The Wald test is a parametric statistical test named after the Hungarian statistician Abraham Wald. Whenever a relationship within or between data items can be expressed as a statistical model with parameters to be estimated from a sample, the Wald test can be used to test the true value of the parameter based on the sample estimate.
Contents
- Mathematical details
- Test on a single parameter
- Tests on multiple parameters
- Nonlinear hypothesis
- Non invariance to re parametrisations
- Alternatives to the Wald test
- References
Suppose an economist, who has data on social class and shoe size, wonders whether social class is associated with shoe size. Say
A Wald test can be used in a great variety of different models including models for dichotomous variables and models for continuous variables.
Mathematical details
Under the Wald statistical test, the maximum likelihood estimate
Test on a single parameter
In the univariate case, the Wald statistic is
which is compared against a chi-squared distribution.
Alternatively, the difference can be compared to a normal distribution. In this case the test statistic is
where
Test(s) on multiple parameters
The Wald test can be used to test a single hypothesis on multiple parameters, as well as to test jointly multiple hypotheses on single/multiple parameters. Let
The test statistic is:
where
Nonlinear hypothesis
In the standard form, the Wald test is used to test linear hypotheses, that can be represented by a single matrix R. If one wishes to test a non-linear hypothesis of the form:
The test statistic becomes:
where
Non-invariance to re-parametrisations
The fact that one uses an approximation of the variance has the drawback that the Wald statistic is not-invariant to a non-linear transformation/reparametrisation of the hypothesis: it can give different answers to the same question, depending on how the question is phrased. For example, asking whether R = 1 is the same as asking whether log R = 0; but the Wald statistic for R = 1 is not the same as the Wald statistic for log R = 0 (because there is in general no neat relationship between the standard errors of R and log R, so it needs to be approximated).
Alternatives to the Wald test
There exist several alternatives to the Wald test, namely the likelihood-ratio test and the Lagrange multiplier test (also known as the score test). Robert F. Engle showed that these three tests, the Wald test, the likelihood-ratio test and the Lagrange multiplier test are asymptotically equivalent. Although they are asymptotically equivalent, in finite samples, they could disagree enough to lead to different conclusions.
There are several reasons to prefer the likelihood ratio test or the Lagrange multiplier to the Wald test: