Wald test Logistic regression

[iuri_leite]

Dear colleagues,

When we run a logistic regression, we assume that the Wald test for a specific parameter has an asymptotic standard Normal distribution. Therefore, when we look at the 95% confidence interval provided for the parameter, before being exponentiated,it is calculated by the following expression: B±1.96SE. But, if the number of observations for the category associated to the parameter estimated is low, then the parameter will not have a Standard Normal distribution. If I am not mistaken, is there a way to change it when estimating a logistic regression?

Thank you in advance.

Best regards,

Iuri

[PaigeMiller]

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@iuri_leite wrote:

Dear colleagues,

When we run a logistic regression, we assume that the Wald test for a specific parameter has an asymptotic standard Normal distribution. Therefore, when we look at the 95% confidence interval provided for the parameter, before being exponentiated,it is calculated by the following expression: B±1.96SE. But, if the number of observations for the category associated to the parameter estimated is low, then the parameter will not have a Standard Normal distribution. If I am not mistaken, is there a way to change it when estimating a logistic regression?

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I'm not sure I understand the question. In particular, when you say "is there a way to change it", what does "it" refer to, what do you want to change?

If I was to guess at what you mean, you need to tell PROC LOGISTIC to compute the confidence intervals for the parameter estimates via the CLPARM= option.

[StatDave · Accepted solution]

You can change the confidence interval from a Wald interval to a likelihood-based interval. Specify CLPARM=PL to request the profile likelihood intervals. For details on their construction, see "Confidence Intervals for Parameters" in the Details section of the PROC LOGISTIC documentation.

[iuri_leite]

Dear @StatDave,

thanks a lot.

It worked just fine.

Best regards,

Iuri 

[Season]

I am also curious about the Wald test. When testing the null hypothesis: β=β0 (in most cases, β0=0), the Wald statistic is the square of the ratio of the difference of β hat and β0 and the estimated standard error of the former (denoted as z) :  , where SE denote the estimated standard error of β hat.

As @iuri_leite has mentioned, the z statistic follows a standard normal distribution given a large sample size. So why bother taking its square and compare it against a Chi-square distribution instead of directly conducting a t-test with the z statistic?

[StatDave]

Because not all tests done are single DF tests on single parameters - for instance, type 3 tests or multi-DF tests you can construct in the CONTRAST or ESTIMATE statements. In general, for testing hypotheses on linear combinations of parameters with 1 or more DF, the Wald test has a limiting chi-square distribution. So, it is used throughout for consistency.

[Season]

Thank you!