403 .811 .913 .967 1.000 197 Probability Distribution of Chain Lengths l3 L‘WNH ODUJNH 1 2 3 4 5 6 .811 .. .154 .028 .005 .001 0 .913 .080 .007 .001 O O .967 .032 .001 O 0 0 1.00 0 0 O 0 0 Observed Chain Length Distribution 56 14 0 1 0 O Weighted Chain Length Distribution 59.765 7.973 O .324 0 0 Mean First Passage Times 1 2 3 4 .MS is undefined Mean First Passage Times (Including Outside) 1 2 3 4 O 3.958 9.493 8.948 22.888 1.231 4.690 8.635 8.813 22.752 1.095 4.629 9.398 8.461 22.691 1.034 4.594 9.364 8.717 22.657 1.000 3.594 8.364 7.717 21.657 2.132 git) gt) (I-Q)’1 ‘>’ 1(1) DWNH 217 74 52 19 198.065 32.042 24.758 9.350 2.029 .334 .120 .078 .080 .043 1.143 .103 .107 .074 1.213 1.262 198 PURE CHAINS, HOUSES ONLY, 1964 — 1965 Unweighted Vacancy Moves 2 3 4 291 116 55 25 6 O 77 10 2 26 12 2 7 13 4 Weighted Vacancy Moves 254.286 98.365 42.835 11.410 2.679 0 36.999 4.909 1.139 12.662 5.366 .730 3.332 6.208 2.114 Transition Probabilities .428 .166 .072 .043 .010 0 .116 .015 .004 .107 .046 .006 .071 .133 .045 Multiplier Matrix .057 .013 0 1.139 .020 .004 .134 1.052 .007 .106 .148 1.049 Mean Chain Length by Stratum 1.266 1.301 1.377 Mean Chain Length 262 285 98 34 220 250 89 33 .112 .883 .420 .135 .827 .787 .761 .708 I: J-‘LANH O-L‘LONH Probability Distribution of Chain Lengths .827 .787 .761 .708 443 472.785 1 1.861 2.795 2.906 3.547 2 .141 .170 .190 .225 164 93.399 199 3 .026 .034 .040 .053 54 21.880 4 5 .005 .001 .007 .001 .008 .002 .011 .002 Observed Chain Length Distribution 12 4 Weighted Chain Length Distribution 3.892 1.108 Mean First Passage Times 2 3.826 2.666 2.920 3.420 3 15.143 14.669 12.853 8.063 4 121.607 118.589 117.267 101.133 0“ 0000 .493 Mean First Passage Times (Including Outside) 5.042 5.299 5.309 5.552 4.550 2 4.505 4.212 4.268 4.462 3.532 3 11.782 11.754 11.422 10.401 10.717 4 29.895 29.831 29.781 28.673 26.691 1.213 1.266 1.300 1.377 2.262 200 PURE CHAINS, APARTMENTS ONLY, 1964 - 1965 Unweighted Vacancy Moves 1 2 3 4 O 0 30 29 24 8 1 3 0 1 0 29 2 0 l 1 0 28 3 0 0 1 0 26 4 0 O 0 0 8 Weighted Vacancy Moves Fit) = 29.862 29.954 25.614 8.538 1.708 0 .405 0 29.457 0 .569 .569 0 29.385 0 O .405 0 26.588 0 O O O 8.538 Transition Probabilities £(t) = .318 .319 .273 .091 ‘p Q. = .054 0 .013 O .933 O .019 .019 0 .963 0 0 .015 0 .985 O 0 0 0 1.000 Multiplier Matrix (ygfl. = 1.057 0 .014 0 O 1.019 .019 0 O 0 1.015 0 0 O O 1.000 Mean Chain Length by Stratum A_ - 1.071 1.038 1.015 1.000 1(t) 1.039 I: ltd bWNH Owal-d 201 Probability Distribution of Chain Lengths 1 2 3 4 5 .933 .063 .004 O 0 .963 .036 .001 O 0 .985 .015 0 0 0 1.000 O 0 0 0 Observed Chain Length Distribution 85 5 1 0 0 Weighted Chain Length Distribution 90.715 2.848 .405 0 0 Mean First Passage Times 2 3 ‘M3 is undefined 6.069 6.384 6.361 6.346 5.346 2 6.430 6.277 6.374 6.359 5.539 3 7.165 7.093 7.098 7.191 6.191 4 4 22.514 22.481 22.458 22.443 21.443 0“ GOOD Mean First Passage Times (Including Outside) 0 1.071 1.038 1.015 1.000 2.039 202 PURE CHAINS STARTING IN NEW HOUSES, 1964 - 1965 Unweighted Vacancy Moves 1 2 3 4 0 0 21 83 42 33 l 8 4 2 O 38 2 13 20 2 0 86 3 6 12 3 2 36 4 4 2 10 3 19 Weighted Vacancy Moves _F(t) = 19.595 77.061 35.131 24.824 2.952 1.949 .975 0 27.343 6.044 9.095 .975 0 78.196 2.924 5.475 1.544 .730 32.452 1.704 .730 4.500 1.544 18.620 Transition Probabilities £(t) = .125 .492 .224 .159 ‘p _ .089 .059 .029 0 .823 Q ‘ .064 .096 .010 0 .829 .067 .127 .036 .017 .753 .063 .027 .166 .057 .687 Multiplier Matrix _1 1.106 .077 .035 .001 (1:0) = .079 1.119 .014 0 .090 .15? 1.045 .019 .092 .064 .187 1.064 Mean Chain Length by Stratum A_ 1.217 1.208 1.306 1.406 Mean Chain Length J(t) - 1.263 P IZ «L‘UJNH ObWNH .823 .829 .753 .687 119 127.001 1 2.414 2.723 2.082 3.509 203 Probability Distribution n: Chiin Lengths 2 i 5 .144 .(1L' ." m ,1 .(101 .140 .025 .005 .001 .200 .039 197 .001 .238 .ChU “13 .002 Observed Chain Lenvft L1 .rrbution 35 20 4 1 Weighted Chain Length DistribuLIUn 19.933 8.104 '.297 .277 Mean First Pass ge Tim: 2 3 i 2.788 8.;28 127.494 2.187 9.’71 _ft 637 2.336 8 =1: 1;9.166 3.423 4 :64 101 043 Mean First Passage limes (loz'nding Outside) 1 10.670 10.938 10.926 11.005 10.578 2 5 4 3.907 8.213 11.717 3.758 8.37; 3.712 3.709 8.219 13.569 4.144 7.352 15.080 2.978 7.280 ‘2.507 A v OOC 1.218 1.208 1.307 1.406 2.263 P(1) £(t) (_1—91‘1 ‘y 1(t) bUJNi-d 204 PURE CHAINS STARTING IN NEW APARTMENTS, 1964 - 1965 U1 CCCC 4.674 CCCC .146 CCCC 1.000 1.090 1.043 Unweighted Vacancy Moves A. 15 CCCC c>eaeae1 4 N CCCC Weighted Vacancy Moves 15.511 CCCC Transition Probabilities .486 CCCC 9.605 .405 .569 .405 0 .301 .087 .037 .037 0 2.134 CCCC .067 CCCC Multiplier Matrix 0 1.000 O 0 Mean Chain Length by Stratum 1.038 .090 .038 1.038 O 1.038 3 0 0 1.000 1.000 4.269 14.941 10.580 2.134 .913 .963 .963 1.000 HZ J-‘LONH C-1--\Lau>l\>1--I 1 2 .913 .083 .10; .963 .035 .001 .963 .036 .001 1.000 0 O 29 1 1 Weighted Chain Lengti 30.588 .569 .405 1 2 3 M*is undefined Mean First Passage Time. ‘1 2 3 13.954 4.256 3.6‘2 13.902 4.204 3 938 13.902 4.205 5.937 13.864 4.167 5. 526 12.864 3.167 5.127 205 Probability Dis!“ 1. ~ 0’ )c: H Lengths \J'I CCCC Observed Chain Length Distribution ’1 J 0 .istribution r \a 0 Mean First Passage Times 4 {including Outside) 30.646 30.594 30.594 39.556 29.356 1.090 1.038 1.038 1.000 2.043 0‘ CCCC PURE CHA INS 3'! 0 196 66 39 13 .SVJNt-I‘ g(1) 178.470 29.090 18.714 6.426 .324 011‘ 7 .408 .125 9 .083 .086 .016 1.149 .112 .116 .045 1.213 1(t) = 1.262 1941: 11\- 01.1) HOUSES, .206 1964 - 1965 nweighted Vacancy Moves 2 3 4 208 74 22 21 4 0 57 8 2 14 9 0 5 3 1 Weighted Vacancy Moves 177.225 63.234 18.011 9.461 1.704 0 27.905 3.935 1.139 7.187 3.822 0 2.602 1.708 .569 Transition Probabilities .406 .145 .041 .041 .007 0 .124 .018 .005 .097 .051 0 .132 .087 .029 Multiplier Matrix .054 .010 0 1.150 .023 .006 .122 1.057 .001 .168 .098 1.031 Mean Chain Length by Stratum 1.291 1.296 1.341 Mean Chain Length 