The chisquare test is used to test the independence of two variables cross classified in a twoway table. For example, suppose we wished to test the hypothesis that calcium supplement use is independent of osteoporosis treatment status and that we have the following observed frequencies obtained as a result of the crossclassification of osteoporosis and supplement use for women.

Osteoporosis Treatment Status  Yes 
Osteoporosis Treatment Status  No 
Total 

Supplement Use  Yes 
155 
566 
721 
Supplement Use  No 
47 
419 
466 
Total 
202 
985 
1187 
In a simple random sample setting (unweighted data), the expected cell frequencies under the null hypothesis that osteoporosis treatment status and calcium supplement use are independent could be obtained by multiplying the marginal total for the ith row by the proportion of individuals in the jth column.
For example, the expected value of supplement users who received treatment for osteoporosis would be 721*(202/1187)=123; the expected value of supplement users who did not receive treatment for osteoporosis 721*(985/1187)=598.
Thus, if Oij = the observed frequency of the ith row and jth column, where i=1,2, … i and j=1,2, … j and Eij = the expected frequency of the ith row and jth column. Then the formula to test the null hypothesis of independence, using the chisquare statistic, would be
This statistic has degrees of freedom equal to the number of rows minus 1, multiplied by the number of columns minus 1.
In a complex sample setting, you would use a statistic similar to equation (1) above, modified to account for survey design with degrees of freedom equal to the number of PSUs minus the number of strata containing observations. This statistic can be obtained through SAS proc surveyfreq (chisq, based on the RaoScott chisquare with an adjusted F statistic). The analogous procedure in SUDAAN version 10.0 (proc crosstab), provides limited chisquare statistics based on Wald chisquare and does not provide an F adjusted pvalue. However, SUDAAN regression models do provide F adjusted chisquare statistics which are recommended for analyzing NHANES data.
The Cochran Mantel Haenzel Test, an extension of the Pearson ChiSquare, can be applied to stratified twoway tables to test for homogeneity or independence in a nonsurvey setting. For a complex sample its analogue can be obtained in SUDAAN proc crosstab (cmh).
Agresti A. An Introduction to Categorical Data Analysis. Wiley Series in Probability and Statistics. 1996. New York.
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