Wald Tests

The Wald test evaluates whether imposing a set of restrictions on estimates significantly reduces the fit of the model. For example, a test might be used to test whether three regression coefficients in a larger model are all equal to zero.

AM currently offers two Wald tests of two sorts--an overall Wald test to evaluate the fit of regression models (including MML regression, probit, logit, etc.), and a post-hoc Wald test that can be used to evaluate the joint signficance of multiple regression parameters. When a regression model includes a constant term, the overall Wald test evaluates the fit of the model against a model with only a constant term (i.e., the constant term is not included in the test). When the regression model includes only a constant term, the significance of the constant term is evaluated. The post-hoc Wald test allows the user to select which estimates are included in the test.

In a complex sample, the variance is estimated as a the stratified, between-PSU variance. Therefore, the total degrees of freedom for the variance estimate equals the number of clusters less the number of strata. The test statistic is based on the complex-sample formula offered by Fellegi (1980).

Currently, AM only tests whether a subset of estimates are all equal to zero. Future releases will add additional capabilities. AM implements an adjusted Wald test, appropriate for complex samples, suggested by Following Fellegi (1980). The basic Wald statistic is given by, where R and q implement a set of linear restructions representing the null hypothesissuch that Rb-q =0. For the restrictions currently tested, q is always a vector of zeros, and R is a k x m matrix where k is the number of restrictions and m is the number of parameters in b. Each column of R includes zeros and a single 1 in the column corresponding to one of the parameters included in the test.

The Fellegi (1980) adjustment implemented in AM approximately follows an F distribution

W' = (d-k+1)/dk~F(k,d-k+1)

where d is the number of degrees of freedom, calculated as the number of PSU less the number strata. (Strata with a single PSU contribute 1 degree of freedom--a degree of freedom is not subtracted off for the stratum). In replication procedures, the degrees of freedom equals the number of replicates that contribute to the variance estimate.

Wald test are available for all regression-type models. To conduct a Wald test , right click on the Completed Run Icon after the model runs. Select "Wald test for multiple parameters." This will bring up a dialog box similar to the dialog box for t-tests.

Click each variable to be included in the tests. Included variables are identified by a blue border (click again to exclude an included variable from the test).

When the intended set of estimates is identified, click OK and results will be sent to your default output device (usually your browser).