Wald Test P-Value Calculator

Calculator

Example Data Table

Formula Used

Wald z statistic:

z = (Estimate - Null Value) / Standard Error

Wald chi-square statistic for one parameter:

χ² = z²

Two-sided p-value:

p = 2 × P(Z ≥ |z|)

Greater one-sided p-value:

p = P(Z ≥ z)

Less one-sided p-value:

p = P(Z ≤ z)

Chi-square right-tail p-value:

p = P(Χ²df ≥ observed χ²)

Ratio estimates:

z = (ln(Ratio Estimate) - ln(Null Ratio)) / Log-Scale Standard Error

How to Use This Calculator

1. Select whether you have an estimate, a z statistic, or a chi-square statistic.
2. Choose raw scale for coefficients, differences, or slopes.
3. Choose ratio scale for odds ratios, hazard ratios, or risk ratios.
4. Enter the null value used by your hypothesis test.
5. Enter the standard error from the same fitted model.
6. Select the alternative hypothesis and reported p-value basis.
7. Press the calculate button to view the Wald statistic and p-value.
8. Use the CSV or PDF button to save the result.

Wald Test P-Value Guide

What the Wald Test Measures

The Wald test checks whether an estimated parameter differs from a chosen null value. It is often used after regression, logistic models, generalized linear models, and survival models. The method is direct. It divides the estimate’s distance from the null value by its standard error. That quotient is the Wald z statistic. A large absolute statistic gives stronger evidence against the null hypothesis.

Supported Input Styles

This calculator supports several common reporting styles. You can enter a raw coefficient, a mean difference, or another estimate with its standard error. You can also enter ratios such as odds ratios, risk ratios, hazard ratios, or rate ratios. For ratio inputs, the test uses the natural log scale. That matches standard model output, because ratio estimates are usually normal after log transformation.

How the P-Value Changes

The p-value depends on the selected alternative. A two-sided test checks for any difference. A greater test checks whether the estimate is above the null value. A less test checks whether it is below the null value. When a single parameter is tested, the squared Wald z statistic equals a chi-square statistic with one degree of freedom. Larger models may report a direct chi-square Wald statistic with more degrees of freedom.

Input Quality and Limits

Good input quality matters. The standard error should come from the same model as the estimate. Robust, clustered, or sandwich standard errors can be used when the model output provides them. Very small samples can make Wald tests unstable. Boundary estimates can also cause misleading results. In those cases, likelihood ratio tests or exact methods may be safer.

Interpreting Results

Use the confidence interval as a companion to the p-value. It shows practical range, not just significance. A small p-value can occur with a tiny effect in a large sample. A large p-value does not prove equality. It may show limited precision. Always report the estimate, standard error, statistic, p-value, alternative, and confidence level together.

Joint Wald Testing

Advanced use also includes checking multiple coefficients through a reported Wald chi-square value. This is useful for model terms, groups of dummy variables, and joint restrictions. Enter the statistic and degrees of freedom when software already provides them. The calculator then returns the right-tail probability from the chi-square distribution, which is the joint Wald p-value during quick model review.

FAQs

1. What is a Wald test p-value?

It is the probability of seeing a Wald statistic as extreme as the observed value, assuming the null hypothesis is true. Smaller values give stronger evidence against the null hypothesis.

2. What inputs do I need?

You usually need an estimate, a null value, and a standard error. You can also use a known z statistic or a known chi-square statistic from model output.

3. What is the default null value?

For raw coefficients, the usual null value is zero. For ratios, such as odds ratios or hazard ratios, the usual null value is one.

4. Why does ratio input use logs?

Ratios are commonly tested on the natural log scale. This makes the Wald approximation match standard regression output for odds ratios, hazard ratios, and similar measures.

5. Should I use one-sided or two-sided p-values?

Use two-sided testing when any difference matters. Use one-sided testing only when your research question clearly focuses on one direction before seeing the data.

6. What does the chi-square option mean?

The chi-square option uses the right tail of a chi-square distribution. It is common for single-parameter squared z tests and joint Wald tests.

7. Is the Wald test always reliable?

No. It can be unstable with small samples, large standard errors, rare events, or boundary estimates. Likelihood ratio tests may perform better in those cases.

8. Can I export my result?

Yes. After calculation, use the CSV or PDF button above the form. The export includes inputs, statistics, confidence level, and p-values.