# Wald test

The Wald test is a parametric statistical test named after the Transylvanian 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.

Suppose an economist, who has data on social class and shoe size, wonders whether social class is associated with shoe size. Say $\theta$ is the average increase in shoe size for upper-class people compared to middle-class people: then the Wald test can be used to test whether $\theta$ is 0 (in which case social class has no association with shoe size) or non-zero (shoe size varies between social classes). Here, $\theta$, the hypothetical difference in shoe sizes between upper and middle-class people in the whole population, is a parameter. An estimate of $\theta$ might be the difference in shoe size between upper and middle-class people in the sample. In the Wald test, the economist uses the estimate and an estimate of variability (see below) to draw conclusions about the unobserved true $\theta$. Or, for a medical example, suppose smoking multiplies the risk of lung cancer by some number R: then the Wald test can be used to test whether R = 1 (i.e. there is no effect of smoking) or is greater (or less) than 1 (i.e. smoking alters risk).

A Wald test can be used in a great variety of different models including models for dichotomous variables and models for continuous variables.[1]

## Mathematical details

Under the Wald statistical test, the maximum likelihood estimate $\hat\theta$ of the parameter(s) of interest $\theta$ is compared with the proposed value $\theta_0$, 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

$\frac{ ( \widehat{ \theta}-\theta_0 )^2 }{\operatorname{var}(\hat \theta )}$

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

$\frac{\widehat{\theta}-\theta_0}{\operatorname{se}(\hat\theta)}$

where $\operatorname{se}(\widehat\theta)$ is the standard error of the maximum likelihood estimate. A reasonable estimate of the standard error for the MLE can be given by $\frac{1}{\sqrt{I_n(MLE)}}$, where $I_n$ 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.[2] A common use for this is to carry out a Wald test on a categorical variable by recoding it as several dichotomous variables.

## Alternatives to the Wald test

The likelihood-ratio test can also be used to test whether an effect exists or not. The Wald test and the likelihood ratio test often give similar conclusions (as they are asymptotically equivalent), but they could disagree enough to lead to different conclusions.

There are several reasons to prefer the likelihood ratio test to the Wald test.[3][4][5] One is that the Wald test can give different answers to the same question, depending on how the question is phrased.[6] 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). Likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R. The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is chi-squared), whereas the likelihood ratio test uses one approximation (that the distribution is chi-squared).

Yet another alternative is the score test, which has the advantage that it can be formulated in situations where the variability is difficult to estimate; e.g. the Cochran–Mantel–Haenzel test is a score test.[7]

## Asymptotic properties

Robert F. Engle showed that the Wald test, the likelihood-ratio test and the Lagrange multiplier test (also known as the score test) are asymptotically equivalent.[8]