A SEAM - MARKOV MODEL or momuw m \Nuusnm » SOC\EHE$ I‘ Dissertation for the Degree of Ph. D. ' ” MlCH\GAN STATE UNIVERSHY PAUL HOWARD TRESS , 1976' ' ' A llllllllllAll]lllflllllllllljllll _‘ 1. Q}, 2171 3 1293 01730 87 . frigid} -‘-.. .‘ Ufiigtiafly This is to certify that the thesis entitled A SEMI-MARKOV MODEL OF MOBILITY IN INDUSTRIAL SOCIETIES presented by Paul Howard Tress has been accepted towards fulfillment of the requirements for Ph. D. degree in Sociology New Major professor Date June 14, 1976 0-7639 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE DEC 1 0 3 6/01 c:/C|RC/DIIDMO.p$-¢L15 ABSTRACT A SEMI-MARKOV MODEL OF MOBILITY IN INDUSTRIAL SOCIETIES By Paul Howard Tress One of the characteristics of advanced industrial soci- eties is the existence of high rates of social mobility. Social mobility studies have usually concentrated on the con- sequences of mobility and have attempted to give a formal rep- resentation of the mobility process. However, the antecedents of mobility are rarely explored, and there has been little or no attempts to incorporate the effects of these antecedents into formal representations of the mobility process. This research has three aims: (1) to see how aspects of the status-role of the individual, especially economic aspects, effect the propensity of the individual to be mobile, (2) to attempt to represent these aspects of the status-role of the individual in a stochastic model of mobility, and (3) to see if this approach is feasible by applying the model to data collected from a longitudinal study in an industrial society. It is first shown that‘most models of mobility are not cognizant of the dynamics of the social structure where mo- bility occurs. In addition, the models usually inadequately Paul Howard Tress represent aspects of the status-role of the individual that may affect mobility. The status-role of the individual effects the life-chances of the individual. In advanced in- dustrial societies, one of the life-chances of the individual is the tendency to be mobile. A stochastic model, the semi-Markov model, is described. The model represents mobility in terms of two processes: movement among a set of graded occupational states, and the time spent in a given occupational state before a move to a different occupational state occurs. The movement among a set of graded occupational states is common to most stochastic models of mobility. We are interested in the second process, the time spent in a state before a move occurs. This is termed the waiting time. The waiting time is represented by a probability distribution function that usually has a Gamma form. We argue aspects of the status-role of the individual affect the parameters of this distribution, which, in turn, would affect the propensity to be mobile. Three aspects of the status-role of the individual are investigated: sex, whether or not the individual is self- employed, and whether or not the industry in which the occupa- tion is located has been growing in terms of the number of full time workers. The model was applied to a closed system of 511 individuals in Great Britain from 1963 to 1970. Specifically, we looked at individuals who changed occupations and asked how long the individual was in the 1963 occupation Paul Howard Tress before the individual moved to a different occupation. This information provided the information needed to estimate the parameters of the semi-Markov model, especially the waiting time distribution. Previous models of mobility that have incorporated the idea of a waiting time distribution have assumed the veracity of the axiom of cumulative inertia. The axiom states the longer an individual stays in a state, the harder it is to move out of that state. Our data do not support the axiom for the entire sample, or for any specific type of move, or for any conditions of the status-role of the individual. It was observed, with one exception, the average time until a move occurs is longer for upward than downward moves. In addition, the time until a move occurs is longer for movesbetween non- adjacent states than for moves between adjacent states. Fin- ally, there is no relationship between the frequency of a move and the average time until that type of move occurs. waiting time patterns were also examined for the status— role variables. The sex of the individual did not seem to affect the waiting time distribution. The self-employment status of the individual affected the waiting time distribution. Compared to self-employed individuals, non-self-employed in- dividuals seemed to wait shorter periods of time for upward moves and longer periods of time for downward moves. Finally, the industry in which the occupation is located affected the waiting time distribution. Compared to individuals in Paul Howard Tress non-growing industries, individuals in growing industries waited shorter periods of time for upward moves and longer periods of time for downward moves. The ease of our analyses suggests it is feasible to in- corporate other aspects of the status-role of the individual into a mobility model. We indicate this should be followed by the deve10pment of a research program to develop a formal theory of mobility. A SEMI-MARKOV MODEL OF MOBILITY IN INDUSTRIAL SOCIETIES By Paul Howard Tress A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Sociology 1976 Cc) Copyright by PAUL HOWARD TRESS 1976 TABLE OF CONTENTS LIST OF TABLES . INTRODUCTION . Chapter I. PURPOSES AND SCOPE OF RESEARCH . 1.1 Why Study Mobility . . . . 1. 2 Scope Conditions of Research . . 1. 3 Mobility in Industrial Societies . 1.4 Ways to Study Mobility . . . II. MODELS OF MOBILITY . 2.1 Causal- static Models . . . . . 2.2 Stochastic Models: An Overview . 2. 3 The SemieMarkov Model and Waiting Time Distributions . III. SAMPLE AND METHOD OF ANALYSIS 3.1 The Butler and Stokes Data: Great Britain from 1963 to 1970 . . 3. 2 Problems of Using Pre-Collected Panel Data . 3. 3 Operationalization of Variables 3. 4 Methods of Analysis. 3.5 Computation of Parameters of the Waiting Time Distribution. IV. FINDINGS OF THE RESEARCH . 4.1 Analysis of the waiting Time Distribution . CONCLUSION . APPENDIX A - MATHEMATICAL MODELS . .APPENDIX B - QUESTIONS AND CODING 0F RESPONSES . .APPENDIX C - OCCUPATIONAL CODES AXPPENDIX D - THE GAMMA DISTRIBUTION B IBLIOGRAPHY . iii iv 12 23 28 32 34 40 53 53 63 68 77 83 92 94 108 118 123 126 128 130 [.4 O \OCDN O\ UT b LA.) N I-‘ LIST OF TABLES Distribution of the Sample Among Occupational Grades in Terms of Proportions. . . Indices of Industrial Production in Great Britain, 1964- 1970 . . Distribution of the Civilian Labor Force of Great Britain: 1939-1970. Growth Rates of Industrial Categories in Great Britain: 1939-1970.. . Distribution of the Sample as of 1963 by Background Factors of Mobility. . . Number of One- Time Moves Made by Different Time Points . . Means of the Waiting Time Distribution. Variances of the Waiting Time Distribution. Time at Which the Maximum Number of Moves Occurs of the Waiting Time Distribution in Years Post 1963 . . Observed Average waiting Times in Years . iv 20 57 59 62 76 84 88 89 9O INTRODUCTION Independently of their political basis, modern societies may be characterized by movement toward capital intensive and non-agricultural employment, or industrialization; movement toward greater concentrations of population, or urbanization; and movement toward formalized social relationships, or bureau- cratization. This research concerns one part of the process, the movement among ordered occupational grades that is charac- teristic of industrialization. We define social mobility as the movement of individuals or family lines in a social structure over a period of time. The social structure is defined as the web of institutions and organizations that contribute to the fulfillment of the on-going needs of collectives of individuals. Implicit in the analysis of social mobility in modern industrial societies is the domi- nance of the economic institution. This dominance is due to the concentration of activities in this institution. Conse- quently, mobility is analyzed in economic terms, such as the occupational grade of the individual or family line at various time points. The first chapter of the dissertation will discuss a specific type of social mobility in industrial society, intra- generational mobility. By intragenerational mobility we mean 1 2 the movement of individuals among ordered occupational cate- gories or occupational grades. Initially, the chapter dis- cusses the nature of mobility in industrial societies, the importance of studying mobility, and types of mobility. Then the next portion presents the research problem. The first chapter concludes with a discussion of ways to analyze mobility utilizing mathematical models. The graded occupations will be represented as the 'states' of our model. In terms of the model, mobility is then represented as movement among this set of states over a period of time. Hence the analysis will focus on career patterns. Subsequent chapters of the dissertation discuss models of mobility concentrating on a specific class of models, stochastic or probabilistic models, and a specific type of stochastic model, the semi-Markov model. The semi- Markov model is applied to a set of data representing mobility in an industrial society, Great Britain. We assume that both the population and the social system are closed. This is, for the period being investigated, we assume that no individuals enter into or exit from the popula- tion. Further, we assume that no new occupational grades are created nor are any old occupational grades eliminated. By dealing with a closed system over a short period of time we can minimize the confounding effect of the rapidity of change in an industrial society which may result in 'fictitious mobil- ity' due to the increased size of occupational groups, espe- cially manufacturing, technical, and service groups. 3 Our major interest is to ascertain whether or not the sociological factors of the sex of the working individual, the employment status (whether or not the individual is self- employed), and the industrial location of the occupational grade affect the life-chances of the individual and level of job satisfaction of the individual. As these two factors may contribute to the prOpensity to be mobile, they are assumed to influence the career trajectory or occupational mobility of the individual. Therefore, our basic unit of analysis is the working individual and our basic process is movement among occupational categories as modified by background factors. Social structures may be compared on the basis of their permeability. One extreme, immobility, would be a caste struc- ture in which movement is not allowed between segments of the social structure; the other extreme, perfect mobility, (Prais, 1955), in which movement between segments of the social struc- ture is completely unrestricted. Another basis of comparison is exchange between segments of the social structure, for we can ask to what extent segments of the social structure ex- change equally sized cohorts (Berger and Snell, 1963). In an equal exchange type of society, the same absolute number of the units of analysis move between states. The unit of analysis is usually family lines or individuals depending on whether or not one is talking about intergenerational or intragenerational mobility respectively. The absolute number of units of analysis is a function of the proportion of the population in a state and the size of the pOpulation in all 4 states. Hence two states may be in equal exchange with each other, but have different proportions of the total population. This is because equal exchange refers to the exchange of units of analysis, not the similarities of the odds of movement be- tween two states. Because we are dealing with mobility in urban, industri- alized societies, we will concentrate on the non-caste type of social structure. The segments of the social structure that interest us are surrogates for social class. Due to prob- lems of measurement, social mobility studies in urban, indus- trialized societies usually employ occupation as the surrogate. Social mobility studies in urban, industrialized societies usually employ occupation as the measure of social class and life chances, because income is highly variable. Industrial- ization has resulted in the increased importance of occupation as a factor in differentiating individuals from each other and in the movement from differentiation based on membership in a caste that usually resulted from birth, to differentiation based on membership in an economic class that usually results from holding a specific occupation. The political ideologies occurring with increased in- dustrialization, urbanization, and bureaucratization have been also associated with changes in the economic structure. These changes have produced an emphasis upon the ideal nature of ex- change between social groups. Societies are viewed as moving from a caste type to an equal exchange type. However, due to changes in the economic structure, especially the growth of 5 service, government, and technical sectors, coupled with a liberalization of the availability of education, the effect upon mobility is unclear. We must entertain the possibility of fictitious mobility (Hauser, et a1., 1975). That is, we must ask: "Is mobility due to changes in the social structure by a growth of some segments and decline of others, for ex- ample the growth of civil service and decline of agriculture in the economic structure, or is it due to changes inside a fixed social structure?" "Are movements away from a caste and towards equal exchange due to actual patterns of mobility or growth of a segment of the social structure?" The rapidity of change in urban, industrialized societies would make any long run prediction about mobility ludicrous. Hence we will focus on a relatively short period of time, seven years, for the same cohort of individuals. In other words, we are selecting a problem that is not concerned with changes of the social structure itself, but tries rather to explain the reasons for changes within the social structure. Our main con- cern will be to incorporate structural factors to initially explicate, and then possibly explain the reason for various degrees of occupational mobility. Our main argument is that most changes in mobility pat- terns and in the distribution of individuals among occupational groups can be explained by structural factors, for example the sex of the individual worker, whether or not the worker is self-employed, and the industrial location of the occupation. 6 Central to this argument is the idea that mobility is considered to be a life-chance of the individual in advanced industrial societies (Miller, 1971). Life-chance is used in the weberian sense to refer to the odds an individual will fulfill his or her full potential as a member of a society, as determined by economic factors. The achievement orientation of individuals in an advanced industrial society has resulted in the expectation the individual, in the world of work, will be mobile, usually in the upward direction. If mobility is a life-chance, it should be a function of the components of the economic systems of advanced industrial societies, especially the component which directly affects the individual, the labor market. The labor market is the part of the economic system that identifies the work role, and, in turn, provides the re- wards for the performance of the work role. We assume individ- uals try to maximize these rewards, which, in turn, result in the individual trying to maximize his or her life-chances and life-style. Hence we postulate mobility is a function of aspects of the world of work. ‘Mobility may be a function of how fixed characteristics of the individual are evaluated in the world of work. An example of this is sex. Mobility may be a func- tion of the investment of the individual in the occupational role and how this modifies the individual's world of work. An example of this is whether or not the individual is self- employed. Finally, mobility may be a function of the size of the labor market. An example of this is whether or not the 7 specific industry in which the individual is employed is ex- panding or contracting. Note these examples are not exhaus- tive and do not consider possible interactions. We do not intend to see mobility as a cause of social behavior, such as mental illness rates or voting behavior. We intend to see the feasibility of viewing mobility as a consequent of the economic structure of a society. This is an extension of a long tradition in sociological theory that began with work on circulation of elites. CHAPTER I PURPOSES AND SCOPE OF RESEARCH This chapter of the dissertation discusses the reasons why mobility is studied. We then proceed to review substan- tive findings about intragenerational mobility in industrial societies and argue a need exists to codify these findings to facilitate the determination of the causes of mobility. Fin- ally, we compare ways to analyze mobility. 