https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1977.tb01600.x

Maximum likelihood from incomplete data via the EM algorithm

Hypotheses involving mediation are common in the behavioral sciences. Mediation exists when a predictor affects a dependent variable indirectly through at least one intervening variable, or mediator. Methods to assess mediation involving multiple simultaneous mediators have received little attention in the methodological literature despite a clear need. We provide an overview of simple and multiple mediation and explore three approaches that can be used to investigate indirect processes, as well as methods for contrasting two or more mediators within a single model. We present an illustrative example, assessing and contrasting potential mediators of the relationship between the helpfulness of socialization agents and job satisfaction. We also provide SAS and SPSS macros, as well as Mplus and LISREL syntax, to facilitate the use of these methods in applications.

https://link.springer.com/content/pdf/10.3758/BRM.40.3.879.pdf

Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models

The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

Generalized Linear Models

For comments on an earlier draft of this chapter and for detailed advice I am indebted to Robert M. Hauser, Halliman H. Winsborough, and Toni Richards, several anonymous reviewers, and the editor of this volume. I also wish to thank John Raisian, Nancy Rytina, and Barbara Mann for their comments and Mark Wilson for able research assistance. The opinions expressed here are the sole responsibility of the author.

Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models

Job Demands, Job Decision Latitude, and Mental Strain: Implications for Job Redesign

"At last, after a decade of mounting interest in log-linear and related models for the analysis of discrete multivariate data, particularly in the form of multidimensional tables, we now have a comprehensive text and general reference on the subject. Even a mediocre attempt to organize the extensive and widely scattered literature on discrete multivariate analysis would be welcome; happily, this is an excellent such effort, but a group of Harvard statisticians taht has contributed much to the field. Their book ought to serve as a basic guide to the analysis of quantitative data for years to come." --James R. Beninger, Contemporary Sociology

"A welcome addition to multivariate analysis. The discussion is lucid and very leisurely, excellently illustrated with applications drawn from a wide variety of fields. A good part of the book can be understood without very specialized statistical knowledge. It is a most welcome contribution to an interesting and lively subject." --D.R. Cox, Nature

"Discrete Multivariate Analysis is an ambitious attempt to present log-linear models to a broad audience. Exposition is quite discursive, and the mathematical level, except in Chapters 12 and 14, is very elementary. To illustrate possible applications, some 60 different sets of data have been gathered together from diverse fields. To aid the reader, an index of these examples has been provided. ...the book contains a wealth of material on important topics. Its numerous examples are especially valuable." --Shelby J. Haberman, The Annals of Statistics

Discrete multivariate analysis: theory and practice

Discrete Multivariate Analysis

Three methods of fitting log-linear models to multivariate contingency-table data with one dichotomous variable are discussed. Logit analysis is commonly used when a full contingency table of s dimensions is regarded as a table of rates of dimension s -1. The split-table method treats the same data as two separate tables each of dimension s -1. We show that the full contingency-table method can be regarded as a generalized approach: models which can be fitted by it include both the mutually exclusive subsets that can be fitted by the other two methods. Even when the logit method permits the model of choice to be fitted, the full contingency-table method of iterative proportional fitting to the set of sufficient configurations has the advantage of requiring neither matrix inversion nor substitution of an arbitrary value in empty elementary cells.

Full Contingency Tables, Logits, and Split Contingency Tables

This paper examines the estimation of expected cell counts in two-dimensional contingency tables when some of the cells are known a priori to be empty. A log-linear model, corresponding to the model of independence or homogeneity for 'complete' tables, is fitted to the cell counts. The resulting cell estimates can always be computed by an iterative proportional fitting method, although some tables can be fitted directly. The direct method of fitting is applied to a set of medical data. It is pointed out that the iterative proportional fitting method can readily be extended to tables of higher dimensionality.

Incomplete two-dimensional contingency tables.

The conditions are defined under which collapsing multidimensional contingency tables, by adding over variables, will affect the apparent interaction between the remaining variables. This leads to a simple method of distinguishing those log-linear models for which the cell estimates may be obtained by direct multiplication, from those requiring iterative fitting. The implications of fitting over-parametrized models are discussed with particular reference to the 'partial association' model used implicitly (a) when information from separate two-dimensional tables is combined to test the association between the two variables, and (b) when rates are adjusted by indirect standardization.

Yvonne M. M. Bishop

Papers

Discrete multivariate analysis: theory and practice

Discrete Multivariate Analysis

Full Contingency Tables, Logits, and Split Contingency Tables

Incomplete two-dimensional contingency tables.

Effects of collapsing multidimensional contingency tables.