How to Perform a Wald Test in R

In this article, we will discuss What is the Wald Test and How to Perform a Wald Test in R Programming Language.

What is the Wald Test?

The Wald test is a statistical hypothesis test used to assess whether parameters in a statistical model are significantly different from hypothesized values. It is widespread in the context of regression analysis, where it tests the significance of individual coefficients or groups of coefficients in a regression model.

How does the Wald test work?

There are some steps to explain how the Wald test works.

Step 1: Formulate Null and Alternative Hypotheses

• The null hypothesis (H0) typically states that certain parameters in the model have specific values, often zero (indicating no effect).
• The alternative hypothesis (H1) generally states that the parameters have values different from those specified in the null hypothesis.

Step 2: Estimate the Model

• Fit the statistical model to the data using methods like least squares estimation in linear regression or maximum likelihood estimation in logistic regression.

Step 3: Compute the Test Statistic

• The Wald test statistic is computed by taking the estimated coefficient divided by its standard error and squaring the result.
• This test statistic approximately follows a chi-square distribution under the null hypothesis.

Step 4: Compare the Test Statistic to the Critical Value

• With the test statistic computed, compare it to the critical value from the chi-square distribution with appropriate degrees of freedom.
• If the test statistic exceeds the critical value, you reject the null hypothesis in favor of the alternative hypothesis, indicating that the parameter(s) of interest are significantly different from the hypothesized values.

Mathematically, Wald test statistic 𝑾 is calculated as

[Tex]W = \left(\frac{\theta – \theta_0}{\text{var}(\theta)}\right)^2 [/Tex]

Where:

• 𝑾 is the Wald test statistic.
• 𝜽 is the estimated parameter value. This is what we get from our data, like the estimated size of a relationship or a coefficient.
• 𝜽₀ is the hypothesized value of the parameter under the null hypothesis. This is what we assume the value is under the null hypothesis (usually 0 if we’re testing if something has an effect).
• var(𝜽) is the estimated variance of the parameter estimate. It tells us how uncertain our estimated value is.

We compare the value of 𝑾 we calculate to a critical value from a chi-square distribution. This tells us if our estimated value is far enough from the hypothesized value to be considered significant.

Chi-square Distribution

The chi-square distribution helps us figure out how likely different values of 𝑾 are under the null hypothesis. We compare our calculated 𝑾 value to values from this distribution to decide if our result is significant or not.

How to Perform a Wald Test in R?

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

1. lmtest: This package provides the waldtest() function, which can be used to perform Wald tests on coefficients in linear regression models.
2. car: The car package provides the linearHypothesis() function, which can perform various types of hypothesis tests, including Wald tests, for linear regression models.
3. 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

1. Hypothesis Testing: The Wald test is often used to test specific hypotheses about the coefficients in a regression model.
2. Comparison of Nested Models: The Wald test can be used to compare nested models, where one model is a special case of another.
3. Test Statistic: The Wald test statistic is calculated by squaring the ratio of the estimated coefficient to its standard error.
4. P-value: The p-value associated with the Wald test measures the likelihood of observing such a test statistic under the null hypothesis.
5. 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.

Conclusion

In summary, the Wald test is a useful statistical tool for determining the significance of coefficients in regression models. It assesses whether specific predictors have a meaningful impact on the outcome. By comparing coefficients to their standard errors, it helps researchers understand which variables contribute significantly to the model’s predictive ability. Overall, the Wald test is a straightforward and widely-used method for hypothesis testing in regression analysis.