1.1 Why Study Mobility Sociologists study social mobility for a variety of reasons. The first reason is mobility analysis indicates the nature of the structural basis of a society. Mobility is an indicator of the degree of achievement orientation in a society due to the continuous operation of economic factors. For ex- ample, Blau and Duncan (1967) have indicated the existence of separate and non-equal systems of mobility for Blacks and Whites in the United States and the positive effects of educa- tion and small family size in overcoming structural barriers. If we interpret mobility in the traditional Weberian class framework, movement among occupational grades or occu- pations is an indicator of the flexibility of class barriers in an industrial society, and in turn may be used as a valid indicator of the stratification system. Since the stratification 8 9 system affects the life style and life chances of the individual, we believe an analysis of occupational mobility is fundamental to the analysis of differentiated social collectives. In an industrial society, the occupational role of an individual is the key determinant of that individual's position in the strat- ification system (Caplow, 1954; Hall, 1975). The result is that an individual's occupation not only is the key symbol of one's social status but, more importantly, it carries with it expectations for behavior both by and toward the holder of the status. If we can identify the possible determinants of mobility we may study the intensity of specific variables in determin- ing the 'stratification space' of a society (Hope, 1972). The 'stratification space' is the set of axes or dimensions which determines the social distance between social groups. Hence, if the key axes of the 'stratification space' could be identi- fied, then the extent of stratification in a society and the degree to which specific variables determine the extent of stratification in a society could be evaluated. Consequently, mobility analyses should attempt to determine the 'stratifica- tion space' of a given society. For example, if the mobility patterns of males were found to be different from those pat- terns of females, we would conclude that sex is an axis of the 'stratification space.' we have selected for study here the variables of sex, self-employment status, and the industrial location of the occupation as possible axes of the 'stratifi- cation space.' 10 The second reason why sociologists study mobility is that it leads to a world view, for we are forced to think in terms of process rather than structure independent of process. By examining the dynamics of a society over a period of time of substantive interest, we can conceive of the movement of a society as the movement from a caste system towards an equal exchange system. Comparative sociologists, especially those dealing with social change and economic development, often compare societies along these dimensions in terms of the time it takes to reach a certain point on these dimensions. In addition, a social process orientation appears to be a more realistic orientation. Social institutions and organi- zations are assumed to be loci of information processing units involving memory, delay, feedback, and decision making affect- fing a social cohort. The urban, industrialized societies we are dealing with can be envisioned as complex decision making structures governed by bounded growth and restrained by pur- poseful maintenance of the social structure. Social mobility also allows us to evaluate the results of these decisions, es- pecially to assess the efficacy of planned growth of a segment of the social structure. For example, the forthcoming repli- cation of the Blau and Duncan work of a decade ago will show the results of the programs of the "Great Society" in the l960's_(Featherman, 1974). In economic terms, a mobility ap- proach allows us to display the dynamics of a society's oppor- tunity structure and extent of underemployed talent. 11 This reason why sociologists study mobility follows from the substantive importance of the processual orientation. Social mobility is a metric to compare different societies or the same society at different periods of time. By being a large scale metric, we can compare societies in terms of the movement from caste and the degree of equal exchange, indepen- dent of the different historical and cultural bases of a given society's institutions. For example, capitalist and noncapi- talist societies can be compared, or the same society can be compared before and after the implementation of social welfare programs. One of the major intellectual traditions in sociology is identification of the common characteristics of societies so as to isolate the unique characteristics that differentiate societies. This is a major theme of the Weberian school. Social mobility, by being a social metric and a social indi- cator, allows us to operationalize this tradition. Social mobility enables us to appreciate the efficacy of social and economic change in record-keeping societies. Featherman (1974) reports a series of studies that will com- pare the same nation at two points in time to discover patterns of change. In short, due to its power of replication, social mobility could be a social indicator, or a'statement of direct normative interest which facilitates concise, comprehensive and balanced judgments about the conditions of major aspects of society" (Duncan, 1969;3). 12 The third and final reason why we study social mobility concerns the potential policy implications of sociological research. Sociologists tend to be increasingly viewed as cen- tral and non-ancillary to social planning and decision making. This is evidenced by an interest in social indicators. Social mobility provides a means to see the structural effects of our policies, especially in the economic sector. By providing a metric to compare different societies or to compare the same society at different points in time, social mobility allows us to evaluate the reality of a classless Eastern European society or the flexibility and opportunity in an Atlantic community democracy. Hence the analysis of social mobility for a spe- cific society may reinforce or depreciate the non-monarchical political themes in the Western world since the French revolu- tion, especially Marxist socialism and Keynesian capitalism. Thus, independent of an individual's political orientation, the study of mobility is a means to observe the structural effects of social policies, especially in the labor market. 1.2 Scope Conditions of Research The types of mobility can be classified along three di- mensions (Caplow, 1954). The first of these dimensions is the time period in which mobility occurs. This usually defines the type of unit that is mobile. Time is classified as inter- generational or intragenerational. Hence mobility studies can be dichotomized as the movement of individuals over their oc- cupational career or the movement of family lines along various l3 economic classes determined by the occupation of the head of the family. This is called intragenerational and intergener- ational mobility respectively. The second dimension is the dimension of space, which is usually covered in work on migration. Migration may be caused by changes in economic opportunity in a given geograph- ical region or by differential fertility. Ladinsky (1967) has shown the degree of geographical movement between jobs is re- lated to material investment of an individual in the tools of his occupation. Thus, physicians in private offices who have a large capital investment in equipment have a lower rate of geographical mobility than physicians associated with a medi- cal school. For purposes of analysis, we will not consider migration to be significant since we will deal with movements among occupational categories and the structural variables behind these movements. The data set we will use is a panel study for which the assumption was made that a respondent was located at the same place of residence at all points in time. Migration can be dichotomized as being germane or not germane for purposes of analysis. The final dimension of mobility is direction inside the social structure. Are we interested in vertical mobility, movement among ranked segments of the social structure, usually occupations, or are we interested in movement among different organizationsbut in the same ranked occupation, horizontal mobility? Usually industrialized social structures are par- titioned along occupational lines that are ranked. A line 14 worker becoming a foreman in the same factory would be mobile in a vertical sense in an occupational scheme but would not be mobile in an industrial category scheme. But this does not prohibit us from examining all line workers who become fore- men to compare their mobility rates for workers in manufactur- ing and non-manufacturing organizations to see if any factors inherent in the social structure of these organizations may explain different mobility rates. Hence the final dimension can be dichotomized as horizontal or vertical mobility. Therefore, our focus of interest shall be on intragen- erational vertical mobility in an ordered occupational system that reflects the social structure during a period of time selected such that the number of segments in the order and the nature of the ordering remains constant. We shall ignore migration factors. Also, we shall assume that we are dealing with a closed population system. That is, we assume that all individuals in our analysis are continuously employed and no individual enters or leaves this population. Despite the substantive limit of our scope conditions, we cannot avoid considering previous mobility studies of the other types of mobility. These studies have made us aware of some of the determinants of mobility. Mobility studies have developed certain types of techniques, especially models. The universal ideas that appear in these studies and the types of analysis used in them allows us to continue in the work of the past, making mobility one of the few areas of cumulative research in the social sciences. Methods of analysis and 15 techniques to study mobility are not necessarily limited to a specific historical time period, the size or nature of the cohort, or the geographical location of the social structure being analyzed. 1.3 Mobility in Industrial Societies Industrialized societies have most of their economic activity concentrated in the processing of raw materials needed for the maintenance and bounded growth of a collective of individuals. Related to industrialization is differentia- tion, or the division of labor, and bureaucratization, the rise of complex, formal organizations. The process of indus- trialization results in a further growth of certain occupa- tional categories, especially those involved in manufacturing. Advanced stages of industrialization deal with the manipulation of information in addition to the manipulation of raw materials, and deal with problems of social cohesion created by differen- tiation. The advanced industrialized society's differentiation results in a bifructation of highly skilled decision makers and low level white collar workers (Meyer, 1972). Hence in our study we have a problem of selecting a time unit that mini- mizes the extent of differentiation of an industrialized society. This would minimize the problem of identification of lines of differentiation in a society that might result in 'fictitious mobility.' We assume our period of 1963-1970 in Great Britain minimizes problems of differentiation since it is a short period of fairly constant and stable growth (Whitaker's A1- manack, 1965, 1971, 1974). 16 By fairly constant growth we mean growth occurs at a ' constant, bounded rate. That is, there are no fluctuations in the growth rate. Sometimes this is called stable growth. A series of examples will explain this seemingly contradiction in terms. In a mathematical sense we mean the derivative of the growth rate is zero; that is, the growth does not change as time changes. An example of this is a car moving at a constant speed. The car is moving, but there is no change in the rate of movement. In demographic terms, stable growth means the difference between the crude birth rate and crude death rate is constant. If this difference is positive, p0p- ulation would grow, just as the car moved, but the growth and movement would be at a constant rate, well defined, and bounded, ceritus paritus. Complex organizational structures have developed to con- trol and integrate the differentiation in industrialized so- cieties. Research in this area has concentrated on how the work setting may govern the pace of work and degree of worker alienation which in turn determines the propensity to be mobile (Blauner, 1964); how the type of product being produced or the size of the formal organization governs the organizational structure (WOOdward, 1965; Blau and Schoenherr, 1971); and how the structure of an organization may result in opportun- ities for promotions or demotions since positions or vacancies must be filled (White, 1970; Stewman, 1975). A specific organizational structure has differential rewards commensurate with location in the organization that 17 results in inducements for mobility. These rewards are usually salary and monetary rewards associated with seniority but may also include informal status allocation and peer group rela- tionships. We will assume that the organizational structure will determine these reward levels. We will also assume that the individual wants to maximize these rewards and will be mobile if given an Opportunity to be mobile. Differentiation and bureaucratization set parameters on career mobility and the worker's orientation toward his col- leagues, his occupation, and his strategy for advancement (Thompson, Avery, Carlson, 1962). Complete analysis of this problem would result in a very complex analysis involving how structural factors result in various social psychological at- titudes of workers toward mobility. Mobility may be characterized by the time it takes for advancement in a certain type of career. Is there an early or late ceiling? An early ceiling occupation, like a nurse, reaches a rapid upper bound and has a limit to more status in the future. A late ceiling occupation, like an engineer be- coming a manager, has an upper bound that takes a long time to reach. Perhaps the occupations that have early ceilings manifest their mobility in horizontal forms more than the oc- cupations with late ceilings. Occupations with little room for advancement may manifest mobility in a horizontal rather than vertical manner (Hall, 1975). Mobility may also be determined by the orientation of the worker toward his occupation or toward his current place 18 of work. An orientation to the occupation would result in mobility patterns being determined by the need of an organi- zation for individuals with specific types of training and ex- perience. An orientation toward the organization would result in mobility patterns determined by the structural factors in- herent in the specific organization, especially the number of advanced positions. Different strategies toward advancement may result in different lengths of duration in a specific oc- cupation because one occupation may be a preparatory stage for subsequent occupations (Hall, 1975). Consequently, a complete analysis of mobility should be able to distinguish mobility in the same formal organization from other formal organizations and immobility in a formal organization from a horizontal movement to the same occupation in a different organization. In addition, a complete data analysis would involve questioning the impact of industrial setting of the occupation and the specific organizational structure where the occupation is located. The impact of these aspects of the world of work incur different levels of rewards that result in various inducements for mobility. Unfortunately, the data problems involved in such an undertaking are beyond the limited resources of this thesis. Hopefully, the results of our research will be preparatory to a study of these problems at a future time. However, as men- tioned above, we will examine the impact of the industrial setting of the occupation on mobility patterns. 