Growing up in Times of Social Change [Reprint 2012 ed.] 9783110810110, 9783110165005

Citation preview

International Studies on Childhood and Adolescence 7

m 1749

I

1999

g

International Studies on Childhood and Adolescence (ISCA) The aim of the ISCA series is to publish theoretical and methodological studies on the social, cultural, economic, and health situation of children and adolescents. Almost all countries worldwide report increased risks and problems in the development of children and adolescents. Many pedagogic, psychosocial, and medical institutes as well as education and training centers are trying to help children and adolescents deal with problematic situations. They step in to help with existing difficulties (intervention) or to avoid problems in advance (prevention). However, not enough is known about the causes and backgrounds of the difficulties that arise in the life course of children and adolescents. There is still insufficient research on the effectiveness and consequences of prevention measures and intervention in families, pre-school institutions, schools, youth service, youth welfare, and the criminal justice system. The ISCA series addresses these issues. An interdisciplinary team of editors and authors focusses on the publications on theoretical, methodological, and practical issues in the above mentioned fields. The whole spectrum of perspectives is considered: analyses rooted in the sociological as well as the psychological or medical and public health tradition, from an economic or a political science angle, mainstream as well as critical contributions. The ISCA series represents an effort to advance the scientific study of childhood and adolescence across boundaries and academic disciplines.

Editorial Board Prof. Klaus Hurrelmann (Coord.), Faculty of Health Sciences, University of Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Tel.: (49-521)-106-3834, Fax: (49-521)-106-2987; Prof. Günter Albrecht, Faculty of Sociology; Prof. Michael Brambring, Faculty of Psychology; Prof. Detlev Frehsee: Faculty of Law; Prof. Wilhelm Heitmeyer, Faculty of Pedagogics; Prof. Alois Herlth, Faculty of Sociology; Prof. Dietrich Kurz, Faculty of Sports Sciences; Prof. Franz-Xaver Kaufmann, Faculty of Sociology; Prof. HansUwe Otto, Faculty of Pedagogics; Prof. Klaus-Jürgen Tillmann, Faculty of Pedagogics; all University of Bielefeld, Postfach 10 01 31, D-33501 Bielefeld Editorial Advisors Prof. John Bynner, City University, Social Statistics Research, London, Great Britain; Prof. Manuela du Bois-Reymond, University of Leiden, Faculty of Social Sciences, Leiden, The Netherlands; Prof. Marie Choquet, Institut National de la Santé, Paris, France; Prof. David P. Farrington, University of Cambridge, Institute of Criminology, Cambridge, Great Britain; Prof. James Garbarino, Erikson Institute, Chicago, USA; Prof. Stephen F. Hamilton, Cornell Human Development Studies, Ithaca, USA; Prof. Rainer Hornung, University of Zurich, Institute of Psychology, Zurich, Switzerland; Prof. Gertrud Lenzer, Graduate School CUNY, New York, USA; Prof. Wim Meeus, University of Utrecht, Faculty of Social Sciences, Utrecht, The Netherlands; Prof. Ira M. Schwartz, University of Pennsylvania, School of Social Work, Philadelphia, USA; Prof. Giovanni B. Sgritta, University of Rome, Department of Demographic Sciences, Rome, Italy; Prof. Karl R. White, Utah State University, Logan, USA

Growing Up in Times of Social Change

Edited by

Rainer Κ. Silbereisen Alexander von Eye

W G DE

Walter de Gruyter · Berlin · New York 1999

Rainer Κ Silbereisen, Prof. Dr., Department of Psychology, University of Jena, Germany Alexander von Department Eye, Professor Dr., of Psychology, Michigan State University, USA With 47 figures and 37 tables

® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Growing up in times of social change / edited by Rainer Κ. Silbereisen, Alexander von Eye. p. cm. - (International studies on childhood and adolescence ; 7) Includes bibliographical references. ISBN 3-11-016500-7 (cloth : alk. paper) 1. Teenagers - Germany. 2. Germany - History - Unification, 1990. I. Silbereisen, Rainer Κ. II. Eye, Alexander von. III. Series. H0799.G5 G76 1999 305.235'0943-dc21 99-19730 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Growing up in times of social change / ed. by Rainer Κ. Silbereisen ; Alexander von Eye. - Berlin ; New York : de Gruyter, 1999 (International studies on childhood and adolescence ; 7) ISBN 3-11-016500-7

© Copyright 1999 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany Printing and Binding: WB-Druck GmbH, Rieden am Forggensee. - Cover Design: Johannes Rother, Berlin.

Series Editor's Foreword

This volume is the last of the series "International Studies on Childhood and Adolescence". The series was established in 1987 under the heading "Prevention and Intervention in Childhood and Adolescence" (PICA) and included 20 volumes. The first volume "Social Intervention — Challenges and Constraints" was a collection of papers from the first International Symposium of the Research Center "Prevention and Intervention in Childhood and Adolescence" at the University of Bielefeld in Germany. Since 1987 this Center has organised the edition of the series. All volumes were designed to cover interdisciplinary research on developmental problems of children and adolescents in western industrialised countries, and to promote new approaches in psychological, sociological, political, and criminological intervention. The series included such volumes as "Crime Prevention and Intervention", "The Social World of Adolescents", "How to Organise Prevention", "Individualization in Childhood and Adolescence", "Health Hazards in Adolescence", and "Diversion and Informal Social Control". The volumes in the PICA series were mainly published by members of the Bielefeld Research Centre. The series, however, became recognised as an interdisciplinary forum by scholars all over the world and the Editorial Board began to receive proposals by scholars from research centres in the US, Europe, and Israel. Therefore, the Editorial Board and the publisher decided to extend the series to international contributions, and for it to be continued under the heading "International Studies in Childhood and Adolescence". The present volume is the sixth in this new series and the 26th altogether. In modern industrialised countries, disorders and impairments affecting the development of children and adolescents give rise to serious social concern. However, research rarely succeeds in linking the analyses of causal conditions to the analyses of the impact of interventions. Thus, little reliable evidence is available relating to the effectiveness and efficiency of intervention measures taken. Still, most current research is of a cross-sectional nature, thus clearly restricting causal analyses. Despite a very small and slowly growing number of longitudinal studies, there is an urgent need for this kind of research strategy. We need this type of study to cover the subsequent stages in the development of

VI

Series Editor's Foreword

children and adolescents in particular, and to know more about the short- and long-term impact of intervention strategies. The present volume was designed to reduce such research deficits. The editors, Rainer Κ. Silbereisen and Alexander von Eye, have collected papers by established scholars in the field, as well as by younger scholars who are still trained in innovative research strategies. The papers in this volume focus on an interesting topic for longitudinal research: They assess the impact of the political unification of the two German states that took place in 1990 when, after more than 40 years of separation, the state of West Germany, with some 63 million inhabitants, and the state of East Germany, with approximately 16 million inhabitants, were integrated into one nation. The long separation of the two German states was accompanied by different life styles and contexts for the psychological development of children and adolescents. The political, social, and economic systems of the two Germanys produced two highly different contexts for individual biographies. In particular, children and adolescents experienced decisive and formative steps of their life courses in totally different settings, culturally as well as ideologically. The contrast between growing up in a planned economy within a totalitarian political system in the East, and starting life in a market and consumer society and pluralistic political system seems to be extreme. During the 1980's, symptoms of a deeply rooted economic crisis in East Germany became manifest, finally leading to the breakdown of the "German Democratic Republic". Since 1989, Germany has faced the challenge of uniting the country. For German adolescents in both parts of the country this historical event also brought insecurity concerning their personal future. While many young people in West Germany are afraid of sharing their wealth and prosperity with those in East, young East Germans have lost many of their social traditions and have faced a thorough change of the educational and economic systems which affect their perspective for the future. In this volume, Rainer Silbereisen and Alexander von Eye have edited a selection of papers in which social scientists, particularly psychologists, sociologists, and demographers, present reports about their research on the impact of German unification on the life of adolescents. Most of the papers were presented at an international conference that centred on the exchange of new methodological techniques for longitudinal analysis. This volume will bring the international book series of the Bielefeld Research Centre to a close. As the director of this centre and the chairman of the Board of Editors of the series I want to express my gratitude to all contributors, to the German Research Association (Deutsche Forschungsgemeinschaft) for the 12 year grant for the Research Centre, and to all the colleagues who contributed to

Series Editor's Foreword

VII

the volumes of this series. Last but not least I want to express my greatest appreciation to Dr. Bianka Ralle from the editorial staff of the publisher of this series, Walter de Gruyter Inc., Berlin and New York, for her invaluable advice, help and encouragement in supporting the series.

Klaus Hurrelmann Professor of Sociology and Public Health University of Bielefeld Germany

Preface

1 The Motives for this Volume This volume is carried by a large number of motives, two of which stand out. The first of these is rooted in the current political situation in Germany and in the recent past. In 1990, the two Geimanys were unified. What had seemed a Quixottesque impossible dream, that is, German unification, suddenly became reality. The New York Times used the headline "Flag at Reichstag marks start of new Era at Center of Europe" (1990, October 3, p. 1). The news were abundant with concerns related to peace, with "dire predictions" (New York Times, 1990, October 3, p. 1) related to the economy, and with reports of jubilant German citizens. Sub-headlines and reports covered such prominent topics as beer and the opposing Republicans (a right-wing party in Germany). Most interestingly from our social science perspective is that none of the predictions was of psychological or sociological nature. Concerns had to do with the feared overpowering of Europe's economy by Germany but not with the personal experiences and feelings of the citizens. The present volume seeks to fill this gap. This volume contains a selection of chapters in which social scientists, specifically psychologists, sociologists, and demographers present reports about their research on the human side of the German unification. The reports cover consequences of unification, as they were and still are experienced by individuals and groups of individuals. Most of the respondents can be assumed to have welcomed the changes that came with unification. Others, however, may have felt differently, because they were scared of the unknown new system, or perhaps they had established themselves in a political landscape that now did not exist any longer. The focus of this volume is on development of adolescents and young adults. This period of life is complicated even without radical political changes. Physical, mental, and societal changes can make life for adolescents and young adults stressful. Adding even a peaceful revolution to these sources of turmoil can only add to the stress. Fall-out is to be expected. This volume presents reports on psychosocial development in this time of radical political change.

χ

Preface

Most interesting is that there is a lack of theoretical psychological and sociological backgrounds of these changes. Historically, individuals and groups of individuals have always subjected themselves and were subjected by others to situations where adaptation to change was a necessity. Examples of forcefully induced change include slavery, expulsion, or migration of peoples that have virtually always taken place. Voluntary change is induced when individuals, families, or groups of people move, for instance, across town or across the Atlantic. The situation after the German unification is unique. The need for adaptation to change can be felt by everybody in Germany and many if not most in Europe. This applies equally to those who longed for this change and those who were afraid of it. In addition, the change came to the people. There was no journey needed, no relocation. The change arrived, quasi over night, and all individuals had to deal with it. Thus, a standard characteristic of typical psychological experiments does not apply: unification-induced change affects everybody, not just volunteering respondents. To the best of our knowledge, there exists no single theory that guides thinking about change that affects everybody in a society and many beyond. The current volume will shed light on adaptation to change under these unique conditions. The second main motive that fueled the production of this volume concerns the methods used to describe change. Considering, for instance, (1) the characteristics of measures used in the social sciences; (2) the confoundedness of such central variables as time of measurement, cohort, and age; (3) the specificity of questions that need to be answered; or (4) the lack of replicability of investigations of unification-induced change in development, it is of utmost importance to do it right. The proper methods need to be devised and applied, in design and in particular in statistical data analysis. Therefore, this volume consists of two parts. The first part contains psychological, sociological, and demographic reports about unification-induced change on development. The second part presents new methods and useful facets of already existing methods for analysis of data that describe change. The methods were selected based on the following three criteria: (1) their usefulness for developmental research; (2) the degree to which they are available in form of computer programs for the most common platforms (all are available); and (3) scope, that is, we tried to present methods that overlap as little as possible so that a wide range of substantive questions can be analyzed.

