Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES

Wickens (1989) gives a good treatment of multiway contingency tables. His basic chapters require study, but seem essential for anyone who wishes to do serious work with frequency data. Usefully different perspectives appear in Fleiss (1981) and in Agresti (1996), who give many illustrative sets of data as well as helpful exercises.

10.1.1a
The polio cases in Table 10.1 include both paralytic and nonparalytic cases listed in Meier's (1989) Table 1. The frequencies of paralytic cases were 33 and 115 in the Vaccine and Placebo groups, respectively.
10.1.4a
Homogeneity and Independence. Contingency tables are mainly of two kinds. In one kind, one variable (row or column) represents different groups of subjects, as with the polio vaccine experiment, and the other variable represents the subject classification. The null hypothesis says that the proportion of each class of response is the same for each group—that the groups are homogeneous.

In the other kind of contingency table, there is only a single group. Row and column variables represent two characteristics of the same subjects, as with the height-happiness example of Table 10.3. The null hypothesis says that the two subject characteristics are uncorrelated, or independent.

For two-way contingency tables, fortunately, the X2 test is exactly the same for both homogeneity and independence. With more than two variables, however, homogeneity and independence may yield different E values and hence different values of X2 for the same numerical data. This is one complication that arises with more than two variables.

10.1.4b
The chi-square distribution was introduced in 1900 by Karl Pearson, the major figure in the early development of statistics. Pearson, however, thought that the df for a contingency table equaled rows × columns 1. In Pearson's view, a 2 × 2 table would thus have 3 df, not 1. When Fisher solved this statistical problem, obtaining Equation 2, Pearson gave him a hostile reception, quoted by Agresti (1996, pp. 251 f):

I hold such a view [Fisher's] is entirely erroneous.… pardon me for comparing him with Don Quixote tilting at the windmill; he must either destroy himself, or the whole theory of probable errors.

To which Fisher replied (rather mildly for Fisher; see Fisher, 1956, pp. 2–3):

If peevish intolerance of free opinion in others is a sign of senility, it is one which he [Pearson] has developed at an early age.

Pearson's substantial contributions to the early development of statistics would shine more brightly today had he been more tolerant, for Fisher was mercilessly correct. Agresti concludes his historical survey of chi-square (p. 265) by saying:

And so, it is fitting that we end this brief survey by giving yet further credit to R. A. Fisher for his influence on the practice of modern statistical science.

10.1.5a
The theoretical chi-square distribution 2) is almost as important as the normal distribution in statistical theory. In fact, χ2 is distributed as the square of a unit normal variable. The χ2 distribution thus appears in Anova: Under the null hypothesis, numerator and denominator of the F ratio have (independent) χ2 distributions.

-300-

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