19 Analysis of data about mobility reveals no definite in- crease in size of prestigious occupational groups when the growth of non-agricultural occupations is included. Shifts in the distributions among occupations have remained virtually non-existent in recent years (Hauser and Featherman, 1973; Hauser, et a1., 1975). Recent reanalysis of mobility data reveals no great change in the patterns of intergenerational mobility in the United States independent of changes in the occupational distribution, i.e., the growth of specific occu- pational categories (Hauser, et a1., 1975). In short, the de- pendence of a son's occupation on a father's occupation has been stable for the past fifty years. We would expect changes in the OCCUpational distribution to be characteristic of an advanced industrial and post- industrial society. However, due to the short time period of our proposed research, seven years, we do not consider this to be problematic. Studies of mobility should address them- selves to how government policies of state capitalism or state socialism affect mobility patterns. Stewman's (1975) analysis of the Michigan State Police and Tuma's (1972) analysis of Mexican-Americans have hinted at the effect of such policies on the stability of an occupational system. Our research at- tempts to formally represent determinates of mobility that may be generalized to incorporate the effect of such exogenous factors on mobility in industrial and post-industrial societies. The data set we propose to analyze reveals a constant distribution among occupational categories (see Table l). 20 Work using intergenerational mobility data, which compares occupational distributions at the same time point for dif- ferent aged individuals, would not allow us to deduce the nature of intragenerational mobility since different life- experiences have happened to each group being at different stages of their working careers. TABLE 1 DISTRIBUTION OF THE SAMPLE AMONG OCCUPATIONAL GRADES IN TERMS OF PROPORTIONS Year Occupational Grade 1963 1964 1966 1970 Professional-managers .143 .147 .164 .204 White collar .278 .264 .280 .270 Blue collar .579 .589 .556 .526 NOTES: Computed from edited data tape, as described in Chapter 3. Sample size equals 511. Occupational shifts in industrial society are generally from manual to non-manual, from being self-employed to being a salaried worker, and from low to high status occupations within a gross occupational category. This leads to the main questions: What are the reasons for such mobility, and what patterns occur when social groups are compared? For example, how do mobility patterns of males and females or self-employed and salaried workers in the same occupation differ? One of the critical questions in such an analysis is how the length 21 of time at a job, job tenure, affects mobility and how social factors affect job tenure. We have information on parts of this process. For ex- ample, we know that professionals are unwilling to change their occupations since such a change would result in a loss of edu- cational investment (Hall, 1975). We also know that lower boundaries exist to mobility between white and blue collar occupations and between blue collar and agricultural occupa- tions (Blau and Duncan, 1967). Finally, we know that self- employed workers exhibit less geographical mobility than sal- aried workers since the self-employed workers, independent of occupation, must invest in capital goods and develop a clien- tele (Ladinsky, 1967). Tuma (1972) has studied the mobility of Mexican-Americans in terms of their occupation, industrial setting of the occupation if the individual is an operative, age of the individual when working life commenced, level of education of the individual, geographic area of origin of the individual, and duration of an individual in an occupation. Tuma argued that mobility is composed of two subprocesses: the process of leaving a job and the process of being attracted to an alternative job. (A job is defined as an occupation in a social location, where a social location is usually the in- dustrial or geographic location of the occupation.) Tuma con- cluded that the length of duration in a job is a determinant both of the rate of job termination and of the attraction to alternative jobs. With respect to leaving a job, Tuma found education to be negatively related to job duration. The level 22 of education of the individual is the key factor affecting duration in a job. The age when the individual commenced work and geographical origin of the individual are of little or no importance. Time measured as duration in a job is a better indicator of mobility than time measured as the age of the individual. With respect to being attracted to alterna- tive jobs, Tuma found rates of attraction are dependent on the actual age of individuals, are ESE necessarily related to pre- vious occupation, and, surprisingly, are independent of the level of education of the individual. Recently, Tuma (1975) has re-examined the same set of data and showed the rate of leaving a job declines with dura- tion in the job. This rate also depends on the initial level of job rewards, the level of individual resources, especially education, and the socially defined value of these resources. Duration in a job was found to increase as median occupational earnings increased. Duration in a job was found to decrease as the individual's educational level increased. The skill level of the current job relative to the previous occupations of the individual and the age of entry into the labor force had no effect on the rate of mobility. However, the number of previous jobs held by the individual affected the mobility rate, albeit in a curvilinear pattern. In general, Tuma found the rate of mobility declined at decelerating rates as duration in the job increased. Finally, Hall (1975) has noted that individuals with specialized training are unlikely to move to occupations where 23 the training is irrelevant. In addition, he feels there is less occupational inheritance for females than males, and in- dividuals with advancement possibilities reinforce these pos- sibilities by anticipatory socialization. No attempts to construct a theory which discusses mobil- ity in terms of ascribed and achieved characteristics of in- dividuals in dynamic economic structures exists. We propose to lay the basis for the development of such a theory. 1.4 Ways to Study Mobility Using empirical data, there are three ways to study mo- bility. The first way is to compare the same society at two points in time or compares different societies at the same or at multiple points in time. Commonly, the proportion of the work force engaged in gross occupational categories, usually agriculture and non-agriculture, is contrasted. The second way is to compare societies in a more complicated way by using a statistic or index that summarizes the movement between seg- ments of the social structure. The third way is to construct mathematical models. In all three cases, we can test the ade- quacy and validity of our work only if we have the appropriate empirical data. The crudest way to represent mobility at the national level is to report the proportion of a nation engaged in crude occupational categories at two points in time and to compare the proportions to other nations. The data is usually some permutation of the number of individuals or family lines in a 24 given occupational category. Since the data is from demographic sources, there is nonuniform categorization, different sampling procedures, and an inability to deal with structural changes. For example, it would be hard to compare the increase in the number of white collar workers in France and Poland in the last twenty years due to different definitions of white collar workers used in the collection of data in France and Poland. Such data reveals little or no mobility into more pres- tigious occupational groups when the growth of non-agricultural positions is considered. Perhaps this is because this type of analysis is inappropriate for nations undergoing rapid economic growth and structural change. Comparative data is a crude index since it gives us summary data about the social structure indicating only a change in the distribution of workers in various occupational categories. Unlike basing comparative data on the distribution of individuals in the social structure, indices are usually con- cerned with patterns of movement among partitions of the social structure. An index of social mobility is a permutation of the movement between social categories rather than among the distribution of a population of individuals or family lines across social categories. This can be clarified if we use an n by n matrix, where n is the number of social categories of interest. The rows of the matrix represent the segments of the social structure at one time point and the columns of the rnatrix represent the segments of the social structure at an- CDther time point. The entries of the matrix represent the 25 movement from one segment of the social structure, represented by the row, to another, represented by the column, during a time interval that is the difference of the two time points. The rows and columns are arranged in such a manner that read- ing down for the rows and across for the columns the nominal and ordinal nature of the social categories is preserved. This type of matrix is called a transition or mobility matrix. Observations are made about the location of individuals or family lines in the social structure at two or more time points. In our set of data we shall use information about the OCCUpations of individuals at time points that are 1 year, 3 years, and 7 years after our initial observation. The problem of developing and testing indices has been called a problem of measurement (Boudon, 1973). Manipulations are done on the entries in the matrix, usually comparing the true entry to the expected entry based on the underlying mul- tinominal distribution, somewhat analogous to the use of a chi-square distribution to analyze contingency tables. The expected entries, computed this way, reflect perfect mobility, the Opposite of the caste type society. In a vacuous sense, the simple permutations in comparative data are base-line in- dices. A recent detailed review of indices of mobility con- cludes there is "no unique best index of mobility" (Boudon, 1973). Indices seem to beget the major problem of the study of {nobility, the explication and explanation of mobility. Even ‘though structural changes may be incorporated into the 26 construction of indices (Boudon, 1973) this incorporation fails to give any insight into how structural factors modify mobility processes. We conjecture that we can gain more in- sight from mathematical models. The use of models to study mobility exhibits more work of a cumulative nature than the work on comparative data and indices. As we shall show in the next chapter, the models we develop are built upon the results of twenty years of accumu- lated research and ideas. we define a model as the result of the intellectual pro- cess of translating from one language to another and manipu- lating ideas and thoughts in terms of the second language in order to gain insight about the phenomena we are studying. Specifically, we are concerned with the translation from one ' natural language, English, to a formal language, that of stochastic processes, a type of mathematics which is processu- ally oriented. The formal language is usually more parsimon- ious and exact than the natural language. Due to these characteristics, the model results in exact communication among the students of a specific phenomena, in this case social mobility. This exactness adds to the cumulation of knowledge resulting in a series of works that explicate and try to ex- plain the phenomena of movement in the social structure classified as social mobility. In a formal language, as in a natural language, the irules of syntax are not restricted to a specific subject area. Iience, models from other subject areas may have applications 27 in sociology. The models we propose originated in an area of operations research that deals with the failure of electronic or mechanical components in a system called renewal theory or point processes (Cox and Miller, 1965). This chapter has presented an introduction to the sub- ject of mobility. We have stressed the relationship of mo- bility to the area of stratification, focusing on the nature of mobility in complex industrial societies. We stressed the need to determine the factors that cause mobility and argued that the interdependencies of the processes of industrializa- tion and bureaucratization give insight into some of the pos- sible determinants of mobility. We then listed unorganized findings about mobility and concluded that a need exists to attempt to codify these findings to facilitate the determina- tion of the causes of mobility. Finally, ways to analyze mobility were compared, focusing on descriptive data, indices, and mathematical models. The next chapter concentrates on the deve10pment and use of mathematical models in mobility studies. CHAPTER II MODELS OF MOBILITY This chapter of the dissertation recapitulates the mathematical development of mobility models. The two main types of models, causal and stochastic models, are introduced, stressing the advantages of the latter over the former. Then, the cumulative development of stochastic models is mentioned, stressing how the awareness of the substantive aspects of mo- bility aided this development. Finally, the argument is made that these considerations lead to the semi-Markov model. In this context, the utilization of a semi-Markov formulation has led us totflfink in terms of the time spent in an occupation before a move to a different occupation is made, which is called the waiting time. The argument that the waiting is a probability distribution which can represent substantive fac- tors of mobility is made as the conclusion of this chapter. Models provide a formal language to organize and manipu- late our observations about social reality. They serve as symbolic analogies which facilitate our thinking and communi- cation about social phenomena. Despite the existence of several different typologies of the dimensions of models, the causal-stochastic dimension is a dominant theme. The causal or deterministic-stochastic idea 28 29 has been a dominant theme in works on models in the philoso- phy of science since the Continental versus English schools represented by Duhem and Campbell, respectively, over seventy years ago. More importantly, this dimension is germane to the refinement of mobility models. We will use the word modeling for the process of con- structing a model. The ideal result of modeling, the exposure of the necessary and sufficient mechanism underlying the ob- served phenomena, is called theory construction. The use of models in social mobility allows us to think in terms of longitudinal factors, delays, accumulations of effects, and feedback. By applying mathematical analysis, es- pecially the limiting properties of certain mathematical re- lationships, we can see how mobility represented as the pat- terns of movement of a population among occupations will look at any point in time, ceritus paritus. We can also see how these patterns will look if there is no change in the system in a time interval, that is, when the system is in a stable state. Finally, we can see how the stability is affected when disturbed. Modeling occurs in four steps. The first step, called feasibility analysis in applied science, like engineering, in- volves gaining insight into the problem and a substantive knowledge of the problem. We have attempted to do this in the first chapter. The