107 0091 ABSTRACT STOCHASTIC MODELS OF SOCIAL MOBILITY: A COMPARATIVE ANALYSIS AND AN APPLICATION TO JOB MOBILITY OF MEXICAN-AMERICAN MEN By Nancy Brandon Tuma This research has two main aims: (l) to analyze and compare previously-proposed stochastic models of social mobility possessing a Markovian interpretation, and (2) to apply these models, insofar as possible, to job mobility of a sample of 584 Mexican-American men with a high school education or less who resided in selected Midhigan counties during the winter of 1967—1968. It is first shown that one can separate the process of indi- viduals leaving social positions from the process of individuals entering new social positions, given that they are leaving other social positions. This division facilitates the comparative anal- ysis of mobility models. The innovative features of most previous models pertained to the process of leaving a social position. The natural assumption is that an individual's probability of leaving a social position is a time-independent constant (designated Probability Law I). Past predictions based on this law have consistently been poor. To improve predictions, three approaches have been used: the assump- tions of nonstationarity (time-dependence), of population hetero- geneity, and of stages within a social position. Each approach has been used to construct various models of the leaving process; how— ever, several models imply the same probability law. In all, six Nancy Brandon Tuma probability laws are implied by previous models. Model construction is only part of an explanation of the leav- ing process. Since mathematical models are intended to describe equivalent events, i.e., equivalent individuals leaving equivalent social positions, it is also necessary to identify variables affect- ing the magnitude of the parameters of a model. In the application of these models to job mobility of Mexican—American men, the vari- ables examined include calendar year, certain fixed attributes of jobs and jobholders, and the jobholder's age, duration in the job, and previous work experiences. For respondents leaving jobs as Operatives, nonfarm laborers, and farm laborers during 1935-1967, educational level and occupation generally affected most the rate of leaving jobs. Job duration was very important; the jobholder's age at entering the job was not. Previous work experiences had some effect but need further study, especially because of their relevance to the validity of the Markov assumption. Other variables were influential in special cases. The identification of variables affecting the leaving process gave twenty-one sets of data for evaluating probability laws. Probability Law I was satisfactory for four data sets, but predic— tions for the other seventeen data sets were noticeably improved with one or more of the other probability laws. The probability law derived from a two-stage model of a job consistently gave the best predictions, although improvement was slight. Substantively, the two-stage model leading to this probability law appeared par— ticularly worthy of further investigation. In previous mobility models the process of individuals Nancy Brandon Tuma entering new social positions has received limited attention. This study considers simple models of the conditional probability of in- dividuals moving among jobs. Fundamental quantities in the models are the distribution of job vacancies and the attraction of jobs with certain attributes for individuals with certain attributes. Ways of approximating and estimating job vacancies are also con- sidered. The application of these models to job mobility of Mexican- American men focuses on the impact of different variables on the attraction of different jobs in selected geographical locations. The most significant variables were the geographical location and occupation of the jobs entered and left, and the jobholder's educa- tional level and earlier occupation. Results suggested that the distribution of filled jobs is a rough approximation to the distri- bution of job vacancies. STOCHASTIC MODELS OF SOCIAL MOBILITY: A COMPARATIVE ANALYSIS AND AN APPLICATION TO JOB MOBILITY OF MEXICAN-AMERICAN MEN By Nancy Brandon Tuma A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Sociology 1972 Copyright by NANCY BRANDON TUMA 1972 Reproduction by the U.S. Government in whole or in part is permitted for any purpose. ACKNOWLEDGEMENTS A number of persons have assisted me in completing this disser- tation, and I am pleased to have this opportunity to acknowledge their contributions. In suggesting the topic of this dissertation, Thomas Conner introduced me to an area of research with which I was not previously acquainted and led me to explore a research problem entirely different from my previous plans. Our discussions while I was engaged in the theoretical analysis found in this document were especially helpful. I am also very appreciative of the generosity of Grafton Trout and Harvey Choldin in furnishing me with the raw data used in this study and in permitting me to direct the editing and coding of employment-residence histories so that the coded data would be in the form most useful to me. I also extend my thanks to David McFarland and Peter Morrison for supplying me with extensive bibliographies when I was becoming acquainted with the field, to Hans Lee for suggesting the use of computerized plotting in my data analysis, to Jim Miller for acting as my liason with the CDC 3600 at Michigan State University during the two years that I lived in San Francisco, to the staff of the Michigan State Computer Center for their excellent service, and to Harry Perlstadt, Anne McMahon, and Michael Hannan for particularly useful comments on various written drafts of this document. In addition, I am deeply thankful for the loving and patient support ii of my husband George, my daughters Katie and Clare, and my parents. Finally, I am very grateful for the financial support of the Manpower Administrations 'The material in this project was prepared under Grant No. 91-24-69-12 from the Manpower Administration, U.S. Department of Labor, under the authority of title I of the Manpower Development and Training Act of 1962, as amended. Researchers under- taking such projects under Government sponsorship are encouraged to express freely their professional judgment. Therefore, points of view or opinions stated in this document do not necessarily represent the official position or policy of the Department of Labor." iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . xii INTRODUCTION . . . . . . . . . . . . . . . . . . . 1 CHAPTER 1 The Scope of the Research . . . . . . . . . . . . 4 1.1 The objects which move . . . . . . . . . . . 4 1.2 The positions which the objects may occupy . 7 1.3 The points in time when the objects may move 9 1.4 The conditions under which movement occurs . 10 (a) Open or closed system . . . . . . . . . . 10 (b) Dependence or independence of movement . 10 1.5 The probability law which describes the move- ment of the objects among the positions through time . . . . . . . . . . . . . . . . 12 CHAPTER 2 The Process of Individuals Leaving a Social Posi- tion: Approaches, Models, and Probability Laws . . 21 2.1 Probability Law I . . . . . . . . . . . . . . 24 2.2 Approach A: The assumption of nonstationarity 29 2.3 Approach B: The assumption of population het- erogeneity . . . . . . . . . . . . . . . . . 29 2.4 Approach C: The assumption of stages within state j . . . . . . . . . . . . . . . . . . . 31 2.5 Probability Law II . . . . . . . . . . . . . 35 2.6 Probability Law III . . . . . . . . . . . . . 37 2.7 Probability Law IV . . . . . . . . . . . . . 40 2.8 Probability Law V . . . . . . . . . . . . . . 45 2.9 Probability Law VI . . . . . . . . . . . . . 50 2.10 Probability Law VII . . . . . . . . . . . . . 52 CHAPTER 3 The Process of Individuals Leaving a Social Posi- tion: Basic Issues . . . . . . . . . . . . . . . . 54 3.1 The correspondence between the theoretical construct state j and observable phenomena . 54 3.2 The identification of equivalent social po- sitions . . . . . . . . . . . . . . . . . . . 57 3.3 The identification of equivalent individuals 61 3.4 The correspondence between the theoretical construct time t and observable measures of time . . . . . . . . . . . . . . . . . . . . 62 iv CHAPTER 4 CHAPTER 5 CHAPTER 6 93000.) O \lO‘U‘I The validity of the Markov assumption . . . . The validity of ApproaChes A, B, and C . . . The validity of Probability Laws I through VII 0 O O O O O O I O O O O O O O O O O O The Process of Individuals Entering a New Social Position: Basic Issues . . . . . . . . . . . . . . 4.1 4.2 UTU'IU" to “NH 504 The correspondence between the theoretical construct state j and observable phenomena Models of the conditional transition proba- bility qjk(t) . . . . . . . . . . . . . . . . The identification of equivalent social posi- tions . . . . . . . . . . . . . . . . . . . . The identification of equivalent individuals The correspondence between the theoretical construct time t and observable measures of time . . . . . . . . . . . . . . . . . . . . The validity of the Markov assumption . . . . Research Design . . . . . . . . . . . . . . . Collection, editing, and coding of the data . Selected characteristics of the respondents . Investigation of the process of individuals leaving a social position . . . . . . . . . . (a) A design for identifying equivalent indi- viduals in equivalent jobs . . . . . . . (b) A design for evaluating probability laws Investigation of the process of individuals entering a new social position . . . . . . . The Process of Individuals Leaving a Social Posi- tion: The Identification of Equivalent Respondents and EqUivalent Jabs 0 O O O O O O O O O O O O O O 6.1 6.2 The findings by occupation . . . . . . . . . (a) Nonfarm laborers . . . . . . . . . . . (b) Nonmigratory farm laborer . . . . . . (c) Migratory farm laborers . . . . . . . . . (d) Operatives . . . . . . . . . . . . . . . The findings by issue . . . . . . . . . . . . (a) Identification of equivalent social posi— tions . . . . . . . . . . . . . . . . . . (b) Identification of equivalent individuals (c) The correspondence between the theoreti- cal construct time t and observable meas- ures of time . . . . . . . . . . . . . . (d) The validity of the Markov assumption . . (e) The validity of Approaches A, B, and C V 63 65 7O 72 73 74 85 92 93 95 97 97 106 107 108 110 116 121 121 121 130 132 137 144 144 146 147 150 152 CHAPTER 7 The Process of Individuals Leaving a Social Posi- tion: Evaluation and Interpretation of Probability CHAPTER 8 Laws The Findings concerning Probability Law I . . . . Findings concerning Probability Law II . . . . Findings concerning Probability Law IV . . . . Findings concerning Probability Law V . . . . Findings concerning Probability Laws VI and VII 0 O O O O O O O O O O O O O O O O O O O 0 Discussion . . . . . . . . . . . . . . . . . . Process of Individuals Entering a New Social Position: The Identification of Equivalent Respond- ents 8.1 8.2 8.3 8.4 8.5 8.6 and Equivalent Jobs . . . . . . . . . . . . . The equivalence of crude vacancy rates of dif- ferent occupations in a given geographical lo— cation . . . . . . . . . . . . . . . . . . . . Findings concerning the identification of equivalent destinations . . . . . . . . . . . Findings concerning the identification of equivalent origins . . . . . . . . . . . . . . Findings concerning the identification of equivalent individuals . . . . . . . . . . . . Findings concerning the correspondence between the theoretical construct time t and observa- ble measures of time . . . . . . . . . . . . . Findings concerning the validity of the Markov assumption . . . . . . . . . . . . . . . . . . CONCLUSION O O O O O O O O O O O O O C O O O O O O Slmmary O O O O O O O O O O O C O O O O O O O conCltls ions 0 O O O O O O O O O O O O O O O 0 REFERENCES 0 O O O O O O O O O O O O O O O O O O 0 APPENDIX A: Interview Schedule for Work Histories . APPENDIX B: Rules for Editing and Coding Job His- tories O O O O O O O O O O O O C O O 0 APPENDIX C: Least Squares Parameter Estimates . . . APPENDIX D: Observations and Predictions for Data Sheets . . . . . . . . . . . . . . . . 