Marginal Model

Statistical models that impose constraints on some (or all) marginal distributions of a set of variables as well as, eventually, on their joint distribution are called marginal models. In a multinormal distribution, modeling the marginal distributions does not create special problems because the first- and second-order moments of any marginal distribution are equal to their counterparts in the joint distribution. For categorical variables, such a one-to-one correspondence between the parameters of the joint and the marginal distributions does not exist in general, and one has to resort to specially developed procedures for analyzing data by means of marginal models.

Table 1 contains data for a response variable Y that is measured at three time points.

Table 1 Observed Frequency Distribution
	Y3 = 1		Y3 = 2
	Y2 = 1	Y2 = 2	Y2 = 1 Y2 = 2
Y1 = 1	121	11	41	44
Y1 = 2	4	2	1	12
SOURCE: Data from the National Youth Survey from the Inter-University Consortium for Political and Social Research (ICPS), University of Michigan.

The data reported here were collected on 236 boys and girls who were repeatedly interviewed on their marijuana use. Here we use the data collected in 1976 (Y1), 1978 (Y2), and 1980 (Y3) with responses coded as 1 = never used marijuana before and 2 = used marijuana before.

The corresponding joint probability distribution of Y1, Y2, and Y3 will be represented by π123 (y1, y2, y3); the bivariate marginal distribution of Y1 and Y2 is then given by

and the univariate marginal distribution of Y1 by

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Other marginal distributions can be defined similarly.

The hypothesis that Y3 and Y1 are independent given Y2 imposes constraints on the joint distribution of the three variables, and these can be tested by fitting the LOG-LINEAR MODEL [Y1Y2, Y1Y3] in the original three-dimensional contingency table. If a model only imposes constraints on the joint distribution, it is not a marginal model.

However, instead of formulating hypotheses about the structure of the entire table, one could be interested in hypotheses that state that some features of the marginal distributions of the response variable do not change over time. The hypothesis of univariate marginal homogeneity states that the univariate marginal distribution of Y is the same at the three time points:

In our example, the corresponding observed univariate marginal response distributions are presented in Table 2.

	Table 2 Univariate Marginals
	Y 1	Y 2	Y 3
1	217	167	138
2	19	69	98
	236	236	236

For these data, the hypothesis that all three distributions arise from the same probability distribution has to be rejected with L2 = 82.0425 and df = 2 (p < .001). Hence, in this panel study, there is clear evidence for a significant increase in marijuana use over time. Note that this hypothesis cannot be tested by the conventional CHI-SQUARE TEST for independence because the three distributions are not based on independent samples.

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