Wald Log-Linear Chi-Square Test :: SAS/STAT(R) 13.1 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 $(R-1)(C-1)$ 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 use the DF= option in the TABLES statement to specify the denominator degrees of freedom.

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 use the DF= option in the TABLES statement to specify the value of s. For $2 \times 2$ 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, $F_{\mathit{Adj\_ L}}$ 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