154 157 166 175 177 185 186 208 208 218 229 234 239 244 247 247 253 260 264 266 268 272 5.2 5.3 5.4 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6 LIST OF TABLES Occupational classification scheme with frequency of the categories in respondents' histories. . . . . . . . . . . Industrial classification scheme with frequency of the categories in respondents' histories. . . . . . . . . . . Respondents' educational level. . . . . . . . . . . . . . Estimates of the number of filled positions in different occupations by year and location, for males. . . . . . . . Variables examined for nonfarm laborers with categories and number of observations. . . . . . . . . . . . . . . . Variables examined for nonmigratory farm laborers and for migratory farm laborers in Michigan with categories and number of observations. . . . . . . . . . . . . . . . . . Variables examined for migratory farm laborers in Texas with categories and number of observations. . . . . . . . Variables examined for operatives in MiChigan with cate- gories and number of observations, by industry. . . . . . Description of data sets of apparently equivalent indi- viduals in equivalent jobs with symbolic designation. . . Statistics pertaining to maximum likelihood parameter estimates for Probability Law I. . . . . . . . . . . . . . Statistics pertaining to maximum likelihood parameter estimates for Probability Law II. . . . . . . . . . . . . Statistics pertaining to maximum likelihood parameter estimates for Probability Law IV. . . . . . . . . . . . . Statistics pertaining to maximum likelihood parameter estimates for Probability Law V. . . . . . . . . . . . . . Values of h (0) and h (w) calculated for data sets Ca, Cb, and Cc usin Models A.II, A.IV, and A.V and the corre- sponding maximum likelihood estimates of parameters. . . . vii 102 103 107 120 122 131 133 139 156 158 167 176 178 191 7.7 Values of the mean and variance of h (0) calculated for data sets Ca, Cb, and Cc using Model; B.II, B.IV, and B.V and the corresponding maximum likelihood estimates of parameters. . . . . . . . . . . . . . . . . . . . . . . . 194 7.8 Assumptions and implications of special cases of Model COVOOO O O O O O O O O O O O I O O O O O O O O O O O O O O 202 7.9 Solutions for selected two stage models of respondents leaving jobs categorized as nonfarm laborer, by educational level. 0 O O O O O I O O O O O O O O C O I O O O O O O O O 203 8.1 Estimates of the rate of growth of sets of equivalent individuals in equivalent jobs. . . . . . . . . . . . . . 213 8.2 Estimates of crude vacancy rates assuming Probability Law I describes the process of leaving a job. . . . . . . . . 215 8.3 Estimates of crude vacancy rates assuming Probability Law V describes the process of leaving a job. . . . . . . . . 217 8.4 Proportion of intra-Michigan transitions to eight occupa- tional categories by previous occupation and by year. . . 219 8.5 Attraction of intra-Michigan transitions to eight occupa- tional categories by previous occupation and by year. . . 220 8.6 Proportion of intrastate transitions to four occupational categories by year, previous occupation, and geographical location 0 O O O O O O O O O I O O O O O O O O O O C O O O 224 8.7 Attraction of intrastate transitions to four occupational categories by year, previous occupation, and geographical location. 0 O O C O O O O O O O O O O O O O O O O O O O O 225 8.8 Proportion of transitions from Texas to four occupational categories by year, previous occupation, and geographical location of destination. . . . . . . . . . . . . . . . . . 227 8.9 Attraction of transitions from Texas to four occupational categories by year, previous occupation, and geographical location of destination. . . . . . . . . . . . . . . . . . 228 8.10 Proportion of transitions to four occupational categories in Michigan by year, previous occupation, and previous location 0 O O I O O O I I O O O O O O O O 0 I O O 0 O O I 230 8.11 Proportion of intrastate transitions to four occupational categories by geographical location, year, and previous occupation. O C O C O O I O O O O O O O O O l O O I O O O 232 viii 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 C.l C.2 C.3 C04 D01 D02 Proportion of intra-Michigan transitions tional categories by previous occupation indUStry, 1945-19670 0 o o o o o o o o 0 Proportion of intra-Michigan transitions tional categories by previous occupation tional level, 1945-1967. . . . . . . . . Proportion of intra-Michigan transitions tional categories by previous occupation tion in which the respondent was reared, Proportion of intra-Michigan transitions tional categories by previous occupation farm baCkgrOLlnd, 19 45 -196 7 o o o o o o o to four occupa— and by previous to four occupa- and by educa- to four occupa- and by the loca- 1945 -1967. o o to four occupa— and by migrant Proportion of males employed in different occupations in Michigan, by year. . . . . . . . . . . . Attraction of intra-Michigan transitions tional categories by previous occupation year. 0 O O O O O O O O O O O O O O O 0 Proportion of intra-Michigan transitions tional categories by previous occupation 1967. O I O O O O O O O O O O O O O O I Proportion of intra-Michigan transitions tional categories by previous occupation in previous occupation, 1945-1967. . . . Proportion of intraéMichigan transitions tional categories by previous occupation pation yi held prior to yj, 1945-1967. . Statistics pertaining mates for Probability to least squares Law I. I O O O O 0 Statistics pertaining mates for Probability to least squares Law II. . . . . . Statistics pertaining mates for Probability to least squares Law IV. . . . . . Statistics pertaining to least squares mates for Probability Law V. . . . . . . Data set Aa: migratory farm laborers in 1967), 9-12 years of completed schooling. Data set Ab: migratory farm laborers in 1967), 4-8 years of completed schooling ix to four occupa- and by calendar to four occupa- and by age, 1945— to four occupa- and by duration to four occupa— yj and by occu- parameter esti- parameter esti— parameter esti- parameter esti- Michigan (1935- Michigan (1935- 233 235 237 238 239 241 242 243 245 268 269 270 271 272 273 D.3 Data set Ac: migratory farm laborers in Michigan (1935- 1967), 0-3 years of completed schooling . . . . . . . . . 274 D.4 Data set Ba: migratory farm laborers in Texas (1935-1967), previous sequence nonagricultural . . . . . . . . . . . . 275 D.5 Data set Bb: migratory farm laborers in Texas (1935-1967), previous sequence not nonagricultural . . . . . . . . . . 276 D.6 Data set Ca: nonfarm laborers in Michigan and Texas (1935-1967), 9-12 years of completed schooling . . . . . . 278 D.7 Data set Cb: nonfarm laborers in Michigan and Texas (1935-1967), 4-8 years of completed schooling . . . . . . 279 D.8 Data set Cc: nonfarm laborers in Michigan and Texas (1935-1967), 0-3 years of completed schooling . . . . . . 281 D.9 Data set Da: nonmigratory farm laborers in Michigan and Texas (1935-1967), 9-12 years of completed schooling . . . 283 D.10 Data set Db: nonmigratory farm laborers in Michigan and Texas (1935-1967), 4-8 years of completed schooling . . . 284 D.ll Data set Dc: nonmigratory farm laborers in Michigan and Texas (1935-1967), 0-3 years of completed schooling . . . 285 D.12 Data set Ea: operatives in fabricated metal manufacture in Michigan (1945-1967), reared in Michigan, 9-12 years of completed schooling . . . . . . . . . . . . . . . . . . . 286 D.l3 Data set Eb: operatives in fabricated metal manufacture in Michigan (1945-1967), reared in Michigan, 0-8 years of completed schooling . . . . . . . . . . . . . . . . . . . 287 D.14 Data set Ec: operatives in fabricated metal manufacture in Michigan (1945-1967), reared in Texas, 9-12 years of completed schooling . . . . . . . . . . . . . . . . . . . 288 D.15 Data set Ed: operatives in fabricated metal manufacture in Michigan (1945-1967), reared in Texas, 0-8 years of completed schooling . . . . . . . . . . . . . . . . . . . 289 D.16 Data set Ee: operatives in fabricated metal manufacture in Michigan (1945-1967), reared in Mexico, 0-12 years of completed schooling . . . . . . . . . . . . . . . . . . . 291 D.17 Data set Fa: operatives in motor vehicle manufacturing in Michigan (1945-1967), reared in Michigan, 0-10 years of completed schooling . . . . . . . . . . . . . . . . . . . 292 D.18 Data set Fb: operatives in motor vehicle manufacturing in Michigan (1945-1967), reared in Texas, 0-10 years of completed schooling . . . . . . . . . . . . . . . . . . . 293 D019 D020 D.21 Data set Fc: operatives in Michigan (1945-1967), of completed schooling . Data set Fd: operatives in Michigan (1945-1967), completed schooling . . Data set Fe: operatives in Michigan (1945-1967), completed schooling . . in motor vehicle manufacturing reared in Michigan, 11-12 years 0 O O O O O O O O O O O O O O O 296 in motor vehicle manufacturing reared in Texas, 11-12 years of O O O O O O O O O I O O O O O O O 297 in motor vehicle manufacturing reared in Mexico, 0-12 years of O O O O O I O O O O O O O C O 298 xi 2.1 2.2 2.3 2.4 2.5 3.1 6.1 6.2 6.3 7.1 7.2 7.3 LIST OF FIGURES General structure of models with all stages in series. . . General structure of models with all stages in parallel. . Structure of Mayer's (1968) model. . . . . . . . . . . . . Structure of Herbst's (1963) model. . . . . . . . . . . . Structure of Conner's (1969) model. . . . . . . . . . . . Structure of the process of leaving an occupation. . . . . Proportion of respondents remaining in a job as a nonfarm laborer in Michigan or Texas (1935-1967) after duration t, by respondent's educational level. . . . . . . . . . . . . Proportion of respondents remaining in a job as a nonfarm laborer in Michigan or Texas (1935-1967) after duration t, by the location in which the respondent was reared. . . . (a) Proportion of respondents with 9-12 years of completed schooling remaining in a job as a nonfarm laborer in Michigan or Texas (1935-1967) after duration t, by the location in which the respondent was reared. . . . . . (b) Proportion of respondents with 4-8 years of completed schooling remaining in a job as a nonfarm laborer in Michigan or Texas (1935—1967) after duration t, by the location in which the respondent was reared. . . . . . (c) Proportion of respondents with 0-3 years of completed schooling remaining in a job as a nonfarm laborer in Michigan or Texas (1935-1967) after duration t, by the location in which the respondent was reared. . . . . . Proportion remaining in the job after duration t for data sets Aa, Ab, and Ac. Values observed and predicted by PtOb abili ty Law I O O O O O O O O O O O O I O O O O O I O 0 Proportion remaining in the job after duration t for data sets Ba and Bb. Values observed and predicted by Proba— bility Law I. O O C O O O O O O O O O O O O O O O O O l 0 Proportion remaining in the job after duration t for data sets Ca, Cb, and Cc. Values observed and predicted by PIObability Law I O O O O I O O O O O I O O C O O O O O O O xii 33 34 49 51 53 55 124 125 127 128 129 159 160 161 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 Proportion remaining sets Da, Db, and Dc. Probability Law I. . Proportion remaining in the job after duration t for data Values observed and pre- sets Ea, Eb, Ec, Ed, Proportion remaining in the job after duration t for data in the job after duration t for data Values observed and predicted by and Be. dicted by Probability Law I. . . . . . . . . . . . . . . sets Fa, Fb, Fc, Fd, and Fe. dicted by Probability Law I. Proportion remaining in the job after duration t for data Values observed and predicted by sets Aa, Ab, and Ac. Probability Law II. Proportion remaining in the job after duration t for data Values observed and predicted by Proba- sets Ba and Bb. bility Law II. . . . Proportion remaining sets Ca, Cb, and Cc. Probability Law II. Proportion remaining in the job after duration t for data Values observed and predicted by sets Da, Db, and Dc. Probability Law II. Proportion remaining in the job after duration t for data in the job after duration t for data Values observed and predicted by sets Ea, Eb, Ec, Ed, and Be. dicted by Probability Law II. Proportion remaining in the job after duration t for data sets Fa, Fb, Fc, Fd, and Fe. dicted by Probability Law II. Proportion remaining after duration t for data sets Aa, Values observed and predicted by Probability Ab, and Ac. Law V O O O O O O O 0 Proportion remaining in the job after duration t for data Values observed and predicted by Proba- sets Ba and Bb. bility Law V. . . . Proportion remaining sets Ca, Cb, and Cc. Probability Law V. . Proportion remaining in the job after duration t for data Values observed and predicted by sets Da, Db, and Dc. in the job after duration t for data Values observed and predicted by PrOb ability Law V O O O O O O O O O O xiii Values observed and pre- Values observed and pre- Values observed and pre- 162 163 164 168 169 170 171 172 173 179 180 181 182 7.17 Proportion remaining in the job after duration t for data sets Ea, Eb, Ec, Ed, and Be. Values observed and pre- dicted by Probability Law V. . . . . . . . . . . . . . . . 183 7.18 Proportion remaining in the job after duration t for data sets Fa, Fb, Fc, Fd, and Fe. Values observed and pre- dicted by Probability Law V. . . . . . . . . . . . . . . . 184 7.19 Model C.V.O, the most general two stage model of equiva- lent individuals in equivalent jobs j. . . . . . . . . . . 