Wald Log-Linear Chi-Square Test :: SAS/STAT(R) 12.3 User's Guide

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. See the section Wald Chi-Square Test for more information about Wald tests.

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 $\widehat{Y}_{rc}$ ,

$\widehat{Y}_{rc} = \log \widehat{N}_{rc} ~ - ~ \log \widehat{N}_{rC} ~ - ~ \log \widehat{N}_{Rc} ~ + ~ \log \widehat{N}_{RC}$

where $\widehat{N}_{rc}$ is the estimated total for table cell (r, c). The null hypothesis of independence between the row and column variables can be expressed as $H_0\colon Y_{rc} = 0$ for all $r = 1, \ldots (R-1)$ and $c=1, \ldots (C-1)$ . This null hypothesis can be stated equivalently in terms of cell proportions.

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

$Q_\mi {L} = \widehat{\mb {Y}}’ ~ \widehat{\mb {V}}(\widehat{\mb {Y}})^{-1} ~ \widehat{\mb {Y}}$

where $\widehat{\mb {Y}}$ is the (R – 1)(C – 1)-dimensional array of the $\widehat{Y}_{rc}$ , and $\widehat{\mb {V}}(\widehat{\mb {Y}})$ estimates the variance of $\widehat{\mb {Y}}$ ,

$\widehat{\mb {V}}(\widehat{\mb {Y}}) = \mb {A} ~ \mb {D}^{-1} ~ \widehat{V}(\widehat{\mb {N}}) ~ \mb {D}^{-1} ~ \mb {A}’$

where $\widehat{\mb {V}}(\widehat{\mb {N}})$ is the covariance matrix of the estimates $\widehat{N}_{rc}$ , which is computed as described in the section Covariance of Totals. $\mb {D}$ is a diagonal matrix with the estimated totals $\widehat{N}_{rc}$ on the diagonal, and $\mb {A}$ is the by $RC \times RC$ linear contrast matrix.

Under the null hypothesis of independence, the statistic $Q_\mi {L}$ 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

$F_\mi {L} = Q_\mi {L} ~ / ~ (R-1)(C-1)$

Under the null hypothesis of independence, $F_\mi {L}$ 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 specify the denominator degrees of freedom with the DF= option in the TABLES statement.

For tables larger than $2 \times 2$ , PROC SURVEYFREQ also computes the adjusted Wald log-linear F statistic as

$F_{\mathit{Adj\_ L}} = Q_\mi {L} ~ (s - k + 1) ~ / ~ (k s)$

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 specify the value of s with the DF= option in the TABLES statement. Note that for $2 \times 2$ tables, k = (R – 1)(C – 1) = 1, and therefore the adjusted Wald F statistic equals the (unadjusted) Wald F statistic, with the same numerator and denominator degrees of freedom.

Under the null hypothesis, $F_{\mathit{Adj\_ L}}$ approximately follows an F distribution with k numerator degrees of freedom and (s – k + 1) denominator degrees of freedom.

The SURVEYFREQ Procedure

Wald Log-Linear Chi-Square Test