224 199 62 15 192 172 56 14 .769 .688 .969 .515 .822 .770 .766 .736 I: 11"" «L‘WNH OJ-‘UJNH .822 .770 .766 .736 324 345.784 1.777 2.809 2.764 3.719 Mean 4.241 4.475 4.463 4.809 3.659 207 2 .140 .182 .185 .203 129 73.467 Mean First Passage Times 2 4.095 2.784 3.214 2.593 2 4.750 4.404 4.535 4.377 3.777 3 .026 .039 .039 .048 34 13.776 3 17.794 16.740 14.458 11.768 3 13.830 13.736 13.283 12.794 12.749 4 .005 .008 .008 .010 Observed Chain Length Distribution 8 Weighted Chain Length Distribution 2.595 4 127.269 123.174 126.388 111.995 4 51.507 51.298 51.372 50.115 50.309 Probability Distribution of Cnain Lengths .001 .002 .002 .002 .831 First Passage Times (Including Outside) 0 1.213 1.291 1.296 1.341 2.262 .493 208 PURE CHAINS STARTING IN OLD APARTMENTS, 1964 - 1965 Unweighted Vacancy Moves l 2 3 4 0 0 25 14 15 6 l 3 0 0 0 25 2 0 1 0 0 14 3 O 0 0 O 15 4 0 O 0 O 6 Weighted Vacancy Moves git) = 25.188 14.444 16.009 6.403 1.708 0 0 0 25.188 0 .569 0 0 14.444 0 0 0 0 16.009. 0 0 0 0 6.403 Transition Probabilities git) = .406 ' .233 .258 .103 .2 .064 0 0 0 .936 0 .038 0 0 :962 0 0 0 0 1.000 0 0 0 0 1.000 Multiplier Matrix (17g)‘1 = 1.068 0 0 0 0 1.039 0 0 0 0 1.000 0 0 0 0 1.000 Mean Chain Length by Stratum A. 1.068 1.039 1.000 1.000 lit) - 1.037 209 Probability Distribution of Chain Lengths l 2 3 4 5 .936 .059 .004 O 0 .962 .036 .001 O O 1.000 0 0 0 0 1.000 0 O 0 0 Observed Chain Length Distribution 56 4 0 O O Weighted Chain Length Distribution 59.765 2.278 O O 0 Mean First Passage Times 1 2 3 4 ‘Mf is undefined GOOD I: book)!" ObUJNH Mean First Passage Times (Including Outside) 1 4.698 4.989 4.949 4.949 3.949 2 8.777 8.417 8.710 8.710 7.710 3 7.961 7.933 7.894 7.894 6.894 4 19.803 19.774 19.735 19.735 18.735 0 1.068 1.039 1.000 1.000 2.037 £02) £(t) (3:9)"1 1(t) buNH 210 CHAINS WITHOUT STUDENTS, 1964 - 1965 236 88 48 19 209.729 39.193 22.316 9.350 2.029 .327 .139 .067 .068 .038 1.167 .090 .094 .063 1.243 1.260 Unweighted Vacancy Moves 2 302 28 78 29 6 3 134 8 13 16 13 4 61 Uwuahic> Weighted Vacancy Moves 268.014 114.761 12.464 37.488 13.799 2.843 3.408 6.128 7.234 6.208 49.320 0 .569 1.299 2.683 Transition Probabilities .418 .044 .112 .100 .053 Multiplier Matrix .060 1.133 .125 .080 1.249 .179 .012 .018 .053 .115 .016 .023 1.060 .131 1.290 .077 0 .002 .009 .050 0 O .010 .053 Mean Chain Length by Stratum 1.327 227 268 106 40. 272 303 117 41 .550 .108 .057 108 .805 .801 .770 .745 211 Probability Distribution of Chain Lengths I 1 2 3 4 5 6 .805 .156 .031 .006 .001 0 .801 .159 .032 .006 .001 0 .770 .182 .038 .008 .002 0 .745 .198 .045 .009 .002 0 Observed Chain Length Distribution 481 173 58 15 4 2 Weighted Chain Length Distribution 513.340 98.525 23.501 4.865 1.108 .493 Mean First Passage Times I: #WNH ObWNH l 2 3 4 1.889 .4.130 13.563 148.277 3.046 2.701 12.864 146.007 3.218 3.073 10.896 140.039 3.827 3.720 7.030 116.373 Mean First Passage Times (Including Outside) 5.132 5.534 5.555 5.752 4.746 2 4.648 4.335 4.412 4.643 3.664 3 10.950 10.879 10.530 9.829 9.876 4 28.282 28.234 28.047 26.924 27.046 0 1.243 1.248 1.290 1.327 2.260 APPENDIX B MAPS This appendix contains the maps showing the study area and the housing sub-areas. 