Graphical and structural equation models
Invited talk by Nanny Wermuth, Department of Biostatistics, Chalmers University, Sweden.
Chair: Roger Millsap, Tuesday 21st July, 12.20 -13.05, Uppercroft, School of Pythagoras.
Graphical and structural equation models (SEM) can both be viewed as extensions of path analysis, proposed by geneticist S. Wright some 80 years ago. Wright used directed acyclic graphs to describe how data could have been generated and checked whether the implications of corresponding linear models were compatible with what he observed in his data. Econometrician T. Haavelmo recognized, in Nobel prize winning work of 1943, that separate linear least squares regressions cannot always be appropriate when more than one response is affected
by changes in a set of explanatory variables. Consequently, SEM's were developed and introduced to sociology and psychometrics about 50 years ago, notably by O.D. Duncan, K. Joreskog and A. Goldberger.
Different generalizations of the linear regression framework were developed separately: log-linear and logit regression models for categorical data (Y. Bishop; L. Goodman); survival analysis (D.R. Cox); and seemingly unrelated regressions (SUR, A. Zellner), a class of linear regression models not covered by the theory of general linear models. Later, A.P. Dempster developed the theory of Gaussian covariance selection. These models as well as a subclass of log-linear models define the same types of independence structure for undirected graphs in which each missing edge corresponds in exponential families to certain canonical parameters taking value zero.
Graphical Markov models today combine both directed and undirected edges of two different types and thereby model sequences of joint and single responses of arbitrary kind. But, it is only with very recent work by M. Drton, K. Sadeghi, G. Marchetti and M. Luparelli, M. Wiedenbeck, D.R. Cox and myself that a general framework has become available to study in depth similarities and differences between SEM's and multivariate regression chains, which include SUR models as special cases. In this talk, we describe results of this type for symmetric binary variables and for Gaussian distributions.