second step is the actual translation of the relevant parts of the problem in an isomorphic manner from a natural language to a formal language. This is called the 30 design phase in applied science. The third step involves deductions and solutions in terms of the formal language, the process of analysis in applied science. Finally, these de- ductions and solutions are empirically tested by means of statistics or computer simulations. That is, we attempt to verify the accuracy of our translation. Models are not fixed by time, space, or the size of the phenomena being studied as long as the conditions of applica- bility or scope conditions of the model are fulfilled. Hence our translation and model of mobility should be an adequate translation for mobility in England from 1963 to 1970 (Butler and Stokes) 1969) or other sets of data that meet the condi- tions-of applicability, like the National Longitudinal Survey which covers the United States for 1966 to 1971 (Hauser, et a1., 1975). Models of occupational mobility are usually one of two types, causal-deterministic-static or stochastic-dynamic, a processually oriented model. Recent work in a third type of model, purposeful theory or game theory models (Coleman, 1973) may have some potential for determining the utilities avail- able to a worker which may determine his prOpensity to be mobile. However, these models, as currently developed, cannot incorporate how changes in the social structure may determine changes in the utilities and hence the propensity to be mobile. The first of the two models, which we will term the static model, is atheoretical and fails to explain the rela- tionship between covarying factors. The best known example 31 of this model is Blau and Duncan's path analysis (Blau and Duncan, 1967). The model is causal only in the sense that the variables antecedent to the end result are uniquely forwardly ordered in time. The sequence is usually postulated by gain- ing some substantive knowledge about the problem and tries to give weights to the specific links between variables in the sequence. This is done by multivariate techniques, seeing how the variance in one variable in the sequence may be explained by other variables in the sequence. To say one variable is related to another variable due to the amount of variance in one variable explained by the variance in the other variable is not the same as explaining the relationship between the variables. Quite often intervening factors occur that may amplify a false relationship between the variables, yielding spurious correlation. The path analysis model of Blau and Duncan should not be termed atheoretical since this qualifier can be equally applied to our proposed stochastic model or any type of model. All models are atheoretical since they are only conceptualizations of reality in ideal terms, void of empirical substance. In this perspective, at most, a model may indicate the form of the theory to the scientist, not the substantive and empirical content of the theory. The second main type of model used to study mobility, stochastic or dynamic models, which we will call stochastic models, has a processual orientation. We argued in the first chapter of this dissertation that the analysis of social 32 mobility forces us to think in terms of a processual orienta- tion. we now discuss both families of models. 2.1 Causal-static Models The causal model of mobility can only state the existence of a relationship between variables. At most, the model can only indicate if one variable is a necessary or a sufficient cause of another variable. Usually these relationships are linear algebraic functions of correlation coefficients. The dominant causal model in mobility is path analysis (Blau and Duncan, 1967; Hope, 1972). The model derives weights for the postulated path linking sequences of variables from correlation coefficients. The sequence is constructed by assuming that some'of the variables are temporarily anterior to others. Frequently, consideration is given to the theoreti- cal relationship between the variables. For example, a major link in Blau and Duncan's analysis is father's income and level of education to self's first occupation, thus implying the dominance of achieved over ascribed factors as determinants of mobility. Usually the model indicates the sequence by a line with an arrow showing the temporal order of events. The base of the line is an anterior variable leading to a posterior variable occurring later in time at the tip of the arrowhead. The line may represent residual effects if no variable is specified at the base of the line, a causal relation if there is one arrowhead and a variable specified at the base of the line, or a simple correlation between variables if the line has arrowheads at either end. 33 The causal model ignores changes in the social structure over time. Ideally, the paths should show the influence of structural variables on the mobility process. The path values are constants in a set of structural equations representing the variables in the mobility system. However, the values in these structural equations are derived from how much of the variance in one of the variables is related to the variance in other variables at a specific point in time. The model does not allow us to ask if the variances, and hence the effects of the variables, represent changes in the effects of social structure on the mobility process over time. Thus path analy- sis results in confounding the level of analysis. For example, do anterior variables represent initial conditions at one point in time or constant factors that originate in a previous time period? In addition, how comparable are the path weights at different points in time? We claim that the causal model is too inflexible because it cannot represent structural changes during a time interval. Since path weights are not identical to correlation coefficients, a change in a weight does not necessarily mean a change in the degree of association between variables (HOpe, 1972). The path model assumes that the social structure and all effects of the social structure on the mo- bility process is in a state of equilibrium. Therefore changes in the social structure or its effects on the mobility process are unidentifiable. In addition, we claim that causal models are 'snap—shot' models, one picture of the social structure at one point in 34 time. This implies that the mobility analysis represents a stable system or that all variables Operate on each other at the same time instant or at the same rate (Leik and Meeker, 1975). Just as a single snap-shot is not a motion picture, a causal model is not a dynamic analysis. When path models are combined with the time lags of research,the result is an unrepresentative picture of the social structure. 2.2 Stochastic Models: An Overview ' Due to the unpredictable nature of human behavior and due to the need to explicate and explain the variability of human behavior over time in social structures, we maintain that social mobility is a dynamic process subject to uncertainty. Statements about mobility made in English may be trans- lated into a processual oriented mathematical language such that the random nature of human behavior is not lost in trans- lation. The result is called a stochastic process. When ap- plied to mobility, the result is a stochastic model which represents movement among a set Of occupations. The movement is governed by probabilistic laws. Thus, movement in the social structure is represented as a set of functions that give the probability of movement between occupational states in a given time period. Since mobility is subject to uncer- tainty due to the unpredictable nature Of humans, we prOpose to use a stochastic representation of mobility. Although stochastic models are probability models indexed by time or space as we are concerned with movement between 35 partitions in the social structure at various time points we will index our model by time. Further, we shall incorporate the social structure in terms of occupations which will be represented as "states" in the model. Hence, the question Of asking what determines mobility patterns becomes the question of what determines the probability of moving between states in a specific time interval. The main inadequacy of the stochas- tic approach to date has been the lack of work determining how these probabilities are modified by sociological factors (Boudon, 1973). We propose to try to overcome this inadequacy by investigating the possibility of representing mobility by a specific type of stochastic model that appears to be able to incorporate sociological factors and that seems to be a logical extension of previous mobility models. In order to do this we need to outline the cumulative development of stochastic models of mobility. The mathematical details of the models mentioned in this section are presented in Appendix A. The simplest stochastic model of mobility is the Markov chain (Blumen, Kogan, and McCarthy, 1955; Prais, 1955). The model assumes moves between states are dependent only on the state from which the move originates and is independent of the previous sequence of moves. This is called the Markov or one- step dependency assumption, common to all Markov processes. By Observation of the movement between states and the distri- bution of the population among states, the probabilities of movement and distribution at any subsequent time point can be 36 computed. This model has two assumptions in addition to the Markov assumption. First, the model assumes the probability of moving between two states is the same for all individuals. This is the assumption Of homogeneity of movers. The second assumption is the probabilities of movement are identical for any two time periods of equal length. This is called the sta- tionarity assumption. The relaxation of these assumptions has produced more advanced models of mobility. Blumen and his associates concluded that the simple model inadequately predicted the number of non-movers. The predicted proportion of non-movers was smaller than the actual proportion of non-movers. Hence non-movers are underrepresented in the simple model. This led Blumen and his associates to question the assumptions of the simple model, especially the homogeneity assumption. Consequently, they develOped the mover-stayer model where the assumption of homogeneity was relaxed: two distinct types of individuals are postulated, those prone to- ward mobility, movers, and those prone to immobility, stayers. This model resulted in a better prediction Of movement. Re- cent work by McFarland (1970) has extended this idea by assum- ing each individual is governed by a unique Markov chain between the ideal types of movers and stayers, resulting in a disaggre- gation Of the simple model. This approach seems to lead to a dead end since mobility may appear as a result of a statistical artifact of the disaggregation (Morrison, 1973). The approaches mentioned so far in this section are termed demographic approaches by Stewman (1975) since mobility 37 is conceptualized as a flow of manpower. An alternative stochastic model of mobility is the vacancy chain. Vacancy chains have been successful in predicting mobility in well defined, highly formalized, autonomous and hierarchical author- ity systems such as church groups (White, 1970) and state police groups (Stewman, 1975). The vacancy chain conceptual- izes mobility in terms of the flow of interrelated job vacan- cies in a formal organization. Vacancies are created when a new job is created, or an individual in the organization dies, quits, or is fired. Vacancies are filled when the job is abolished or a new recruit fills the job. The advantage of the vacancy chain model is a conceptualization of the internal dynamic interrelationships of an organization. The main dis- advantage Of the vacancy chain model is its atemporal nature. White (1970), Tuma (1972), and Stewman (1975) contend the model is stationary since vacancies are rapidly filled, but they point out the model varies in degree of heterogeneity with respect to external economic conditions that create the vacancies. The vacancy chain model can be interpreted as conforming to the Markov assumption if one is interested in the average time until a vacancy is filled or the time a vacancy stays within a set of states. However, Boudon (1973) indicates that the Markov assumption is violated if the vacancy chain model is adopted to analyze the overall transition matrix of in- dividuals over time among a set of states. 38 In substantive terms, the vacancy chain model may be limited in its sc0pe since multiple formal organization set- tings Of different occupations result in the necessity of consideration of temporal factors in the forms of time delays due to the complex interdependencies of modern economics. For example, vacancies in an organization engaged in automo- bile production may determine vacancies in an organization engaged in steel production. The relaxation of the stationarity assumption is aided by utilizing pre-existing mathematical work on continuous time stochastic processes. Mayer (1972) uses instantaneous rates of transitions between states to deduce the transitions be- tween states in probabilistic terms. The result is the prob- abilities take on different values at different points in time. The probabilities usually take the form of a modified decay function. Mayer has developed three models that elaborate on these ideas (see Appendix A). The first allows instantaneous moves only to adjacent states, the second allows moves to any state weighed by a decay factor, and the third associates con- ditions of permanent retention or some degree of non-retention for each state. Though these models relax the assumption of stationarity, they still assume homogeneity since the transi- tion rates are identical for all members of the population. Conner (1969) has used a continuous time model to see how an individual's degree of commitment to an occupation, after being in a state of indecision as to remaining in the occupation, results in the probability of leaving the 39 occupation. In substantive terms, Conner is asking how social psychological factors induce the propensity to be mobile. Sim- ulations involving the analysis of Mexican-Americans whose first occupation is farm labor revealed a fit between the ac- tual and predicted proportion of the sample still in agricul- ture as a function of time. The significance of Conner 5 work is the attempt to rep- Ad resent substantively the determinates of the parameters of stochastic models of mobility as a function of social psycho- D logical states. This attempt has been generalized and extended 53 by Tuma (1972) by seeking macro level sociological factors that determine the parameters of stochastic models. Tuma (1972) analyzed mobility in terms of two subprocesses, leaving a job and being attracted to an alternative job. By relaxation of the homogeneity, stationarity and Markov assump- tions, Tuma showed most models of mobility are composed of specific mathematical functions termed probability laws which are special forms of basic equations for the two subprocesses. TUma compared the laws in terms of identification of future states, subsets of homogeneous individuals, best representation of time (duration in a state, age of the individual, or time when a move is made), and validity of the Markov assumption. She found the probability laws which best fitted her data im- ply a heterogeneous, non-stationary, non-Markovian process. As we are concerned with how substantive factors deter- mine mobility, we view our work as an extension of Tuma's re- search. First, we extend the list of possible factors to 40 include some aspects of the status-role of the individual and the industrial location of the individual's occupation for all occupations. Then we argue the status-role of the individual continuously affects the experiences of the individual which affect the parameters of the mobility process. Consequently, our research is an example of the cumulative nature of model- building, since, like Tuma, we argue for the importance of 'studying how time-varying sociological factors result in a heterogeneous population. Our approach involves a formaliza- tion known as the semi-Markov model. 2.3 The Semi-Markov Model and WaiEing Time Distributions The use of the semi-Markov model has been proposed for studies of migration (Ginsberg, 1971), housing turnover (Gil- bert, 1972) and occupational mobility (McGinnis, 1968). The semi-Markov model seems fruitful for the construction of a formal theory of mobility since it allows us to see how the time spent in a state modifies transitions between states. Mobility is conceptualized to be a series Of moves be- tween any two states within a set of states. Hence the career of an individual is composed of a series of moves between pairs of states, such that the state of destination for a given move becomes the state of origin for the next move, if any subse- quent moves occur. The only requirement is that the states of origin and destination are different states. We are especially interested in the duration in the state of origin, the time an individual waits in the state of origin before a move to 41 the state of destination occurs. We feel that since mobility is a life-chance, aspects of the status-role of the individual relevant to the world of work affects the duration in the state of origin. The semi-Markov model postulates that movement among a set of states is a function of the probability of movement be- tween any two states and contingent on the time spent in the t prior, original state. Two sets of relationships are involved: the simple Markov Chain or conditional probability Of moving between states and a set of functions specifying the proba- 5 bility of making a move between any pair of states given a specific length of duration in the state where the move orig- inates or waiting time in a state. The set of functions is a mathematical representation of the status-role and world Of work of the individual. The semi-Markov model, when compared to the vacancy chain model, allows us to investigate the determi- nants of the waiting time till a move is made. These factors, especially aspects of the status-role that reflect duration at the job and seniority, may determine the rate at which vacan- cies are created and filled (Tuma, 1972). If we stress the substantive advantage of the vacancy chain approach, namely the consideration of the structural determinates of mobility, we see the semi-Markov model's use of waiting times allows us to incorporate this set of considerations in terms of the in- dustrial setting of the occupation and the status-role of the individual. That is, the semi-Markov model will allow us to ask theoretical questions about the specification of conditions 42 of change of the mobility process, a need in mobility research stressed by Stewman (1975). TUma's (1975) re-analysis of data on Mexican-Americans employs a semi-Markov model to attempt to model the causes of social mobility. Mobility is viewed by Tuma to be composed of two subprocesses: leaving a state and being attracted to a new state. Specifically, Tuma mentions the duration of an in- dividual in a position does not depend on the previous history of the individual, but may depend on characteristics of the person's present position and his destination. Tuma also re- alizes the pattern of duration in a state need not necessarily be a monotonically decreasing pattern as duration increases. Tuma tried to oversome two faults of her model, the omission of the effects of population heterogeneity, and the Markovian nature of part of the model, the embedded chain. In contrast, we will argue that in the semi-Markov model the an- alysis of the duration of time in a state, which we term the waiting time, shows the effects of population heterogeneity. In addition, the Markovian nature of the embedded chain is modified by the nature Of the mathematical distribution of the waiting time (see Appendix A). Tuma developed a multivariate, linear model to analyze her two subprocesses, leaving a state and being attracted to a new state. She also investigated the impact of the duration in a state on these subprocesses. Our approach is similar since we explicate how factors antecedent to mobility affect 43 the distributions Of waiting times in the state of origin be- fore a move to the state of destination. Since the semi-Markov model is closely related to Tuma's analysis, we should explicate the differences between the two approaches. With respect to the type of formalization involved, the semi-Markov model does not represent mobility in terms of two subprocesses of duration in a state and moving among a set of states, but incorporates both subprocessesin the waiting time function and the embedded Markov chain. The unit Of analy- sis in the formalizations differ since Tuma is concerned with how attributes of a specific job affect rates of mobility, while we examine how the attributes of the status-role of an individual in the occupational structure affect rates of mo- bility. In addition, we are explicitly concerned with the state of destination and the state of origin. Tuma's ignores the implicit link between current and future states. This ignores the process implicit in the individual's decision to change his state: a decision to move implies the comparison Of the current and future occupational state, which implies at least a vague knowledge of the future occupational state. Fin- ally, the existence Of vacancies is non-problematic in the semi-Markov formulation since the factors behind a move, if a move occurs, depend on the state where the move commences and terminates. Therefore, we assume that a vacancy must have existed in the state where the move terminates. In substantive terms, the scope of Tuma's model and the semi-Markov model differ. In the former, one can change a job 44 without changing an occupation, but in the latter a change in an occupation implies a change in a job. In addition, our ap- proach is more comparable to traditional sociological approaches to studying mobility because we conceive mobility in terms of a state of origin and of a state of destination. Analysis of the waiting time distributions in the semi-Markov process allows us to examine the determinants of duration in a state, which, given a state of destination, governs the prOpensity to be mobile. These considerations, however, do not mean the semi- Markov model solves all problems of formalization. The idea of waiting time distributions in the model has a caveat: though the distributions need not be identical, the distributions are independent of each Other. In substantive terms this means waiting in one occupation does not affect waiting in other oc- cupations, probably a very strong assumption given the various mutual effects of technology on white and blue collar occupa- tions in industrial societies and the modern organization of the industrial state. The possible interactions of the dis- tributions may result in alternative interpretations of our findings, but this is beyond the scope of our research. At the same time, the model does not make the assumption of sta- tionarity since the mathematical properties of the model re- sult in a non-stationary process. Finally, the semi-Markov model, due to thetse of waiting time distributions, incorpo- rates heterogeneity into the model despite the homogeneity of the underlying Markov chain of the model. 45 In this dissertation we limit ourselves to a simple case, a closed occupational system with three states: professional- manager, non-manual workers, and manual workers. Rather than representing mobility in terms of the probabilities of moving among these states we are interested in the determination of these probabilities. Although our unit of analysis is the general occupational category of the individual, we are con- cerned with how sociological factors, such as the status-role of the individual and the economic milieu of the occupation determine these probabilities. We also argue that patterns 3 of inequality in industrial societies set parameters which af- fect the level of job satisfaction of the individual which in turn determines the life chances or the propensity to be mobile of the individual over time (i.e., in his career). These pat- terns of inequality are represented in our study by three background factors. The first factor, sex, represents some of the ascribed characteristics of the status-role. If the waiting time dis- tribution of males differs from females we conclude sex gives different opportunities for mobility. If the distributions are similar we conclude sex is not relevant for the given data. The second factor, self-employment status, reflects the achieved characteristics of the status-role. If the waiting time dis- tribution of self-employed workers differs from salaried workers we conclude self-employment gives different opportun- ities for mobility. If the distributions are similar, we con- clude that this factor is not relevant. The third factor, 46 industrial growth rate of the location of the occupation, in- corporates the effect of economic milieu on mobility. If the waiting time distribution of expanding industries differs from contracting industries we conclude the type of industry gives different Opportunities for mobility. If the distributions are similar we conclude that this factor is not relevant. We are not examining the interactions of these factors, nor are we including other factors in the analysis. Of the three factors, the industrial growth rate of the . .-.:‘- r - __—'V-V' location of the occupation needs the most clarification. v Speaking in terms of the growth rate of the industrial loca- tion of an occupation may sound confusing since location is usually used to refer to the site of employment. We do not mean location in the spatial sense. We feel location is a component of the occupation of an individual, since, in many cases, occupations are situated in more than one industry. For example, an engineer may be employed in a steel mill, an automobile factory, a government agency, or in research and deve10pment organizations. The extent to which each of these locations of employment expands or contracts may affect the mobility of the engineer (Keyfitz, 1973). Growth may be measured in terms of industrial output or the size of the labor force. We are concerned with growth as measured by the increase or decrease of the size of the labor force over time. Hence the location of the industry in which the individual is employed refers to how the economic milieu is represented in the semi-Markov model. We are concerned 47 with the part of the economic milieu termed the labor market. Continuing with our example, the number of engineers in the automobile industry may be increasing, while the number in the steel industry decreases. This is postulated to result in differential mobility patterns. Hence we are concerned with the relative growth rate of the industry where the occupation is located, which we term the growth rate of the industrial location of the occupation. This variable, like sex and self- employment status, takes on one of two values, growth and non- growth. By a process Of formal theory construction using the semi-Markov model we will assess the feasibility of explicating and explaining observed empirical regularities of mobility, for example increasing immobility with increased tenure in a state (Morrison, 1973; TUma, 1972). This would permit us to predict the nature Of mobility in industrial societies. In other words, we are asking if mobility can be represented by a stochastic process such that the time till a move is made is contingent on the occupation from which the move originates and in some cases also contingent on the occupation to which the move will be made, given information about factors that result in vary- ing degrees of retention of the individual in the initial occu- pation. Our reasoning is as follows: mobility is determined by the time spent in an occupation which reflects commitment to the world of work of the individual and salient features of social differentiation. For example, we want to see if it is 48 possible to represent the status-role of the individual in a dichotomous manner, male and female, and then to see how sex determines the length of time spent in an occupation. Hence, in our example, we are asking the probability of moving be- tween occupations given a length of duration in the initial occupation that reflects and depends upon the sex of the in- dividual rather than simply asking the probability of moving between occupations. Since we are processually oriented, we need to stress the various meanings of time in our analysis. we are assuming that mobility over a time period is contingent on the time ' spent in the state from which the move originates. We called the time spent in a state the waiting time. we distinguish among the chronological age of an individual, the time period in which a move may occur, and the waiting time in a specific state before a move occurs. IdeaUy'Our data would be a continuous monitoring of the individual during his working life, looking at ascribed factors, such as sex, age, or race, and achieved factors, such as level of education, level of income, supervisory status, and economic factors. With this type of continuous data we could examine the waiting time till a move is made and see how these factors modify the waiting time. We must be content with existing data sets despite their inadequacies. One of these sets, on Great Britain, is described in Chapter 3. As waiting time is the time till a move is made, we need to observe the occupa- tion from which a move occurs and the occupation to which a 49 move occurs, contingent on the time spent in the original occupation. (The idea of waiting time between events is called time till a failure occurs, or failure time, in a branch of probability theory called renewal theory. The idea of waiting time does not incorporate an idea of multiple-step dependency. Only duration in the current state is involved. Perhaps the label semi-Markov model is misleading since this type of model is actually a multi-state renewal model. Some- times waiting time is called duration time.) If we talk in terms of waiting times we can introduce determinants of mobility into the model by specifying the nature of the mathematical representations that reflect the idea of waiting time. For example, the mathematical function may represent the idea of cumulative inertia, increased immo- bility as length of stay in a state continues (McGinnis, 1968). Ideally we may consider our data to be a set of individ- uals where the individual is represented as a vector, a set of multidimensional elements, such as age, sex, race, and level of education. Our analysis can be conceptualized as a rearrange- ment of the elements of these vectors for all individuals in our sample. The specific elements of the vectors that interest us are the sex, self-employment status, industrial location of the occupation, occupation, and length of time in the oc- cupation of the individual. From this information, we could compute the embedded Markov chain of the semi-Markov model and plot the distribution of waiting times. However, our main concern is what determines these distributions. '50 We will be concerned with a simple case, consisting of three OCCUpational states, professional-manager, white-collar, and blue-collar workers. Since we are concerned with what a move looks like, if a move occurs, we need to specify the num- ber of different moves that may occur among a set of states. A move means going from a state Of origin to a state of des- tination in a countable period of time. Hence the number of different moves among a set of states is equal to the number of possible combinations between pairs of dissimilar states. This is formally computed, in counting theory, as the number of combinations of a finite number of objects taken two at a time. In our simple case, there are six possible transitions between any pair of dissimilar states. Our first step in re- ordering the elements of the vectors is to determine whether a move occurs, and, if it does, to ascertain what type of move occurs. Then we have the waiting time, the time in the first state before moving to the second state. This is plotted for all individuals with this pair of states. This type of analy- sis is detailed in the next chapter. We assume no periods of unemployment betWeen moves and, as we are interested in the move itself, we ignore any intermediate moves. We realize these intermediate moves may be preparatory for future moves, but the issue of intermediate moves is a tOpic for future analysis if we conclude that the semi-Markov model is worth further consideration. 