How to Perform a Wald Test in R?

In R, there are many packages that helps to performing Wald tests.

lmtest: This package provides the waldtest() function, which can be used to perform Wald tests on coefficients in linear regression models.
car: The car package provides the linearHypothesis() function, which can perform various types of hypothesis tests, including Wald tests, for linear regression models.
aod: It is known for functions related to overdispersion analysis but the aod package also provides the wald.test() function, which can perform Wald tests for coefficients in regression models.

Here we perform a Wald test using the lmtest package with the ‘mtcars’ dataset from R.

R

# Load necessary library
install.packages("lmtest")
library(lmtest)

# Fit regression model
model <- lm(mpg ~ disp + hp + wt, data = mtcars)

# Perform Wald test to determine if the coefficients of disp and hp 
wald_result <- waldtest(model, terms = c("disp", "hp"))

# Print the result
print(wald_result)

Output:

Wald test

Model 1: mpg ~ disp + hp + wt
Model 2: mpg ~ wt
  Res.Df Df     F   Pr(>F)   
1     28                     
2     30 -2 5.983 0.006863 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Fit a linear regression model to predict mpg using the disp, hp, and wt variables from the mtcars dataset.

The waldtest() function from the lmtest package to perform a Wald test to determine if the coefficients of disp and hp are simultaneously equal to zero.
The result of the Wald test is printed, indicating whether the null hypothesis (both coefficients are zero) is rejected or not.

Model 1: This represents the more complex model, which includes the predictors disp, hp, and wt to predict mpg.

Model 2: This represents the simpler model, which includes only the predictor wt to predict mpg.
Res.Df: This indicates the residual degrees of freedom, which is the difference between the total number of observations and the number of parameters estimated in the model.
Df: This represents the change in degrees of freedom between Model 1 and Model 2. In this case, Model 2 has 2 fewer parameters estimated compared to Model 1 because it includes fewer predictors.
F: This is the test statistic for the Wald test. It follows an F-distribution under the null hypothesis that the parameters in the reduced model (Model 2) are equal to zero. In other words, it tests whether the additional predictors in Model 1 contribute significantly to the model.
Pr(>F): This is the p-value associated with the F-test statistic. It represents the probability of observing an F-statistic as extreme as the one calculated under the null hypothesis. In this case, the p-value is 0.006863, which is less than 0.05, suggesting strong evidence against the null hypothesis.

Significance codes: These asterisks provide a quick indication of the level of significance of the test. In this case, ** indicates significance at the 0.01 level.

In this example we create a dataset with a binary outcome variable and several predictor variables, and then perform a logistic regression analysis with a Wald test using the aod package.

R

# Load necessary library
install.packages("aod")
library(aod)

# Generate synthetic data
set.seed(123)  # for reproducibility
n <- 100  # number of observations

# Predictor variables
x1 <- rnorm(n)  # continuous predictor
x2 <- sample(0:1, n, replace = TRUE)  # binary predictor
x3 <- rnorm(n)  # continuous predictor
x4 <- sample(0:1, n, replace = TRUE)  # binary predictor

# Outcome variable (binary)
y <- rbinom(n, 1, plogis(-1 + 0.5 * x1 + 0.8 * x2 - 0.3 * x3 + 0.6 * x4))

# Combine variables into a dataframe
data <- data.frame(y, x1, x2, x3, x4)

# Fit logistic regression model
model <- glm(y ~ x1 + x2 + x3 + x4, data = data, family = "binomial")

# Perform Wald test to determine if the coefficients of x3 and x4 
wald_result <- wald.test(b = coef(model), Sigma = vcov(model), Terms = c(4, 5))

# Print the result
print(wald_result)

Output:

Wald test:
----------

Chi-squared test:
X2 = 2.1, df = 2, P(> X2) = 0.36

Set a seed for reproducibility then generate synthetic data with 100 observations.

x1 and x3 are continuous predictors sampled from a normal distribution.
x2 and x4 are binary predictors sampled from a Bernoulli distribution.
y is a binary outcome variable generated from a logistic regression model.
Fit a logistic regression model using the glm() function.
The formula y ~ x1 + x2 + x3 + x4 specifies the model with predictors x1, x2, x3, and x4.
We specify family = “binomial” to indicate logistic regression for binary outcomes.
Use the wald.test() function to perform a Wald test.
b = coef(model) specifies the coefficient estimates obtained from the logistic regression model.
Sigma = vcov(model) specifies the variance-covariance matrix of the coefficients obtained from the logistic regression model.
Terms = c(4, 5) specifies the indices of the coefficients corresponding to x3 and x4 that we want to test simultaneously.

Chi-squared test: This indicates that the test statistic follows a chi-squared distribution.

X2: This is the value of the chi-squared test statistic. In this case, it is 2.1.
df: This represents the degrees of freedom associated with the chi-squared distribution.
P(> X2): This is the p-value associated with the chi-squared test statistic. It represents the probability of observing a chi-squared statistic as extreme as the one calculated under the null hypothesis. In this case, the p-value is 0.36.

Uses of Wald Test in R

Hypothesis Testing: The Wald test is often used to test specific hypotheses about the coefficients in a regression model.
Comparison of Nested Models: The Wald test can be used to compare nested models, where one model is a special case of another.
Test Statistic: The Wald test statistic is calculated by squaring the ratio of the estimated coefficient to its standard error.
P-value: The p-value associated with the Wald test measures the likelihood of observing such a test statistic under the null hypothesis.
Decision: If the p-value is small then we reject the null hypothesis, indicating significance. If it’s larger, we fail to reject the null, suggesting non-significance.

Uses of Wald Test in R

How to Perform a Wald Test in R

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