212 213 paw-m 6 x / n .x'z/ \e/ . \éha Twp.— Maid-9n Congressional Delta Twp. Lansing—East Lansing (U.S. Bureau of the Census, 1971) ..r / , lensing Taup. Pan / ED 3038 /. pa“ {ED/522} ..z 2 \ 7 ... 9 1 Delhi Twp, 214 .Qm Mamm><3 MapZ Housing Sub -Areas L ow 1. §§ 2. Lower Middle no 11 Au .4. v; e nr hr nu DJ :o%}J.o. 9.....6951 "V é. . ;.u 09 $¢w 9A .u:~ ”on .4 éuvnwav9n .66 PPS; aw!” ..VP P Scan}. P2 BIBLIOGRAPHY BIBLIOGRAPHY Abu-Lughod, Janet and Mary Mix Foley 1970 "The Consumer Votes by Moving." Pp. 460-78 in Robert Gutman and David Popenoe (eds.) Neighbourhood, City and Metropolis. New York: Random House. Adams, John S. 1969 "Directional Bias in Intra-Urban Migration." Economic Geographyg45:302-23. Albig, William 1936 "A Method of Recording Trends in Urban Residential Mobility." Sociology and Social Research 21:120-27. Anderson, Theodore W. and Leo A. Goodman 1957 "Statistical Inference about Markov Chains." Annals of Mathematical Statistics 28:89-110. Arminger, Louis Earl Jr. 1966 Toward a Model of the Residential Location Decision Process: A Study of Recent and Prospective Buyers of New and Used Homes. Chapel Hill: Center for Urban and Regional Studies. Berry, Brian J.L. and Duane F. Marble (eds.) 1968 §P3tial Analysis: A Reader in Statistical Geography. Englewood Cliffs: Prentice Hall. Beshers, James M. 1962 Urban Social Structure. Glencoe: Free Press. and Edward.0. Laumann 1967 "Social Distance: A Network Approach." American Sociological Review 32:225-36. Blumen, Isadore, Marvin Kogan and Philip McCarthy 1966 "Probability Models for Mobility." Pp. 318-34 in Paul F. Lazarsfeld and Neil W. Henry (eds.) Readings in Mathematical Social Science. Cambridge: M.I.T. Brown, Laurence A. and Frank E. Horton 1970 "Functional Distance: An Operational Approach." Geographical Analysis 2:76-83. 215 216 and Eric G. Moore 1970 "The Intra-Urban Migration Process: A Perspective." Geografiska Annaler, Series B 52:1—13. , Frank E. Horton and Robert I. Wittick 1970 "On Place Utility and the Normative Allocation of Migrants. Demography 7:175-83. and John Holmes 1971 "Intra-Urban Migrant Lifelines: A Spatial View." Demography 8:103-22. Butler, Edgar W., F. Stuart Chapin, George C. Hemmans, Edward J. Kaiser, Michael Stegman and Shirley F. Weiss 1969 Moving Behavior and Residential Choice: National Cooperative Highway Research Program, Report 81. Highway Research Board. Cave, P.W. 1969 "Occupancy Duration and the Analysis of Residential Change." Urban Studies 6:58f69. Chayes, F. 1962 "Numerical Correlation and Petrographic Variation." Journal of Geology 70:440—52. Cochran, William G. 1963 Sampling Techniques, 2nd Edition. New York: Wiley. Duncan, Otis D., Ray P. Cuzzort and Beverly Duncan 1961 Statistical Geography. Glencoe: Free Press. Fisher, Ernest M. and Louis Winnick 1951 "A Reformulation of the Filtering Concept." Journal of Social Issues 7:47-58. Folger, John K. 1958 "Models in Migration." Proceedings of the 34th Annual gpnfarenge of the Milhgnk Memorial FundLl957L Part III. aw York. Foote, Nelson N., Janet Abu-Lughod, Mary Mix Foley and Louis Winnick 1960 _flgusing Choices and Housing Constraints. New York: McGrsw-Hill. Forrester, Jay W. 1969 gggban Dynamics. Cambridge: M.I.T. Gibbs, Jack P. (ed.) 1961 Urban Research Methods. Princeton: Van Nostrand. 217 Gilbert, G. 1972 "Two Markov Models of Neighbourhood Housing Turnover." Environment and Planning 4:133-46. Goldscheider, Calvin 1971 Population, Modernization and Social Structure. Boston: Little, Brown. Goldstein, Sidney 1954 "City Directories as Sources of Migration Data." American Journal of Sociology 60:169-76. 1964 "The Extent of Repeated Migration: An Analysis Based on the Danish Population Register." Journal of the American Statistical Association 59:1121—32. Goodman, Leo 1961 "Statistical Methods for the Mover—Stayer Model." Journal of the American Statistical Association 56:841-68. Grigsby, William G. 1963 Housing Markets and Public Policy, Philadelphia: University of Pennsylvania. Haenszel, William 1967 "Concept, Measurement and Data in Migration Analysis." Demography 4:253-61. Hawkes, Roland K. 1972 ”Spatial Patterning of Urban Population Characteristics." American Journal of Sociology 78:1216—35. Hodge, Robert W. 1966 "Occupational Mobility as a Probability Process." Demography 3:19-34. Hoyt, Homer 1939 The Structure and Growth of Residential Neighbourhoods in American Cities. Washington: Government Printing Office. Hua, Changi 1972 Modelling HousingyVacancy Transfer in the Study of Housing Sector Interaction. Harvard University: Unpublished Thesis. Ianni, Francis A.J. 1957 "Residential and Occupational Mobility as Indices of the Awulturutiun of an I‘ltlmlu ()l“m:p." :lfl'lal Forces 36:05—72. 218 Johnson, R.J. 1969 "Some Tests of a Model of Intra-Urban Population Mobility: Melbourne, Australia." Urban Studies 6:34-57. Jones, F. Lancaster 1968 "Social Area Analysis: Some Theoretical and Methodological Comments Illustrated with Australian Data." British Journal of Sociology 19:424—44. Kemeny, John G. and J. Laurie Snell 1960 Finite Markov Chains. Princeton: Van Nostrand. Kristof, Frank S. 1965 "Housing Policy Goals and the Turnover of Housing." Journal of the American Institute of Planners 31:232-45. Kuznets, Simon and Dorothy S. Thomas 1958 "Internal Migration and Economic Growth." Proceedings of the 34th Annual Conference of the Milbank Memorial Fund, 1957, Part III:196-211. Land, Kenneth C. 1969 "Duration of Residence and Prospective Migration: Further Evidence.". Demography 6:133-40. Lansing, John 3., Charles Wade Clifton and James N. Morgan 1969 New Homes and Poorer People. Ann Arbor: Institute for