197 xiv INTRODUCTION As a person approaches physical and social maturity, he typically begins to engage in activities which are socially denoted as work or labor. In modern societies work is highly differentiated: it is socially significant not only whether or not a person works, but also what kind of work he does (his occupation) and in what social and physical setting he does it (his work situation). Together occupation and work situation identify a person's job. From time to time a person may change his job;1 that is, he may change his occupation, his work situation, or both. Occasionally a person may be unemployed, i.e., temporarily between jobs. And even- tually everyone's work history ends: each worker either retires or dies while holding a job. The aim of this research is to develop an improved explanation of movement into and out of jobs for a specified segment of the labor force. While the structure of the explanation should, to a great extent, be applicable to job mobility of members of a labor force in any modern industrialized society, interest in the present investiga- tion focuses on the job mobility of Mexican—American men with a high school education or less who resided in selected Michigan counties during the winter of 1967—1968. Work histories of a sample of 584 1While some individuals have two or more jobs at the same time, this complication is ignored in models considered in the present study. 1 2 persons with these characteristics are the data used in this study. The main stimulus of sociological research on job mobility has been concern for its consequences with regard to both the production and distribution of scarce, valued goods in a society. First of all, work organization and the characteristics of jobholders affect the level and efficiency of the production of goods and services. Productivity will be less than what is technologically possible if job mobility occurs so rapidly that a significant prOportion of jobholders are newcomers, or if it occurs so slowly that changes in demands for goods and services cannot be met, or if it fails to match the competencies of individuals with job requirements. Secondly, in modern societies a person's job determines to a great extent both his present life style and his future life chances. Possession of such scarce and generally desired values as wealth, health, education, social prestige, and social power are highly correlated with occupa— tion, although the extent of variation is often considerable within an occupation. Work situation has a secondary but nonnegligible effect: in most occupations in the United States, direct rewards from work tend to be greater in cities than in rural areas, in the North- east or West than in the South, in a large corporation than in a small company. Thus rates of movement into and out of jobs of particular types indicate the likelihood of improvement, stagnation, or decline in the quality of life for members of the society. Yet concern for the consequences of job mobility does not necessarily provide a sound basis for understanding the mobility process. Of much more relevance are explanations of other forms of intragenerational social mobility. In the present study social 3 mobility is used as a generic term indicating movement of individuals among social positions; it includes geographic, occupational, indus— trial, and social prestige mobility, as well as job mobility. Previous investigators have been moderately successful in explaining social mobility by viewing it as a stochastic process, i.e., as a process which deve10ps along one or more dimensions, such as time, according to probabilistic laws. Possibilities for improving explanations of this type appear promising and are pursued in this research. CHAPTER 1 The Sc0pe of the Research Imagine a system with a finite number of objects and a finite nume ber of positions matched in such a way that each object is associated with ("occupies") at most one position and each position is occupied by at most one object. Not all objects necessarily occupy a position, nor are all positions necessarily filled. If from time to time an object instantaneously changes location according to a probability law, then change in the system, i.e., mobility, is a stochastic process. A stochastic model of social mobility can be constructed by describing for the system above (1) the objects which move, (2) the positions which the objects may occupy, (3) the points in time when the objects may move, (4) the conditions under which movement occurs, and (5) the probability law which describes the movement of the objects among the position through time. In this chapter I discuss major alternatives for each of these five features and indicate reasons for restricting the present study to models with particular characteristics. 1.1 The objects which move For the purpose of constructing stochastic models of social mobility there are at least four possible ways of conceptualizing the object which moves: (1) as an individual moving from one social position to another; (2) as a social position moving from one individual to another; (3) as a vacancy moving from one social position to another; 4 5 (4) as unattachment moving from one individual to another. Which conceptualization is chosen depends on the object of explanation and the type of data available for testing models. The first is apprOpriate when one wants to explain the changing social positions of certain individuals, for example, the Mexican- Americans in the present study. The second is appr0priate when one wants to explain the changing composition of the incumbents of certain social positions possessing an existence and identity independent of an occupant, for example, seats in the U.S. Senate or House of Repre- sentatives. The third is appropriate when one wants to explain movement in the system generated by sequences of vacant SOCial positions "pulling" individuals out of social positions. This conceptualization has primarily been developed by Harrison C. White (1968, 1969, 1970) and tested by him using data on the movement of clergy among pastorates. The fourth conceptualization is appropriate when one wants to explain movement in the system generated by sequences of unattached individuals "pushing" or ”bumping" other individuals out of their social positions, which occurs, for example, when men in unionized factories whose current job has been terminated "bump" less senior men from their jobs (cf. White, 1970). Since the type of data required to test models constructed on the basis of these different conceptualizations varies, research may be constrained to a particular conceptualization because of limitations on the type of data available. Information on the histories of individuals is required in the case of the first conceptualization anti on the histories of social positions in the case of the second. IniFormation for testing models based on the third and fourth 6 conceptualizations is generally more difficult to obtain as one must be able to identify both the social destination of an individual's predecessor and the social origin of his successor. Consequently histories of a sample of either individuals or social positions do not suffice as data for testing models of either a vacancy or unat- tachment chain. Since work histories of a sample of Mexican-American men are both the data and object of explanation in the present study, further discussion is limited to stoChastic models in which the moving objects are people. The population of moving objects in a stochastic model may be homogeneous or heterogeneous. Homogeneity of the population implies that movement among positions in the system occurs according to the ggmg_probability law with the sgmg_parameters for every member of the population. Perfect homogeneity of the population is a markedly unrealistic assumption in modeling social mobility, and some degree of heterogeneity has generally been recognized, at least implicitly, in empirical work with stochastic models of social mobility. Assuming a particular probability distribution of the population on parameters of the model, which permits one to formulate an explicit mathematical expression of the stochastic process for the entire popu— lation, is one approach to population heterogeneity. However, the most common approach has been to assume that the same probability law h01ds throughout the population but that values of parameters vary W1thsex, ethnicity, educational level, or other relevant variables cmIstant throughout the duration of the mobility process. Thus a het- erogeneous population is conceived as being composed of subpopulations 7 that are homogeneous with respect to variables influencing mobility. In the extreme case each individual defines a separate subpopulation, but it then becomes impossible to estimate parameters for a model. One faces the dilemma of identifying a minimal set of variables which disaggregates the total population into relatively homogeneous subpop— ulations while retaining sufficient individuals in the subpopulations to permit parameters to be estimated.1 In the present study I assume that the Mexican-American popula- tion examined is heterogeneous and explore both techniques mentioned above for dealing with the problem of population heterogeneity. 1.2 The positions which the objects may occupy In choosing to construct models of social mobility in which the moving objects are conceptualized as people, one concurrently chooses to conceptualize the positions which they may occupy as social posi— tions. Like the objects which move, the positions which the objects may occupy may be homogeneous or heterogeneous. Heterogeneity of the social positions is virtually always assumed in work with stochastic models of individuals moving among social positions. The total set of heterogeneous social positions is conceived as being composed of sub- sets of homogeneous social positions. In the extreme case each social position defines a separate subset, but as in the case of extreme pop- ulation heterogeneity, it then becomes impossible to estimate param- Eters for a model. One again faces the dilemma of identifying a minimal set of n variables which disaggregates the total set of 1Regression analysis with parameters of a stochastic model as deI>endent variables has been proposed by James S. Coleman (1970) as anéé- technique for handling this problem. 8 heterogeneous social positions into relatively homogeneous subsets while retaining enough social positions in each subset to permit parameters to be estimated. The subsets of homogenous social posi- tions form the states in the state space of the stochastic process. The dimension of the state space is n, the minimal number of variables needed to disaggregate the total set of positions into homogeneous subsets. Most stochastic models of social mobility have assumed a unidimensional state space, but this is not assumed a priori in the present research. Each dimension of the state space may be discrete or continuous. The set of points or elements in a discrete dimension is finite or countably infinite (in one-to-one correspondence with a set of integers); the set of points in a continuous dimension corresponds to an interval on the real line. Furthermore, the set of points on a particular dimension may be treated as a nominal, ordinal, interval, or ratio scale. The first two are usually associated with a discrete dimension and the last two with a continuous dimension. Movement on a dimension can be treated with much more mathematical elegance if one assumes that the set of points on the dimension represents a ratio scale rather than one of the other types. Even the assumption of an ordinal scale allows construction of mathematically more interesting models, e.g., prestige mobility has been modeled as a simple birth and death process (Coleman, 1964; Mayer, n.d.). But in the present study discussion is restricted to models in Which dimensions of the state space are discrete, finite, and form nc“ninal scales. This eliminates models in which ranking or ordering 0f? social positions forms an intrinsic property of the stochastic 9 process. I impose this limitation on the discussion because I think that adequate bases for ordering social positions, and in particular for ordering jobs, have not yet been devised. 1.3 The points in time when the objects may move A stochastic process is defined over a set of ordered points in time. The process occurs in discrete time if the set of time points is finite or countably infinite (in one—to-one correspondence with a set of integers). In a discrete time process movement may be assumed to occur, for example, only on the first day of the month or perhaps on a person's birthday. The process occurs in continuous time if the set of time points corresponds to an interval of the real line. In a continuous time process movement may occur at any moment between the lower and upper boundaries of time. Phenomena that occur continuously in time can be approximated by discrete time models, and vice versa. Often discrete time models are easier to analyze numerically, while continuous time models are more likely to have simple analytical solutions. Most investigators who have viewed social mobility as a stochastic process have used discrete time models; however, the discussion of stochastic processes is limited hereafter to continuous time models. In particular, discrete time models previously proposed by other investigators are discussed in the continuous time form. This reveals the basic mathematical structure of these models much more clearly than the discrete time form so that similarities and differences of the various models are more easily detected . .-. 1F.. v.‘ .hi fif- 10 1.4 The conditions under which movement occurs (a) Open or closed system In an open system moving objects enter and leave the state space, and the total number of objects in the system ordinarily fluctuates with time. The number of objects entering or leaving the system may be expressed either as an explicit function of time or as a time— dependent random variable. Models of Open systems are appropriate when behavior of the entire system, its overall growth, as well as internal structural changes, is the object of explanation. In a closed system objects may neither enter nor leave the system. Clearly people do not move forever, but permanent exit from the system, e.g., retirement from the labor force or death, can be handled by including appropriate absorbing states2 in the state space. Entrance into the system, e.g., birth or entry into the labor force, can be treated by including a state in the state space that cannot be entered. Interim periods between occupation of social positions can be handled by including unattachment, e.g., unemployment, as a state in the state space. Models of closed systems are suitable when behavior of a seg- ment of the population within the overall system is to be explained. As the present research falls within this category, further discussion centers on closed systems. (b) Dependence or independence of movement Under the condition of dependence of movement, a transition from One state to another by one person affects the transition probabilities 0f others in the system. This happens whenever the number of social 2By definition the probability of leaving an absorbing state is ZEJ:-o. 11 positions available for occupancy is limited. An analogous situation occurs when a large number of people tries to sit in a small number of chairs: when one person sits in one of the chairs, the chances are reduced that one of the persons still standing will find a seat. Dependence of movement is a realistic assumption for models of labor or geographic mobility in an entire social system: not everyone can be employed as a welder, and not everyone can live in San Francisco. Unfortunately, the amount of data required to test such models is formidably large. One must know not only how many positions of each type are filled at any given time, but also how many positions of each type are vacant at any given time. Under the condition of independence of movement one assumes that the transition from one state to another by one person does not affect the transition probabilities of any other person in the system. The data available for this study necessitate making this assumption. This assumption may not be a bad approximation when one considers social mobility for only a relatively small segment of the total population of the system. Under these circumstances it is unlikely that the individuals under consideration actually compete for the same position. It is not clear to what extent these conditions apply to the Mexican- American men whose work histories provide the data for this research. Assuming that transitions are independent does not mean that var- iables affecting the number of vacancies, and thus transition probabil- ities, cannot be incorporated into a model. For example, Matras (1960, 1961, 1967) included changes in the distribution of occupations in his d1ficrete time models of occupational mobility. Possibilities of this tYI>e are explored in Chapter 4. 12 1.5 The probability law which describes the movement of the objects among the positions through time The fundamental assumption is that location in the state space, i.e., the social position of an individual in the social system, is a time—dependent random variable, S(t). Let i, j, and k represent typi— cal elements in the state space. One would like to know the probability that an individual will be in state k at time t when he is in state j at an earlier time u. That is, one wants to know: (1.1) pjk(u,t) s prob[S(t) = k [S(u) = j] for all j and k and u < t. Since matrix notation is sometimes conven— ient, let P(u,t) represent the matrix of transition probabilities: (1.2) P(u,t) E {pjk(u,t)}. Several assumptions follow naturally from our intuitive under— standing of a probabilistic process and from the restrictions imposed earlier in this chapter. Clearly we want to assume that the lower and upper limits of transition probabilities are zero and one, respectively: < < (1.3) 0 - pjk(U.t) —1. Because the present discussion is limited to closed systems, we want to assume that the probability of moving from a state j at time u to some state in the system at time t is exactly one: 1. = . ( 4) E pjk(u,t) 1 131 addition it is reasonable to assume that (l - = 5) pjj(.t.t) 1 13 and (1'6) ij(t,t) = O for j # k, or in matrix notation: (1.7) P(t,t) = I where I is the identity matrix (a square matrix with ones on the diagonal and zeroes everywhere else). An important question remains: what assumptions should be made concerning the relationship between an individual's location in the state space at one time t and his location at some earlier time u? Mathematically the simplest assumption is that the value of S(t) is independent of the value of S(u) for u < t: (1. 8) prob[S(t) = k | S(u) = j] = prob[S(t) = k] for all j and k and u < t. Under this assumption an individual's pres ent location in the state space gives no clues concerning his locladtion at any future time. Obviously this is an unrealistic assump- tion for a theory intended to explain social mobility. A slightly more complex assumption is that an individual's location in the next small interval of time depends on his present location, but not On the path taken to get to this present location. This is the Markov assumption and can be expressed as follows: (1.9) prob[S(t) = k | S(u) = j] = prob[S(t) = k | S(u) j. S(v) i,..] f or all i, j, and k and for v < u < t. It follows from this assumption 14 that the Chapman-Kolmogorov equation holds: (1.10) pik(v,t) = g pij (WU) ij(U,t). or in matrix notation, (1.11) P(v,t) = P(v,u) P(u,t). From a mathematical point of view the Markov assumption is very attractive. The theory of Markov processes has been rather extensively deVeloped and is sufficiently complex that Markov models can approxi- mate actual mobility processes, but not so complicated that sociologists must deal with intractable mathematical problems. Nevertheless, ilil’srestigators modeling social mobility have complained from time to time that the Markov assumption is unrealistic and inadequate for Cons tructing models of social mobility. The force of this criticism can be mitigated by using known tech- ni‘llles for transforming non-Markovian processes into Markov processes. Although these techniques have previously been used, they have not been ConSidered in any systematic fashion by those developing stochastic models of social mobility. Three procedures (cf. Cox and Miller, 1965) particularly applicable to the modeling of social mobility are: (a) the inc-lusion of supplementary variables, (b) the device of stages, and (Q) the use of an imbedded Markov process. The first and second are the basis for two techniques used to develop improved descriptions of the process of an individual leaving a social position; these are is‘Cussed as Approaches B and C, respectively, in Chapter 2. The third technique, in which a subset 'of time points are selected so that the resulting process is Markovian, is applied in the overall 15 strategy of this research whereby the process of leaving a social position and the process of entering a new position are examined separately (one may have a Markovian interpretation even if the other does not), and also in the rationale given in Section 3.1 for concep- tualizing a job, rather than some other social position, as a state j. Prudence demands that the use of these techniques be thoroughly exPlored before the Markov assumption is discarded. The fact that this exploration has not been completed provides ample justification for limiting the present research to stochastic processes either manifestly Markovian or else open to a Markovian interpretation through some trans formation technique. In most analyses of social mobility as a Markov process, investi- gatiOn has been restricted to stationary processes, that is, Markov proC-esses whose transition probabilities are constant through time. But as many have recognized, it is unrealistic to assume that the prohabilities of moving from one social position to another do not change with time whether one conceptualizes time as an individual's age, his duration in a position, or the date on the calendar. Hence the Present research assumes in general that transition probabilities are nonstationary (time-dependent). Two additional assumptions are needed to develop the mathematical theory of Markov processes. While these assumptions are reasonable, their necessity is perhaps not as intuitively apparent as that of the four assumptions represented by equations (1.3) through (1.6), and (1‘10). The first additional assumption (cf. Feller, 1968) is that for every state j, there exists a continuous function hj (t), :- hj (t) < 0°, such that: l6 1 - .. t,t + At pJJ( ) (1.12) lim = h.(t). At+0 At 3 In other words hj(t), which can be interpreted as the instantaneous rate of leaving state j, exists and is a nonnegative, finite, and continuous function everywhere within the time span being considered. The second additional assumption is that for every pair of states j and k, j 7‘ k, there exist transition probabilities qjk(t)’ in general depending on time, such that: P.k(t,t + At) (1.13) 11111 J = h.(t) q.k(t) At+0 At 3 J with passage to the limit uniform with respect to j for fixed k, and (1 . ._. 14) E qjk 1 (l . 15) qjj(t) = 0- The function qjk(t) can be interpreted as a conditional transition probability, i.e., the probability that state k is the destination if State j is left between t and t + At. It is also useful to define a function rjk(t) such that (1.. 916) rjk(t) hj(t) qjk(t) for j 34 kand (l. 17) rjj(t) - hj(t). The function rjk(t) for j 75 k can be interpreted as the instantaneous r ate of transition from state j to state k. Note that 17 (1.18) 2 rjk(t) = —-rjj(t) k#j which simply means that the rate of loss from state j exactly balances 'ttie: sum of the rates at which the other k states gain from state j. Two systems of partial differential equations, called the forward and backward Kolmogorov equations, can be derived from the six assump- t:irxns represented by equations (1.3) through (1.6), (1.10), and (1.12) tfirxaugh (1.15) (cf. Feller, 1968). The forward Kolmogorov equation can be written as follows: (1.1 __.__. = 9) at g pij(U.t) rjk(t) or in matrix notation 8P(u,t) (1.20) —— = P(u,t) R(t) 3t Where 3P. (mt) 3P(u,t) 1k (1-21) ————— a { } at at and (1.22) R(t) .=. {rjk(t)}. Similarly the backward Kolmogorov equation can be written: 3p1k(11,t) (1.23) ——-—-—————— = - X rij (11) ij(uat) 1 Eu 0 O O r in matrix notation 18 3P(u,t) (1.24) ——-——— = -IKu)POhtL Bu The significance of the Kolmogorov equations lies in the fact that, given the assumptions that have been made in this section, they have a unique solution. Usually investigators modeling social mobility have been interested in obtaining P(0,t), the matrix of probabilities of moving from state i at time 0 to state j at time t, when R(t), the matrix of instantaneous transition rates, is postulated to have a SPeCified form. The apprOpriate equation to use in order to solve for P(O, t) is (1.20), the forward Kolmogorov equation, which becomes a System of ordinary linear differential equations when time u is set e(111611 to zero: dP(0,t) (1.25) -———— = H0¢)RQL dt This equation can in turn be used to derive a system of ordinary linear differential equations for the expected number of individuals in given states at a given time t. Let (1:26) N(t) E {nj(t)} b . . . . e a row vector of the actual number of 1nd1v1duals in each state j at time t,3 and let (1° 27) mt) s {53.0)} \ Va For mathematical convenience n (t) is treated as a continuous riable, whereas actual numbers of individuals must quite obviously 1rlteger quantities . 19 be a row vector of the expected number of individuals in each state j at time t. We intuitively expect that on the average pij (0,t) of the ni(0) persons in state i at time 0 are in state j at time t. Hence we write: (1.28) R(t) = N(O) P(0,t). Differentiating this equation with respect to time t yields d'fi(t) dP(0,t) = N(0) dt dt (1.29) Subs tituting according to equation (1.25) and then applying equation (1-28) , we obtain: dmt) (1.30) = 150;) R(t) . dt This equation is equivalent to deu) (1.31) —— = Z hi(t) rij“) dt 1 or a l ternatively dnj(t) (1.32) -H.(t) h.(t) +2 R(t) h.(t) q..