51 Information about the waiting time distribution is an important piece of information in studying mobility in a so- ciety. Even if a society appears to have a steady state dis- tribution among occupations or appears to be non-fluid with respect to movement between social strata, different patterns of moves may be occurring and the steady state or non-fluid as- pects of the mobility system may be artifacts of the statisti- cal aggregation of two or more different patterns. It is for this reason that we ask what background factors may determine mobility patterns. In terms of our model, we are asking what happens when we disaggregate the waiting time distribution by using the various elements of the vectors of individuals. It may very well be that disaggregation results in dissimilar patterns of waiting times. This would indicate the importance of the variables along which the disaggregation was conducted. This point can be illustrated by an example. Suppose we dis- aggregate self—employed from salaried individuals among in- dividuals moving from non-manual to professional-manager states. Self-employed individuals may stay in the non-manual state longer than salaried individuals since the former may leave their occupation only after failure is evident which may involve a long period of time, while salaried individuals may have a constant rate of leaving non-manual occupations. To conduct this type of analysis we use the set of in- dividual vectors to determine the time till a specific type of move is made. Then we search the vector for sociological fac- tors of interest, such as sex, self-employment status, and the 52 industrial location of the occupation as represented by the growth rate, and compute the disaggregated waiting time dis- tribution. If different distributions occur for the two states or conditions of the disaggregated variable, we conclude these factors are relevant in determining the waiting time distri- bution and therefore a possible reason why mobility occurs. If different distributions do not occur we can conclude only that the factor is not relevant for the data being analyzed for the given time period. This chapter has discussed the mathematical aspects of our research. we began with an overview of types of mathe- matical models of mobility, concentrating on the stochastic model. After discussing the cumulative development of the stochastic model we focused on the semieMarkov model, which represents mobility as the probability Of moving between oc- cupations given a waiting time spent in the occupation from which the move originates. we stressed how the determination of the waiting time distribution may lead to a theoretical representation of mobility if we conceive of the effects of the world of work and of the status role of the individual as influencing the waiting time distribution. This, in turn, im- plied the semi-Markov model is a cumulative model of mobility and is also inclusive in a substantive sense. The next chap- ter of the dissertation discusses the sample used to test these ideas and the empirical procedures to be used. CHAPTER III SAMPLE AND METHOD OF ANALYSIS This chapter of the dissertation describes the data which was collected in Great Britain from 1963-1970. Notes are given on the uses and misuses of precollected data, with an emphasis on time-dependent data, since our data set is of this type of data. Finally, the computer procedure employed to generate the waiting time distributions is discussed, con- centrating on the creation of a subsample that meets our scope conditions and operationalizes our background variables of sex, self—employment status, and the industrial location of the occupation. We also discuss our operationalization of the occupational state. The original wording of the questions to gather this information and ways this information is coded is located in Appendix B. we do not claim that the background factors are exhaustive. we only assume that they are exclusive of each other and that they are sufficient to permit an assess- ment of the semi-Markov model for mobility analysis. 3.1 The Butler and Stokes Data: Great Britain from 1963 to 1970 The Butler and Stokes study is a longitudinal behavioral analysis of voting in Great Britain. Individuals were inter- viewed over a seven year period commencing in 1963. Subsequent 53 54 interviews were conducted in the election years of 1964, 1966, and 1970. The main advantage of this data is that it has in? formation on the world of work of the individual providing information about factors that impinge on the mobility process at multiple time points. The primary sampling unit is parlia- mentary constituencies. Interviews were sought with 32 in- dividuals in each constituency according to a random procedure which resulted in 2009 interviews in 1963. No interviews were collected from Northern Ireland which has its own parliament. The first wave of the sample is a self-weighting, multi-stage, stratified random sample of the adult population of England, wales, and Scotland. Butler and Stokes (1969) argue that this multi-stage random sample yields smaller sampling error than a simple ran- dom sample since the former type of design reduces the cost of interviewing, yielding a larger sample with a smaller sampling error. For a sample of this size, an interval the width of_ two sample errors (where the maximum sample error is approx- mately 3.8%) contains the true value 95% of the time. As one of our scOpe conditions is a closed system, we are interested only in information about individuals who are interviewed at all time points, 1963, 1964, 1966, and 1970. The result is a high rate of attrition. Out of the initial sample size of 2009 in 1963, 718 remain in 1970, which results in a sample size of 511 after individuals who violate additional scope conditions are eliminated. The procedure employed to create the subsample of 511 individuals who meet the sc0pe 55 conditions of our research is discussed in the third section of this chapter. The major sources of attrition in panel studies include the death of an interviewee and/or the failure of the inter- viewer to obtain subsequent interviews with the same inter- viewee because the interviewee was out of the country, did not leave a forwarding address after a move, or refused to be .reinterviewed. Unfortunately, Butler and Stokes (1969) do not indicate the rates of attrition due to each of these categories. It is necessary to survey the economic history of Great Britain during this period since exogenous economic forces may give us insight into alternative interpretations of our analysis. Hence we will give a general economic survey of Great Britain from 1963 to 1970 and a detailed examination of the composition of the labor force by industrial categories since 1939, concentrating in the 1963 to 1970 period. The labor force analysis is a necessary part of our study since this provides the information needed for our Operationaliza- tion of the industrial growth rate. Analysis of Great Britain has a few substantive advan- tages for an exploratory analysis of mobility. First, Great Britain is more racially homogeneous than other nations. With the exception of racial disturbance against non-white immi- grants from overseas in 1968, the period under investigation has minimal racial strife. More importantly, the non-whites in our data are less than one percentage of the sample. Be- cause Of this, we eliminated non-white from analysis. This reduced the sample from 718 to 714. 56 . Second, the 1963 to 1970 period exhibits no dramatic change in the distribution of the labor force among industrial categories although between 1950 and 1965 the proportion of manual occupations declined by only 5% (Butler and Stokes, 1969). Third, the period 1963 to 1970 exhibited relatively low unemployment. Great Britain during this period is representa- tive Of an industrial nation near full employment. However, from 1963 to 1970 wage and salary increases were greater than the increase in productivity, resulting in demand-push infla- tion and a foreign exchange crisis. This resulted in an im- balance Of payments and a devaluation of the pound by 14% in 1967. This may have a secondary effect on the distribution and growth rates of workers in industrial categories and partly ex- plain the decrease in the growth of individuals in distributive trades from 1964 to 1970 as presented in Table 4 of this chapter. Fourth, Great Britain is almost an ideal type industrial society. Manufacturing and trade are the largest segments of the industrial work force with the majority in metal-related industries producing heavy durable goods, especially tools, machinery, and transportation equipment. Fifth, and finally, due to social planning Great Britain does not have its work force as concentrated in specific areas as other industrial nations. Since we are not studying how geographic factors may modify mobility processes and since the British work force is geographically diffuse, the interaction 57 effect of these two factors is minimized. For example, tex- tiles are concentrated in Lancashire, coal and electricity in the Scottish Highlands, heavy industry in the Midlands, light and middle industry in the suburbs of London and Liverpool, etc. There is one major characteristic of Great Britain from 1963 to 1970 that may confound our analysis and interpretation of the rate of expansion presented in this chapter: industrial production in the 1963 to 1970 period is undergoing decline or is at best stagnant. As Table 2 shows, only one index, crude steel does not decline during this period. This may be a re- sult of nationalization of 90 percent of the steel industry. TABLE 2 INDICES OF INDUSTRIAL PRODUCTION IN GREAT BRITAIN, 1964-1970 Year' Index 1964 1967 1970 Growth Ratea Coalb 185.4 172.2 150.5 -o.19 Crude steelb 26.2 23.9 26.4 0.00 Automobilesc 1868 1552 1717 -0.08 SOURCE: Whitaker's Almanack, 1965, 1971 (London, Eng.: Whitaker's, 19657 19717. NOTES: aGrowth rate is 1970 figure minus 1964 figure divided by 1964 figure. bIn millions of tons. CIn thousands of units. 58 The negative growth rates may reflect the less than one percentage growth rate of the entire labor force during the 1963 to 1970 period. This would imply a stagnant economy in terms of per capita production. The growth of the number Of workers in an industrial category is our Operationalization of the growth rate of the industrial location of the occupa- tion. One of the possible problems of our analysis is the possibility of a delayed effect of the expansion of an indus- try. For example, upward mobility in the 1963 to 1970 period may be due to possession of the states of the status-role that were Operating prior to 1963. Hence our analysis is open to multiple interpretations. Because of the possibility of de- layed effects, it is necessary to present data before the 1963 to 1970 period. In addition, the older segment of the labor force had the unique experience of the post WOrld war Two re- covery. Hence data is presented for a series of years. Table 3 presents data for the industrial composition of the civilian force for 1939, 1953, 1960, 1964, and 1970. The cited date is taken to be as of 30 June for each year. 1939 was selected as a pre-war, post-depression year. 1953 was selected as a point midway between 1939 and 1970 and long enough after the conclusion of WOrld war Two to minimize the effects of a post war reconstruction. 1960 was selected to see the entire decade of the 1960's and to detect whether or not any short run delay factor may have existed in the years immediately be- fore 1963. 1964 was used as data for 1963 was unavailable. 59 TABLE 3 DISTRIBUTION OF THE CIVILIAN LABOR FORCE OF GREAT BRITAIN: 1939-1970 as of 30 June) in thousands) Year IndUStrY 1939 1953 1960 1964 1970 Agriculture, horticulture, and fishing 950 1092 971 886 391 ‘Mining and quarrying 873 876 765 657 419 National b government 539 595 501 539 580 Local b government 846 725 740 813 805 Gas, water, and electricity 242 373 370 398 386 Transportation and communication11233 1726 1652 1617 1552 Manufacturing 6815 8723 8834 8831c 9388C Building and construction 1310 1448 1541 1720 1343 Distribution trades 2887 2641 3265 3404 2702 Professionals, finance and miscellaneous d services 2252 3991 __,4954 5375 4851 Males 15478 15798 a 1746 Females 8115 8442 Total P7947 22190 23593 24240 24267 SOURCE: Whitaker's Almanack, 1954, 1957, 1961, 1965, 1971, 1974 (London, Eng.: Whitaker's, 1954, 1957, 1961, 1965, 1971, 1974). 6O NOTES: 8Employers and self-employed. bIn 1967. 1970 is an aggregation of National and Local Government of 1378. cSum of all manufacturing categories since data is dis- aggregated. dSum of all professionals et a1. categories since data is disaggregated. An examination of Table 3 reveals a picture similar to I most industrial nations. The agriculture, mining and related industrial categories seem to decline in their number of i workers while manufacturing, energy producing, professionals 1 and related industrial categories increase. Surprisingly, building and construction, and government workers are rela- tively constant in the period although the fluctuation in the former may be due to completion of the "new towns'by the early 1960's. The sharp decline in distribution trades at 1970 may be a result of the balance of payment problem and devaluation of the pound in 1967. (The reader of the table should note that transportation and communication includes the government run railroad, telephone, and postal services. The figures given for government workers do not count double government workers in other industrial categories.) The growth rate of industrial categories provides the information needed to compare rates of expansion. The growth rate is equal to the size of the labor force in the year at the end of the period minus the size of the labor force at ‘the beginning of the period divided by the size of the labor 61 force at the end of the period. The size of the labor force for a given industry is in Table 3. Growth rates for the industrial categories are presented in Table 4. The rate given means (1 - g) x 100% of the figure at the end of the period is equal to the figure at the begin- ning of the period, where g is the growth rate. Thus, unlike Table 2, we are using the final year as the base year, which makes our growth rates in Table 4 oriented forward in time. Examination of Table 4 reveals the 1964 to 1970 period exhibited no expansion. The entire labor force for this period exhibited no measurable growth. Agriculture and min- ing exhibit sharp declines. Gas, water, and electricity, transportation and communication, building and construction, distribution trades, and professionals, exhibit moderate to slight declines. The slight decline in professionals may re- flect the disaggregation of the 1970 data. National govern- ment and manufacturing exhibit slight increases. Looking at Tables 2, 3, and 4 together seems to imply most movement was to manufacturing and most new entrants to the labor force were to manufacturing. Table 4 gives other growth rates due to the possible delay factor. The 1939/70 rate is given to see the long run pattern of growth. Long run growth is exhibited in utilities, transportation and communication, manufacturing, and professionals. The 1939/70 period is split into 1939/53, 1953/64, and 1964/70 periods. This analysis reveals the majority of the 62 TABLE 4 GROWTH RATES OF INDUSTRIAL CATEGORIES IN GREAT BRITAIN: 1939-1970 Time Perioda Industry 1939/70 1939/53 1953/64 1964/70 1939/60 1960/70 Agriculture, horticulture and fishing -1.42 .13 -.23 -1.26 .02 -1.48 Mining and quarrying -1.08 .00 -.33 - .56 -.14 -.82 National government .07 .09 -.10 .07 -.O8 .14 Local government -.06 -.17 .11 .00 -.14 .08 Gas, water,axi electricity .37 .35 .06 -.O3 .39 .04 Transportation and communication .20 .28 -.04 -.O4 .25 -.06 Manufacturing .27 .22 .01 .06 .23 .06 Building and construdjon .02 .10 .16 -.28 .15 -.15 Distribution trades -.07 -.O9 .22 -.26 .12 -.21 Professionals, finance and miscellaneous . services .54 .44 .25 -.11 .54 -.02 Total .26 .19 .08 .00 .24 .03 NOTE: aThe / means the period; for example, 1964-7O means the 1964 to 1970 period. 