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Wald Log-Linear Chi-Square Test

If you specify the WLLCHISQ
option in the TABLES statement, PROC SURVEYFREQ computes a Wald test for independence based on the log odds ratios. For more
information about Wald tests, see the section Wald Chi-Square Test.

For a two-way table of R rows and C columns, the Wald log-linear test is based on the (R – 1)(C – 1)-dimensional array of elements ,

where is the estimated total for table cell (r, c). The null hypothesis of independence between the row and column variables can be expressed as for all and . This null hypothesis can be stated equivalently in terms of cell proportions.

The generalized Wald log-linear chi-square statistic is computed as

where is the (R – 1)(C – 1)-dimensional array of the , and estimates the variance of ,

where is the covariance matrix of the estimates , which is computed as described in the section Covariances of Frequency Estimates. is a diagonal matrix with the estimated totals on the diagonal, and is the by linear contrast matrix.

Under the null hypothesis of independence, the statistic approximately follows a chi-square distribution with (R – 1)(C – 1) degrees of freedom for large samples.

PROC SURVEYFREQ computes the Wald log-linear F statistic as

Under the null hypothesis of independence, approximately follows an F distribution with (R – 1)(C – 1) numerator degrees of freedom. PROC SURVEYFREQ computes the denominator degrees of freedom as described in the section
Degrees of Freedom. Alternatively, you can use the DF=
option in the TABLES statement to specify the denominator degrees of freedom.

For tables larger than , PROC SURVEYFREQ also computes the adjusted Wald log-linear F statistic as

where k = (R – 1)(C – 1), and s is the denominator degrees of freedom, which is computed as described in the section Degrees of Freedom. Alternatively, you can use the DF=
option in the TABLES statement to specify the value of s. For tables, k = (R – 1)(C – 1) = 1, and therefore the adjusted Wald F statistic equals the (unadjusted) Wald F statistic and has the same numerator and denominator degrees of freedom.

Under the null hypothesis, approximately follows an F distribution with k numerator degrees of freedom and (s – k + 1) denominator degrees of freedom.