(t): J j , i 1 1:} dt 1 1743' w hich simply states that the rate of change in the expected number of i ndividuals in state j equals the rate of change in the expected number 1 eawing state j plus the sum of the rates of change in the expected n umber entering state j from each of the other 1 states. Solving a system of differential equations ~- either equation ( 1' 25) for the matrix of transition probabilities, P(0,t), or equation 20 (1.30) for the vector of the expected number of individuals in each state, R(t) -- is mathematically difficult unless the matrix of the instantaneous rates of transition among the states, R(t), has a simple form. Yet it has been necessary to rely on this approach in most studies of social mobility because available data has consisted 0f information on the states occupied by individuals at discrete Points in time, rather than information on the exact sequence of States occupied by each individual and the length of time Spent in each state. But since continuous time work histories are available in this study, I take a different tack which is derived from the prob ability theory develOped above but does not require solving a m of differential equations. The essence of this tactic is to exanline the mobility process on a more fundamental level -- namely, to Study separately the process of individuals leaving a social Position and the process of individuals entering a new social posi— tion, given that they are leaving some other social position.4 The former is examined in Chapters 2 and 3 and the latter in Chapter 4. \ 4After this research began, I learned that Morrison (1970) advo- c ates essentially the same strategy for develOping models of migration. CHAPTER 2 The Process of Individuals Leaving a Social Position: A Review of Approaches, Models, and Probability Laws A stochastic model of individuals leaving a social position con— Sis ts of not only the probability law describing the leaving process but; also the set of assumptions used to derive the probability law. This is important because a particular probability law can often be detived from more than one set of assumptions. In such cases testing a InOdel includes, but is not synonymous with, testing a particular Probability law. While previous investigators have pr0posed and tested SeVen different probability laws intended to describe the process of individuals leaving a social position,1 they have, properly speaking, pr0p<>sed somewhat more than seven models because some probability laws Were derived from more than one set of assumptions. \ 1Previous investigators (Rice, Hill, and Trist, 1950; Mayer, 1968) to deproposed and tested two other mathematical formulations intended am (1123212238iiiiisieiiuiidi‘éiifiiisf2:331:32S°2“i.§°ii§ii“' mi. 8 8 n, S 8 Y P EOSed, satisfies the assumption that 0 5 h.(t) < on. In the Rice, Hill, nd Trist formulation, the expected density of individuals leaving have s tate j is a hyperbolic function of duration t, f.(t) = a t‘b W111 J + Ch implies that hj(t) + 00 as t + 0 . In the Mayer formulation hj(t) = a (1 - bt) exp[ - bt] w thigh is less than zero for t > l/b. Both preposals can be made to fit restrictions on h.(t) by suitable limitations on the time span un- 1r Consideration. In view of the atheoretical basis of both propos- S a this task does not presently seem worthwhile. 21 22 The seemingly natural assumptions of stationary (time—independ- ence) and population homogeneity lead to the first probability law discussed below, which I have designated Probability Law 1. Yet obser- vations on individuals leaving several different types of social posi- tions have deviated consistently from predictions based on this proba- bility law. In order to develop an improved description of the process of individuals leaving a social position, previous investigators have used three sets of general assumptions or approaches; the presentation of these three approaches follows my discussion of Probability Law I. This chapter then concludes with a review and analysis of the six probability laws that have been derived from models constructed through the use of these three approaches. With one exception the order of discussion of these probability laws follows the chronology of their original published proposal. In the present chapter I do not undertake to discuss or resolve several important issues relevant to an explanation of social mobility. These include establishing correspondences between the theoretical con— struct "state j" and observable social positions and between the theo- retical construct "time t" and observable measures of time, as well as identifying equivalent individuals (i.e., homogeneous subpopulations) and equivalent social positions. In discussing the work of previous investigators in this chapter I report without critical comment their decisions concerning these issues. My own thoughts on these problems are given in Chapter 3. In order to understand and compare models of individuals leaving a social position, it is necessary to develop further the mathematical theory presented in Chapter 1. The starting point is equation (1.32): 23 dnj(t) __ __ (2.1) T = - nj(t) hj(t) + E ni(t) hi(t) qij(t). ii‘j If we ignore those who enter state j after some point in time arbitrar- ily defined as time 0, this equation becomes: dnj(t) (2.2) -——EE—— = - nj(t) hj(t). The solution of this differential equation is easily found. Separating the variables, we obtain dnj(t) (2.3) —_'—'—— = - hj(t) dt, 11. t J() and after integrating both sides, t (2.4) 1n 51(t) - 1n h.(O) = - J h (u) du. J J 0 3 Let C (t) be the expected proportion remaining in state j at time 3 t. The preceding equation implies that . t (2.5) C.(t) = J = exp[ - J h.(u) du]. J nj(0) O J In addition, let F3(t) be the prOportion expected to have left state j at time t, that is 2.6 it = 1-E.t, ( ) J() J() and therefore applying equation (2.5), t (2.7) -F (t) = 1 - exp[ - J h.(u) du]. J 0 J Differentiating F3(t) with respect to time t, we obtain: 24 _ dFj(t) t 2.8 f. t = -——-—— = h. t ex - h. u du ( ) J( ) dt J( ) p[ [0 J( ) 1, where f3(t) is the probability density function for the expected number of leavers at time t. Equations (2.5) and (2.8) together imply that (2.9) h (t) = __ j G (t) Note that h (t), the instantaneous rate of leaving state j, J uniquely determines 63(t), F3(t), and f3(t). In other words, a proba- bility law describing the process of individuals leaving a social po- sition may be specified by postulating hj(t),'Gfi(t),.F3(t), or f3(t). In order to ensure clarity of presentation and to facilitate compari- sons, I indicate hj(t), Gj cussed below. I omit F3(t) because it can be obtained very simply from (t), and f3(t) for each probability law dis- E%(t) using equation (2.6). In the equations for these functions Greek letters represent parameters of the probability law. These parameters are assumed to be fixed for individuals in a given state j who are homogeneous on all variables affecting the process of their leaving state j. 2.1 Probability Law I The mathematical equations defining Probability Law I are: (2-10) hj(t) = K (2.11) 63(t) = exp[ - Kt] (2.12) f5(t) = K exp[ - Kt] where K 2 0 and time t Z 0. Probability Law I arises from the assumption that members of a 25 homogeneous population (which presumably has been identified) leave a given state j at a fixed rate K as given by equation (2.10). This assumption seems natural to us because we fundamentally assume that the probability of a well-defined event (e.g., leaving state j) does not change from moment to moment. If evidence supports the fact that the probability of an event changes with time, then we ordinarily doubt that the event is "well-defined," i.e., that instances of the same phenomena have been observed. Consequently Probability Law I, in spite of its simplicity, serves as the standard basis of comparison as well as the foundation from which other models are constructed. The stationary Markov process, a special case of Probability Law I often discussed as a model of social mobility, requires the additional assumption that q (t), the conditional probability of a move from jk state j to state k, is independent of time for all states j and k. With equation (1.16) this implies that 2.13 r, . ( ) J 3k for all states j and k. In other words, the matrix of instantaneous transition rates, R(t), is independent of time: (2.14) R(t) = R. Given these assumptions it can readily be shown that equation (1.25) has the solution (2.15) P(0,t) = exp[Rt] and that equation (1.30) has the solution (2.16) '§(t) = N(0) exp[Rt] where by definition 26 w z (2.17) exp[Rt] = Z (Rt? z=0 z and (2.18) R0 2 I. If the total number of states in the system is finite, as I assumed in Section 1.2, the above expressions for P(0,t) and R(t) converge to a limit. Blumen, Kogan, and McCarthy (1955) tested the discrete time ver- sion of equation (2.15) using records of the Bureau of Old Age and Survivor Insurance to determine the industry in which a sample of workers were employed during twelve consecutive quarters. They used the matrix of the proportion of transitions between industries in one quarter to predict the matrix of the prOportion of transitions between industries in four and eight quarters for both males and females in three different age groups. As the length of time of the prediction increased, the predicted values increasingly differed from the ob- served values. In particular, the authors found that the model over- estimated the number of workers who had changed their industry after a long period of time. Further investigation showed that workers with a long period of continuous employment in an industry were more likely to remain in the same industry for one more quarter than were workers with a short period of continuous employment in the same industry. The magnitude of this effect differed among industries. Since equa- tion (1.12) implies that (2.19) pjj(t,t + At) % 1 — hj(t) At where pjj(t,t + At) is the probability of remaining in state j at time v...- n-— .... 'Y' u‘s ,114 27 t for an additional small interval of time At, the observations of these investigators are consistent with the conjecture that hj(t), the instantaneous rate of leaving state j, is a declining function of time t rather than independent of time as in Probability Law 1. Probability Law I has also been tested with data on labor mobility without the added assumption that qjk(t)’ the conditional probability of a move from state j to state k, is independent of time for all states. Silcock (1954) used data on the distribution of the length of employment of workers who had left two British factories to estimate parameter K for Probability Law I. When he used his estimate of K to predict the distribution of the length of employment of the workers, he found large discrepancies between the actual and predicted distri- butions. The predicted number of leavers in an interval At, approxi- mately nj(0) f3(t) At, was too low for short durations of employment and too high for long durations. In order to achieve a better predic- tion, h (t), the instantaneous rate of leaving state j, needs to be 1 larger than the estimate of K for short durations and smaller than the estimate of K for large durations. In other words, Silcock's results also suggest that hj(t) declines with time. Data obtained by Hedberg (1961) on the completed length of em— ployment of male workers hired during 1949-1952 in sixteen Swedish factories also provide a test of Probability Law I. Hedberg plotted the logarithm of the observed proportion remaining in a factory versus duration t for seven age categories. Equation (2.5) implies that t (2.20) 1n Cl(t) = - J h.(u) du J 0 J so that for Probability Law I, (2.21) 1nEj(t) = -. stage 1 > stage 2 > stage d > L Figure 2.1 General structure of models with all stages in series Erlangian distribution, for which the parameter AZ equals minus the instantaneous rate of leaving stage 2, (2.45) AZ - “z for z = 1, 2, . . ., d, and the parameters 82 are as follows: d wél uw / (uw - Hz)- w¢z (2.46) Bz If the instantaneous rate of leaving each stage in state j is the same, (2-47) Hz = U 0“ It .5. b 1 «~- 4 I... . U»... y... a C-‘ '1‘». -.. L I . 34 for z = l, 2, . . ., d, then the probability law is known as the spe- cial Erlangian distribution and the expression for fj(t) becomes: (2.48) ?j(t) = 11d td-l exp[ - ut] / (d-l)! If we abandon the notion of distinct stages and permit d to take on any positive real value, then equation (2.48) becomes the probability density function of the gamma distribution, which we can consider a generalization of the special Erlangian distribution. While a simple analytic expression for h (t), the instantaneous rate of leaving state 3 j, cannot be written for either the special Erlangian or gamma distri- bution, it can be shown that hj(t) peaks at some time t greater than zero for d greater than one. As I indicated in the discussion of Probability Law I in Section 2.1, this may be a desirable preperty for a model of social mobility to have. To my knowledge no model of sub- type (a) has yet been tested using data on social mobility. stage 1 stage 2 -ri~‘~‘* / stage d Figure 2.2 General structure of models with all stages in parallel For probability laws derived from models of subtype (b), which have all stages in parallel (cf. Figure 2.2), (2.49) AZ = - u and the parameters Bz are in the range (0,1) for z = l, 2, . . ., d. We can think of a probability distribution existing for the set {82} with Bz equal to the probability that an individual enters stage 2. 