Preface

XI

2 Contents: An Overview The first part of this volume contains nine scientific reports about unificationinduced effects on development in German youth. Hans Bertram contributed the first chapter. This chapter reflects many of the problems specific to research on the effects of unification. Re-definitions of districts are needed that are not based on political but also on sociological and psychological arguments. Migration needs to be taken into account. As a result, the frame of reference is no longer a constant but a moving target itself. The author does an impressive job considering all these changes when he presents his description of familial forms of life. The second chapter, authored by Klaus Boehnke, is concerned with social change. This research uses a rarely employed yet obviously highly productive qualitative method, specifically, the interpretation of contents of photographs to depict social change. Results suggest that the perception of change, even at the level of large-scale political change, is age-dependent. Respondents can reflect change only when equipped with a frame of reference. The third chapter, written by Peter Noack, contains a report on adolescent peer relations in times of social change. In a fashion similar to the second chapter, peer relations in adolescence seem to reflect social change only under certain conditions. Adolescents select peers and interact with them when times are stable and also when the political landscape is in turmoil. Effects of political changes are based on perceptions of social change. The fourth chapter, co-authored by Bernhard Nauck and Magdalena Joos, focuses on the important topic of child poverty. More specifically, the authors investigate the interaction of institution transfer and family type in the transformation process. In other words, the authors investigate the effects that changes in the structure of institutions have on families' and individuals' well being. The fifth chapter describes qualities of children's friendships in middle childhood in East- and West Berlin. It is contributed by Lothar Krappmann, Harald Uhlendorff, and Hans Oswald. In unison with the three chapters before, the results of this study suggest that, rather than reflecting politically-induced educational programs, children seem to behave in response to general needs of children in modern societies. Interestingly, however and in contrast to overt behavior, the concepts of friendship seem to be reflective of ideologies (and age). The sixth chapter, co-authored by Hubert Sydow, Christine Wagner, BerndRüdiger Jülisch, and Hagen Kauf, reports results of a study on future oriented control and subjective well being of students in East- and West Berlin. Overall, this study suggests that earlier observations in other studies, according to which children in East- and West Germany differ in control belief systems cannot be replicated. Assuming validity of both results one can hypothesize that (1) dis-

XII

Preface

tance to the moment of unification can explain this difference, and (2) children are highly flexible in their adaptation to environmental characteristics. Chapter 7 is concerned with risk. It is co-authored by Wolfgang Ihle, Günter Esser, Martin H. Schmidt, Bernd Blanz, Olaf Reis, and Bernhard MeyerProbst. More specifically, this chapter compares longitudinal data on risk conditions and developmental patterns of mental disorders from childhood to early adulthood in Rostock (former East Germany) and Mannheim (former West Germany). The interesting and surprising results of this study suggest that, in contrast to commonly held expectations, adolescents and young adults in the former East Germany are better off rather than worse than their counterparts in the former West Germany. The American-German team of Fred W. Vondracek, Matthias Reitzle, and Rainer Κ. Silbereisen contributed the eighth chapter. It describes results from an investigation of the important topic of the influence of changing contexts and historical time on the timing of initial vocational choices. There can be no doubt that the change from a socialist to a capitalist political system can have massive impact on career planning of adolescents and young adults. Results suggest that the relationship between personal identity and professional identity seems to be stronger in adolescents from the former West. The authors interpret this finding from a hypothesis of Eastern Adolescents' self-protection against the effects of economic circumstances beyond their control. The ninth chapter is concerned with both substance and methods. Gisela Trommsdorff presents a methodological critique of concepts and research on social change and individual development in East Germany. The author discusses data characteristics, comparability of data gathered before and after unification, and raises the important issue of the function that observed differences in ability and preferences may have when dealing with environmental demands. Because this chapter is of interest both in the substantive and in the methodological domains, it concludes the substantively oriented Part I of this volume and leads readers to tjie methodologically oriented Part II. Part II of this volume contains six chapters on methods of statistical analysis of developmental processes. The first chapter, or Chapter 10 over all, is coauthored by H. Joachim Bretz, René Weber, Gerhard Gmel, and Bernhard Schmitz. This chapter has two aims, a substantive and a methodological one. Substantively, the authors present results from a study on TV viewing habits around the time of German Unification. Specifically, the authors ask how the East German news channel, the Aktuelle Kamera, formerly an outlet of the communist party in East Germany, has tried to transform itself and how this transformation affected usage. Methodologically, the authors describe methods of transfer-function analyses, an ARIMA time-series method, and use the substantive results for illustration.

Preface

XIII

Chapter 11 discusses methods for analysis of change in categorical variables. Alexander von Eye uses the design matrix approach to present log-linear models of symmetry, describes new developments, and illustrates application. These methods are of importance because categorical longitudinal data often remain under-explored, for lack of readily available methods of analysis. Christiane Spiel introduces in Chapter 12 a research strategy for the reanalysis of longitudinal data. Considering the issues of data comparability and variable meaning raised in Chapter 9, this strategy may prove very useful. The author delineates the six elements of the strategy and presents sample results. Applications of this strategy to data collected in the context of German unification are discussed. Michael J. Rovine, Peter C.M. Molenaar, and Sherry E. Corneal contribute Chapter 13. The authors describe the very useful yet under-utilized method of P-technique. This is a variant of factor analysis where repeated observations of several variables are analyzed for single individuals. Sample results are presented. This chapter is important not only because it presents a lucid description of P-technique, but also because it provides protocols of sample runs that enable researchers to tailor their own applications. Chapter 14, provided by Peter C.M. Molenaar, Michael J. Rovine, and Sherry E. Corneal, extends the possibilities depicted in Chapter 13 to the domain of dynamic factor analysis. Whereas P-technique is applied mainly in exploratory contexts, dynamic factor analysis has chiefly explanatory applications. Other differences concern the number of observation points. P-technique requires large numbers of observation points. In contrast, dynamic factor analysis requires only a few. Application of the method is illustrated using data and sample command file protocols. Together, Chapters 13 and 14 present an outstanding, state-of-the art rendering of the factor analytical possibilities to depict the space of repeated observation in both exploratory and explanatory research. The last, the 15th chapter in this volume provides another high light. Kasuo Yamaguchi and Lei Jin describe the most useful method of event history analysis, its most recent developments, and indicate future developments of the method. Event history analysis allows researchers to model repeated observations of categorical variables. The method is widely known in Sociology. Chapter 8 presents one of the rare applications of event history analysis in Psychology. The chapter also contains sample applications and sample command file protocols.

XIV

Preface

Acknowledgements Transatlantic communication is not the easiest of endeavors, even in a time of electronic shuffling back and forth of manuscripts. Therefore, we are greatly indebted to Annett Weise for keeping this communication fluent and stylish. A subset of the chapters of this book resulted from contributions to a conference with the same title as this volume, organized by the editors in 1996 at the University of Jena. This conference was conducted as part of the research priority program Childhood and Adolescence before and after German Unification of the German Research Foundation (DFG). The conference consisted of two parts. The first part was an expert meeting and the second part was a methods workshop. We owe our thanks to the DFG for financial support of this conference, to the University of Jena for its hospitality, to Matthias Reitzle, Verona Christmas, Annett Weise, and Katrin Müller for helping organize the conference and edit this volume. All of the papers underwent an extensive process of review and revision. Many friends and colleagues supported this project by reviewing and commenting, identifying problems, and proposing solutions. As two placeholders for all, we would like to thank Verona Christmas who edited many chapters, and Christof Schuster who did more than his share by reading and reviewing many chapters. We would also like to thank the contributors to this volume. They all exhibited their unique variants of patience and responded graciously and professionally to requests for editor-induced and, perhaps also, unification-induced change in their drafts. Some of the chapters took several rounds of reviewing and revising. We appreciate the authors' efforts and are convinced that the time and energy invested were worth it. We feel we should also thank the publishers. Specifically, we are indebted to Elisabeth Abu Homos for her cooperation and guidance through the production process. There are many more we have to thank, too many to be listed here. However, and most of all, we would like to thank those who endure us. These are the ones who love us and live with us. Together, we grow, change perspective, and watch the times change, and we enjoy it. We cannot ask for more.

Jena and East Lansing, July 1998 Rainer Κ. Silbereisen and Alexander von Eye

Contents

Series Editor's Foreword

V

Preface

IX

Part I Social and Behavioral Sciences Reports

Regional Diversity and Familial Forms of Life — Structural and Social Conditions of Child Socialization Hans Bertram

3

Is There Social Change? Photographs as a Means of Contrasting Individual Development and Societal Change in the New States of Germany Klaus Boehnke

31

Adolescent Peer Relations in Times of Social Change Peter Noack

51

Child Poverty in East Germany — The Interaction of Institution Transfer and Family Type in the Transformation Process Bernhard Nauck and Magdalena Joos

73

Qualities of Children's Friendships in Middle Childhood in East- and West Berlin Lothar Krappmann, Harald Uhlendorff, and Hans Oswald

91

Future Oriented Control and Subjective Well-being of Students in East- and West-Berlin Hubert Sydow, Christine Wagner, Bernd-Rüdiger Jütisch, and Hagen Kauf 107

XVI

Contents

Risk Conditions and Developmental Patterns of Mental Disorders from Childhood to Early Adulthood — Results from Two Longitudinal Studies in Rostock and Mannheim Wolfgang Ihle, Günter Esser, Martin H. Schmidt, Bernd Blanz, Olaf Reis, and Bernhard Meyer-Probst

131

The Influence of Changing Contexts and Historical Time on the Timing of Initial Vocational Choices Fred W. Vondracek, Matthias Reitzle, and Rainer Κ. Silbereisen

151

Social Change and Individual Development in East Germany: A Methodological Critique Gisela Trommsdorff

171

Part Π Methods The Time of the Wende: Social Change and the Reception of Political Broadcasting in the German Democratic Republic. Time-Series Analyses of News Usage H. Joachim Bretz, René Weber, Gerhard Gmel and Bernhard Schmitz..

203

Standard and Non-Standard Log-linear Models for Analyzing Change in Categorical Variables Alexander von Eye

225

A Strategy for Data Reanalysis in Longitudinal Studies Christiane Spiel

245

Analysis of Emotional Response Patterns for Adolescent Stepsons Using P-Technique Factor Analysis Michael J. Rovine, Peter C. M. Molenaar, and Sherry E. Corneal

261

Dynamic Factor Analysis of Emotional Dispositions of Adolescent Stepsons Towards Their Stepfathers Peter C. M. Molenaar, Michael J. Rovine, and Sherry E. Corneal

287

Event History Analysis in Human Developmental Research Kazuo Yamaguchi and Lei Jin

319

Contributors

341

Parti Social and Behavioral Sciences Reports

Regional Diversity and Familial Forms of Life — Structural and Social Conditions of Child Socialization* Hans Bertram

1 Plurality in the Forms of Life Due to Regional Diversity When the participants in the Leipzig Monday demonstrations chanted 'We are one people', they called the political order of the GDR into question, not, however, their regional attachment with Leipzig or Saxony. It is an essential feature of German society that the regionally differing cultural traditions, life experience and political orientation patterns are part of the German culture. The scope for action of the individual, the education chances, the career options, the possibilities of participating in forms of socialisation, as for instance in associations, the opportunities of taking part in cultural events — all this is closely linked to the development of the respective towns, federal states and regions of the Federal Republic of Germany. When Braudel (1989) postulates for France that the French cultural identity is only to be fathomed from the diversity of its provinces, then this is certainly also true for the Federal Republic. There is no national ultimate center in which the economic, political and cultural elites define uniform standards, e.g. for the education system. On the contrary, the regional diversity of the Federal Republic contributes to standards and behavioral expectations being formulated that are of significance only for a certain regional context. This economic cultural and social poly-centralism of the Federal Republic leads to the situation where, for instance, no politician, leading figure in industry and commerce or scholar, be they from Swabia, Bavaria or Saxony, has to give up the regional origin in their language and attire in favor of a neutral and refined language or a non-identifiable cultural self-depiction. Rather, it is expected of a politician or manager, as a member of a regional culture, that he/she can mix the refined language, accent and self-depiction in such a manner that the regional attachment remains recognizable. This is not only true for the elevated groups in our society but for all groups of the population. The life of an industrial skilled worker in Berlin-Wedding is maybe comparable with regard to income, training and job with the life of an industrial skilled worker in Din* I thank the reviewer of this chapter for many good ideas and criticism.

4

Hans Bertram

golfing but this comparability stops already at the gates of the firm as far as mentality, orientation and life styles are concerned. Whereas this regional diversity poses no problem for us in everyday life, it is difficult for sociologists to come to grips with such a diversity of regional structures. This is because most of their theoretical models for the description of social structure are designed vertically. When empirical analyses, however, show, that this model of a vertical structure of the patterns of views, the scope of activity and mentality of individuals does not correspond with the empirical facts, many sociological authors tend to give up such models of vertical inequalities totally, and to design new models of society or to talk of the dissolution of the classical environment. Schelsky's "society of the middle class" (1960), just as Geiger's "class society in a smelting pot" (1949) were nothing else than the insight that not all opportunities of action, expressions of life and mentalities of individuals may be jammed into a vertical pattern. Also Beck's "society of risk" (1986) or Schulze's "society of experience" (1993) are founded on an easily understandable and correct observation that the mentality of individuals and their life style in a society such as the Federal Republic cannot only be defined with the help of three indicators: the educational level reached, professional prestige and income. There have only been few exceptions, at least in German sociology, which have tried to break up this theoretical limitation of sociological analysis of society. The attempt undertaken by Peisert (1967) and Dahrendorf (1965a,b) to show that educational chances are partly influenced more by the region than the professional position of their parents, seems to be the politically most successful one. The "Catholic working class girl from the countryside", who differs considerably in her educational chances to Protestant boys in the urban regions of South Germany and West Germany then and today (Bertram, 1991b), was not identified in this discrimination on the basis of a demanding theory. However, in contrast to many theories of stratification, this approach had the advantage that it also described social facts. In the following the attempt is undertaken to identify such regions in the whole of Germany, which differ to a maximum extent with regard to the private forms of life. As seductive as it may be on the one hand to make the area and the actions of individuals, in space and time, a subject of theoretical elaboration (Giddens, 1984), the empirical translations of most of the theoretical area concepts in the sociological field (Pareto, 1955; Simmel, 1908; Halbwachs, 1946; Durkheim, 1981; Werlen, 1988) have been little convincing.