63 long run growth in utilities, transportation and communica- tion, manufacturing and professionals occurred before 1953. Alternatively, the 1939/70 period is split into 1939/60 and 1960/70 periods. This reveals the bulk of the decline in agriculture and mining occurred in the early 1960's. However, the national government seems to be undergoing a rapid growth in the same period. The bulk of the increases in utilities, transportation and communication, manufacturing, building and construction, and professionals occur before 1960 or in the early 1960's. The result of this analysis of the industrial labor force of Great Britain is information that enables us to Op- erationalize one of our variables that may impinge on the mo- bility process, the variable of the industrial location of the occupation. This will be discussed later in the chapter, along with the discussion of the Operationalization of the other variables. 3.2 Problems of Using Pfe-COIlecteleanel Data Although the data we prOpose to use is a voting study of political change in Great Britain, the data provides in- formation about factors that impinge on the mobility process as information on background variables was collected to ex- plicate the voting behavior. The use of the panel technique involves the administration of the same set of questions on the same group of individuals at multiple time points. In other words, panel studies are replicated interviews. The -.-n’ - n: 64 panel technique is superior to one step observations or col- lecting data at one point because we can see the change in patterns of responses or the type of phenomena being studied at various time points. This allows us to make longitudinal statements about social phenomena without having to make strong assumptions about the regularity of the phenomena. The panel technique is employed here to identify the dynamics of i- the regularities. By having data representing change over a period of time, especially net change, one can hOpefully begin to explain social processes such as mobility in terms of dy- L namic regularized and recognized patterns. Panel studies tend ' to minimize the problems of induction of dynamic social processes (Galtung, 1967). There are no techniques other than panel techniques to see the dynamics of intragenerational mobility. There are some techniques, usually involving cohort analysis, that at- tempt to synthesize a dynamic process. For example, Mayer (1972), used the Blau and Duncan (1967) data to develOp data amenable to dynamic analysis by dividing the data into four- groups of individuals aged 25, 35, 45, and 55 years of age. The older groups are considered to be the younger groups after a duration of time equal to the differences between the ages of the two groups. Individuals aged 35, 45, and 55 are con- sidered to be the individual aged 25 after 10, 20,and 30 years respectively. For example, the pattern of moves observed for the 45 year old group would be considered to be the pattern of moves for individuals who have a duration of 20 years in their 65 state of origin, given the state of origin is the state where the individual is located when the individual is 25 years old. By this method, the same group can be synthetically analyzed at future time points, simulating the effects of duration. Any changes in the social structure are negated by assuming social forces of change are equally salient to all individuals at all time points. This negates the intent of processual 1 analysis in sociology; namely, studying how changes in the social structure affect social processes. The results of any ‘1 mobility study using cohort analysis probably would be errone- L ous, especially in periods of economic change and different ' rates of expansion of industries. The panel technique is not without its disadvantages (Galtung, 1967). Since the panel study is a replicated inter- view, the disadvantages of the panel combine the disadvantages of all interviews and all longitudinal studies. With respect to the disadvantages of interviews, the panel study does not have pre-specified independent or antecedent variables, and does not have a control group. Usually the types of questions asked and manner responses are recorded are incongruent with the needs of the user of the data. The user of panel data for secondary analysis must compromise his need for data with the data available. Usually this compromise involves using the pre-collected data to attempt a falsification of the postu- lated relationship. To do this, the postulated relationship should have a level of abstraction such that its sc0pe includes the observations represented by the pre-collected data. The 66 pre-collected data, not the postulated relationship, is in a fixed time and space. In addition, the panel study involves a social psycho- logical process since it is an obtrusive form of research and subject to experimental effects that may invalidate the data collected. This may be amplified in the panel study, since subjects may attempt to recall previous responses (Galtung, 1967). Finally, the panel study usually has an attrition factor due to the death of some individuals and the failure to obtain subsequent interviews. Hence analysis of data collected by a panel study may be a confounding of real changes and a sample bias, since the individuals who contribute to attrition may have characteristics distinct from the entire sample population. These problems can be partly overcome. The lack of control and pre-specified independent variables can be mini- mized by multivariate analysis. The extent of experimental effects can be studied by examining the degree of incongruity in answers to questions concerning non-varying factors, like the level of education of one's spouse, or the age difference between a respondent's parents. Experimental effects can be minimized by triangulation. The extent of the discrepancy of real change and sample bias due to attrition may be measured by assessing whether or not the part of the sample that con- tributes to attrition possesses characteristics different from the rest of the sample population. Attrition can be minimized 67 by introducing a panel weight factor to the remaining part of the sample. With respect to the problems of longitudinal analysis, the panel study results in an aggregation problem. Specif- ically, the problem when applied to mobility, is asking whether Observed changes in the transition matrix are due to structural factors acting on all individuals or statistical , artifacts due to the behavior of individuals. This problem of panel studies is a specific example of one type of response uncertainty (Coleman, 1973). i This type of response uncertainty asks if individuals analyzed at multiple time points may give erroneous answers to questions, i.e., be uncertain of their responses. Hence Ob- served change, for example transitions between states, may not be an index of the expected proportion of all individuals who make a specific transition but may be due to an uncertain response as to current and/or past location in a specific state. Studies of perceived social class and class conscious- ness continually exhibit such response uncertainty (Landecker, 1963). Response uncertainty confounds actual change and human forgetfulness resulting in an apparent heterogeneity of move- ment at the macro level being partly due to response uncertain- ty at the micro level. This contributes to the problem of aggregation in mobility models: the heterogeneity of mobility patterns may be due both to response uncertainty (Coleman, 1973) and possible actual different patterns of behavior 68 (McFarland, 1970). This results in a dilemma: the degree of change can never be completely identified at the structural level. Out of all the problems that arise in using pre-collected data we believe the major problem is the researcher must com- promise his desires since the way a variable is recorded may not always be in a desired or usable format. For example, the Butler and Stokes data fails to differentiate between public and private workers in all industrial categories, or individuals who work in small or large organizations, two fac- tors that affect the world of work of the individual. We also note we only know the individual's occupation is his or her chief occupation; we do not know if it is a full time or part time occupation. 3.3 Operationalization of Variables The scOpe conditions of our research are determined by our interests. Since we are interested in the feasibility of using the semi-Markov model, we are dealing with the simplest case of applicability of the model: an adult, civilian, full- time employed working population in an industrial society at multiple time points. As mentioned above, we are interested in whites only. All non-whites were excluded from analysis. This resulted in a reduction of our original sample of 718 to 714. The Butler and Stokes data is recorded so that it is possible to differentiate the respondent from the head of 69 household. This led us to believe we may be able to increase our sample size; for, if interviews were conducted during the day, it is likely the respondent would be a "non-worker" and out of the scOpe of our research. The data is also recorded such that we can easily examine only individuals interviewed in 1963, 1964, 1966, and 1970. We assume if the head Of household is not the respondent, the head of household is the same individual at all time points. This ensures the continu- ity of our panel. We also assume that all individuals are continuously employed in years between the interviews. Since information about changes in marital status and periods of un- employment is unobtainable, these assumptions are untestable. Moreover, the head of household, if not the respondent, is the respondent's father or husband, and is not retired. All other heads of household who are not respondents were excluded from subsequent analysis since it is impOssible to determine the sex Of individuals in this group, as it may include wives, siblings, and mothers, and the sex of any sibling is not given. Occupational states are operationalized as the occupa- tional grade of the respondent and head of household. If a ' or does respondent is a state pensioner, or a "non-worker,' not give information as to his or her occupational state in any year, the individual was excluded from our sample. We constructed three graded occupational states, the states of our model, from this information, by using criteria suggested by Goldthrope and Hope (1974). “I JS-L 7O GoldthrOpe and HOpe contend that any set of graded oc- cupations is a scale of recognized prestige differences such that the elements of this scale are large homogeneous sub- groups that are presumed to be similar with respect to socio- economic characteristics. We note certain occupations may overlap these gross states such as a skilled craftsperson who sells wares in his or her shop, while some occupations are L heterogeneous, especially white collar occupations in an ad- . vanced industrial society like Great Britain. Goldthrope and a Hope recommend that when one collapses categories one should '17‘ maintain differences among states, collapse adjacently ranked states, try to maintain the symbolic meaning of a state, and try to collapse states such that individuals are involved in similar work tasks. Using these guidelines, we ended up with three states: professional-manager, non-manual, and manual. This is discussed in detail in Appendix B. The head of household is operationalized as the indi- vidual who is financially responsible for the place where the interview is conducted. Knowing if the respondent is different from the head of household enables us to prevent double count- ing of individuals who may be housepersons and heads of house- holds. WOrking heads of households may be males or females. However, these individuals if not the respondent,and if not the respondent's father or husband, are all lumped together. Since this category is only 6.3% of all heads of households in 1963, we excluded this category from further analysis. Note it is possible to be female and in this category, such 71 as a wife, mother, or sister. The result of this procedure is all heads of households, if not respondents, are males, while respondents can be males or females. The first background variable to be operationalized is sex. Butler and Stokes do not present any information on the Operationalization of the sex of the respondent. we assume this is operationalized by visual inspection since all data was collected in face to face interviews. One of our interests is if sex reveals different mobil- ity rates. Since sex is a multiple value status characteris- tic in all societies, that has a preferred state, male, we would expect individuals of different sexes to exhibit dif- ferent mobility patterns. Since mobility is a life-chance in modern societies, possession of the preferred state, male, should affect the degree to which the individual is mobile. For example, if males and females exhibit different rates of mobility for the same type of move such that males wait shorter for an upward move to be made compared to females, we would have evidence of inequality of opportunity with respect to sex in the population being analyzed. The second background variable to be operationalized is self-employment status. Butler and Stokes term this variable economic status. Economic status asks if the individual is self-employed with or without employees, or a manager, or a foreman/supervisor, or any other employee with no direct managerial or supervisory responsibility. We would expect different rates of mobility for self-employed individuals since 72 being self-employed usually reflects monetary investment in equipment and the deve10pment of a clientele (Ladinsky, 1967). Goldthrope and Hope's guidelines, used in developing occupa- tional states, were applied to economic status. This resulted in two conditions, self-employed and non-self—employed or salaried individuals. Individuals who were unemployed, not in the labor force, or not giving information with respect to their economic status in any year were excluded from sub- sequent analysis. The final background variable tO be operationalized is the relative rate of growth of the industry in which respon- dent is employed and the location of the occupation. Butler and Stokes term this information the individual's occupation in contrast to the occupational grade which provided the in- formation to determine the states in our model. Our opera- tionalization is done with the aid of Table 4, which presents the growth rate of occupational categories by industrial loca- tion, as measured by the increase or decrease of full-time workers. The Operationalization involved using this informa- tion in order to dichotomize the industrial setting of the occupation into growth and non-growth categories, which reflect the information termed the individual's occupation by Butler and Stokes. The industries are dichotomized as follows: 223- growth (farmers, foresters, fishermen, miners, quarrymen, construction workers, painters, decorators, non-civil service clerical workers, and sales workers) and growth (gas, coke, chemical, glass, ceramic makers, furnace, forge, foundry, 73 rolling mill, electronic, electrical and engineering workers, woodworkers, leatherworkers, textile and clothing, food, drink, tobacco, paper and printing workers, drivers, service, sports and recreation workers, professional and technical workers, transportation and communication workers). Occupations were coded by using the General Registrar's Office Classification of Occupations, the British analogue of the U.S. Department of Labor Occupational Classification. Butler and Stokes note when the occupation is coded the code reflects the individual's usual occupation, or if multiple occupations are given, the oc- cupation with the higher social grade is coded. They also note if the description of the occupation is ambiguous the simplest interpretation of the occupation that seems compatible with remaining details is employed. Finally, professionals with managerial responsibilities are coded as professionals if in an organization germane to the profession, and coded as mana- gers otherwise. Appendix C lists the 25 gross groups of occu- pations that were used in our analysis of the industrial growth rate of the setting of the occupational state. When determin- ing whether or not an individual is in