Wald Test - Mathematical Details

Mathematical Details

Under the Wald statistical test, the maximum likelihood estimate of the parameter(s) of interest is compared with the proposed value, with the assumption that the difference between the two will be approximately normally distributed. Typically the square of the difference is compared to a chi-squared distribution. 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 is the standard error of the maximum likelihood estimate. A reasonable estimate of the standard error for the MLE can be given by, where is the Fisher information of the parameter.

In the multivariate case, a test about several parameters at once is carried out using a variance matrix. A common use for this is to carry out a Wald test on a categorical variable by recoding it as several dichotomous variables.