Table Of ContentAcademic Press Rapid Manuscript Reproduction
Stochastic Modelling
of Social Processes
Edited by
Andreas Diekmann
Peter Mitter
Institute for Advanced Studies
Vienna, Austria
1984
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Main entry under title:
Stochastic modelling of social processes.
Includes index.
1. Social sciences-Mathematical models-Addresses,
essays, lectures. 2. Stochastic processes-Addresses,
essays, lectures. I. Diekmann, Andreas. II. Mitter,
Peter.
H61.25.S76 1984 300\724 84-6315
ISBN 0-12-215490-8 (alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
84 85 86 87 9 8 7 6 5 4 3 2 1
Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Gerhard Arminger (245), Department of Economics, Universität Gesamt-
hochschule Wuppertal, D-5600 Wuppertal, Federal Republic of Germany
James S. Coleman (189), Department of Sociology, The University of Chi-
cago, Chicago, Illinois 60637
Andreas Diekmann (123), Institute for Advanced Studies, A-1060 Vienna,
Austria
Michael T. Hannan (39), Department of Sociology, Stanford University,
Stanford, California 94305
Marcus Hudec (283), Institut für Statistik und Informatik, Universität Wien,
A-1010 Vienna, Austria
Kenneth C. Land (215), Population Research Center and Department of
Sociology, The University of Texas at Austin, Austin, Texas 78712
Peter Mitter (123), Institute for Advanced Studies, A-1060 Vienna, Austria
Anatol Rapoport1 (7), Institute for Advanced Studies, A-1060 Vienna, Austria
Aage B. S0rensen (89), Department of Sociology, University of Wisconson-
Madison, Madison, Wisconsin 53706
Gilg U. H. Seeber (155), Institut für Statistik, Universität Innsbruck, A-6020
Innsbruck, Austria
Present address: 38 Wychwood Park, Toronto, Ontario M6G 2V5, Canada.
Vll
Preface
There has been in recent years a rapid growth in the field of stochastic
modelling and its applications in the social sciences. Progress stems from such
diverse disciplines as mathematical statistics, demographics and actuarial
methods, medical statistics and biometrics, as well as from applied research by
economists, sociologists, and psychologists. The stochastic modelling ap-
proach is therefore highly interdisciplinary in its nature.
New developments in statistics, computer software, and mathematical
modelling allow for more realistic applications in the social sciences today than
in the days of early social mobility research in the fifties. Those early models
were based on the simple homogeneous Markov model with discrete time.
Extensions of the basic Markov model refer to the use of continuous time
scales, the relaxation of the assumption of time independence, the allowance
for heterogeneity by introduction of covariates, and the extension to multistate
models. Powerful statistical estimation methods and the availability of modern
computer facilities open the route to empirical estimation of model param-
eters. In addition, techniques of survival analysis provide for robust nonpara-
metric estimation procedures particularly useful for exploratory data analysis.
This volume demonstrates that stochastic models can fulfill the goals of
explanation and prediction. Furthermore, their practical value for social sci-
entists is that they, combined with statistical estimation techniques, are ex-
tremely useful tools for analyzing career data, waiting times, time intervals
between events, event-history data, etc. There are numerous examples of
potential applications: unemployment episodes, length of marriage, time
interval data in psychological experiments, survival times of organizations,
occupational careers, recidivism intervals, and time spans of membership in
groups or political parties. Its instrumental approach to analyzing time-related
data will be especially helpful in increasing the popularity of stochastic models
in empirical social research.
All chapters in this volume are original contributions and are written pri-
marily by statisticians and sociologists. They document progress in statistical
methods and modelling as well as progress in concrete applications. The con-
tributions result mainly from a series of lectures held by guest professors and
staff members at the Institute for Advanced Studies in Vienna. These lectures
IX
X PREFACE
were organized in an academic, scientifically stimulating atmosphere, during
the period when Anatol Rapoport was director of the Institute. The editors
owe a considerable debt to this great man of science.
We take pleasure in thanking Mrs. Beatrix Krones for performing the diffi-
cult task of typewriting, and Mr. Robert Davidson and Dr. Eckehart Köhler
for proofreading parts of the manuscript.
ANDREAS DIEKMANN
PETER MITTER
INTRODUCTION
Very often progress in knowledge is rooted in solutions to
apparently unconnected problems. Practical problems of gambling
in the upper classes inspired the growth of the mathematical dis-
cipline of probability theory and the theory of stochastic pro-
cesses. For example the French Chevalier de Mere posed the follow-
ing problem to the philosopher and mathematician Pascal. Two
players agree to play a game with several rounds. The first winner
of a certain number of rounds, say one-hundred, takes the pot, but
if the game is aborted before the end, what is a fair division of
the pot between the players? How should they divide the sum if one
player wins 90 and the other player 80 rounds? In a letter to Fer-
mât dated July 1654 Pascal proposed dividing the sum according to
the principle of the expected values of the players, nowadays a
key concept in the theory of stochastic processes. The name of
another central term, the hazard rate, also has its origin in
games of chance. The Arabian word "asard" originally denoted a
1
certain "difficult" combination of eyes (3 or 18 with three dice) .