a non-growth or growth category, if the individual was in the armed forces, or a houseperson, or a student, or unemployed, or retired, or refused to answer, or inadequately described his or her occupation in any year, the individual was excluded from further analysis. Once our variables were Operationalized we were able to conduct some simple analyses by standardizing and editing the Butler and Stokes data tape. Note the Operationalizations of 74 our variables involved data standardization in order to make the data meet our sc0pe conditions. Our editing started with the creation of a file using relevant data from the Butler and Stokes voting study. This study has 1244 variables due to its longitudinal nature. This was reduced to 45 variables of interest to us as students of mobility, that include only white individuals interviewed at all stages of the panel. Af- ter this file was created, we created a new working file to prevent double counting of heads of household and respondents, that also eliminated non-civilian and non-continuously employed workers. The final data base is presumed to be an adult, ci- vilian working population, continuously employed from 1963 to 1970. This is the closed occupational system that meets the scope conditions of our research. The end result is a sample Of 511 individuals. Finally, a working file for purposes of analysis of the waiting time was created. This file consists of information on the sex, 1963 dichotomized self-employment status, 1963 dichotomized industrial setting of the occupation, and col- lapsed occupational states of the individual, using data from previous files. Since we are examining the distribution of waiting times for moves where the period of observation starts in 1963, we think this information is necessary and sufficient for our analyses. Table 5 gives the distribution of the 511 individuals among the three occupational states as of 1963. The disaggre- gation in Table 5 is with respect to each Of the dichotomized 75 background variables. Hence we can see the way individuals having a specific condition of the background variables are represented in the three occupational states. The raw data in Table 5 may be employed to show the proportion of each con- dition of the three background variables in each of the occu- pational states, independent of other occupational states. For example, 65 of the 73 professional-managers are males, or 89.0%, compared to 11.0% that are female. Since we are in- terested in the distribution of our sample of 511 individuals among occupational states, contingent on background variables, we are not interested in this type of analysis. Hence in substantive terms, Table 5 represents the prOportion of males, females, self-employed, non-self-employed individuals, and in- dividuals in non-growth and growth industrial locations of the occupation among graded occupational states for our sample as of our initial time, 1963. Table 5 allows us to see if the various groups repre— sented by conditions of the background variables are over- or underrepresented in a given occupational state. A group is overrepresented if a larger prOportion of its members is in a given occupational state than the prOportion of the total pop- ulation in that state. A group is underrepresented if a smaller proportion of its members is in a given occupational state than the proportion of the total population in that state. Hence, in the professional-managerial state, males and females are equally represented, but self-employed individuals and individ- uals in growing industrial locations of the occupation are 76 .mumxomun CH maOHuHanum "mHoz mom new use as ea ass Ham amuse Amme.v com Aams.v oa Aame.v cam aema.v e Aems.v am “dam.v sea Aaem.v new asses: Aaaa.v am Amad.v moH AaeN.v NNH Anna.v om Aamd.v om Asm~.v «Ha AmeN.V asa assene-soz Asma.v an Akeo.v ea AmHH.V mm Amos.v ma AsHH.V m Aeea.v no Amsa.v me sanded: uamnowmmomoum cpzouu £u3ouwuaoz cohoHQEm cohoHQEw moHdEwm mead: mammacoz— 1.36m 1309 madam HOaOHummsooo manmfium> UfinOwaomm >HHAHmoz.hO mmOHuOHQEO OHOEmm OHOZ Meow nHmenaoz anmm HODOH >m mo>oz OHQOHHO> mHzHOm MZHH HzmmmmmHO >m MQoz.mZHH|mzo mo mmmZDz 0 mafia. 85 Since we are interested in movement over time, the events represented by the random variable is whether or not a move occurs by a given time. In turn, Table 6 can be inter- preted as the Observed frequency with which each of the random variables occur for different types of moves, disaggregated with respect to the background variables of sex, self-employment status, and the industrial location of the occupation. The random variable, Xi’ has the value of l, 3, and 7 years res- pectively for the index i having the value 1, 2, or 3. The value is the number of years since the initial point of obser- vation, 1963. For example, among males moving from the non- manual to professional-manager state the frequencies are 4 for X1, 5 for X2, and 13 for X3. The information given in Table 6 is necessary and sufficient to compute the maximum likelihood estimates of the waiting time distribution. Table 7 lists the estimated maximum likelihood mean for each of the distributions assuming the distribution takes the form Of a Gamma function. Knowing the value of each random variable, and the frequency of each random variable from Table 6, the mean value, of X can be computed. Next the nat- ural logarithm of each value of the random variable is computed. These values and R are then used to compute the right hand side of the equation in Appendix D which gives the value of E. This computed value falls within the range of 0 to 1. This value is compared to different values of E that are manipulated to compute the left hand side of the same equation. Some of these values are listed in Appendix D. The value of E is 86 000.H 0N0.0 5H0.0 000.N 000.0. me.N 000.0 Hanna: 0» Hwfinmeucoz 00N.H 000.H 00m.H rrrrr nuns: 0mm.H 0mm.H Hanan: on Hmwmcme uHchHmmmmoum 000.0 000.H me.0 000.H N00.0 N00.0 wa.0 Hasnmauaoz on Hmwdnme qunOHmmomoum 0H03fl300 000.0 000.0 000.0 000.H 003.0 mm0.0 0m0.0 HancmEucoz on Hanan: 00n.0 0N¢.0 050.0 uuuuuuuuuu 050.0 050.0 Hmwoame nHOnOHmmmwoum on Henna: mwn.0 N00.0 0H0.0 0N¢.0 N00.0 000.0 000.0 Mommame rHOOOHmmmmoum on Hdscmeraoz 0HO3mD EHSOHU £u3ouwucoz Umonme womoHQEo oHdamm OHM: uMHmmuGOz IHHom HeuOH o>oz_mo make oHannm> ZOHHDmHmHmHQ MZHH UZHHH<3 mmH ho mzoz we make mHanHd> m ”0ng ZOHHDmHmHmHO MZHH UZHHH<3 mmH mo mmuz 89 000.N 0H¢.N 000.N 000.H 000.0 05¢.N NHO.N Hanan: on Handmauaoz 00H.0 000.N 000.0 uuuuuuuuuu 000.0 000.0 Hanan: on Hmwmnme uHMSOHmmwwonm 000.N 000.N 000.N 000.N 00N.N 00N.N 00N.N HanamEuaoz on Hmwmcma uHMGOHmmowoum 0u03c3oo 000.0 000.0 q0N.0 000.N 000.0 00H.0 05H.0 Hdncdeucoz ou Hanan: 000.N N00.¢ «00.0 uuuuuuuuuu «00.0 000.0 umwdcme uHmaOHmmwmonm on Hanan: emn.N 030.3 mmm.m Nee.a aeo.m mmm.m smm.m newness nHQSOHmmmmoum on Hadcweunoz Unmzmo £u3ou0 £u30uwuaoz Um>OHQEm vmhoHaEm deEmm oHdz umeeroz uMHmm HmuOH m>oz.mo 0008 mHannw> 000H Hmom mm ZH ZOHHDmHmHmHQ MZHH UZHHH<3 0 MHmoz_mo mmmzbz ZDZHxOH050 wmonQEm mHMEOm mez uMmeunoz umem HauOH m>oz mo wake mHanum> mm zH mMZHH GZHHH<3 mu< Qm>mmmmo OH mHm t> 118 119 The McGinnis and Cornell school of models introduces the idea of duration in a state, duration of period r. Hence rP(t) is the transition matrix for people in the t-th interval who have been in their states for r periods. Using a notation like in the mover-stayer model, rP(t) = r8 + fM There is no general equation for the model although a unique equilibrium exists. Boudon (1973) presents iterative techni- ques for the model. The techniques recognize at time t there are 2t types of people who move or stay in different patterns. In the vacancy chain model, Q1 + QO = 1, where each eler ment of matrix Q is the probability of a vacancy moving from state i to state j, Qo is a vector representing the probability of a vacancy in a given state moving outside of the system, and 1 is a vector consisting of all elements equal to one. The probability distribution vector giving the probability a chain starting in state i terminates in n moves is n-l Pn = Q Qo' In Mayer's continuous time models, P(t) = eAt In the first model, the birth-death model, moves only to ad- jacent states are allowed. In this model, I Z 0 if li-jl = 1 a.. = s 0 if i=j t = 0 if li-ji > 1 120 In the second model, each term of A is weighted by a decay factor, so -ct A(t) = e A In the final model each state is decomposed into a transient and absorbing state so 2n states exist. In the model, EC A= , no where C is an absorbing matrix with only diagonal entries and the entire matrix has the properties of an instantaneous tran- sition matrix. In Tuma's model, the basic factors of interest are fj(t), the instantaneous rate of leaving state j, and kij(t)’ the probability an individual of type i is attracted to state j at time t. The leaving process is uniquely determined by fj(t), since if Gj(t), is the expected proportion of the population in state j at time t, and Hj(t) = 1 - Gj(t), then d H.(t) t J = Hj(t) = fj(t) X exp. (Bffj(u)du). dt The attraction process is determined by a time—varying attrac- tion factor, aij(t)’ and the number Of vacancies in state j at time t, vj(t), such that V t j<> n Evm(t) m=l kij(t) = aij(t) Note if fj(t) = fj for all t, Tuma's model is the instantan- eous rate model. 121 The semi-Markov model consists of two sets of factors, the embedded Markov Chain, P, and the waiting time distribu- tion. The latter will be discussed in detail in Appendix D which concentrates on the Gamma Distribution. The waiting time distribution for the relationship between states i and j, Fij(t)’ is defined as the time spent in state 1 before a move to state j, or Fij(t) = Pr{T(t) - T(t-1).<_:TIX(t-1)=i, X(t)=j} The semi-Markov model uses the idea of a renewal den- sity, or a transition from state i to state j associated with a duration in state i, and a sequence of events leading to alln states at time t-u followed by a transition to state j in an interval Of length u. This can be written as the matrix equation, H(t) = m(r)+ {)tH(t-u)M(u)du, where M(t) is the term by term concatenation of P and f(t), where f(t) =.§§é32 . Note if fij(t) = fi(t), irrespective of the state of destination, j, the semi-Markov model reduces to the leaving part of Tuma's model. H(t) has a solution that is found by Laplace Transforms since a convolution is involved. The solution, in terms of Laplace Transforms, is, H(e) = <>>‘1. The intent of the semi-Markov model is to compute P(t), whose elements are the probability of moving from state i to state j given a length of time t in state i. The equation for P(t) is similar to the renewal density equation, the 122 probability of moving from i to j after being in i for a period Of time t and surviving in j plus a sequence of events leading to all n states at time t-u followed by surviving a transition to state j associated with an interval of length u. This can be written as a matrix equation, P(t) = N(t) + at H(t-u)N(u)du, where ”W = 6%..) dv. The matrix equation has a solution in terms of Laplace d Transforms Of, P(S) = (1+H(S))(N(S))- All the results can be combined to get the equation, P = <1/s) <1-M>'1>. which does not involve computing the renewal density function. Hence P(t) can be computed directly from P and F(t). Finally, due to a property of Laplace Transforms, the equilibrium distribution can be computed, since P(w) = lim sP(s) s~0+. APPENDIX B QUESTIONS AND CODING OF RESPONSES The part of the survey entitled "Household Composition and Occupational Details" provides the information used in our analysis. The relevant Open-ended questions of the inter- view include: - can you tell me who else there is in your household living here besides yourself? (give relationship to respondent, sex, age, whether or not in job, marital status) - which member of your family living here is actually the owner/ is responsible for the rent? - what type of firm or organisation (sic) does (house- holder) work for? - what job does (householder) actually do? does he/she hold any particular position in the organization? - (if in public service) what is his/her rank or grade? - (if proprietor or manager) how many employees are there? If the respondent is not the head of household, the questions were repeated. Occupational Grade gives information about the occupa- tional states. After editing out individuals Who were not always in the panel, or state pensioners, or housepersons, 123 124 or failed to give information, the following coded categories remained: First, for interviews conducted in 1963 and 1964, for respondents only, 1. higher managerial, administrator, or professional 2. intermediate managerial, administrator, or professional 3. supervisory or clerical and junior managerial, administrator, or professional 4. skilled manual workers 5. semi-skilled and unskilled manual workers Second, for interviews conducted in 1966 and 1970, for respondents only, and for heads of household in all years, 1. higher managerial or professional 2 lower managerial or professional 3 skilled or supervisory non-manual 4. lower non-manual 5 skilled manual 6 unskilled manual In both scales, the five item and six item scale, the first two categories were grouped for the professional-manager state. Category three and categories three and four of the five and six item scales respectively were grouped for the non-manual state. The remaining states in each scale were grouped for the manual state. Economic Status gives information about the background variable of self-employment status. After editing out 125 individuals who were not always in the panel, or unemployed in any year, or housepersons, or failed to give information, the following coded categories remained: 1. 2 3 4. 5 self-employed with employees self-employed without employees managers foremen/supervisors other employees (includes all employed persons who have no direct managerial or supervisory responsi- bility) Categories one and two were combined to designate self- employed individuals. The remaining categories designate non-self—employed individuals. APPENDIX C OCCUPATIONAL CODES The variable of occupation is used to determine the in- dustrial location of the occupation to find the effects Of growtheuulnon-growth on mobility patterns. After editing out individuals who were in the armed forces, or housepersons, or students, or unemployed, or failed to give information, or inadequately described their occupation in any year, the following gross categories of occupations remained: CATEGORY NUMBER OCCUPATION I Farmers, Foresters, Fishermen II Miners and Quarrymen 111 Gas, Coke, and Chemical Makers IV Glass and Ceramics Makers V Furnace, Forge, Foundry, Rolling Mill Workers VI Electrical and Electronic Workers VII Engineering and Allied Trade Workers VIII Woodworkers IX Leatherworkers X Textile WOrkers XI Clothing Workers XII Food, Drink, and Tobacco Workers XIII Paper and Printing Workers 126 CATEGORY NUMBER XIV XV XVI XVII XVIII XIX XXII XXIII XXIV XXV 127 OCCUPATION Makers of other Products Construction Workers Painters and Decorators Drivers of Stationary Engines, Cranes, etc. Other Laborers Transportation and Communication Workers Warehousemen, Storekeepers, Packers, Bottlers Clerical Workers Sales Workers Service, Sport and Recreation Workers Administrators and Managers Professional, Technical Workers, Artists APPENDIX D THE GAMMA DISTRIBUTION In Appendix A we noted the semi-Markov model requires a distribution of waiting times, F(t). F(t) is a monotonic probability distribution function. In the Gamma distribution, 0:) = a _ r'1 (r-l)! d -1 an F(t) = 1 - ES e'at(at)K K=O K! If r = l we get the exponential distribution. If used in the semi-Markov model, the result is the continuous time model, P(t) = eAt. Using maximum likelihood techniques, we can get estimates foraand r, denoted éand E, from the observed time till a move occurs, Xi' E is the solution to the equation n 1n r - F'(r) = 1n X’- (l’n) E ln.Xi, where F(r) K=1 X is the mean time till a move occurs, and F'(r) is - F r 1 <> r-1 1 22K" K=l L a 128 129 where 713 Euler's constant ( = .572157...) The following is a partial table Of r and values of 1n r - F'(r) F (r) r 1n r - F' r I' gr; 1 O 2 .7295377 5 .1033124 10 .0508276 20 .0252067 50 .010.. Knowing E, we estimate 8 aseg- . Hence we need information X about the time till a move occurs for our estimation. 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