Concept Architecture
Concept
Theoretically, Wald Test is a statistical hypothesis test used to determine whether one or more estimated model parameters differ significantly from specified values, most commonly zero. It is based on the asymptotic distribution of maximum likelihood estimators and evaluates whether an estimated coefficient is sufficiently large relative to its sampling variability. In health economics, the Wald test is widely used in regression modelling, survival analysis and generalised linear models to assess the statistical significance of explanatory variables used to estimate costs, utilities, treatment effects and disease risks.
Mathematically, the Wald test compares the estimated parameter with its hypothesised value by standardising the difference using its estimated standard error. For a single parameter, the resulting test statistic follows an asymptotic standard normal distribution when squared, yielding a chi-square distribution with one degree of freedom. For multiple parameters, the Wald statistic follows a chi-square distribution with degrees of freedom equal to the number of jointly tested parameters.
In practice, analysts perform Wald tests after estimating likelihood-based statistical models to determine whether individual predictors or groups of predictors should be retained. In health economics, Wald tests are routinely applied during regression model development, survival model estimation and health outcome prediction. Results are interpreted alongside likelihood ratio tests, confidence intervals and model diagnostics because Wald tests may perform less well when sample sizes are small or parameter estimates are near the boundary of the parameter space.
Purpose
Used to test whether estimated model parameters differ significantly from hypothesised values, thereby supporting variable selection and statistical inference in health economic models.
Mathematical Formulae
Primary Formula
For a single parameter,
W = [(?? ? ??) / SE(??)]�
where:
- ?? = estimated parameter
- ?? = hypothesised parameter value
- SE(??) = standard error of the estimate
Under the null hypothesis,
W ? ?�?
Supporting Formulae
For multiple parameters,
W = (?? ? ??)? V?? (?? ? ??)
where:
- V = estimated covariance matrix of the parameter estimates
Related Mathematical Methods
- Maximum likelihood estimation
- Likelihood ratio test
- Score test
- Chi-square hypothesis testing
- Logistic regression
- Cox proportional hazards regression
- Generalised linear models
Example
A health economist estimates a logistic regression model predicting hospital admission.
The coefficient for previous hospitalisation is
?? = 0.80
with
SE(??) = 0.20
Testing
H?: ? = 0
gives
W = (0.80 / 0.20)� = 16
With one degree of freedom, the p-value is less than 0.001, indicating that previous hospitalisation is a statistically significant predictor of admission.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER((B2-C2)/D2,2) | Calculates the Wald statistic from the parameter estimate and standard error. |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(E2,1) | Calculates the p-value for a one-parameter Wald test. |
| ABS | =ABS(B2/D2) | Calculates the absolute Wald z-statistic before squaring. |
| IF | =IF(CHISQ.DIST.RT(E2,1)<0.05,""Significant"",""Not significant"") | Determines whether a predictor should be retained based on the Wald test. |
VBA (Optional)
Automate Wald tests for all model parameters and generate tables of coefficients, standard errors, Wald statistics and p-values.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Agresti A. Categorical Data Analysis. 3rd ed.
- Dobson AJ, Barnett AG. An Introduction to Generalized Linear Models. 4th ed.
- Harrell FE. Regression Modeling Strategies. 2nd ed.
- Wald A. Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations Is Large. Transactions of the American Mathematical Society. 1943;54:426?482.
Related Concepts (2)
Library
Publications
1
Modeling Good Research Practices — Overview: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-1 — Caro, Briggs, Siebert & Kuntz, Task Force Report 1 ed., 2012 (Value in Health / Medical Decision Making)
The overview paper of the seven-part ISPOR-SMDM modelling good-practice series, setting out best-practice recommendations across model design, technique selection, implementation, validation, parameterisation, uncertainty and use in decision making.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the Wald test?
A statistical test assessing whether an estimated parameter differs significantly from a specified value, based on the ratio of estimate to standard error.
Source: Wald 1943
How does the Wald test judge whether a parameter matters?
The Wald test judges whether an estimated parameter differs meaningfully from a reference value, usually zero, by comparing the size of the estimate against its standard error. A coefficient that is large relative to the uncertainty around it yields a big test statistic and is taken as unlikely to be zero, whereas one small relative to its error could plausibly be zero. The ratio of estimate to its uncertainty is what the test turns into a probability. Harrell (2015) describes the test.
Source: Harrell 2015
How does the Wald test work?
The Wald test works by taking the estimated parameter, subtracting the value specified under the null hypothesis, and dividing by the estimate's standard error, giving a statistic that under the null follows a standard normal distribution, or whose square follows a chi-squared distribution. A large statistic, and correspondingly small p-value, indicates the parameter differs significantly from the specified value. The test thus judges significance from how many standard errors the estimate lies from the hypothesised value, relying on the estimate's approximate normality.
Source: Wald 1943
When is the Wald test used?
The Wald test is used to assess the statistical significance of estimated parameters, such as regression coefficients, testing whether each differs significantly from zero, which indicates whether the corresponding variable has a significant effect. It is applied in many models, including regression and survival models, because it requires only the estimate and its standard error from the fitted model. This convenience makes it a standard test of coefficient significance, providing a quick assessment of whether parameters differ from a specified value.
Source: Wald 1943
How does the Wald test relate to other tests?
The Wald test relates to the likelihood ratio test and the score test, which are three standard approaches to testing hypotheses about parameters in likelihood-based models. The Wald test uses the estimate and its standard error, the likelihood ratio test compares the likelihoods of nested models, and the score test uses the gradient of the likelihood. The three are asymptotically equivalent but can differ in finite samples. The Wald test is convenient, needing only the fitted model's estimates, though the likelihood ratio test is sometimes preferred for accuracy.
Source: Neyman & Pearson 1933
What are the limitations of the Wald test?
The Wald test relies on the approximate normality of the parameter estimate, which may not hold well in small samples or near boundaries of the parameter space, where it can be inaccurate or misleading. Its results can also depend on how the parameter is expressed, since transformations affect the estimate and standard error. In such cases the likelihood ratio test may be more reliable. These limitations mean the Wald test, though convenient, is used with awareness that it is an approximation, best in large samples.
Source: Wald 1943
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 17 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/wald-test
- Term code
- HE-EM-MV-091
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