35 Models of subtype (b) of Approach C closely resemble models con- structed through Approach B. In fact, these two types of models have only one significant difference: the variable y in Approach B ordinar- ily refers to attributes of individuals, whereas the variable 2 in Approach C, subtype (b), refers to attributes of positions. If vari- able 2 is substituted for variable y, then equations (2.28) and (2.29) can be used to define the probability law for a model with all stages in parallel. No general statements can be made about the parameters 82 and AZ of probability laws derived from models of subtype (c) of Approach C, which have stages both in parallel and in series. All previously pro- posed models of individuals leaving a social position constructed through the use of Approach C have been of this subtype. These models have assumed that everyone in state j begins in the same initial stage, sometimes called the undecided stage, and eventually leaves this stage to become more or less committed to state j. In other words, unlike models of subtype (b), these models have postulated that at time 0 all positions in state j are truly equivalent and occupied by truly equiv— alent individuals. 2.5 Probability Law II The mathematical equations defining Probability Law II are: (2.50) hj(t) = 6/(0 + t) (2.51) Ej(t) = e/(e + t)‘8 (2.52) ‘fj = wane/(e + mm where 6 > 0, O > 0, and t Z 0. -! . r y ( I) 36 Silcock (1954) prOposed Probability Law II after finding large discrepancies between the actual distribution of the length of employ— ment of workers who had left two British factories and the distribution predicted by Probability Law I. He postulated that individuals vary in their proneness to leave, or in terms of the mathematical theory devel- oped in the presentation of Approach B in Section 2.3, that (2.53) hj(Y.t) = Y. and that initially they are gamma—distributed on y, (2.54) cj = 95 y‘S‘l exp[ - ey1/ 0 and time t Z 0. Equation (2.57) is the probability density function of the lognormal probability distribution. Lane and Andrew (1955) originally proposed this probability law after examining the distributions of the length of employment of work- ers who had left branches of a British steel factory. They postulated (1955: 297) "that the probability of an employee's leaving is strongly ' an assumption of duration- dependent upon his length of service,‘ dependent nonstationarity. However, the authors suggested no substan- tive reasons for selecting the lognormal probability law. Furthermore, their sole test of the probability law was to observe that plots on I I 0". a 1 Cl- .,‘- 9‘... -.. cs. — bu. o{_ -$.. h” '4'- (I, i I 38 logarithmic probability paper of the observed preportion remaining in a branch of the factory versus duration t appeared to be straight lines. An attractive feature of Probability Law III is that hj(t), the instantaneous rate of leaving state j, increases to a maximum at some time t greater than zero and then decreases to zero as time t ap- proaches infinity. As I mentioned in the discussion of Probability Law I in Section 2.1, for Morrison's (1967) data on 18-24 year olds the semiannual probability of migrating gradually increased as dura- tion of residence increased to two years and then declined with fur- ther increases in duration of residence. Likewise Lane and Andrew's data indicate that the probability of leaving the factory was very low during the first week of employment, then gradually increased during the first month or two of employment, and finally declined with fur- ther increases in duration of employment. In the discussion following Lane and Andrew's paper (1955), Spratling reported that the probability of quitting work during the next month peaked at three months in data on Station porters. Probability Law III is one of the few previously proPosed probability laws that is consistent with observations indi— cating that h (t) peaks at some time t greater than zero. :1 Iiowever the substantive interpretations which have been proposed for Probability Law III seem less than satisfactory. In a brief com- ment on Lane and Andrew's paper (1955: 318), Aitchison, coauthor with BrOWn of The Lognormal Distribution (1957), offered the following \———— in 2Logarithmic probability paper has the ordinate scale (durati-on (inthis instance) graduated logarithmically and the abscissa 8031: gradthis case the proportion remaining in a branch of the factory uated as the equivalent normal deviates. 39 interpretation of Probability Law III: . . . suppose that Kapteyn's law of proportionate effect holds, namely, that the chance of an employee with x years service remaining in service for another y years depends only on the ratio y/x; such a system if allowed to operate freely would generate a stable lognormal pOpulation of lengths of service . . I understand this to mean that pjj(x, x + y) depends only on y/x, a conjecture of duration-dependent nonstationarity deserving considera- tion. However my attempts to derive Probability Law III from a formal expression of this conjecture have not yet been successful. 0n the other hand, in a modification of Aitchison's interpreta- tion, Bartholomew (1967) has demonstrated that a lognormal distribution 0f lengths of jobs arises under the following assumption: (a) let tx’ the length of an individual's xth job, be a random multiple of the length of his previous job, (2.58) tx = ux tx-l Where x = 2, 3, . . ., 0° and {u2, u3, . . .} is a sequence of random vartléfloles with a known joint probability distribution, and (b) let the cent:I‘al limit theorem hold for the summation of ln ux, x = 2, 3, . . .. m' Jit.then follows that the logarithm of the xth job, 1n tx’ ap- proaClies a normal distribution as x approaches infinity. To Bartholo- mewr (11967: 130) this suggested "that if particular care is taken to ensure satisfaction in a person's first job then their subsequent rate of t‘-l?r:nover will be reduced." Note that the interpretation proposed by Bartholomew is non-Markovian (length of present job depends on pre- Vious experience) and arises from an assumption of poPulation hetero— geneity (Approach B) rather than nonstationarity (15413131133Ch A) ' - f Although mathematically correct, Bartholomew's interpretation 0 40 Probability Law III seems unsatisfactory from a substantive point of View. Bartholomew does not explain why the length of an individual's second (third, fourth, etc.) job should be a random multiple of the length of his previous job, and the prima facie evidence for this pos— tulate is weak. For example, this assumption implies that if the length of the first job was forty years for one man (as occurred in my sample) and six months for another, then the probability that the sec- ond job lasts more than ten years is greater for the first person than the second, an implausible conclusion given that a working career rarely lasts more than fifty years. Even if this postulate is correct, it does not mean that Probability Law III describes the process of h job when x is small, and in particular, when x equals (t) . l e aving the x one . While Probability Law III may have a desirable feature in hj the instantaneous rate of leaving state j, peaking at some time t greater than zero, further work on the underlying model from which it is derived is needed. 2 - 7 Probability Law IV The mathematical equations defining Probability Law IV are: (2-59) hJ.(t) = w eXPI - 9t] + 5 (2 . 60) Ej(t) = exp[ - Et - (11(1 - eXp[ " ¢t])/¢] (2- 61) ?j (t) = (w exp[ _ @t] + g) exp[ - Et - 03(1 - exp[ " ¢t])/¢] W herew>0,¢>0,€20,3ndtimet20. Note that as time t increases, the instantaneous rate of leaving If 5 equals zero, then the s tate j, hj(t), asymptotically approaches 5. e xpected pr0portion remaining in state j after infinite time is nonzero: 41 (2.62) lim 03(t) = exp[ — w/¢] t—HIO Contrarily, if E is greater than zero, the expected proportion remain- ing in state j after infinite time is zero. Given that everyone even— tually dies, the latter is a more realistic assumption for a model of individuals leaving a social position. Since the instantaneous rate of leaving state j, h (t), must al- j ways be nonnegative (cf. equation (1.12)) and appears to decline with time on the basis of most previous observations, one of the simplest postulates is that its prOportionate rate of decrease is constant: dhj(t) (2.63) dt = - mhj(t). The solution to this differential equation is equation (2.59) with parameter 5 equal to zero. Equation (2.59) is obtained exactly by assuming that dhj(t) (2.64) T = - [hj(t) ‘ E]. Probability Law IV can be constructed through the assumption of p0pulation heterogeneity (Approach B) or of stages within state j (Approach C, subtype (b)) as well as through the assumption of nonsta— tionarity (Approach A) as described in the preceding paragraph. In this case we postulate that there exists a variable y (an attribute of individuals in Approach B or of social positions in Approach C, sub- type (b)) such that (2.65) hj(y.t) = ¢y +-€. where y = 0, 1, 2, . . ., w and t Z 0, and that the initial distribu— tion on y is a Poisson distribution with parameter m/¢, '- ‘eh (I; . ': s 42 (2.66) cj(y,0> = exp[ - w/¢1y / y! Substituting equations (2.65) and (2.66) into equation (2.29) and sim- plifying yields equation (2.60) of Probability Law IV. Insofar as studies of social mobility are concerned, Hedberg (1961) appears to be the first to have prOposed Probability Law IV.3 0f the researchers who have discussed this probability law with refer- ence to social mobility, only Hedberg postulated that the parameter a is greater than zero. While the analysis of his apparently excellent data on the length of employment of workers in sixteen Swedish facto- ries led Hedberg to propose that the instantaneous rate of leaving declines with duration t as given by equation (2.59), he unfortunately did not choose to use these data to estimate the parameters of Proba- bility Law IV or to test the probability law in any way. Mayer (1968), the only person actually to test a version of Prob— ability Law IV with data on social mobility, postulated that the in— stantaneous rate of leaving declines with age t as given by equation (2.59) with parameter 6 zero. This test is not conclusive because of the a priori assumption that g is zero and because of the nature of Mayer's data, which consisted of occupational status scores of the first job and the job in March of 1962 for 20,700 U.S. males aged 20 to 64 years. Because Mayer had data on only two points in a man's career, he did not know when an individual left any particular status and therefore could not look directly at the process of individuals leaving a single state j. Instead he investigated the overall mobility 3Probability Law IV has a much lengthier history in the study of human mortality than in the study of human mobility (cf. Gompertz, 1825, and Makeham, 1860, as cited by Chiang, 1968). 43 process, solving equation (1.25) for P(0,t). In order to solve this equation he assumed that the parameter ¢ in equation (2.59) is the same for all occupational statuses and that qjk(t), the conditional probability of moving from state j to state k, is independent of time for all states j and k. Furthermore, he found it necessary to con- struct a synthetic cohort, i.e., to treat data on different age groups gathered at one point in calendar time as observations on a single birth cohort at different points in its life history. Using a syn- thetic cohort presents no problems if the structure of the mobility process remained relatively constant between 1916 (when the 64 year old men in his sample presumably entered the labor force) and 1962; however, the validity of this presupposition is highly dubious. Mayer rated the predictive ability of his version of Probability Law IV as "not terribly impressive," but given the limitations of this test, Probability Law IV should not be eliminated from further consideration. McGinnis's Axiom of Cumulative Inertia (1968: 716), which states that "the probability of remaining in any state of nature increases as a strict monotonic function of duration of prior residence in that state," is an assumption of duration—dependent nonstationarity and therefore one form of Approach A. This axiom has been incorporated by McGinnis and his coworkers into what has been named the Cornell Mo- bility Model. In his discussion of the Cornell Mobility Model, McGinnis (1968) defined the mobility process with two matrices, termed { by him R E {rij} and dS dsii ing from state j to state k, qjk(t),4 is equivalent to rij in McGinnis's }. The conditional probability of mov- 4 . . . My own notation is used unless otherWise indicated. 