Regional Diversity and Familial Forms of Life

5

2 Regions, Historical Events and Cultural Traditions 2.1 Regions as Aggregates Politological analyses show again and again that in spite of many changes the conservative parties CDU and CSU gain mostly in those election districts, where the catholic Zentrum won 70 or 100 years ago (Falter, 1994). In the Ruhr district which nowadays is less a region of coal, iron and steel industry and whose leading steel companies have changed to communication companies, you find still today the focus of the social democrats. In spite of deep changes in the education system as well in West as in East Germany even today the regions with historically provable high quota of high education levels show above average values. The protestant regions of Erlangen/Nürnberg within the catholic Bavaria have high quota of the Abitur ( = high school diploma; Peisert, 1964), which did not change till today (Bertram, 1991b). Even the GDR was not able to break such traditions. In spite of its centralist school system, the education participation in regions such as Jena, Halle or Wittenberg was always above the GDR average (Hartmann, 1996). It was never possible to bring even the city regions of Mecklenburg-Vorpommern to the level of these originally protestant regions. Without doubt even in secularized countries as the GDR the education traditions lived on, which were originally related to religion. The comparison between the protestant Bavarian Erlangen with the originally protestant Jena shows that such traditions head themselves against quite different political systems. The election behavior, too, is widely influenced by such traditions in spite of all social change. While education differences and regional variations in the elective behavior are interpreted rather evidently as being influenced by religious and cultural traditions, you will not find such analyses for other aspects of social life in psychology and sociology. There is wide ignorance that in the catholic south of Germany the birth rate is significantly higher than in the protestant north, the divorce rate, however, substantially lower, family size far greater, children leave the family later, and so on (Bertram/Bayer/Bauereiß, 1993, Bertram, 1995). These variations are mentioned without asking, if and how these regional differences influence the socialization conditions of children. The actual research is asking in a more global way for differences between East and West, though the living conditions are homogenous neither in West Germany nor in East Germany. There is rare discussion that even today the socio-economic situation of a city as Potsdam, measured by the purchasing power of the population, can hold the comparison with certain West German regions, whereas even in West Germany in some coast regions the unemployment rate, the economic development and the purchasing power of the population lead to tendencies with high probability for further poverty.

6

Hans Bertram

Regions in its relevance for the living conditions of children are considered only in the analysis of certain city districts, where the differences seem more clearly and are often assumed as the result of a combination of social disadvantages of certain social groups of the society and disadvantaging living conditions. In contrary it is here considered that, even independently from social classes and cumulatively working socio-ecological factors, regions are of considerable importance for the socialization of children, because they represent the historical-cultural traditions of a society. They influence the living conditions of children in form of structural effects. When children grow up in a region were more-children-families are usual, where the neighborhood contacts interact with the child's development, where the participation in higher education is usual, then even children from disadvantaged social groups will take more profit from the development chances than in settings with less fostering conditions. The little consideration of regional factors in socialization research is yet astonishing, because since the socio-ecological approach of Urie Bronfenbrenner (1974) the relevance of the context for the development of children was founded theoretically convincing. But even new sociological theories with high dissemination, such as James Coleman's "asymmetric society" (1974), give evidence for the importance of regional contexts. It is especially the merit of Glen Elder (1974) to have proved that the combination of historical-economic developments and certain cultural traditions of the farmer families in Iowa deeply influence their handling and thus the living chances of children and juveniles. The rare finding of regionally oriented analyses can supposedly be founded less theoretically, but more methodically. An index of social class within an empirical investigation will be easy to be constructed and to be measured. Regional variations, however, can be measured only by means of data of the official statistics, e.g. the census. For describing regions by indicators always means to differentiate a society on the macro level by means of structural data. Information about the birth rate, the purchasing index, the unemployment rate or other indicators cannot be taken within an empirical investigation on the socialization of children; here you need official data. At least as difficult as the gaining of the data naturally is the development of a typology of regions. Should they be built especially on the basis of economic factors or are religious traditions of more importance? You find lots of economic typologies of regions (Gensko, 1996), but no one with socio-cultural indicators.

Regional Diversity and Familial Forms of Life

7

2.2 The Construction of the Regions of Germany: Methods Checking the official data of the Federal Bureau of Statistics for possible indicators, you first must consider, that for reasons of data protection the smallest empirical unit of the official data set is the district with more than 100 000 inhabitants, both in East and in West Germany. At this level you find lots of indicators, so you must find in a preliminary selection those indicators which help to combine certain districts in such a way that regional differences will be visible. This selection will be described in detail in discussing the results. First it is oriented to indicators which point to variations in the familial life form, such as birth rate, divorce rate, number of children, always relative to 100 000 inhabitants. As till now there is no valid, theoretically differentiated description of the regions, here an inductive method of analysis for the official data was taken, which has proved itself in the politological research of election behavior, the "automatic interaction detector" 1 , which works like a hierarchical analysis of variance. With an interval-scaled dependent variable many analyses of variance are iteratively done with various numbers of independent variables. At each step those groups of the independent variables are taken which maximize the variance of the dependent variable. This method may calculate for each combination of aspects of each independent variable and then split into two groups that differ mostly with respect to the dependent variable, measured in mean differences. In the further analysis the groups of the first step are worked out in the described way, so that you get a hierarchically organized tree (therefore "tree analysis"), which helps to identify all interactions of the independent variables. This is a rather mighty method with high demands to the quality of the data. You especially need an immense number of cases, because the hierarchical iterative process very quickly leads to small groups. The dependent variable must be intervalscaled and the independent variables should be categorical. The data of the census (65 millions persons) and the other official statistics fit these criteria. In contrast to other methods like cluster analysis or factor analysis this procedure considers a specialty of official data, which normally is ignored by social scientists working within the socio-ecological framework. Sociologists and psychologists usually conceptualize their variables being more or less independent from context, for example, a certain parental educational behavior leads to a certain behavior of the child. In contrast, structural data must be interpreted in a context-specific way. A high rate of unemployment, a low purchasing rate and a high number of children are usually associated with a low ownership of ground and houses. This relationship is often true, but not always. In certain rural regions the rate of ground and house ownership is especially high for reasons of a broad neighborhood assistance, with high unemployment rate, low purchasing 1

AID: Automatic Interaction Detector (also: tree analysis).

8

Hans Bertram

index and high number of children. But this is not true for all rural regions. If one wants to investigate the relevance of the regions for the living conditions of the children, one define regions by means of empirical methods that correspond with the real conditions. The aim is not the construction of abstract types of regions, but the reconstruction of really existing regions with homogenous indicators. I know that the here sketched procedure seems plausible, but hard to follow, especially when the empirical analyses are based on individual data. Mostly those indicators, which show different effects with other indicators with respect to their different characteristics, are eliminated. But just these uneven effects are special for the data from the census and the official statistics, because they represent a piece of historical development, which is not systematical. If one wants to clear contextual effects, one must work like this. To make the findings understandable, the results of the tree-analyses are presented spatially and finally shown in a map to make it easier to identify them as typical regions of Germany.

2.3 The Construction of the Regions of Germany: Results It is known from numerous investigations (Huinink, 1993) that families with children differ in how they lead their private lives from those households without children, just as families with many children differ from those with few children. Thus the share of 6 to 14 year-old children in the respective districts of the Federal Republic were chosen as a dependent variable on whose basis the regions of the Federal Republic are to be homogenized. Both in the old and the new federal states2, these children were born before 1989. Their share of the respective population documents the reproduction behavior of adults between the ages of 28 and 50 without the influence of the social development over the past six to seven years. As this age group of adults from the new federal states have migrated much less from East Germany than for instance the young adults between 18 and 24 years, one can also assume that both in the old and the new federal states this group corresponds with the share of children who were born in the respective region 6 to 14 years ago. The homogenization and identification of regions on the basis of this share of 6 to 14 year-olds of the entire population was not conducted by simply using the different shares as a scale of differentiation of individual regions, as one would presumably not arrive at historically definite regions in this way. Certain regions in Bavaria with a traditionally high birth rate and regions in Mecklenburg2

Former West Germany is referred to as the 'old federal states' and former East Germany is referred to as the 'new federal states' (the translator).

Regional Diversity and Familial Forms of Life

9

Vorpommern with a similarly high birth rate cannot be subsumed in one region as despite similarities with regard to birth rates, they are culturally very separate regions. The homogenization of the regions is conducted within the framework of contrast group analyses. For a more exact differentiation of different regions in the Federal Republic, a number of aggregate variables of the official statistics at district level were used, of which it may be assumed that they document the different development of individual regions of the Federal Republic to a sufficient degree. The results of the 1990 national elections were taken as an indicator for the political development. The 1993 index of purchasing power (Gesellschaft fur Konsumforschung GFK, 1993) was used for the economic development. Apart from the results of the 1990 national elections, which were used to indicate the political climate, the 1992 unemployment rate3 was also considered as an expression of the economic development. Over and above politics and the economy, the age structure of the population between 0 and 24 years, the age of the mothers at the birth of their children, the marriage and divorce figures were also drawn on as indicators for the development of the population and the family in the respective districts of the Federal Republic. Due to the focus on the familial way of life, it is especially the services of youth welfare such as the share of crèches, kindergartens and after-school centers as well as social security for children and young people that were utilized in the area of infrastructural services. This is how altogether 14 regions, which differ clearly with regard to their share of 6 to 14 year-olds, were identified in the Federal Republic of Germany, with the help of a number of contrast group analyses. There are regions, for instance the West and South German service centers, with a share of 6 to 14 yearolds per 1,000 of the total population of only 48 children and young people. In contrast there are more than 90 children and young people between the ages of 6 and 14 per 1,000 of the total population in other regions of the Federal Republic, for instance in the rural regions of Mecklenburg-Vorpommmern and Brandenburg. The empirically derived pattern of the 14 regions in Germany is easily comprehensible. On the one hand there is a clear differentiation between the new and the old federal states. Already the first step of the contrast group analysis expresses clearly the very different levels of purchasing power between the old and the new federal states, a clear difference between the two parts of Germany. The further steps show considerable variations also internally in both parts of the country. The densely populated urban regions of both the old and the new federal states are clearly distinguished. In the East, the Saxon, Saxon-Anhalt 3

All aggregated variables of the official statistics are either from the regional data pools of "Deutsches Jugendinstitut" or from the research project "Living conditions of children" (Nauck/Bertram/Klein 1994), sponsored by German Research Society, DFG.

Hans Bertram

10

and Thuringian towns are separated from the northern towns in MecklenburgVorpommern and Brandenburg. There is a similarly clear differentiation in the old federal states, the Northern German service centers such as Hamburg, Bremen, Kiel and Berlin and the West German and South German service centers such as Dusseldorf and Cologne or Stuttgart and Munich. Apart from that there exist further differentiations between the towns, such as the exclusion of the towns in the Ruhr area or the clear demarcation of the South German towns of Nuremberg, Augsburg or Heilbronn from the university towns, especially in South Germany, such as Freiburg, Heidelberg and Tübingen. Table 1 : Region - Results of the Contrast Group Analysis

Types of Regions

Abbrevations

Number of Districts

Share of 6-14 years olds

West/South German service centers North Germna service centers South German cities University towns Ruhr area Wealthy Suburbs North German protestant region South German protestant region South German catholic region North German catholic region Saxonian, Thuringian cities Saxon, Saxon-Anhalt, Thuringia-region Brandenb.Mecklenburg, cities Brandenb.Mecklenburg, region

W/SÜDDIENST NORDDIENST SÜDSTDT UNI RUHRSTÄDTE REICHEVORO NORDPLAND SÜDPLAND SÜDLAND NORDKLAND SÜDNBLGRST SASTH-LAND NORDNBLSTA MVBR-LAND

9,00 5,00 29,00 8,00 14,00 16,00 45,00 21,00 63,00 15,00 13,00 101,00 14,00 61,00

6,42 6,45 7,19 7,24 7,51 8,01 8,25 8,82 9,17 9,25 10,43 10,95 11,97 11,95

Apart from these differentiations within the urban regions, which may be clearly determined also with regard to the population density, the rural areas of the new federal states were divided into Mecklenburg-Vorpommmern and Brandenburg on the one hand and Saxony-Anhalt, Thuringia and Saxony on the other hand. The differentiation of the rural areas in the old federal states, was conducted in a four-fold way, that is to say Southern/Catholic, especially the regions in Bavaria and Baden-Württemberg, as well as Southern/Protestant, again especially Württemberg and Bavaria, also in North Germany as Northern/Protestant and Northern/Catholic (Bertram, 1992, Bertram, Bayer, & Bauereiß, 1993).