Social Research. LaPiere, Richard T. 1934 "Attitudes and Actions." Social Forces 13:230-37. Lazarsfeld, Paul F. 1959 "Problems in Methodology." Pp. 39-78 in Robert K. Merton, Leonard Broom and Leonard S. Cottrell, Jr. (eds.), Sociology Today. New York: Basic Books. Levin, Yale and Alfred Lindesmith 1961 "English Ecology and Criminology of the Past Century." Pp. 14-21 in George A. Theodorson (ed.), Studies in Human Ecology. Evanston: Row, Peterson. Lingoes, James C. 1973 The Guttman-Lingoes Nonmetric Program Series. Ann Arbor: Msthesis Press. Lowry, Ira S. 1960 "Filtering and Housing Standards: A Conceptual Analysis." Land Economics 36:362-70. 219 Metres, Judah 1973 ngulations and Societies. Englewood Cliffs: Prentice-Hall. McAllister, Ronald J., Edward J. Kaiser and Edgar W. Butler 1971 "Residential Mobility of Blacks and Whites: A National Longitudinal Survey." American Journal of Sociology 77: 445-56. McFarland, David D. 1970 "Intragenerational Social Mobility as a Markov Process: Including a Time-Stationary Markovian Model that Explains Observed Declines in Mobility Rates over Time." American Sociological Review 35:463-76. McGinnis, Robert 1968 "A Stochastic Model of Social Mobility." American Sociolggical Review 33:712-22. and John White 1967 "Simulation Experiments on a Stochastic Attraction Medal." Unpublished Manuscript, Cornell University. Michelson, William 1970 Man and His Urban Environment. Menlo Park: Addison-Wesley. Moore, E.G. 1966 "Models of Migration and the Intra-Urban Case." Australian and New Zealand Journal of Sociology 2:16-37. 1969 "The Structure of Intra-Urban Movement Rates: An Ecological Model." Urban Studies 6:17-33. Morrison, Peter A. 1967 "Duration of Residence and Prospective Migration: The Evaluation of a Stochastic Model." Demography 4:553-61. 1971 "Chronic Movers and the Future Redistribution of Population: A Longitudinal Analysis." Demography 8:171-84. Myers, George C., Robert McGinnis and George Masnick 1967 "The Duration of Residence Approach to a Dynamic Stochastic Model of Internal Migration: A Test of the Axiom of Cumulative Inertia." Eugenics Quarterly 14:121-26. Olds, Edward B. 1961 "The City Block as a Unit for Recording and Analyzing Urban ”“15“." PP: 1‘98'65 ‘43 «1‘30“ P, 01»be (ed.), Urban Research Methods. Princeton: Van Nostrand. 220 Pickvance, C.G. 1973 "Life Cycle, Housing Tenure and Intra-Urban Residential Mobility." Sociological Review 21:279—97. Ratcliff, Richard U. 1949 Urban Land Economics. New York: McGraw-Hill. Robinson, John P., Robert Athanasiou and Kendra B. Head 1969 Measures of Occupational Attitudes and Occupational Characteristics. Ann Arbor: Institute for Social Research. Rogers, Andrei 1966 "A Markovian Analysis of Migration Differentials." Proceedings of the American Statistical Association, Social Science Sectionz452-66. Rossi, Peter H. 1955 Why Families Move. Glencoe, 111.: Free Press. Schmid, Calvin F. and Earle H. McCannell 