44 notation; this is not pertinent to the present discussion of models of individuals leaving a social position. However, what McGinnis termed dsii is equivalent to the probability of remaining in a state j at time t for the next interval of time At, p. JJ (t), the instantaneous rate of (t, t + At), which is di- rectly related by equation (2.19) to hj leaving state j. In particular, McGinnis reported that computer simulation experi- ments had been carried out under the assumption that "with each incre— ment of time, the probability of staying is increased by a fixed frac- tion, l/a, of the remaining range, I S" (McGinnis, 1968: 719). ’ d—l For a given state this implies (in McGinnis's notation) that (2.67) dsii = 1 - (1 - 1/6.)d"l (1 - 1311) for a > 1. In my notation this implies that (2.68) hj(t) At m (1 - 1/6)t (1 - pjj(0, o + At)). If McGinnis's parameter a is large, then (2.69) (1 - 1/a)t = [(1 - 1/a>’“1't/“ m exp[ - t/a].5 and l/o is approximately equal to ¢ in equation (2.59). Moreover, (l - lsii) in McGinnis's notation is approximately equal to m At, where w is the parameter in equation (2.59). Thus the computer simu- lation experiments of the Cornell group were based on a discrete time version of Probability Law IV with parameter 5 equal to zero. This group has not yet published data to test its version of this law. In their respective versions of Probability Law IV both Mayer and 5 lim (1 - 1/x)"x = e lx +8 45 McGinnis assumed that the parameter E in equation (2.59) is zero, that the parameter ¢ is the same for all states j, and that only the param— eter m and the conditional transition probability qjk vary with state j. The fundamental difference between their versions of this probabil- ity law is the reference scale for time t. Mayer treated time t as age, or more exactly, as years in the labor force: time t is zero only at the moment that a person enters the labor force, which Mayer apparently took as 18 years of age for the purpose of his calculations. On the other hand, McGinnis treated time t in Probability Law IV as duration in a state j: time t is zero at the first moment in each state that a person enters in his lifetime. Which conceptualization of time t is (t). better is at least as fundamental as the question of the form of hj the instantaneous rate of leaving state j, but as I mentioned at the beginning of this chapter, I am deferring discussion of this question until Chapter 3. 2.8 Probability Law V The mathematical equations defining Probability Law V are: 8 A1 exp[ - Alt] + (l - 8) A2 exp[ - Azt] (2.70) 11.(t) = 3 B eXp[ - Alt] + (1 - 8) exp[ - Azt] (2.71) 03(t) = 8 exp[ — Alt] + (1 - 8) exp[ - 5 t] (2.72) f3(t) = e 11 exp[ - Alt] + (1 - s) 12 exp[ - Azt] where 8 ¢ 0, A1 > 0, A2 3 0, and time t Z 0. Probability Law V can be derived from models constructed through Approaches A, B, or C. In order to derive this probability law by using Approach A, we postulate that for a homogeneous papulation in state j the instantaneous rate of leaving state j, hj(t), declines with 46 time t as given by equation (2.70). Because of the complexity of this equation, it is not surprising that previous investigators have not employed this route to Probability Law V. Originally Probability Law V was derived from a model assuming heterogeneity of the population in state j (Approach B). The neces- sary postulates for such a model are as follows. For prOportion B of the population in state j the instantaneous rate of leaving is Al: (2.73) cj(y,0) = B (2.74) hj(yl,t) = A1. For the remaining proportion (l — B) the instantaneous rate of leaving state j is A 2: (2.75) cj(y2,0) = l - B (2.76) hj(y2,t) = AZ. Substituting these expressions into equation (2.29) gives equation (2.71) as the expression for 53(t), the expected proportion remaining in state j at time t. In the best known version of Probability Law V the parameter 12 is assumed to be zero. As time t approaches infinity, GJ expected proportion remaining in state j, approaches (1 - 8); these (t), the are called "stayers." The other pr0portion B of the original popula— tion are called "movers" as all of them eventually leave state j. This version of Probability Law V, with the added assumption that qjk(t), the conditional probability of moving from state j to state k, is independent of time, has been named the Mover-Stayer Model. After Blumen, Kogan, and McCarthy (1955) found that Probability Law I provided an unsatisfactory explanation of their data on 47 inter-industry mobility, they proposed the discrete time version of the Mover—Stayer Model to explain movement in the system. Let B be a square matrix with diagonal elements Bj, the proportion of "movers" in state j at time 0, and with zeros for all elements off the diagonal. Let R be a time-independent matrix of instantaneous transition rates with elements 2.77 r,, = - A. ( ) JJ J and 2.78 r. = A . ( ) 3k j qu where Aj is the movers' instantaneous rate of leaving state j and qjk is the conditional probability of a move from state j to state k. Then the solution of the Mover-Stayer Model is (2.79) P(0,t) = I - B + B exp[Rt] where time t is calendar time. Blumen, Kogan, and McCarthy used the inter—industry transition matrix for eight quarters to estimate parameters and then used these estimates to calculate expected transition matrices for four, eight, and twelve quarters. The agreement between the actual and predicted matrices was quite good for eight quarters; however, for four quarters too few men were predicted to be in the same industry as at time 0, and for twelve quarters too many men were predicted to be in the same industry as at time 0. As the authors pointed out, the discrepancies for twelve quarters might result from the unrealistic assumption that some people never move. In another version of this model, both parametersA1 and A2 are 48 assumed to be greater than zero with Al > A2: everyone is a mover, but one group leaves at a more rapid rate than the other group. As time t approaches infinity, 53(t), the expected prOportion remaining in state j at time t, approaches zero. Bartholomew (1959) originally prOposed this version of Probability Law V without giving it any interpretation but later (1967: 127) re- ferred to it as if it were derived by assuming population heterogeneity (Approach B). Using data gathered by Silcock (1954) and by Lane and Andrew (1955) on the length of employment of workers in three firms, Bartholomew estimated the parameters of the probability law by equat- ing the observed pr0portion remaining in state j at three, six, and eighteen months with the value of 03(t) at these times. Then he used his estimates of the parameters to calculate the predicted distribution of the length of employment of workers, nj(0) f3(t) At, and on the basis of inspection concluded that the predictions were adequate. In 1967 Bartholomew noted that Probability Laws 11 and V predicted these data about equally well. Probability Law V can also be derived from Approach C. Any model postulating two distinct stages within state j and Markovian movement between these stages and out of state j leads to the equations speci- fying Probability Law V. Mayer (1968) constructed a model of this type in order to explain the dependency on age of occupational pres- tige mobility. In his model (of. Figure 2.3), when an individual leaves the initial stage U, he either enters an absorbing stage PC within state j or leaves state j, i.e., enters L. Like the Mover-Stayer Model of Blumen, Kogan, and McCarthy (1955), Mayer's model implies that parameter A of Probability Law V equals 2 *Av. v Ihtv 0 b... _ 6'. : a»... . g “a... _ '1. - v.4, ,A.'_ ‘ ~~..:_ l J I. o“ '4‘ r. W): is) 1; “ [.Lp (n l p).- .' .v n n Figure 2.3 Structure of Mayer's (1968) model zero. However the interpretation of the probability law differs in two significant ways. First Mayer interpreted time t as age, or more exactly as years in the labor force, while Blumen et a1. interpreted time t as calendar time. More importantly, in using Approach C Mayer constructed his model so that the mover-stayer dichotomy is a property of the relationship between an individual and a social position; thus each person entering a new position has a chance of becoming a stayer. In using Approach B Blumen et a1. postulated that the mover-stayer di- chotomy is a pr0perty of persons. According to this conjecture Proba- bility Law V describes the process of individuals leaving the state occupied at time 0, but since anyone leaving this state is a mover, Probability Law I describes the process of leaving any state entered after time 0. Mayer tested his version of Probability Law V using the same data as in his test of Probability Law IV. Once again he used a synthetic cohort and solved equation (1.25) for P(0,t), requiring him to assume that qjk(t)’ the conditional probability of moving from state j to state k, is independent of time for all states j and k. Mayer's disr cussion of his estimation procedures indicates that he never obtained an analytic solution for P(0,t) but only an approximation to it. Al- though Mayer encountered difficulties in estimating parameters, he did calculate approximate values for the expected mobility to various prestige categories at given ages, and he compared these with the .. .. .‘IO- :3 I n1- . 0 5 ‘u. s.) n H r- - ‘$ \- 50 observed values. Using the index of dissimilarity as a measure, Mayer found that his version of Probability Law V did not predict his data as well as Probability Law IV. Mayer justifiably retained his interest in his version of Proba- bility Law V because it represents an alternative to assuming nonsta- tionarity or population heterogeneity. However the considerable dif- ficulties involved in solving equation (1.25) for P(0,t) with this structurally simple model argue forcibly for studying the process of individuals leaving one particular state j before trying to explain the entire process of movement among states in a system. 2.9 Probability Law VI The mathematical equations defining Probability Law VI are: 8 AZ exp[ — Azt] 4 E z (2.80) hj(t) = Zgl 221 82 exp[ - Azt] __ 4 (2.81) Gj(t) = 221 82 exp[ - Azt] __ 4 (2.82) fj(t) = 221 82 AZ exp[ - Azt] where 82 # 0, 81 + 82 + B + 84 = 1, AZ 2 0, and time t Z 0. 3 Like Probability Law V, whose mathematical form it closely resem- bles, Probability Law VI can be derived from models constructed through Approach A, B, or C. In using Approach A, one postulates that for a homogeneous population in state j the instantaneous rate of leaving state j, hj(t), declines with time t according to equation (2.80). In 51 using Approach B, one postulates that the population in state j con- sists of four groups in preportions 81’ 82, B3, and 84 for which the instantaneous rate of leaving state j is Al, A2, A3, and A4, respec— tively. In using Approach C, one postulates that state j consists of four distinct stages and that movement among these stages and out of state j is Markovian. PC Figure 2.4 Structure of Herbst's (1963) model Herbst (1963), the sole prOponent of Probability Law VI, utilized Approach C. In his model (cf. Figure 2.4), he postulated that (1) upon entering state j all individuals are in an undecided stage U, (2) those leaving U go either to a temporarily committed stage TC or to an uncom- mitted stage UC, (3) temporarily committed persons may enter either a permanently committed (absorbing) stage PC or the uncommitted stage UC, and (4) everyone in the uncommitted stage UC eventually leaves state j, i.e., goes to L. Probability Law V1 with time t referring to duration in state j and with parameter 14 equal to zero results from these assumptions. Using data collected by Hedberg (1961) on the distribution of the length of employment of workers in two Swedish steel factories, Herbst estimated the parameters of Probability Law VI through an undescribed Procedure. On the basis of inspection he obtained an excellent fit between the observed prOportion remaining in the two firms after 52 various durations and the proportion predicted using the parameter estimates. However Herbst's version of Probability Law VI has six independent parameters to be estimated giving it an advantage (of indeterminate magnitude) over those probability laws previously dis- cussed, which have at most three independent parameters. Herbst used his parameter estimates to calculate the instantaneous rates of transition among the stages of the model. While these values seem reasonable, Herbst's data are not sufficient to rule out other models with four stages within state j or models constructed through Approach A or B. 2.10 Probability Law VII The mathematical equations defining Probability Law VII are: 3 .221 82 AZ exp[ - Azt] (2.83) hj(t) = 3 221 82 exp[ - Azt] __ 3 (2.84) Gj(t) = 221 82 exp[ - Azt] __ 3 (2.85) fj(t) = 2:1 82 AZ exp[ - Azt] = Z Z . where 82 ¢ 0, 81 + 82 + 83 1, AZ 0, and time t 0 Like Probability Laws V and VI, Probability Law VII can also be derived from models constructed through Approach A, B, or C. In using Approach A, one postulates that for a homogeneous population in state j the instantaneous rate of leaving state j, h (t), decreases as time j t increases as given by equation (2.83). In using Approach B, one postulates that the population in state j consists of three groups in 53 proportions 81’ 82, and 83 for which the instantaneous rate of leaving state j is A A2, and A3, respectively. In using Approach C, one 1’ postulates that state j consists of three distinct stages and that movement among these stages and out of state j is Markovian. UC U/7 \_(L \iTc/7 Figure 2.5 Structure of Conner's (1969) model Employing Approach C, Conner (1969) made assumptions similar to those of Herbst (1963) except that he reasoned that no one could ever stay permanently in a'state j, which eliminated Herbst's stage PC. In addition he argued that temporarily committed persons left state j di- rectly without passing through the uncommitted stage UC. The structure of Conner's model is diagrammed in Figure 2.5 with the labels of the stages to be interpreted as in Herbst's model. Probability Law VII with time t referring to duration results from these assumptions. Conner tested Probability Law VII with data on the number of con- tinuous years of employment as a farm laborer from age sixteen for 144 Mexican-American males born in 1920 or later. He estimated the param- eters by trying different values for the instantaneous rates of tran- sition among the stages of the model and adjusting them to improve the accuracy of their prediction. 