11

Regional Diversity and Familial Forms of Life

3 Family and Population in the Regions The regions identified here are so different in their demographic structure that it is extremely difficult, on the basis of these data of the official statistics, to speak of 'the family' rather than of a plurality of familial forms of life in the Federal Republic. When for instance nearly 225 children and young people per 1,000 persons of the total population live in Mecklenburg-Vorpommern and Brandenburg, as compared to just 150 children and young people of this age in the West German and South German service centers, then this demographic difference not only means that there are less children in the latter areas, but that the South West German regions are experiencing an enormous aging process, which can only be halted by a migration of young adults to those regions. MVBR-LAND NORDNBLSTADT' SASTH-LAND' SÜDNBLGRSTADT" NORDKLAND' SÜDLAND' SÜDPLAND NORDPLAND' REICHEVORORTE RUHRSTÄDTE UNI SÜDSTDT' NORDDIENST' W/SÜDDIENST'

• m M

Ί 1 1 1 75 100 125 150 Cases in thousands

Figure

1: Children

and Young Adults from

1 175

0 to 24 (Share of the

Γ 200

0-1 Y. 1-2 Y. 3-5 Y.

•

6-14 Y.

m

18-24 Y

225

Population)

Such migratory movements are taking place on a large scale in the Federal Republic with the consequence that the share of single and one-person households is very high, especially among young adults. A comparable pattern is to be found in the North German service centers and the towns of the Ruhr area, whereas in all other regions the share of this young age group is at least 175 persons per 1,000 total population. The chart also shows that it makes sense to have chosen the age group of 6 to 14 year-olds as starting point for the demographic differentiation of the regions as the number of 0 to 1 year-olds shows a distinct difference between the regions of the new and the old federal states, demonstrating the much quoted birth swing-about in the new federal states. There were only seven children between 0 and 1 years per 1,000 total population in 1991, as opposed to eight to ten children in the old federal states. The comparable shares of 18 to 24 year-olds of about 50 to 60 young people and young adults per 1,000 of the total population in the new federal states also documents the migratory movement of the young adults between 1989 and 1991

12

Hans Bertram

from the new to the old federal states (Freitag, 1994). The high proportion of young adults of this age in the university towns but also in the West and South German service centers, for instance, are not the result of a changed reproduction behavior in these areas over the past 20 years but rather a consequence of the heavy influx of these age groups into the towns. Whereas the historical events accompanying the re-unification are clearly reflected in the population structure of 0-24 years in the selected areas, the age of the mothers at the birth of the children per 1,000 live births shows that the reproduction behavior of the mothers in the new federal states had not changed yet between 1989 and 1991. The share of young mothers, that is to say of mothers under 20 years of age and between 20 and 25 is always higher in the four selected regions of the new federal states than in all regions of the old federal states. This statement is true with one exception, the towns of the Ruhr area in which, there are an above average number of young mothers. The age of mothers at the birth of their children still differs considerably between the old and the new federal states. This magnitude of difference would perhaps be expected of two different countries rather within a country. Just as the proportion of young mothers in the new federal state exceeds their proportion in the old federal states, the share of mothers between 25 and 35 is greater in the old federal states than that of the mothers of this age group in the new federal states. One may deduct the hypothesis that the decline in the birth rate in the new federal states is possibly not to be attributed to the mothers between 20 and 25 years or younger already adapting to the conditions in the West and postponing the birth of their children to a later point in time, but rather that they do without a second and third child due to the insecure situation in the new federal states. This first child is still bom at a relatively young age of the mother.

MVBR-LAND' SASTH-LAND' NORDNBLSTADT' ÜDNBLGRSTADT' RUHRSTÄDTE' NORDKLAND SÜDPLAND' SÜDSTDT" W/SÜDDIENST' NORDDIENST' NORDPLAND' SÜDLAND' UNI' REICHEVORORTE' 100 2 0 0 300 4 0 0 500 6 0 0 7 0 0 800 900 1000 Cases of thousands

Figure 2: Age of Mothers

at Birth of Children

in Thousands of Live

Births

13

Regional Diversity and Familial Forms of Life

This corresponds with the results of the demographic development as it was to be observed also after the first and the second world wars. In view of the deep re-structuring and the inherent personal insecurity the decision is not to do without children at all but that small families are favored to larger ones. The often discussed phenomenon of the 'old mothers', that is to say mothers of more than 40 years of age, does not play a decisive part in any of the regions investigated in the old federal states, as the proportion is lower than 20 mothers per 1,000 live births. Neither are there any great regional variations with regard to mothers between 35 and 40 years of age in the old federal states. This may be due to different reasons. It may not be excluded that with older mothers in the rural areas these are women who give birth to their second or third child, whereas in university towns and service centers those mothers may dominate who give birth to their first child. The clear structural pattern evident in the population structure of the 0 to 24 year-olds, as well as the age group of the mothers, is continued in the number of marriages and divorces. First of all there is a marked difference between the old and the new federal states. The number of marriages of 30 per 10,000 of the population is extremely low in the rural areas of the new federal states, just as the divorce rate in the new federal states is well below that of the old federal states. This, too, is to be seen as an expression of the historic re-structuring under way in the new federal states. NORDDIENST W/SÜDDIENST RUHRSTÄDTE SÜDSTDT REICHEVORORTE UNI NORDPLAND SÜDPLAND NORDKLAND SÜDLAND NORDNBLSTADT SÜDNBLGRSTADT MVBR-LAND SASTH-LAND

Cases in tens of thousands f ^ Marriages/10000

Q

Divorces/10000

Figure 3: Marriages and Divorces

In the old federal states the number of marriages per 10,000 of the population varies only slightly with regard to the regional perspective. There are, however, clear differences in the number of divorces. The divorce rate is much lower in the rural Catholic and the rural Protestant regions of South Germany as well as

14

Hans Bertram

the North German Catholic and also Protestant regions than in the service centers of North and South Germany. With 24 divorces per 10,000 of the population the North German service centers exceed the South German Catholic-rural regions, with 15 divorces per 10,000 of the population, by nine. This differentiation makes it clear that in regard to the stability of a marriage or the risk of it failing, there are quite considerable variations in the Federal Republic, similar to the population structure of the young population, which is mainly due to the respective cultural context in which the individuals live. Unfortunately, we were not able to differentiate regionally the everyday life conditions of families on the basis of official data. The only indicator included in the analysis possibly reflecting everyday family life, in order to have at least one criterion, is the proportion of 4-room apartments in the respective region. There is a clear three-stage differentiation in the regional analysis and that is between the towns, the rural areas of the new and the rural areas of the old federal states. In the rural areas of the old federal states, in which the largest families live, we find the highest proportion of 4-room apartments. This is partly due merely to the fact that the share of owner-occupied property is especially high in these regions and families with children have invested in property to a great extent. It is astonishing that in the urban regions, both in the old and the new federal states, there is little difference with regard to the proportion of these apartments, even though the share of larger families in the new federal states is considerably higher also in the towns than in the old federal states.

4 The Social Infrastructure Comparing the proportion of crèche, kindergarten and after school center capacities with regard to the respective age groups between the regions, there is again a clear difference between the old and the new federal states. Specifically, in the new federal states the share of crèche and kindergarten capacities, relating to the children up to an age of six, is more than 80 per cent in all selected regions. In contrast, in the old federal states the share of crèche and kindergarten capacities does not exceed the 40 per cent limit in any region. This is due mainly to the fact that in the old federal states investments were made only in the kindergartens, whereas crèche capacities are more or less non-existent, so that the age group of 0 to 6 year-olds have fewer capacities than in the new federal states. In the old federal states, the offer in this field averages only half of the figure of that in the new federal states. In addition, the differences with regard to afterschool centers are even more striking as such capacities for 6 to 9 year-olds only exist to any significant extent in the service centers of South and North Germany. In all the rural regions, but also in the Ruhr area, there are more or

15

Regional Diversity and Familial Forms of Life

less no after-school centers, so that a comparison with the new federal states is not possible. These marked differences with regard to the offer of an infrastructure for childcare is not mirrored in the field of support for maintaining family life. Related to the 0 to 7 year-old children per 10,000 of the population as well as the 21 to 24 year-old young adults per 10,000 of the population, the greatest proportion of children and young adults who rely on support for maintaining their lives are to be found in the service centers of North Germany. The South and West German service centers also reach a high proportion among the 0 to 7 year-olds, but their share among the 21 to 24 year-olds is much lower, which partly reflects the different job market situation in these regions. One has to say, however, that the regional variation in the means to support the 0 to 7 year-olds in maintaining their lives is extremely high, with DM 1,700 per 10,000 of the population in the North German service centers, it is nearly four times as high as in the South German Catholic-rural regions. At the same time the fact remains that in the regions with the lowest share in 0 to 7 year olds, the highest percentage of this age group receives support for maintaining their lives. RUHRSTADTE SÜDNBLGR NORDNBLSTADT SASTH-LAND MVBR-LAND NORDKLAND UNI SÜDPLAND SÜDSTDT W/SÜ SÜDLAND REICHEVORORTE NORDPLAND NORDDIENST

Crèche+KG/K

with

R =

diag

- R (0 )

diag

- R (I)

diag

- R (I )' -

C/(0)

Cf(\y~

/

A

Cf(\)

C/(0)

A

I

diag -Q

0

(6)

R(oj

0

• ' I

Cf(Q)

A'

0" I

The left-hand side of Equation 6 consists of (2p,2p)- and (2r,2r)-dimensional patterned matrices called block-Toeplitz matrices (where the blocks are, respectively, (p,p)and (r,r)-dimensional matrices). Notice also that cov[e(t),e(t)']=diag-Q, but that cov[e(t-l),e(t-l)']=Cf(0) for the reasons given above. Together, Equations 5 and 6 define a LISREL representation of the statespace model given by Equation 2 (and 3). What remains is the choice of the dimension of the latent factor series f(t), that is the value of r. The LISREL code for r = 5 is given in Appendix C.

3.4 Procedure of the Analysis and Results The analysis of the 28-variate series of each subject proceeds as follows. Firstly, the block-Toeplitz matrix of the manifest y(t) series is determined (see left-hand side of the first matrix equation of Equation 6). The source code of a Fortran program for the estimation of this block-Toeplitz matrix is given in Appendix B. Next the LISREL representation of the state-space model is repeatedly fitted to this block-Toeplitz matrix, starting with a model in which the dimension r of the latent factor series f(t) is r = 1, then a model in which r = 2 , etc. For each repeated model fit (each value of r) Akaike's Information Criterion (AIC) is determined and the state-space model having the minimum AIC value

300

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

is selected. The AIC is defined as the value of the chi-squared goodness-of-fit statistic corrected for the degrees of freedom of this statistic: AIC = chisquared goodness-of-fit — 2df. If a low chi-squared goodness-of-fit value is obtained, but at the expense of many free parameters in the fitted model, then the degrees of freedom associated with this chi-squared goodness-of-fit value will be relatively small and hence the subtracted term 2df will be relatively small also. The chi-squared goodness-of-fit value of a model with few free parameters will necessarily be larger, but the associated degrees of freedom also will be larger, leading to a relatively large correction by 2df. Akaike has shown that a plot of the AIC against degrees of freedom of the chi-squared goodness-of-fit will yield a unique minimum. The model associated with this minimum AIC then is the selected model. The AIC for each LISREL representation of a state-space model has been computed as follows. Remember that the dimension of the manifest y(t) series is ρ=28, while n = 2 . The block-Toeplitz matrix expressing the covariance structure of y(t) is a patterned matrix composed of Cy(0) and Cy(l), each of which are (28,28)-dimensional matrices. But Cy(0) is a symmetrical matrix and hence has only 29*28/2 = 406 free elements. In contrast, C y (l) has 28*28 = 784 free elements. Consequently, the block-Toeplitz matrix concerned has in total 406+784 = 1190 free elements. If the dimension of the latent factor series f(t) is r (where r can be 1,2,...) then the number of free factor loadings in Η (see Equation 2) is p+(p-l) + . . . + ( p - r + l ) . That is, the first column of Η has all ρ entries free; the second column of Η has all ρ entries free save for one; and so on till the r-th column of Η which has all entries free save for r-1 entries. The reason why the second column of Η has to have one entry fixed, etc., will be explained below. For f(t) r-dimensional there are r free autoregressive coefficients akk, k = l , 2 , . . . , r in Equation 3. Also there are r free variances of ek(t) in diag-Q. Finally, the covariance structure of the residual series v(t) includes diag-R(O) and diag-R(l), two (p,p)-dimensional diagonal matrices having a total of 2p free elements. Based on these specifications, and substituting ρ = 2 8 and the value of r under consideration, the correction term 2df in the AIC has the following values for r = 1,2, ..., 5: r r r r r

