Two-Way Tables

The Rao-Scott likelihood ratio statistic is computed from the likelihood ratio chi-square statistic and a design correction based on the design effects of the proportions. Under the null hypothesis of no association between the row and column variables, this statistic approximately follows a chi-square distribution with degrees of freedom. PROC SURVEYFREQ also computes an statistic that can provide a better approximation.

The Rao-Scott likelihood ratio chi-square is computed as

where is the design correction described in the section Design Correction for Two-Way Tables, and is the likelihood ratio chi-square based on the estimated totals. The likelihood ratio chi-square is computed as

where is the sample size, is the estimated overall total, is the estimated total for table cell , and is the expected total for cell under the null hypothesis of no association. The expected total for cell equals

Under the null hypothesis of no association, the Rao-Scott likelihood ratio chi-square approximately follows a chi-square distribution with degrees of freedom. A better approximation can be obtained by the statistic,

which has an distribution with and degrees of freedom under the null hypothesis. The value is the degrees of freedom for the variance estimator and depends on the sample design and the variance estimation method. The section Degrees of Freedom describes the computation of .

One-Way Tables

For one-way tables, the Rao-Scott likelihood ratio chi-square statistic provides a design-based goodness-of-fit test for equal proportions. Or if you specify null proportions with the TESTP= option, the Rao-Scott likelihood ratio chi-square provides a design-based goodness-of-fit test for the specified proportions. Under the null hypothesis, the Rao-Scott likelihood ratio statistic approximately follows a chi-square distribution with degrees of freedom for a table with levels. PROC SURVEYFREQ also computes an statistic that can provide a better approximation.

The Rao-Scott likelihood ratio chi-square is computed as

where is the design correction described in the section Design Correction for One-Way Tables, and is the likelihood ratio chi-square based on the estimated totals. The likelihood ratio chi-square is computed as

where is the sample size, is the estimated overall total, is the estimated total for level , and is the expected total for level under the null hypothesis. For the null hypothesis of equal proportions, the expected total for each level equals

For specified null proportions, the expected total for level equals

where is the null proportion for level .

Under the null hypothesis of no association, the Rao-Scott likelihood ratio chi-square approximately follows a chi-square distribution with degrees of freedom. A better approximation can be obtained by the statistic,

which has an distribution with and degrees of freedom under the null hypothesis, The value is the degrees of freedom for the variance estimator and depends on the sample design and the variance estimation method. The section Degrees of Freedom describes the computation of .