0n the basis of inspection he obtained close agreement between the actual and predicted distributions of the duration of continuous employment as a farm laborer. '1 n1) CHAPTER 3 The Process of Individuals Leaving a Social Position: Basic Issues In the present chapter I discuss the following issues pertaining to the process of individuals leaving a social position: (1) the correspondence between the theoretical construct state j and observable phenomena; (2) the identification of equivalent social positions; (3) the identification of equivalent individuals, i.e., homogeneous subpopulations; (4) the correspondence between the theoretical construct time t and observable measures of time; (5) the validity of the Markov assumption; (6) the validity of Approaches A, B, and C, and (7) the validity of Probability Laws I through VII. 3.1 The correspondence between the theoretical construct state j and observable phenomena In previous empirical investigations pertaining to stochastic models of individuals leaving a social position, a state j has variously been conceptualized as an occupation, industry, occupational prestige category, firm, factory, or geographical area. As I indicated in the Introduction, the explanandum of this research is job mobility, and therefore a job is conceptualized as a state j in this study.1 Is there a reason, other than personal preference, for selecting one particular 1In Chapter 5 I specify in detail the working definition of a state j that I have used in the empirical investigation. For the present discussion it suffices to consider a continuous period (exclud— ing vacations, lay-offs, illnesses, and other temporary interruptions) 54 55 ; conceptualization rather than another? The mathematical theory devel- oped for Approach C in Section 2.4 can be used to support an affirmative answer. 7 Suppose that a particular occupation j is identified as state j. Now an individual may have from one to d consecutive jobs in occupation j where d is some possibly large but definitely finite number. Let Jz represent the 2th job in a sequence of jobs in occupation j, and let L depict the condition of having left occupation j. Then the following diagram illustrates the simplest possible substructure of the process of leaving occupation j. > J l 2 > > Jz > . . . > Jd 4 Figure 3.1 Structure of the process of leaving an occupation Let us assume that the instantaneous rate of leaving the 2th job in occupation j, hj(z,t), is uz, a time-independent constant specific for the 2th job in occupation j, and that the conditional probabilities of transitions from the 2th job in occupation j to the (z + 1)th job in occupation j and to L are also time-independent constants specific for the zth job in occupation j. Since these assumptions parallel those made for Approach C (the 2th job corresponds to stage 2), the equation for G (t), the expected pr0portion remaining in occupation j at time t, J is exactly equation (2.38): spent in one occupation with one employer in one geographical location as a state j. To obtain a closed system, other fulltime activities that may take the place of work (being retired, unemployed, or in school) are also treated as states in the state space. .0. f- e-V van “‘1' 5“ a. v at Q‘h I) c 8,,” CU ».N.\ .H\ L» . t i . .ss :13 ‘1! 56 __ d (3.1) Gj(t) = 221 62 exp[Azt] where Az equals - “z for z = l, 2, . . ., d. Under the special case of Hz equal to K for z = 1, 2, . . ., d, this equation reduces to (3.2) 53(t) = exp[ - Kt] which is equation (2.11) of Probability Law 1. However, as I reported in Chapter 2, evidence in a wide variety of situations has not supported Probability Law I. This suggests that we should tentatively assume that “2 does not equal K for z = l, 2, . . ., d. Consequently if an occupa— tion j is conceptualized as a state j, the simplest expression that we might reasonably expect for Gj equation (3.1). If the instantaneous rates of transition among jobs in (t) is a sum of d exponential terms as in occupation j are n2£_independent of time, then the expression for 53(t) is even more complex than equation (3.1). If an industry, occupational prestige category, firm, factory, or geographical area is conceptualized as a state j, my reasoning parallels that above.2 However, in addition to decomposing a state j into a sequence of d consecutive jobs, I would also differentiate among w = l, 2, . . ., c types of jobs.3 Then the 2th job in state j might be J21, J22, . . ., or ch, and the diagram of the substructure of the process of leaving state j would be correspondingly more complex. 2This type of analysis only applies to mobility among geographical areas sufficiently large to exclude changes of residence that do not necessitate job changes. 3It might be necessary to differentiate among types of jobs when an occupation j is conceptualized as a state j, but I ignored this compli— cation in my discussion above so that the initial presentation of the basic idea would be more clearly understood. u 5 I. ‘l 0- Old n «P. 5..- m». A A129 is P‘OW u: tow. the ”o 5‘. 57 Let us assume that the instantaneous rate of leaving the 2th job of type w in state j, hj(z,w,t) is “zw’ a time-independent constant specific for z and w, and that conditional transition probabilities among jobs within state j and to L are also time-independent constants. Then the equation for Gj state 3 at time t, is still a sum of exponential terms: (t), the expected proportion remaining in II MD: (3.3) 5341:) = w exp[xzwt] E 8 z 1 w=1 z where Azw equals - “zw for w = l, 2, . . ., c, and z = l, 2, . . ., d. The expression for Gj taneous rates of transition among the jobs composing state j are not (t) is even more complex than this if the instan— time-independent constants. In order for the equation for Gj contain as few independent parameters as possible, I conclude that it is (t) to be as simple in form and to desirable to conceptualize a state j as the simplest observable position, which in the present research is a job. Since, as I have indicated, the process of leaving an occupation, industry, occupational prestige cate- gory, firm, factory, or geographical area can be decomposed into the process of moving among a sequence of jobs of possibly different types, information on job mobility can help us to understand these other forms of social mobility. In addition, as I show in the next section, concep- tualizing a job as a state j can assist in evaluating systems of classi- fying social positions and can provide a foundation for a substantive explanation of the leaving process. 3.2 The identification of equivalent social positions While the importance of this issue has previously been recognized (cf. the discussion of "lumpability" by Kemeny and Snell, 1960, and by 58 McFarland, 1970), it has not yet stimulated extensive empirical research. In fact McFarland (1970: 464) has stated: Occupations could be classified into states for a stochastic process in a large variety of ways, and searching for one which would make the process a Markov chain (even if such a classification existed) is somewhat like looking for the proverbial needle in a haystack. Although a classification of jobs which makes job mobility a Markov pro— cess probably cannot be devised,4 I do think that some ways of classify- ing jobs provide a more satisfactory basis for a theory of social mobility than others and that the adequacy of various classification systems can and should be empirically investigated. Such an investigation aims to identify equivalent social positions and is no more like looking for a needle in a haystack than any other scientific research. A research design for such an investigation is given in Section 5.3a. In the present study I examine the effect on the process of indivi— duals leaving jobs of three job attributes for which I have adequate information —- occupational classification, industrial classification, and geographical location. While I do not regard these as completely sufficient attributes for establishing which jobs are equivalent social positions with regard to the leaving process, I do have reasons, dis- cussed below, for considering them important. The substantive, analytic framework outlined below also underlies the discussion in Sections 3.3 through 3.6. Let three types of job terminations be distinguished: those occur- ing unintentionally, those initiated by the employer,5 and those ini- tiated by the jobholder. These are listed in the reverse of the usual 4I doubt that a classification of jobs which makes job mobility a Markov process can ever be devised because I expect the probability of leaving a job to depend on duration in the job (cf. Section 3.4) and I a .\ , .3 .7 e .v .. a TI ~\~ .\ ‘ul 59 order of frequency of occurrence and sociological interest. I treat the last two as instances of intentional job terminations. Death and disability are the primary reasons that jobs are unin- tentionally terminated. These events are associated with the physical hazards and stresses of a job, and consequently with the occupation and occasionally with the industry of the job. As health care varies throughout the United States, the rate of unintentional job terminations may also vary somewhat with geographical location. While a complete explanation of the process of leaving jobs would take account of these effects, their net impact on the instantaneous rate of individuals leaving jobs is ordinarily slight due to the relatively small propor- tion of unintentional job terminations. In practice the magnitude of the instantaneous rate of individuals leaving a job j, hj(t), primarily depends on the level of intentional job terminations. The analytic framework that I employ in explaining intentional job terminations rests on the assumption that the probabil— ity of an individual (jobholder or employer) initiating a change in his situation increases if the individual judges that his present situation is unsatisfactory and that he can probably obtain another more satisfac- tory situation. Therefore, I expect the magnitude of hj(t) to increase with the employer's and jobholder's dissatisfaction with the present situation and with their perceived ability to find a more satisfactory alternative situation. the conditional probability of moving from one job to another to depend on previous work experiences (cf. Section 4.6). 5The proportion of individuals in the present study who reported any periods of self-employment is so small that I am ignoring this possibil- ity and its distinctive problems. .. ~u. ‘nfi ‘v-rb .I\ H 1‘1‘ 26“ Q; 7 \ n a“; L L 4 60 I assume that an employer is highly likely to terminate jobs of enqolxbyees (by firing, transferring, promoting, or retiring them) if he no .113nger foresees a need for their work activities, or if he expects thzrt the continuation of the present assignment of employees to jobs wilLL Llead to unsatisfactory work performance and that he can obtain anotfluear more satisfactory assignment of personnel. The employer's abi]xit:y'to find other satisfactory employees for the jobs in question depends primarily on the supply of qualified job applicants, and thereafiore on the geographical location and occupational classification of the jobs. Ii assume that a jobholder is very likely to terminate his job if he is; (iissatisfied with it and thinks that he can probably obtain a more asartisfactory job (cf. March and Simon, 1958). The social rewards ObtailleKi in a job, and presumably the individual's satisfaction in his 30b: are highly correlated with occupation, and to a lesser extent with indUSttrjy and location. The ability to find other employment depends jointJQY' on the structure of job vacancies (i.e., the number of vacancies in different types of jobs) and on the structure of job-seekers (i.e., the number of individuals with different qualifications who are looking for a .j