Analyzing Collapsed Contingency Tables Without Actually Collapsing
证明，对完整列联表拟合对数线性模型并排除折叠变量的某些参数，可以得到与折叠表相同的结果，尤其适用于分类变量的递归系统分析。
For a variety of reasons, log-linear models are sometimes fit to contingency tables which sum over the categories of one or more variables in the original table. I show here that the same results can be readily obtained by fitting a log-linear model to the full table, without actually collapsing it. All that is necessary is to exclude certain parameters pertaining to the collapsed variable. This approach is particularly useful for the analysis of recursive systems of categorical variables. There are several situations in which it is desirable to take a contingency table with a given number of cells and collapse it into a smaller contingency table by summing over some or all of the categories of one or more variables. It may be that some of the categories are thought to be theoretically or practically equivalent. Or perhaps one of the variables is considered to be irrelevant to the analysis at hand. Most importantly, collapsing is an essential feature of Goodman's (1973) method for the log-linear analysis of recursive systems of categorical variables. For example, suppose we have four categorical variables-A, B, C, and D-and we assume that A is causally prior to B, C, and D; B is causally prior to C and D; and C is causally prior to D. Goodman proposes that one first analyze the fourway table, with D taken as the dependent variable. Then one collapses over D to obtain a three-way table, with C taken as the dependent variable. Finally, one collapses over C to study the effect of A on B. There are disagreements as to the appropriateness of this method. Reynolds (1977) is troubled by the fact that significant associations can appear in the collapsed table even when there are no such associations in the full table, and suggests that this may produce misleading results. However, Gillespie (1978) and Goodman (1979) have countered that in a recursive system it is the collapsed table that more accurately describes the true causal structure. Thus, when looking at the effects of A and B on C, it is misleading to further cross-classify on a subsequent dependent variable D. Let me emphasize at the outset that I completely agree with Goodman and Gillespie. There is simply no way to represent a recursive causal system by a single log-linear model for the full multiway table. The method of successive collapsings, on the other hand, seems entirely appropriate for such recursive systems. Nevertheless, collapsing can be te