# Wald Test

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## Definition: Wald Test

Wald Test is a statistical test used to verify the true values of different parameters such that the statistical relationship between these parameters is to be modeled and the values under verification are derived from samples of a population of these parameters.

Let’s elaborate the use of Wald test in determining whether a relationship exists between social class and size of shirt of a person ( X). Let it is assumed that the size of shirt on an average increases by X0 for the upper class people than the lower class people. An analysis of sample of the population shows that the maximum likelihood estimate of the increasing relationship is given by  X.  Suppose the difference between    X and X0 is normally distributed. Then the Wald Statistic is given by –

(X  - X0 )2

Var (X )

which is compared with a chi-squared distribution to find out if the current value of X0 represents exact relationship between the social class and size of shirt.

There are certain limitations of Wald Test because of which other methods like Likelihood-ratio test, Cocharan-Mantel-Haenzel test etc are preferred. One of the limitations of Wald test is –

i. Wald test gives different answers to same question depending on how the question is framed.

e.g.) Let R is the parameter under test. Suppose Wald Test verifies if R=1. Then the test gives different results for R=1 and log R=0.

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