In cleaning out my INBOX I came across this message, to which I had intended to reply some time ago. A possibly useful reference (if you have access to it), along the lines you asked about, is Goodman, L. Partitioning of chi-square, analysis of marginal contingency tables, and estimation of expected frequencies in multidimensional contingency tables. Journal of the American Statistical Association 66 (1971), 339-344.
An approach that I have sometimes found useful is the following. 1. If H0: [rows & columns are independent] has been rejected, display the standardized residuals for each cell. If these are all about the same size, then the non-independence is, so to speak, distributed more or less evenly throughout the table and it may be difficult to identify useful patterns to extract. Usually, however, there are a few cells with large residuals (and therefore supplying large contributions to the total chi-square value). If this is the case, it may be useful to remove the effects of these cells (one cell at a time, or several cells at a time, as seems convenient and/or appropriate), as follows. 2. For the cell whose effect is to be removed: substitute for the observed frequency in that cell a fictitious frequency obtained from the expected frequency. (Actually, for reasons I can adduce later but won't bother with now, it is usually efficient to overcorrect somewhat.) Using the fictitious frequency, recalculate the chi-square value, and observe the contribution to the total chi-square (or the standardized residual, if you prefer) in the "adjusted" cell. What you want to do is to adjust the fictitious frequency until that standardized residual is zero, or as close to zero as the chi-square routine will permit. (If, as in MINITAB, one is restricted to integer values in the contingency table, you won't get exactly zero, but you can get to a negligible value.) With a little practice, one can get to the optimal adjustment in three or four successive approximations. 3. With the contribution of this cell reduced to zero, or as near to it as you can get, examine the standardized residuals of the other cells (and the total chi-square value) in the adjusted table, to see if there are other potentially interesting effects to be extracted. If so, repeat the procedure for one or more additional cells. (Note that every time you've adjusted the frequency in a cell, the number of degrees of freedom is reduced by 1. Your software will not know that, so you'll have to refer the reported value of chi-square to a table after you start this process. On the other hand, any individual cell contribution can be referred to the critical value of chi-square with one d.f. (which is conservative: since you have a table with r*c cells and only (r-1)*(c-1) d.f. to start with, it can be argued that the initial critical value for one cell is for chi-square with (r-1)(c-1)/rc d.f.; tables of chi-square for fractional d.f. are not common, however). It is demonstrable that a single cell in a r-by-c table, if it departs strongly from the expected value, can make other effects in the table less easy to see. One can think of reducing the effect of the current cell to zero as rather like removing an object from the foreground of a landscape, so that objects in the middle distance or in the background can the more easily be perceived. You may find the following example interesting. Some years ago, a PhD candidate in linguistics was analyzing the frequencies with which aphasic patients mispronounced vowels in ordinary English. This more or less naturally led to a square contingency table in which the rows represented the "target" vowel (the correct sound for the word being spoken) and the columns represented the actual vowel sound produced; it was as I recall an 11-by-11 table. But of course it didn't only contain errors: its major diagonal contained frequencies with which the vowels were correctly pronounced. And as one might expect, the total frequencies on that diagonal represented rather more than 75% of the data. An ordinary chi-square analysis wuld show large positive residuals on that diagonal, and a highly significant chi-square value; while effects occurring in the errors, which were the focus of the study, were pretty well obliterated. And it wouldn't do to substitute zero frequencies on the diagonal; you'd have much the same effect in reverse. Replacing the diagonal observed frequencies with fictitious frequencies, chosen so as to make the contribution of that whole diagonal as near to zero as possible, permitted one to analyze the errors these speakers actually made: one could identify vowel sounds that more frequently tended to be substituted incorrectly as well as those that were less frequently substituted than would be expected. (And one could even interpret the fictitious frequencies, as those errors that accidentally produced the correct vowel, in a manner of speaking.) The candidate did have some physiological reasons for expecting certain kinds of errors to be made preferentially, and was able to show evidence that these errors did occur relatively frequently. -- DFB. On Thu, 30 Oct 2003, VOLTOLINI wrote: > Hi folks, > > My doubt is about the chi-square test. > > I heard from a colleague that.... > > When the Ho of independence is rejected in a contingency table you can > compare the partial chi-square values with the chi-square value from > the table to check if there is one or more partial chi-square values > that could reject the Ho alone. > > Is it true? > > I cannot find this idea in any classic stats text book :( > > Thanks for any help !!! > > Voltolini ------------------------------------------------------------ Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
