The Wald test is a statistical method that evaluates the significance of individual model parameters in regression and other statistical models. It tests the null hypothesis that a given coefficient equals zero (or another specified value), using the estimated coefficient and its standard error. Analysts frequently use the Wald test because it is simple to compute and typically included by default in the output of regression software for linear, logistic, and Poisson regression models. However, it can be less reliable than the likelihood ratio test in certain situations—especially in small samples or when testing boundary parameters (like random effects variances). In those cases, analysts often prefer likelihood-based methods.
The test statistic is calculated as:
W = (Estimate – Hypothesized Value)² / (Standard Error)²
Under the null hypothesis, the test statistic follows a chi-square distribution with 1 degree of freedom (or more for joint tests). A large Wald statistic indicates that the coefficient is significantly different from the hypothesized value, and a small p-value suggests the predictor contributes meaningfully to the model.
For example, suppose you’re modeling house prices and the coefficient for square footage is 90 with a standard error of 20. The Wald test statistic would be:
W = (90 – 0)² / 20² = 20.25, leading to a small p-value.
This result suggests that square footage is a statistically significant predictor of house price.
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