= = = = =

1: 2: 3: 4: 5:

2df 2df 2df 2df 2df

= = = = =

2(1190 2(1190 2(1190 2(1190 2(1190

-

86) = 2208 115) = 2150 143) = 2094 170) = 2020 196) = 1988

Using the correction terms given above in combination with the values of the chi-squared goodness-of-fit statistics yields the AIC of each model for each subject. All models have been fitted to the data according to the unweighted

Dynamic Factor Analysis of Emotional Dispositions

301

least squares (ULS) technique. The main reason for choosing ULS instead of the maximum likelihood (ML) technique is because we expect that the condition of the block Toeplitz matrix associated with the manifest series y(t) may be problematic. Remember that this block Toeplitz matrix has dimension (56,56) and is estimated from 80 serially correlated observations, hence its smallest eigenvalues may be close to zero. The ML method in LISREL minimizes a criterion that depends, among other things, upon the inverse of the product of the eigenvalues concerned and the latter inverse product may become very large in case the smallest eigenvalues are close to zero. If this happens the ML method can yield biased estimates of standard errors and chi-squared goodness-of-fit statistics. In contrast, the ULS method minimizes a criterion that does not depend upon the condition of the input block-Toeplitz matrix and therefore does not present these problems (although it is less efficient than the ML method). The following AIC values were thus obtained for subject 3 and for subject 8:

r= 1 r=2 r=3 r=4 r=5

AIC Subject 3 AIC Subject 8 - 931.44 -1890.81 -1472.27 -1919.74 -1582.36 -1931.65 -1593.61 -1895.75 -1603.71 -1858.37

It is seen that the selected state-space model for subject 3 has r = 5 and for subject 8 r = 3 . Hence the behavior of the dynamic system underlying the manifest emotional disposition scores of subject 3 is characterized by a 5-variate latent factor series, while this system for subject 8 is characterized by a 3-variate latent factor series. It is noted that the minimum AIC value for subject 3 is obtained with the state-space model having the largest value of r in the set of fitted models. We therefore cannot be certain whether this is indeed the global minimum AIC value for this subject. Perhaps the global minimum AIC occurs at r > 5 , but the fit of a state-space model with r = 6 yields numerical problems. Hence we tentatively have to settle for the state-space model with r = 5 for this subject and proceed with a discussion of the selected model for each subject. Subject 3. The estimated matrix of factor loadings H is given in Appendix A. In trying to interpret the obtained dynamic factor solution, the following heuristic guideline will be used: we look for items that have a substantial positive or negative loading on a dynamic factor and that do not have such substantial loadings on other dynamic factors. It then is seen that the first dynamic factor, fl(t), has distinct high loadings on 'excited' (.85) and 'interest' (.58). We therefore will interpret this dynamic factor as Excitement. The second dynamic factor, f2(t), has distinct high loadings on 'ashamed' (.39) and 'irrit' (.38). We therefore interpret this dynamic factor as Shame. The third dynamic factor,

302

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

f3(t), has high loadings on 'loved' (.78), 'liking' (.63), 'close' (.57) and 'content' (.54); it therefore is interpreted as Love. The fourth dynamic factor has distinct substantial loadings on 'deter' (.44) and 'satis' (-.38). Hence f4(t) is interpreted as Determination. Finally, the fifth dynamic factor fs(t) has a distinct substantial loading on 'enthus' (.51) and is interpreted as Enthusiasm. It is noted that our interpretations of the dynamic factors should be regarded as conjectures that can be replaced by other ones that the reade considers to be more appropriate. It is also noted that some dynamic factors can be assigned a more clear-cut interpretation than others, although this is of course a subjective assessment. We consider fi(t) (Excitement), f3(t) (Love) and f4(t) (Determination) more easily interpretable dynamic factors than f2(t) (Shame) and f5(t) (Enthusiasm). For instance the high positive loading of fi(t) on 'active' (.37) is consistent with its interpretation as Excitement. In a similar vein, the relatively high loading of f3(t) on 'accept' (.42) is in our view consistent with its interpretation as Love. And the high positive loading of f4(t) on 'active' (.53) appears to be consistent with its interpretation as Determination. But we find the high negative loading of f5(t) on 'loved' (-.43) difficult to reconcile with its interpretation as Enthusiasm. In particular f2(t) would seem to be difficult to interpret because of the many relatively substantial loadings on various items that do not seem to present an entirely consistent pattern. For instance, f2(t) has high negative loadings on 'excited' (-.50) and 'active' (-.43), while it has high positive loadings on 'accept' (.44) and on 'attent' (.30). Perhaps a more encompassing interpretation of f2(t) would be Retraction. Finally we note in passing that four of the five dynamic factors have substantial loadings on 'active': fi(t) (.37), f 2 (t) (-.43), f4(t) (.53) and f 5 (t) (.39). We now turn to a consideration of the time-dependent properties of the five dynamic factors, where from now on f2(t) will be interpreted as Retraction. According to Equation 3 the five dynamic factor series constitute mutually uncorrected first-order autoregressions. The following estimated autoregressive coefficients ak, k = l, ..., 5, have been obtained (t-values in parentheses): fl(t+l) = f2(t+l) = f3(t+l) = f4(t+l) = f5(t+l) =

-.42 fi(t) + e i ( t + l ) (4.30) .99 f2(t) + e2(t+l) (9.15) ,82f3(t) + e3(t+l) (6.98) .68 f4(t) + e4(t+l) (3.49) -.40 f5(t) + e5(t+l) (2.65)

Dynamic Factor Analysis of Emotional Dispositions

303

It is seen that a relatively high value of Excitement at a particular day tends to be followed by a decreased value of Excitement at the next day, and vice versa, because a l l is - 42. Another way to express this finding is that the autocorrelation of fi(t) at time-lag 1 is -.42. Enthusiasm shows a similar dynamic pattern: a55 is -.40, inducing an autocorrelation of -.40 at time-lag 1 for f5(t). Consequently, both Excitement and Enthusiasm are characterized by high-frequency diachronic oscillations. In contrast, f2(t), f3(t) and f4(t) have positive autoregressive coefficients: a22=-99, a33 = .82 and a44=.68. Hence a relatively high value of Retraction, Love or Determination at a particular day is followed by a similar value at the next day, and vice versa. The autocorrelations of Retraction, Love and Determination at time-lag 1 are, respectively, .99, .82 and .68, and are indicative of stable time-dependent behavior that is characterized by slow, low-frequency oscillations. In particular the time-dependent variation of Retraction is very small, thus inducing a strong diachronic structure in the manifest series. We do not present the detailed results concerning the residual series vk(t), k = l , 2 , ..., 28. Suffice it to say that the degree to which the time-dependent variation of the manifest y(t) series is explained by the common dynamic factors is in general much higher than the degree to which this time-dependent variation in y(t) is explained by the residual series. Moreover, almost all elements in diag-R(l) are nonsignificant, implying that the residual series vk(t) do not account for most of the diachronic structure in y(t). Subject 8. The estimated matrix of factor loadings is presented in Appendix A. It is seen that the first dynamic factor fi(t) has high loadings on 'interest' (.42), 'satis' (.34), 'accept' (.34), 'excited' (.33) and 'deter' (.30). A tentative interpretation of this dynamic factor may be in terms of Positive Interest. The second dynamic factor, f2(t), has high loadings on 'attent' (.43) and 'active' (.41) and is interpreted as Attention. The third dynamic factor, f3(t), has substantial loadings on 'liking' (.44), 'loved' (.38) and 'upset' (-.31); it is interpreted as Love. We note in passing that all three dynamic factors have substantial loadings on 'accept': .34 for Positive Interest, -.21 for Attention and .26 for Love. The time-dependent properties of the three dynamic factors are give by three mutually uncorrelated first-order autoregressions (see Equation 3). The estimated autoregressive coefficients and t-values are fl(t+l) = f2(t+l) = f3(t+l) = (7.14)

.83 fi(t) + e i ( t + l ) (4.42) .67f2(t) + e 2 ( t + l ) (3.95) .89 f3(t) + e 3 ( t + l )

304

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

It is seen that a high value of Positive Interest, Attention or Love tends to be followed by an increased value on the next day, and vice versa, because the estimated autoregressive coefficients have relatively large positive values: ai 1 = .83, a22=-67 and a33 = .89, respectively. Hence the three dynamic factors show rather stable behavior characterized by slow, low-frequency oscillations. We do not present the detailed results concerning the 28-variate residual series v(t). Suffice it to say that the degree to which the time-dependent variation of the manifest y(t) series is explained by the common dynamic factors is in general much higher than the degree to which this time-dependent variation in y(t) is explained by the residual series. Moreover, almost all elements in diagR(l) are nonsignificant, implying that the residual series v(t) does not account for most of the diachronic structure in y(t).

4 Discussion and Conclusion The results of our dynamic factor analyses of the 28-variate emotional disposition series of subjects 3 and 8 show that the latent dynamic systems underlying the observed series differ considerably between the two subjects. This latent dynamic system is 5-dimensional for subject 3, whereas it is 3-dimensional for subject 8. Moreover, there appears to be very little commonality in the interpretation of the dynamic factors for the two subjects. Perhaps the dynamic Love factors come closest to having the same interpretation, although the detailed pattern of loadings still differs considerably between the subjects. It therefore would not have made much sense to pool the data for the two subjects, as is the common procedure in standard multivariate analysis. We then would have ended up with a factor solution that probably does not pertain to any of the two subjects concerned. Any comparison of factor solutions should take into account the rotation problem. A given factor solution may not be unique, but belong to an infinite set of equivalent solutions, the members of which can be obtained through rotation of the factors in the given solution. If this is indeed the case, that is, if a factor solution is not unique, one cannot decide whether such a solution is equal or not to another given factor solution. It then is required that the two given factor solutions are rotated so that their mutual agreement is maximized. Such a factor rotation is called a Procrustes rotation. Only after carrying out a Procrustes rotation can one decide confidently whether or not two given factor solutions, one or both of which being only unique up to rotation, are equal or not. This point raises two questions concerning the dynamic factor analyses of the data of subjects 3 and 8: a) How is factor rotation defined in the dynamic factor model, and b) are the obtained dynamic factor solutions only unique up to rotation?

Dynamic Factor Analysis of Emotional Dispositions

305

To answer the question how factor rotation is defined in the dynamic factor model, we will restrict attention to the state-space model used in the analyses of subjects 3 and 8. Factor rotation in the general dynamic factor model will be dealt with elsewhere. For the state-space model given by Equation 2 factor rotation can be carried out directly on the matrix of factor loadings H. For instance, H can be subjected to varimax rotation, equimax rotation, or any other type of rotation that is deemed relevant. If the rotated matrix of factor loadings is denoted by H", then the effect of the rotation can be described as H* = HT, where Τ is an (r,r)-dimensional rotation matrix. Denoting the inverse of Τ by T"l, it then follows that the matrix of autoregressive coefficients A in Equation 2 is changed by the rotation in the following way: A* = T'^AT, where A* is the rotated matrix of autoregressive coefficients associated with the latent factor series. This implies that factor rotation in the state-space model used to analyze the data of subjects 3 and 8 will change the time-dependent structure of the r dynamic factors fk(t), k = l , 2 , ..., r, comprising the behavior of the latent dynamic system underlying intraindividual variation in emotional dispositions. More specifically, the lack of cross-correlation between the fk(t), k = l , ..., r, as specified by Equation 3, can be destroyed by factor rotation. That is, after factor rotation (whether orthogonal or oblique rotation) the r latent factor series can be mutually dependent. It should be reiterated that the above description of factor rotation in the statespace model only applies to the analyses of subjects 3 and 8 reported in this chapter. For more general versions of the state-space model and for the even more general dynamic factor model the situation is much more complex. But based on the description given here and using the detailed information about the estimated loadings presented in an appendix as well as the estimated autoregressive coefficients given in the previous section, the reader can carry out factor rotations of the obtained results. Regarding the second question concerning the uniqueness of the dynamic factor solutions presented in the previous section, we already provided the answer. These solutions can indeed be rotated and therefore are only unique up to factor rotation. Yet the state-space model was reformulated as a covariancestructure model and then fitted to the data by means of LISREL. And LISREL does not allow factor models in which the factors are unique up to rotation. How then can this be reconciled with our remark that the dynamic factors in the state-space model are unique up to rotation and that therefore the obtained loadings in Η can be rotated at will? The answer is that we have arbitrarily imposed minimal constraints on the loading matrix Η so that a unique solution is guaranteed and the LISREL program can be used. This is accomplished by fixing the first entry in the second column of Η at zero, fixing the first and second entry in the third column of Η at zero, etc. These fixed entries do not affect the goodness-of-fit of the dynamic factor solution, but only serve to arrive at a fully