1955 "Basic Problems, Techniques and Theory of Isopleth Mapping." Journal of the American Statistical Association 5_0_:220-39. Shevky, Eshref and Wendell Bell 1955 Social Area Analysis. Stanford: Stanford. Shryock, Henry S. Jr. 1967 Population Mobility Within the United States. Chicago: Community and Family Study Center. , Jacob S. Siegel and Associates 1971 The Methods and Materials of Demography. Washington: U.S. Government Printing Office. Simmons, James W. 1968 "Changing Residence in the City: A Review of Intraurban Mobility." Geographical Review 58:622-51. Smith, Wallace F. 1970 "Filtering and Neighbourhood Change," pp. 64-89 in Michael A. Stegman (ed.), Housing and Economics: The American Dilemma. Cambridge: M.I.T. Spilerman, Seymour 1972a "The Analysis of Mobility Processes by the Introduction of Independent Variables into a Markov Chain." American Sooiolggical Review 371277-94. ' 3 1972b "Extensions of the Mover-Stayer Model." American Journal of Sociology 78:599-626. 221 Stouffer, Samuel A. 1940 "Intervening Opportunities: A Theory Relating Mobility and Distance." American Sociological Review 5:845-67. Straits, Bruce C. 1968 "Racial Residential Succession." Mimeographed paper presen- ted for discussion at the 1968 Meetings of the Population Association of America. Taeuber, Karl E. 1961 "Duration-of—Residence Analysis of Internal Migration in the United States." Milbank Memorial Fund Quarterly 39:116-31. 1966 "Cohort Migration." Demography 3:416-22. , William Haenszel and Monroe G. Sirken 1961 "Residence Histories and Exposure Residences for the United States Population." Journal of the American Statistical Association 56:824-34. Tarver, James D. and William R. Gurley 1965 "A Stochastic Analysis of Geographic Mobility and Population Projections of the Census Divisions in the United States." Demography 2:134-39. U.S. Bureau of the Census 1971 Census of Housing: 1970. Block StatisticsJ Final Rgport HC(3)-125J Lansing, Mich. Urbanized Area. Washington, D.C.: U.S. Government Printing Office. Watson, Chris J. 1974 "Vacancy Chains, Filtering and the Public Sector," Journal of the American Institute of Planners 40:346-52. Webb, Eugene J., Donald T. Campbell, Richard D. Schwartz and Lee Sechrest 1966 Unobtrusive Measures: Nonreactive Research in the Social Scluncou. Chicago: Rand McNally. White, Harrison C. 1970 Chains of_Qpportuni§y. Cambridge: Harvard. 1971 "Multipliers, Vacancy Chains, and Filtering in Housing." Journal of the American Institute of Planners 37:88-94. Winnick, Louis 1960 "Economic Constraints.” Pp. 3-67 in Nelson N. Foote, Janet Abuubushud. Mary ”is Foley and Louis Winniuk (eds.), Housing Ehgicaa and Housing Constraints. New York: McGraw—HiII: 222 Zipf, G.K. 1946 "The P1P2 Hypothesis: On the Intercity Movement of Persons. D American Sociological Review 11:677-85. H ICHIGRN STRTE UNIV. LIBRRR IES l l ll lllll ll llll ll lllll ll Ill Ill ll l Ill ll llll l l l 31293102518275