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The SURVEYFREQ Procedure

Rao-Scott Likelihood Ratio Chi-Square Test

The Rao-Scott likelihood ratio chi-square test is a design-adjusted version of the likelihood ratio test, which involves ratios between observed and expected frequencies. For two-way tables, the null hypothesis for this test is no association between the row and column variables. For one-way tables, the null hypothesis is equal proportions for the variable levels. Or you can specify null hypothesis proportions for one-way tables by using the TESTP= option.

Two forms of the design correction are available for the Rao-Scott tests. One form of the design correction uses the proportion estimates, and you request the corresponding Rao-Scott likelihood ratio test with the LRCHISQ option. The other form of the design correction uses the null hypothesis proportions. You request this test, called the Rao-Scott modified likelihood ratio test, with the LRCHISQ1 option.

See Lohr (2009), Thomas, Singh, and Roberts (1996), and Rao and Scott (1981, 1984, 1987) for details about design-adjusted chi-square tests.

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 .

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