306

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

identified (and hence unique) dynamic factor model that can be fitted to the data by means of LISREL. But having available the LISREL solution thus obtained we can drop these arbitrary constraints on the loadings in H and subject H to factor rotation. We can conceive of the constraints on H in order to arrive at a fully identified dynamic factor model as the choice of one solution from among the infinite set of equivalent rotated solutions. Because these constraints have been chosen in exactly the same way for subjects 3 and 8, the solutions obtained for both subjects are directly comparable. But if each of these solutions would subsequently be subjected to varimax rotation, say, then the rotated solutions thus obtained are no longer directly comparable. We reiterate that the analyses presented in this chapter address the sources of variation that appear to be the most fundamental in understanding the causes of human behavior. The synchronic and diachronic structure of intraindividual variation is directly linked up with the mechanisms that generate this variation and that were interpreted as latent dynamic systems. Only in special circumstances can interindividual variation provide the same information. Whether these special circumstances indeed occur has to be checked in each application. At least in the present study it turns out that emotional dispositions show patterns of synchronic and diachronic variation that do not warrant the pooling of data across many subjects. This important result has been established by means of an objective analytic technique that meets the standards of proper scientific research. Moreover, this technique leaves intact the individuality of each adolescent participating in the study. The finding of idiosyncratic structures of intraindividual covariation in emotional dispositions corroborates the general point of view concerning the foundational character of single-subject research (cf. section 2). Stated somewhat otherwise, it is clear that the time domain of generalization of emotional dispositions of subjects 3 and 8 (i.e., the population of future and previous time points at which these dispositions are defined) is structured in ways that differ from the structure of interindividual differences in the same emotional dispositions in the relevant population of subjects. In fact, the state-space models characterizing the emotional dispositions of subjects 3 and 8 do not correspond to the factor structure of the emotional disposition scale as assessed in a standard analysis of interindividual variation. This has important implications for application-oriented situations. For instance, if one is interested in predicting emotional dispositions of a particular subject (e.g., in counseling or selection), then results obtained in a standard factor analysis of interindividual variation do not apply! In reverse, if one is interested in assessing the relative stance of a particular subject with respect to other subjects in the relevant population, then one should only use the results obtained in a standard factor analysis of interindividual variation. In sum, for measurements of emotional dispositions obtained

Dynamic Factor Analysis of Emotional Dispositions

307

by the scale used in the present study, generalization to the population of time points occurs according to structural models that are entirely different from those used in generalization to the population of subjects. Both types of population are important in applied settings, dependent upon the details of the particular research questions under scrutiny. Yet it still holds that any determination of the presence of such structural differences between the two domains of generalization can only be carried out by means of single-subject research, which in this sense is more basic than analysis of interindividual variation. The latter conclusion also would seem to have important implications for the overarching topic of this book. The standard approach in comparing the behavior of subjects from the Eastern and Western parts of Germany after unification is based on an analysis of interindividual variation within and between groups of subjects. It is an open question, however, whether significant differences thus identified between subjects from East- and West Germany also constitute valid descriptions of the individual developmental processes of each of the subjects involved. The latter question can only be assessed in appropriate analyses of intraindividual variation, for instance along the lines as sketched in the present chapter. Similar remarks can be made with respect to the assessment of effects of interventions. It may be found that a particular intervention induces a significant change in the mean and/or covariance structure characterizing the interindividual variation within and between subjects from East- and West Germany. But again, it is an open question whether this particular intervention will induce the same kind of changes in the mean and/or covariance structure characterizing each individual subject. The latter question can only be assessed in an appropriate analysis of the effects of the intervention on intraindividual variation (such as is common practice in applied optimal control in the engineering sciences; cf. Goodwin & Sin, 1984). The choice of the proper approach therefore depends upon the intended domain of generalization (either prediction to a population of time points or else generalization across a population of subjects) in combination with the (non-)ergodicity of the processes under consideration. In closing, we would like to draw attention to the fact that the status of singlesubject designs has recently become more important because of the revolutionary developments in applied nonlinear dynamics. Applications of, for instance, chaos theory, have to be based on the analysis of intraindividual variation. In our own work we have concentrated on the use of nonlinear dynamical models explaining stagewise development (e.g., van der Maas & Molenaar, 1992). To apply these models to real time series data, we again employ suitably generalized versions of the dynamic factor model (Molenaar & Hartelman, 1996; Molenaar, 1994). Accordingly, the dynamic factor model considered in this chapter can be conceived of as belonging to a hierarchy of similar models that cover a wide range of increasingly complex developmental processes. Together

308

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

these models constitute powerful analytic tools to understand and guide development at the level of the individual subject.

Appendix A Estimated matrix H of dynamic factor loadings for subject 3 (t-values in parentheses)

interest distres liking upset strong excited hum il scared hostile enthus irrit discour accept ashamed inspired notwant loved nervous

fl(t) 0.58 (9.74) -0.19 (-3.58) 0.27 (4.41) -0.22 (-3.98) 0.05 (0.93) 0.85 (10.05) -0 .22 (-3.87) 0.15 (2.69) -0.12 (-2.18) 0.33 (4.25) -0.22 (-3.77) 0.01 (0.17) 0 .14 (2.23) -0.30 (-4.63) -0.11 (-2.00) -0.23 (-4.00) 0.29 (3.86) -0.15 (-2.62)

f2 (t)

f3 (t)

f4 (t)

f5 (t) -

0 .23 (5.28) -0.39 (-3.51) 0.19 (3.45) 0.01 (0.33) -0.50 (-7.61) 0.20 (4.32) 0.16 (3.76) 0.18 (4.31) -0.38 (-6.72) 0.38 (8.20) 0.35 (7.27) 0 .44 (5.19) 0.39 (8.05) 0.18 (4.21) 0.28 (6.10) 0.32 (2.45) 0.21 (4.92)

-

-

0.63 (5.93) 0.00 (0.06) -0.03 (-0.69) 0.10 (0.83) 0 . 04 (0.61) -0 . 04 (-0.63) 0.03 (0.53) 0.07 (0.70) 0.07 (0.85) 0 . 14 (1.77) 0.42 (4.94) -0 . 08 (-0.93) 0.04 (0.62) -0.07 (-0.99) 0.78 (9.81) -0.02 (-0.29)

0.21 (2.41) 0.02 (0.28) -0 .12 (-0.95) 0.09 (1.06) 0.01 (0.14) -0.03 (-0.42) -0 .14 (-1.32) 0.03 (0.27) 0.01 (0.12) -0.09 (-0.53) -0 . 04 (-0.38) 0.04 (0.54) -0.04 (-0.43) 0.01 (0.04) -0.01 (-0.13)

0 . 05 (0.61) 0.31 (2.45) 0 . 13 (1.53) -0 . 12 (-1.54) 0.15 (1.97) 0.51 (4.80) 0 . 17 (2.01) 0.00 (0.03) -0.17 (-2.06) 0.33 (3.47) 0.10 (1.31) 0.20 (2.34) -0.43 (-4.07) 0 .17 (2.06)

309

Dynamic Factor Analysis of Emotional Dispositions

Estimated matrix H of dynamic factor loadings for subject 3 (t-values in parentheses) f5(t) fl(t) f3 (t) f4(t) f2(t) 0.17 deter 0.44 0.23 0.14 -0.11 (1.93) (3.94) (-1.22) (3.76) (1.52) 0.12 alert 0 .14 0.22 -0.03 0.09 (1.50) (-0.36) (2.49) (4.84) (1.40) 0.00 0.12 attent 0.05 0.30 0.15 (0.00) (1.13) (0.98) (6.02) (1.95) 0 . 01 close 0.57 0.29 0.39 -0.13 (0.13) (2.67) (6.53) (8.14) (-1.36) 0 . 09 satis 0 .24 0.22 -0 .38 0.13 (1.10) (-3.19) (4.16) (1.38) (3.08) proud 0 . 01 0.06 -0.06 0.13 0.03 (0.17) (1.07) (-1.07) (1.81) (0.66) -0 . 04 j ittery -0.10 -0.06 0.01 0 .12 (-0.55) (-1.82) (-1.14) (0.15) (2.73) 0.39 active 0.37 0.53 -0.43 -0.15 (3.17) (3.76) (4.77) (-1.52) (-4.10) -0.02 afraid -0.02 -0.09 -0.04 0.15 (-0.32) (-1.76) (-0.64) (-0.30) (3.46) content 0.00 -0.02 0.54 0.09 0.28 (0.03) (-0.29) (0.63) (3.06) (7.43)

Estimated matrix H of dynamic factor loadings for subject 8 (t-values in parentheses)

interest distress liking upset strong excited humil scared hostile

f 1 (t) 0.42 (5.51) 0.06 (0.95) 0.10 (1.14) 0.07 (0.93) 0 .17 (2.35) 0.33 (4.31) 0.04 (0.63) 0.03 (0.37) 0.07 (0.99)

f2 (t)

0.19 (2.45) 0.06 (0.39) -0.08 (-0.71) -0.20 (-2.43) -0.11 (-1.12) -0.08 (-0.90) -0.03 (-0.37) -0.12 (-1.11)

f3(t) -

-

-

-

0 .44 (6.37) -0.31 (-4.30) -0.01 (-0.11) 0.11 (1.35) -0.19 (-2.80) -0.18 (-2.85) -0.26 (-3.52)

310

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

E s t i m a t e d m a t r i x H of dynamic factor loadings for subject 8 (t-values in parentheses) f 3 (t) fl(t) f2(t) 0.09 enthus -0.11 0.11 (1.29) (-1.58) (1.39) irrit 0.02 -0.18 0.11 (0.32) (-2.59) (1.23) discour -0.25 -0.08 0.11 (-1.02) (-3.59) (1.05) accept 0.34 -0.21 0.26 (-1.79) (2.63) (3.96) ashamed -0.04 -0.19 0.06 (-0.59) (0.67) (-2.90) 0 .22 0.21 0.26 inspired (2.86) (3.18) (1.91) notwant -0 .14 -0.01 -0.25 (-1.41) (-3.46) (-0.18) loved -0 .14 0.38 0.26 (-1.64) (3.88) (1.93) nervous 0.06 0 . 00 -0 .10 (0.96) (-0.05) (-1.63) deter 0.30 -0.01 -0.08 (4.09) (-0.87) (-0.17) alert 0.19 0.25 -0.27 (-2.91) (2.06) (2.11) attent 0 .14 0.43 -0.08 (1.48) (4.72) (-0.63) close 0.04 0.04 0.26 (0.50) (0.41) (4.04) satis 0 . 34 -0.14 0.12 (4.33) (-1.45) (1.43) proud -0.02 0 .17 -0.10 (-0 .28) (-1.16) (2.60) j ittery 0.02 -0.10 -0.11 (0.28) (-1.68) (-1.28) active 0 .28 0 .41 -0.03 (3.01) (4.37) (-0.21) afraid 0.05 -0 .17 -0.20 (0.64) (-1.74) (-2.56) content -0.04 0 .23 -0.06 (-0.60) (2 .92) (-0.66)

311

Dynamic Factor Analysis of Emotional Dispositions

Appendix Β Source code of the Fortran program used to compute the block-Toeplitz matrices associated with the 28-variate time series of subject 3 and of subject 8 program Toep c c c c c c c c c c c c c c c c c c c c c

70

This program computes the input block-Toeplitz correlation matrix to be used in LISREL across an arbitrary number of subjects . It does not require external subroutines The input series are in the file called 'in.y' : for each subject, one subject after the other, there are as many lines as there are time points, each line containing the nv observations at that time point for that subject; after thus completing nt lines for the first subject the same setup is repeated for the next subject, etc. The block-Toeplitz ' sigma' .

correlation

matrix

is

in

the

file

nt = number if time points nv = number of observed component series per subject lag = number of lags (including zero lag) at which Toeplitz covariance marix is estimated nc = number of subjects amiss = missing value parameter(nt=80,nv=28,lag=2,nc=l,amiss=-999) dimension ζ(nv,nt,nc),c(nv,nv,lag),χ(ην),io(nv) open(2,file='in.y') open(3,file='sigma') ntot=0 miss=0 do 70 i=l,lag ntot=ntot+nc*(nt-i+1) continue ntot=ntot/lag do 98 i=l,nv

312

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal d o 98 j = 1 , η ν d o 98

