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This paper is concerned with the description of a specialized form of linear regression analysis commonly known as "dummy variable" regression analysis. Efforts are made to show the relationship between "dummy variable" regression analysis and other multivariate analysis techniques with a specific emphasis on the kinds of problems and data that sociologists deal with where "dummy variable" regression analysis is particularly appropriate and useful. Efforts are also made (1) to give illustrations and examples of problems to which this type of multiple-regression analysis might be applied productively; (2) to show how "dummy variable" regression analysis is both similar to and different from other multivariate techniques in terms of the analytical procedures and the kinds of interpretations that are made possible.

One of the most common procedures in school effects studies is multivariate regression analysis employed in conjunction with an input-output model. In the usual, two-step regression method, student input is controlled by first computing from the input variables an "expected" output for each school. The difference between the observed and the "expected" output is then used as the dependent variable, and the school environment variables as the independent variabls, to determine the influence of the school environment. A heuristically superior alternative to the two-step method is the procedure recommended by McNemar (1962, p. 176) in which both student input and college environment variables enter into a single regression equation. The various components of the predictable variance then can be calculated from equation (i) n n-I n

This article is concerned with the specification and estimation of relationships whose dependent variable is qualitative in nature (such as "yes" or "no"). It discusses logit equations with and without interaction, and the estimation procedure is generalized least squares. Part I deals with dependent variables that take only two values, Par II with variables taking more than two values, and part III describes informational measures for the explanatory power of the determining factors. The discussion of more advanced technical matters is contained in various appendixes.

1. Summary. A "mixed model" is proposed in which the problem of the appropriate assumptions to make about the joint distribution of the random main effects and interactions is solved by letting this joint distribution follow from more basic and "natural" assumptions about the cell means. The expectations of the mean squares ordinarily calculated turn out, with suitable definition of the variance components, to have the same values as those usually found in more restrictive models, and some of the customary tests and confidence intervals are justified, but some aspects appear to be novel. For example, the over-all test found for the fixed main effects and the associated multiplecomparison method require Hotelling's T2. 2. Introduction. We consider K replications of a two-way layout with I rows and J columns (I > 1, J > 1, K _ 1), the rows corresponding to levels of a "Model I" [4] factor A, whose effects we wish to regard as fixed effects, and the columns corresponding to the levels of a "Model II" factor B, whose effects we wish to regard as random effects. We let yijk denote the kth measurement in

For selecting and interpreting appropriate measures of association, a "proportional-reduction-in-error" (P-R-E) criterion is useful. However, efforts to give a P-R-E interpretation to measures of ordinal association have not been successful, especially in delineating the "form" or "shape" of ordinal association. An effort is therefore made in this paper to introduce the notion of relevant forms of ordinal association, such as strict monotonic, monotonic, and nonmonotonic associations, and to suggest a few P-R-E measures that would assess such particular forms of association in ordinal data. When we talk about a measure of association, we mean an index designed to measure the degree to which two or more variables "go together" in some sense. In the loosest sense, any two variables may be said to be associated as long as the two are not statistically independent. In a stricter sense, however, two variables are said to be associated only if they show some patterned covariation. The pattern of our interest may be linear or curvilinear, monotonic or nonmonotonic, to name a few. Although lack of independence can be indicated without any reference to a specific form of relationship, the degree of association can be indicated only with reference to a specific pattern of relationship. Costner (1965) pointed out the importance of making the implied form of association explicit when he advocated the adoption of the "proportional-reduction-in-error" (P-R-E) criterion in selecting appropriate measures of association for sociological research. The P-R-E interpretation, as he cogently argued, will give us not only an exact probabilistic statemenlt about the degree of ani association but also a clear indication of the reference pattern of association under consideration. For instance, it is clear from the P-R-E interpretation of Pearsonian r that it measures the linear relationship, and that the value of r2 will reach unity (maximum) if and only if one variable is a linear function of the other (see Kruskal 1958; Hays 1963).

This paper investigates the errors of prediction of multiple regression equations, in which the regressor variables are regarded as drawn at random from a multivariate normal population. In most theoretical treatments of multiple regression it is assumed that one or more "dependent" variables are to be predicted, given the observed values of a number of "independent" variables. The independent or regressor variables are treated as constants. However, in many applications, it is more reasonable to regard them as random variables, drawn, for example, from a multivariate normal population. In this case the term "independent" variables is confusing, and it seems better to call them regressor variables. It has been suggested by Ehrenberg (1963) that regression or stochastic regressor variables is useless. This paper shows that although such regression has it limitations, especially if the number of regressor variables is large, it may be useful if the limitations are understood. The particular case to be discussed is that in which the regressor variables are drawn from a multivariate normal population. The k-dimensional vectors X1 , X2, ... X *... X, are given as a random

To illustrate the models and methods of the present article, we shall reanalyze those data in the famous study of The American Soldier by Stouffer et al. (1949), subsequently analyzed by Coleman (1964), Zeisel (1968), and Theil (1970). The methods we present reveal how the odds pertaining to a given dichotomized variable (e.g., the odds that a soldier would prefer a Northern to a Southern Camp assignment) are related to other dichotomized variables (e.g., (a) the soldier's race, (b) his region of origin, (c) his present camp location). The usual regression analysis methods do not suit the case considered here, where the dependent variable is the odds pertaining to a given dichotomous variable. Nor do the usual methods suit the case where the dependent variable is a proportion pertaining to the dichotomous variable. This article presents some relatively elementary models and methods suitable for analyzing the odds (or a proportion) pertaining to the given dichotomous dependent variable. Applying these models and methods to the data referred to above, new insights are obtained.

In most empirical work, the investigator's understanding of the economic process under study is only minimally reflected in the econometric methodology. This paper suggests that in many cases, the construction of a small-scale simulation can "prepare" the data for regression in a manner which takes cognizance of the theory of the process. Regression is then used to scale the output of the simulation up to observed magnitudes of the variable to be predicted. The simulation has the function of exploring for the nature of the nonlinearities and interactions and thus replaces the usual search for a form which maximizes R2. The simulation may also be helpful where colinear data are a problem. An example is presented in which the effects of wages, unemployment rates, and labor turnover on poverty are studied through a "prepared" regression. IN THE LAST three decades, regression analysis has become the Procrustean bed into which all economic data are fitted. In the usual empirical paper by an economist, the obligatory "theoretical" discussion which precedes the description of the regressions generally contributes little more to the empirical methodology than an indication of which variables ought to be included in which equation, what the signs of the coefficients might be expected to be, and whether the regressions should be run in linear or logarithmic form. One reaction to this state of affairs has been the commencement of construction of large systems of microsimulation, notably one at the Urban Institute emphasizing demography and the distribution of income [6], one at the National Bureau of Economic Research on urban problems [4], and one at the University of Maryland featuring money flows [2]. While these big microsimulations are designed to describe the processes of the economy in a more natural way than can be done exclusively by usual regression methods, they tend to take years to build and tend to be unavailable to economists not involved in their building. It is possible, however, to occupy a middle ground between the regression runners and the large-model microsimulators. In many cases, improvement over the usual regression procedures can be gained by a combination of a very simple do-it-yourself simulation model with regression. The simulation model has the function of "preparing" the data for regression, in the sense of exploring for the nonlinearities and variable interactions inherent in the phenomenon under study. The regression, which uses the output of the simulation as an explanatory variable, has the function of "scaling" the simulation results up to observed magnitudes of the variable under study.