An important problem was tackled by Daniel Bernoulli about one
hundred years later. He focused on the problem of the change in
mortality rates if a certain disease (smallpox) was abolished. In
modern terms, he presented a "competing risk model" in his famous
lecture before the French Academy in 1760. Today such models are
See Gnedenko (1968, pp. 358-372) for a condensed survey on
the history of probability theory.
STOCHASTIC MODELLING Copyright © 1984 by Academic Press, Inc.
OF SOCIAL PROCESSES 1 All rights of reproduction in any form reserved.
ISBN 0-12-215490-8
Ί INTRODUCTION
applied in social sciences in modelling occupational movements to
different destination states. The basic principles of competing
risk theory and its applications to occupational careers are de-
scribed in the article of Gilg Seeber.
Three main aspects underline the fruitfulness of the stochas-
tic approach in the social sciences. First, the models explicitly
treat dynamic processes in contrast to static models. Secondly,
most "laws" or regularities in social sciences are not determinis-
tic but probabilistic. Therefore, stochastic models seem to be
very appropriate for social science problems. The third aspect is
related to measurement. Time scales or absolute number of event
scales as commonly used in stochastic model building cause less
difficulties in interpretation and assumptions concerning scale
properties than more arbitrary psychological and sociological
scales based on items analysis. Anatol Rapoport's contribution
deals among other things with this latter aspect.
In outlining the "philosophy" of stochastic modelling and
describing its usefulness for the social sciences Rapoport is
also concerned with an interesting idea: the entropy interpreta-
tion of stochastic processes. This approach regards steady-state
probability distributions as the result of maximizing entropy
under constraints. The nature of the constraints-hypothesis de-
termines the form of the distribution as illustrated by Zipf's
law of rank-size distributions.
The simple homogeneous Markov chain in discrete time and the
Markov process in continuous time together with the Poisson dis-
tribution of the number of events and its twin brother, the ex-
ponential distribution of waiting times, serve as the basic models
in applied stochastic theory. These models are points of depar-
3
INTRODUCTION
ture. Much of the progress in stochastic model building consists
in the construction of more elaborate models which allow the re-
laxation of one or the other assumption of the basic model.
Michael T. Hannan's article outlines some of the recent deve-
lopments. Realistic models must take the factum of population
heterogeneity into account. Following the tradition of Coleman
(1964, 1981) and Tuma et al. (1979) heterogeneity can be control-
led by incorporating independent variables or covariates in rate
equations. This line of reasoning leads to stochastic causal
models. While Coleman (1964, 1981) is primarily concerned with
panel data, parameters of stochastic causal models can best be
estimated by more informative event history data. Generalizations
toward multistate models also developed in demography, and allow-
ance for unobserved heterogeneity make the models even more real-
istic. Problems of multistate models are illustrated by their
applications to the analysis of marital stability and employment
status as well as to migration.
Another point of departure from the basic model is the in-
corporation of duration-time effects in transition rate equations,
In early social mobility research in the fifties,the Cornell mobi-
lity model suggested the axiom of "cumulative inertia", i.e. the
conditional probability of a change to another occupational state
decreases with time spent in the present state. Today there are
a variety of hazard models capturing time effects of different
functional type.
Aage B. S0rensen, analyzing job careers with event-history
data, utilizes two parametric models with duration-time dependence,
the Gcmpertz and the log-logistic models. Effects of covariates are
also considered. A strength of S0rensen's research is the explicit
connection of labor market theories, vacancy competition models
4 INTRODUCTION
and other theories of attainment processes like human capital
formulations to transition rate equations.
In the article by Diekmann and Mitter five time-dependent ha-
zard rates are compared with a new non-monotonous "sickle model".
The six models represent rival hypotheses that are confronted in
an empirical test using marriage cohort data. It is assumed
that the time path of the risk of divorce, i.e. the hazard rate,
follows a non-monotonous sickle-type pattern. The more general
theory of semi-markov processes, accounting for time spent in a
state effects, as well as estimation techniques for such models,
are outlined in Gilg Seeber's paper mentioned above.
Obviously the domain of stochastic model construction is the
causal analysis of covariate effects and the identification of
duration time effects on hazard or transition rates determining
the process. However, James S. Coleman demonstrates the applica-
bility of stochastic modelling also in the context of purposive
actor theory, based on the principle of utility maximization. In
his article, Coleman develops a stochastic model of exchange re-
lations in perfect and imperfect markets. Transition rates for
the exchange of goods are regarded as functions of prices, wealth
of the exchange partner, and interest of the actor in the respec-
tive good. By modifications of the perfect market model Coleman
arrives at a "matching market model" illustrated with Swedish
marriage data. Coleman1s model and the entropy model described in
Rapoport's article have one thing in common. They both use a max-
imization principle: maximization of entropy in the latter and
maximization of utility in the former case.
Like Hannan, Kenneth C. Land is concerned with the recent
innovations of multistate demographics and its relations to the
sociological research tradition. Hcwever, in contrast to other
authors in this volume, Land focuses on the analysis of aggre-