98

k=l,lag

c(i, j,k)=0 d o 95

i=l,nc

d o 95

j=l,nt

read(2,4)(io(k),k=l,nv) 4

format(7x,28il) d o 95

k=l,nv

ζ (k, j , i ) = f l o a t ( i o ( k ) ) 95

continue d o 94

i=l,nv

d o 64

j=l,nc

q=0 n=0 d o 93

k=l,nt

if(z(i,k,j).eq.amiss)miss=miss+l if(z(i,k,j).eq.amiss)goto

93

n=n+l q=q+z(i,k,j) 93

continue if(n.ne.0)q=q/n d o 67

k=l,nt

if(z(i,k,j).eq.amiss)goto

67

z(i,k,j)=z(i,k,j)-q 67

continue

64

continue

94

continue ntot=ntot-(miss/nv) write(6,71)ntot

71

format(/,lx,'ntot=i5) d o 99

ii=l,nc

do 20

i=l,lag

do 21

j=l,nv

l=nv if(i.eq.l)l=j do 21 k = l , 1 n=0 q=0 d o 22

m=i,nt

if(z(j,m,ii).eq.amiss.or.z(k,m-i+1,ii).eq.amiss)goto n=n+l q=q+z(j,m,ii)*z(k,m-i+1,ii)

22

Dynamic Factor Analysis of Emotional Dispositions 22

continue

21 20 99

if(n.ne.0)c(j,k,i)=c(j,k,i)+q/n if(n.eq.0)write(6,*)ii,i,j,k continue continue cont inue

80

97 23

40

do 80 i = l , n v x(i)=sqrt(c(i,i,1)) do 97 i = l , l a g do 97 j = l , n v l=nv if(i.eq.l)l=j d o 97 k = l , 1 c(j,k,i)=c(j,k,i)/(x(j)*x(k) ) continue format(6(lx,f10.5)) do 40 i = l , l a g do 4 0 j = 1 , n v do 40 k = l , i m=nv if(k.eq.i)m=j w r i t e ( 3 , 2 3 ) ( c ( j , 1, i - k + 1 ) , 1 = 1 , m ) continue stop end

Appendix C LISREL code of the state-space model for subject 3 State-space model subject 3; dim. latent factorseries r=5. da no=80 ni=56 ma=cm cm sy fi=sigma mo ny=56 ne = 10 ly=fu,fi ps=di,fi be=fu,fi te=sy,fi ft ly(l,l) ly(2,l) ly(3,l) ly(4,l) ly(5,l) fir l y ( 6 , l ) l y ( 7 , l ) l y ( 8 , l ) l y ( 9 , l ) l y ( 1 0 , l ) fr l y ( l l , l ) ly(12,l) l y ( 1 3 , l ) ly(14,l) ly(15,l) fr l y ( 1 6 , l ) l y ( 1 7 , l ) l y ( 1 8 , l ) l y ( 1 9 , l ) l y ( 2 0 , l ) fr l y ( 2 1 , 1 ) l y ( 2 2 , l ) l y ( 2 3 , l ) l y ( 2 4 , l ) l y ( 2 5 , l ) ft ly(26,l) ly(27,l) ly(28,l) ft ly(2,2) ly(3,2) ly(4,2) ly(5,2) ft ly(6,2) ly(7,2) ly(8,2) ly(9,2) ly(10,2)

314

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

fr ly(ll,2) ly(12,2) ly(13,2) ly(14,2) ly(15,2) fr ly(16,2) ly(17,2) ly(18,2) ly(19,2) ly(20,2) fr ly(21,2) ly(22,2) ly(23,2) ly(24,2) ly(25,2) fr ly(26,2) ly(27,2) ly(28,2) fr ly(3,3) ly(4,3) ly(5,3) fr ly(6,3) ly(7,3) ly(8,3) ly(9,3) ly(10,3) fr ly(ll,3) ly(12,3) ly(13,3) ly(14,3) ly(15,3) fr ly(16,3) ly(17,3) ly(18,3) ly(19,3) ly(20,3) fr ly(21,3) ly(22,3) ly(23,3) ly(24,3) ly(25,3) fr ly(26,3) ly(27,3) ly(28,3) fr ly(4,4) ly(5,4) ly(6,4) ly(7,4) ly(8,4) ly(9,4) ly(10,4) fr ly(ll,4) ly(12,4) ly(13,4) ly(14,4) ly(15,4) ly(16,4) fr ly(17,4) ly(18,4) ly(19,4) ly(20,4) ly(21,4) ly(22,4) fr ly(23,4) ly(24,4) ly(25,4) ly(26,4) ly(27,4) ly(28,4) ft ly(5,5) ly(6,5) ly(7,5) ly(8,5) ly(9,5) ly(10,5) fr ly(ll,5) ly(12,5) ly(13,5) ly(14,5) ly(15,5) ly(16,5) fr ly(17,5) ly(18,5) ly(19,5) ly(20,5) ly(21,5) ly(22,5) fr ly(23,5) ly(24,5) ly(25,5) ly(26,5) ly(27,5) ly(28,5) eq ly(l,l)ly(29,6) eq ly(2,l) ly(30,6) eqly(3,l)ly(31,6) eq ly(4,l) ly(32,6) eq ly(5,l) ly(33,6) eq ly(6,l) ly(34,6) eq ly(7,l) ly(35,6) eq ly(8,l) ly(36,6) eq ly(9,l) ly(37,6) eq ly(10,l) ly(38,6) eq l y ( l l , l ) ly(39,6) eq ly(12,l) ly(40,6) eq ly(13,l) ly(41,6) eq ly(14,l) ly(42,6) eq ly(15,l)ly(43,6) eq ly(16,l) ly(44,6) eq ly(17,l) ly(45,6) eq ly(18,l) ly(46,6) eq ly(19,l)ly(47,6) eq ly(20,l) ly(48,6) eq ly(21,l) ly(49,6) eq ly(22,l) ly(50,6) eq ly(23,l) ly(51,6) eq ly(24,l) ly(52,6) eq ly(25,l) ly(53,6) eq ly(26,l) ly(54,6) eq ly(27,l) ly(55,6) eq ly(28,l) ly(56,6) eq ly(2,2) ly(30,7) eq ly(3,2) ly(31,7)

Dynamic Factor Analysis of Emotional Dispositions eq ly(4,2) ly(32,7) eq ly(5,2) ly(33,7) eq ly(6,2) ly(34,7) eq ly(7,2) ly(35,7) eq ly(8,2) ly(36,7) eq ly(9,2)ly(37,7) eq ly(10,2) ly(38,7) eq ly(l 1,2) ly(39,7) eq ly(12,2) ly(40,7) eq ly(13,2) ly(41,7) eq ly(14,2) ly(42,7) eq ly(15,2) ly(43,7) eq ly(16,2) ly(44,7) eq ly(17,2) ly(45,7) eq ly(18,2) ly(46,7) eq ly(19,2) ly(47,7) eq ly(20,2) ly(48,7) eq ly(21,2) ly(49,7) eq ly(22,2) ly(50,7) eq ly(23,2) ly(51,7) eq ly(24,2) ly(52,7) eq ly(25,2) ly(53,7) eq ly(26,2) ly(54,7) eq ly(27,2) ly(55,7) eq ly(28,2) ly(56,7) eq ly(3,3) ly(31,8) eq ly(4,3) ly(32,8) eq ly(5,3) ly(33,8) eq ly(6,3) ly(34,8) eq ly(7,3) ly(35,8) eq ly(8,3) ly(36,8) eq ly(9,3) ly(37,8) eq ly(10,3) ly(38,8) eq ly( 11,3) ly(39,8) eq ly(12,3) ly(40,8) eq ly(13,3) ly(41,8) eq ly(14,3) ly(42,8) eq ly(15,3)ly(43,8) eq ly(16,3) ly(44,8) eq ly(17,3) ly(45,8) eq ly(18,3) ly(46,8) eq ly(19,3)ly(47,8) eq ly(20,3) ly(48,8) eq ly(21,3) ly(49,8) eq ly(22,3) ly(50,8) eq ly(23,3) ly(51,8) eq ly(24,3) ly(52,8) eq ly(25,3) ly(53,8)

316 eq ly(26,3) ly(54,8) eq ly(27,3) ly(55,8) eq ly(28,3) ly(56,8) eq ly(4,4) ly(32,9) eq ly(5,4) ly(33,9) eq ly(6,4) ly(34,9) eq ly(7,4) ly(35,9) eq ly(8,4) ly(36,9) eq ly(9,4) ly(37,9) eq ly(10,4) ly(38,9) eq ly(l 1,4) ly(39,9) eq ly(12,4) ly(40,9) eq ly(13,4) ly(41,9) eq ly(14,4) ly(42,9) eq ly(15,4) ly(43,9) eq ly(16,4) ly(44,9) eq ly(17,4) ly(45,9) eq ly(18,4) ly(46,9) eq ly(19,4) ly(47,9) eq ly(20,4) ly(48,9) eq ly(21,4) ly(49,9) eq ly(22,4) ly(50,9) eq ly(23,4) ly(51,9) eq ly(24,4) ly(52,9) eq ly(25,4) ly(53,9) eq ly(26,4) ly(54,9) eq ly(27,4) ly(55,9) eq ly(28,4) ly(56,9) eq ly(5,5) Iy(33,10) eq ly(6,5) ly(34,10) eq ly(7,5) ly(35,10) eq ly(8,5) ly(36,10) eq ly(9,5) ly(37,10) eq ly(10,5) ly(38,10) eq ly(ll,5) ly(39,10) eq ly(12,5) ly(40,10) eq ly(13,5) ly(41,10) eq ly(14,5) ly(42,10) eq ly(15,5) ly(43,10) eq ly(16,5) ly(44,10) eq ly(17,5) ly(45,10) eq ly(18,5) ly(46,10) eq ly(19,5) ly(47,10) eq ly(20,5) ly(48,10) eq ly(21,5) ly(49,10) eq ly(22,5) ly(50,10) eq ly(23,5) ly(51,10) eq ly(24,5) ly(52,10)

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

Dynamic Factor Analysis of Emotional Dispositions eq ly(25,5) ly(53,10) eq ly(26,5) ly(54,10) eq ly(27,5) ly(55,10) eq ly(28,5) ly(56,10) fr te(l,l) te(2,2) te(3,3) te(4,4) te(5,5) fr te(6,6) te(7,7) te(8,8) te(9,9) te(10,10) fr te( 11,11) te(12,12) te(13,13) te(14,14) te(15,15) fr te(16,16) te(17,17) te(18,18) te(19,19) te(20,20) fr te(21,21) te(22,22) te(23,23) te(24,24) te(25,25) fr te(26,26) te(27,27) te(28,28) fr te(29,l) te(30,2) te(31,3) te(32,4) te(33,5) fr te(34,6) te(35,7) te(36,8) te(37,9) te(38,10) fr te(39,ll) te(40,12) te(41,13) te(42,14) te(43,15) fr te(44,15) te(45,17) te(46,18) te(47,19) te(48,20) fr te(49,21) te(50,22) te(51,23) te(52,24) te(53,25) fr te(54,26) te(55,27) te(56,28) eq te( 1,1) te(29,29) eq te(2,2) te(30,30) eq te(3,3) te(31,31) eq te(4,4) te(32,32) eq te(5,5) te(33,33) eq te(6,6) te(34,34) eq te(7,7) te(35,35) eq te(8,8) te(36,36) eq te(9,9) te(37,37) eq te(10,10) te(38,38) eq te( 11,11) te(39,39) eq te(12,12) te(40,40) eq te(13,13) te(41,41) eq te(14,14) te(42,42) eq te(15,15) te(43,43) eq te(16,16) te(44,44) eq te(17,17)te(45,45) eq te(18,18) te(46,46) eq te(19,19) te(47,47) eq te(20,20) te(48,48) eq te(21,21) te(49,49) eq te(22,22) te(50,50) eq te(23,23) te(51,51) eq te(24,24) te(52,52) eq te(25,25) te(53,53) eq te(26,26) te(54,54) eq te(27,27) te(55,55) eq te(28,28) te(56,56) fr be(l,6) be(2,7) be(3,8) be(4,9) be(5,10) frps(l)ps(2) ps(3) ps(4) ps(5) va 1.0 ps(6) ps(7) ps(8) ps(9) ps(10) st .1 all

318

P. C. M. Molenaar, M. J. Rovine, and S. E. Corneal

st 1 te( 1,1) te(2,2) te(3,3) te(4,4) te(5,5) st 1 te(6,6) te(7,7) te(9,9) te(10,10) st 1 te( 11,11) te(12,12) te(13,13) te(14,14) te(15,15) st 1 te(16,16) te(17,17) te(18,18) te(19,19) te(20,20) st 1 te(21,21) te(22,22) te(23,23) te(24,24) te(25,25) st 1 te(26,26) te(27,27) te(28,28) ou ns xm ad=off it=999 ul

References Goodwin, G.C., & Sin, K.S. (1984). Adaptive filtering, prediction and control. Englewood Cliffs, NJ: Prentice-Hall. Jones, C. (1978). Multivariate autoregression estimation using residuals. In: D.F. Findley (Ed.), Applied time series analysis. New York: Academic Press, 139-162. Molenaar, P.C.M. (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika, 50, 181-202. Molenaar, P.C.M. (1994). Dynamic latent variable models in developmental psychology. In: A. von Eye & C.C. Clogg (Eds.), Latent variable analysis: Applications for developmental research (pp. 155 - 180). Thousand Oaks: Sage Publications. Molenaar, P.C.M., & Hartelman, P.A.I. (1996). Catastrophe theory of stage transitions in metrical and discrete stochastic systems. In: A. von Eye & C.C. Clogg (Eds.), Categorical variables in developmental research: Methods of analysis (pp. 107 - 130). San Diego: Academic Press. Rovine, M.J., Molenaar, P.C.M., & Corneal, S.E. (this volume). Analysis of emotional response patterns for adolescent stepsons using P-technique factor analysis. Van der Maas, H.L.J., & Molenaar, P.C.M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395-417.

