Why Regression Uses the Wald Test and What the P-value Actually Means

Introduction

When researchers examine results from a regression model—such as logistic regression, Poisson regression, or Cox proportional hazards regression—they often see a table like this:

Variable	Coefficient	SE	P-value
Smoking	1.25	0.32	0.001
Age	0.04	0.01	0.003
Sex	0.20	0.27	0.42

A common question is:

Where do these p-values come from, and does this mean the model is testing each variable independently while ignoring the others?

The answer is no.

These p-values typically come from the Wald test, which evaluates whether the coefficient of a variable in a regression equation differs from zero after adjusting for all other variables in the model.

The Basic Structure of a Regression Model

Suppose we study the relationship between smoking and lung cancer using logistic regression.

logit (P (Y = 1)) = β_{0} + β_{1} Smoking + β_{2} Age + β_{3} Sex

Where:

Y = occurrence of lung cancer
Smoking = smoking status
Age = age
Sex = biological sex

When statistical software estimates this model, it calculates all coefficients simultaneously.

Thus the estimates

\hat{β_{1}}, \hat{β_{2}}, \hat{β_{3}}

are derived from the same likelihood function.

Therefore,

\hat{β_{1}}

does not represent the crude effect of smoking on lung cancer.

Instead it represents:

the effect of smoking after adjusting for age and sex

This is what epidemiologists call an adjusted effect.

What the Wald Test Does

After estimating the regression coefficients, we want to test the hypothesis:

H_{0} : β = 0

This null hypothesis means:

the variable has no association with the outcome.

The Wald test evaluates this hypothesis using the statistic:

W = \frac{\hat{β}}{SE (\hat{β})}

Where:

β^ = estimated coefficient
SE(β^) = standard error of the coefficient

This statistic is then used to compute a p-value, which measures the statistical evidence against the null hypothesis.

Interpreting the P-value in Regression

Suppose the regression output shows:

Variable	P-value
Smoking	0.001

The correct interpretation is:

After adjusting for age and sex, smoking remains statistically associated with lung cancer.

Thus the Wald test is not evaluating

Smoking vs outcome

but rather

Smoking vs outcome | age, sex

The vertical bar | means "conditional on" or "holding other variables constant."

Does the Wald Test Examine Variables One by One?

Most regression software reports partial Wald tests, which test each coefficient individually:

H_{0} : β_{1} = 0

H_{0} : β_{2} = 0

H_{0} : β_{3} = 0

However, it is also possible to test multiple coefficients simultaneously, for example:

H_{0} : β_{2} = β_{3} = 0

This is called a joint Wald test, which evaluates whether a group of variables collectively contributes to the model.

How the Wald Test Differs from t-tests and Chi-square Tests

Classical statistical tests are typically used in simpler situations.

Research question	Typical test
Compare means between two groups	t-test
Compare proportions between groups	Chi-square test

However, regression models estimate parameters of an equation rather than directly comparing groups.

Therefore regression models use the Wald test to evaluate whether the estimated coefficients differ from zero.

An Interesting Insight: Many Classical Tests Are Special Cases of Regression

Mathematically, many familiar tests can be expressed as regression models.

t-test

Equivalent to the linear model:

Y = β_{0} + β_{1} Group

Chi-square test

Equivalent to logistic regression:

logit (P (Y)) = β_{0} + β_{1} Group

In these regression formulations, the significance of β1 can also be evaluated using a Wald test.

Thus regression provides a unified framework for many statistical tests.

Limitations of the Wald Test

Although widely used, the Wald test has some limitations. It can perform poorly when:

the sample size is small
coefficients are very large
the data are sparse

In these situations, many statisticians prefer the Likelihood Ratio Test (LRT), which tends to be more stable.

Final Summary

The Wald test is a statistical method used to evaluate whether a regression coefficient differs from zero.

Key points:

Regression models estimate all variables simultaneously.
The Wald test evaluates the effect of one variable after adjusting for the others.
The p-value reported for a variable represents conditional inference, not a crude comparison.

Therefore, when interpreting regression output, the p-value does not mean that the variable is examined in isolation. Instead, it answers the question:

Does this variable still have an association with the outcome after controlling for the other variables in the model?