Interpreting adjusted residuals in Crosstabs cell statistics
I have just used the Crosstabs procedure to test the independence of two categorical variables. The chi-square statistics were significant, indicating that there is an association among the variables, but the table is larger than 2x2 I had requested the adjusted standardized residuals from among the options in the Cells dialog (or /CELLS subcommand). I need some help interpreting these adjusted residuals. I was told that absolute values of 2.0 or higher for the adjusted residual in a cell indicate that "something is going on" in those cells. Does this mean that the adjusted residuals point us to
where the real associations are in a table with a significant chi-square?
Resolving the problem
Under the null hypothesis that the 2 variables are independent, the adjusted residuals will have a standard normal distribution, i.e. have a mean of 0 and standard deviation of 1. So, an adjusted residual that is more than 1.96 (2.0 is used by convention) indicates that the number of cases in that cell is significantly larger than would be expected if the null hypothesis were true, with a significance level of .05. An adjusted residual that is less than -2.0 indicates that the number of cases in that cell is significantly smaller than would be expected if the null hypothesis were true. So, something is going on in that cell in that there are fewer or more cases (depending on the sign of the adjusted residual) than would be expected if the 2 variables were independent. If the crosstab table is a 2x2 table, then all of the adjusted residuals will have the same absolute value, but exactly 2 of them will be negative. The cells in the upper-left and lower-right corners will have one sign and the cells in the upper-right and lower-left cells will have the other sign. Suppose that the variables are called X and Y and that both variables have values 1 & 2. If the probability that Y=2 is higher than expected (from the independence hypothesis) when X= 1, then the probability that Y=2 will be lower than expected when X=2. When the table is larger than 2x2, interpretation of a significant chi-square is more difficult and the adjusted residuals can help here by showing you which cells have larger or smaller counts than expected. Many of the cells may have adjusted residuals close to 0, with a few cells providing most of the contribution to the large chi-square for the table.
There are a few notes on adjusted standardized residuals (under the name Standardized Pearson Residual) in:
Agresti, A. (2002). Categorical Data Analysis (2nd Ed.). New York: Wiley. (the definition is on page 81.)
As noted by Agresti, the standardized residuals (called Pearson residuals in Agresti) that are also provided as cell statistics in Crosstabs, are also asymptotically normally distributed with a mean of 0 under the null hypothesis of independence. However, their asymptotic variances are less than 1, so that comparing standardized residuals to standard normal distributions would lead to conservative estimates of a table cell's lack of fit to the null hypothesis.
Agresti notes that as the degrees of freedom for the chi-square increases (with increasing numbers of rows and/or columns in the cross-tabulation), it will be increasingly likely that at least one of the cells will have a large adjusted standardized residual by chance alone. Refraining from interpreting these residuals when the chi-square test of independence is not significant would be one way of controlling overall Type I error in interpreting the residuals, although Agresti seems to advocate not being overly restricted by the chi-square result in examining the table. The reader is directed to Agresti for a consideration of the issues involved in such interpretations.
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