Event History Analysis in Human Developmental Research Kazuo Yamaguchi and. Lei Jin

1 Introduction While event history analysis has been widely employed in certain areas of social science research — such as social demography, job mobility and labor-market behavior, social epidemiology, and population-ecological studies of organizations — it has been rarely used in human development research. Recent anthologies of longitudinal research methods (e.g., von Eye, 1990a,b; Magnusson et al. 1991), which cover methods for human development research, each include a chapter on event history analysis, but the applications are limited to analyses of job mobility and social mobility. This relative paucity of applications of event history analysis to human development research partly reflects the fact that event history analysis typically relies on data about exact timing of transitions between discrete states for a variable of interest, while (1) many major variables of interest in human development research, such as variables that characterize physical, cognitive, and emotional growth, are considered as continuous-state variables rather than discretestate variables, and (2) although some variables of interests may be regarded as having discrete states, the exact timing of transitions between those states is often difficult to measure. Regarding characteristic (1), a few points should be noted. First, there are some variables of interest in human development research that can be specified as the event in event history analysis. Silbereisen et al. (1995), for example, list the first occurrence of the following transitions or behavioral experiences as events of interest: (1) falling in love, (2) going steady with a romantic friend, (3) having a sexual experience, (4) making a decision about appearance, (5) attending a discotheque, (6) participating in a political debate, and (7) having a concrete occupational plan. In the study of children's acquisition of physical and cognitive abilities, we may study the first occurrence of such behaviors as creeping, crawling, standing without support, uttering a word, and speaking a complete sentence. In the study of drug abuse among adolescents, initiations and relapses of alcohol drinking, cigarette smoking, and use of marijuana or other illicit drugs have been analyzed (e.g., Yamaguchi and Kandel, 1984). In the

320

Kazuo Yamaguchi and Lei Jin

study of crime and delinquency, first commitment of theft, personal attack, and vandalism can be studied by using event history analysis. Not only can events such as the initiation and relapse of drug use or crime be studied using event history analysis, but the timing of reaching a state of recurrence, such as becoming a weekly marijuana user or a heavy drinker, becoming a frequent disco goer, having a partner in regular sexual activity, etc., can be analyzed separately from the initial or experimental occurrence of a particular behavior. In demography, in addition to marriage, marital childbirth, and divorce, such events as puberty, premarital pregnancy, premarital childbirth, premarital cohabitation, and menopause have also been analyzed using event history analysis (e.g., Hogan and Kitagawa, 1985; Yamaguchi and Kandel, 1985). In the study of schooling, dropping out of school or resuming schooling after leaving school can be analyzed (Ferguson, 1991; Yamaguchi, 1991). These events are often interdependent, such that becoming pregnant promotes dropping out of school, failure to continue schooling promotes the chances of deviant behavior, and experiencing deviant behavior promotes the chances of premarital pregnancy and childbirth. Hence, causal mechanisms — especially whether multiple life events had common causes or whether the occurrence of each event affected the occurrence of another event in the observed sequence of events — are an important topic in life course study of transitions from childhood, through adolescence, to young adulthood. Second, the fact that certain variables have a continuous scale does not always imply that treating states of such variables as continuous produces the most fruitful results. The use of linear models for continuous-state variables with correlation in measuring their association fails to allow for the possibility that determinants of downward transition in the state of a variable differ from those of upward transition, which is often the case. Yamaguchi (1996), for example, emphasizes in his analysis of dynamic change in personal efficacy that determinants of the vulnerability of psychological well-being (defined as the increase in the rate of downgrading the discrete level of personal efficacy) differ from determinants of the suppression of psychological well-being (defined as the decrease in the rate of upgrading the level of personal efficacy). If cognitive and emotional growth are nonmonotonic and subject to downward as well as upward "mobility," the use of discrete states and the distinction between upgrading and downgrading their levels will be informative. Event history analysis, or panel data analysis of discrete states with duration dependence, may be employed to analyze determinants of change in such discretized psychological variables over time (Yamaguchi, 1991). The second issue, that transition between states is often latent and measurement of its exact timing is difficult, raises a different point. Suppose, however, that although the exact timing of a particular event — such as becoming a drug addict, becoming HIV positive, acquiring a specific symptom of mental disor-

Event History Analysis in Human Developmental Research

321

der, etc. — is difficult to measure, we know that we can obtain data about whether the event has occurred or not by the time of the survey. Then, we can apply hazard rate models for current status data. These models can handle not only right censoring (i.e., possible event occurrence in the future) but also left censoring (i.e., event occurrence in the past without any knowledge of timing). We describe such models later. Several significant advantages are gained by using event history analysis in human development research. First, time-varying predictors of human development can be taken into account — as long as we have data on the timing of changes in their states. In addition to time-varying characteristics of individuals such as age, schooling, having a steady boy/girl friend, etc., we can include in the analysis such time-varying family-contextual and social-contextual variables as family composition, parents' employment statuses, regional youth crime rate, and the distinction between historical time after versus before the unification in Germany. Certain time variables, such as age at unification and region of residence at unification, are time-constant once their values are realized but are still time-dependent covariates when the unification occurred during the time at risk for an event of interest among subjects. Another advantage of event history analysis will be especially important for human development research. Such research is intrinsically interested in intraindividual variations, and inter-individual differences are of interest inasmuch as they reflect differences in the pattern of intra-individual variations. However, many analyses of survey data, including those of longitudinal data, typically rely on inter-individual differences in estimating intra-individual variations. Thus, unobserved population heterogeneity tends to be confounded with intra-individual variations when the characteristics of human development are analyzed. In event history analysis, this issue is related to (1) modeling unobserved population heterogeneity and (2) distinguishing between causation and selection regarding the effects of covariates, or the issue of controlling for selection bias. We describe related methods and models later. Below, we first review (1) basic event history models and (2) selected advanced models that are particularly relevant in human development research. Then, we illustrate an application of two groups of basic models to an analysis of predictors of age at first sexual intercourse, with a detailed description of the analytical procedure, including the use of software.

322

Kazuo Yamaguchi and Lei Jin

2 Models and Methods: A Review 2.1 Basic Hazard Rate Models — 1: Continuous Time Models Event history analysis is based on models of hazard rates. The hazard rate (or hazard function), h(t), expresses the instantaneous risk of having the event at time t, given that the event did not occur before time t. It is the derivative of the conditional probability of having the event at time t, given the condition that the event did not occur before time t. Let Γ be a random variable for duration of the risk period for an event. Then the hazard rate h{t) is given as

Δ/—>oo

At

S(t)

where P(t+At>T>_t\T>j) indicates the probability that the event occurs during the time [i, i+Är] given that the event did not occur prior to time t, fit) is the probability density function, and S(t) is the survivor function, which indicates the probability of not having the event prior to time t. S(t) is given as S(t) = P(T>t) = cxp[-[h(u)du The expression of h(t) as a function of covariates differs among models. The most frequently used group of models are proportional hazards models and their extensions with time-dependent covariates such that: h{t) = h0(t)e\p(b'\(t)) ,

(3)

where h0(t) is the baseline hazard function that corresponds to the state where χ = 0, χ and b are covariate and parameter vectors, respectively, and covariate states may depend on time. If there are no time-dependent covariates, we obtain a proportional hazards model where the rate of event occurrence for subjects with χ is exp(b'x) times the rate for subjects with the baseline state (x = 0) regardless of time t. If the baseline hazard function h^(t) is specified parametrically, we can employ maximum likelihood estimation based on the full likelihood, L = A¡ h(t)mS¿t), where ä(i) = 1 when the event occurs for the inh observation and ä(;) = 0 if the /th observation is censored. One such subgroup of proportional hazards models includes piecewise constant rate models, which assume that the baseline hazard function as well as values of time-dependent covariates change

Event History Analysis in Human Developmental Research

323

only across time intervals for a given set of time intervals and do not change within each time interval, such as within each age or duration interval. Alternatively, we can leave the baseline hazard function hQ(t) unspecified for proportional hazards models and their extensions and estimate parameters based on the Cox method (Cox, 1975). The Cox method, based on maximization of the partial likelihood (PL) function, takes into account only information about who among subjects at risk has the event, i.e., information about relative risks among subjects, at each time a subject in the sample had the event. In other words, the PL function uses only information about the rank order of the timing of the event among subjects in the sample, but not information about their exact timing. The PL function is given as hk(tk)

PL-Π Li

Z W jeS(k)

=Π ü

exp(b xk(lk))

'

Σ

(4) */(>*))

jeS(k)

where tk is the event time for the person among Κ subjects who had the event. S(k) is the set of subjects whose event time or censoring time ts satisfies ts >_ tk; therefore, the denominator of Equation (4) is the sum of the hazard rates of all subjects who are at risk at tk, including the person who had the event at tk. The PL function does not depend on the baseline hazard function h0(t). When we assume that the unspecified baseline function also depends on a time-constant categorical variable, called the stratifying variable, we have a stratified proportional hazards model. The estimation of parameters for a stratified model is based on maximization of the product of stratum-specific PL functions. Although a stratified model can be used for controlling possible interaction effects of a stratifying variable and time if the effect of this variable is of little interest, it can also be used for controlling unobserved population heterogeneity, as described later. Another group of continuous-time hazard rate models includes accelerated failure- time models (Kalbfleisch and Prentice, 1980). Although such models are used far less often than proportional hazards models in social science research, their use in human development research has potential. Generally, the model assumes that hazards of, say, one year in time for someone in the baseline group, for whom χ = 0, imply hazards of exp(-b'x) years for someone with covariate states χ, and thereby the survivor probability of not having the event for a person with states χ is given as S(i|x) = 50(/e"b x), where S0(t) is the survivor function that corresponds to the baseline hazard function. In other words, the model assumes that some people "mature," or experience life, more rapidly than others, with the speed of maturation proportional to exp(-b'x). All accelerated failure-time regression models can be expressed by regressions on log(7) such that:

324

Kazuo Yamaguchi and Lei Jin

log(7) = b'x + â,

(5)

where Τ is the random variable for duration of time and â is the random error term whose distribution varies from model to model. Hence, for accelerated failure-time models, expected durations of not experiencing the event, rather than hazard rates, are assumed to be proportional among people with different values of x. A major limitation of accelerated failure-time models is that only time-constant covariates can be employed. On the other hand, the advantages of the models include (1) the fact that they can be nicely extended into mover-stayer models with a pair of regressions, one for predictors of event timing and the other for predictors of probability of ultimate event occurrence (Schmidt and Witte, 1989; Yamaguchi, 1992), and (2) the fact that they provide relatively simple nonlinear regression forms for analysis of current status data (Vanderhoeft, 1982). The latter characteristic is further described later.

2.2 Basic Hazard-Rate Models — 2: Discrete-Time Models When the measurement of time in a survey is crude — for example, coded in ages instead of years and months — then we may employ a discrete-time model by treating the set of equally spaced time intervals as a set of discrete time points. Suppose Γ is a discrete random variable that indicates the time of an event. If Τ = t, the event occurs at time t. Suppose that the probability of having an event at each discrete time point in the population is given by fit) such that/fa) = Ρ(Γ = ι,), where th i = 1,2,...,/, indicates the irh discrete time point and satisfies t{ < t2 t{) = O^fiti). The hazard at í¡ is defined as the conditional probability of having the event at t¡ given that the event did not occur prior to time f, such that ë, = P(T=tì\T>tù

=fitòWd·

(6)

Then, we also obtain i-l (7)

Any parametric specification of conditional probabilities for ëj, j = 1,2, ... becomes a discrete-time hazard-rate model.

Event History Analysis in Human Developmental Research

325

A discrete-time logit model assumes that the odds of having the event for a person with states χ at each discrete-time t„ i = 1,2, ..., /, are exp(b'x) times the odds of having the event for a person with the baseline state (x = 0) such that

•;*,·) Ι-λο (ti)

'*