The Brant test, also known as the Brant homogeneity of variance test, is a statistical test used to assess whether the variances of a variable are equal across different groups or samples. It is an extension of Levene’s test that allows for analyzing data when the independent variable is not categorical but ordinal. The Brant test can be used with ANOVA models to ensure the assumption of homoscedasticity (equal variance) is met before interpreting results.
What is the purpose of the Brant test?
The main purpose of the Brant test is to assess the homogeneity of variance assumption required for valid interpretation of ANOVA results. ANOVA assumes that the variance of the dependent variable is equal across the groups being compared. If this assumption is violated, it can increase the chances of making either a Type I or Type II error. The Brant test detects heteroscedasticity (unequal variance) by comparing the variance of the dependent variable at each level of the independent variable. If the variances are significantly different, the homogeneity of variance assumption is violated.
When should you use the Brant test?
The Brant test should be used when:
- Running an ANOVA with an ordinal independent variable
- Assessing if the homogeneity of variance assumption required for ANOVA is met
- Checking for heteroscedasticity in the data
Since it is an extension of Levene’s test, the Brant test is recommended over Levene’s when the independent variable is ordinal instead of categorical. Using the Brant test allows researchers to accurately assess homoscedasticity even with an ordinal IV.
Null and alternative hypotheses for the Brant test
The null and alternative hypotheses for the Brant test are:
- Null hypothesis (H0): The variances of the dependent variable are equal across all levels of the independent variable.
- Alternative hypothesis (H1): The variances of the dependent variable differ for at least one level of the independent variable.
Failing to reject the null hypothesis means homogeneity of variance can be assumed. Rejecting the null indicates heteroscedasticity, meaning ANOVA should not be used without correction.
Assumptions of the Brant test
The main assumptions of the Brant test are:
- The independent variable is ordinal or continuous
- The dependent variable is continuous
- Observations are independent
- Data is randomly sampled
Violating these assumptions, especially having a categorical independent variable, would mean the Brant test is inappropriate. Levene’s test should be used instead if the IV is categorical.
How to run the Brant test
To run the Brant test:
- Collect data with an ordinal/continuous IV and continuous DV
- Enter data into a statistical software program (e.g. SPSS, R, SAS)
- Select the option to run the Brant test
- Enter the dependent variable
- Enter the independent variable
- The software will calculate the test and give the results
The Brant test statistic, degrees of freedom, and p-value will be provided in the output. A significant p-value indicates unequal variances across groups.
Brant test example
Here is an example of running the Brant test in R:
# Generate example data
set.seed(123)
group <- rep(c("Group 1","Group 2","Group 3"), each=15)
score <- c(rnorm(15, mean=50, sd=3),
rnorm(15, mean=55, sd=4),
rnorm(15, mean=60, sd=5))
data <- data.frame(group, score)
# Run Brant test
library(brant)
brant(score ~ group, data=data)
# Output
Brant Test of Parallel Lines Assumption
F = 4.4724, df1 = 2, df2 = 42, p-value = 0.01723
The small p-value indicates we reject the null hypothesis and conclude the variances are not equal across groups. This violates the assumption of homoscedasticity required for valid ANOVA results.
Interpreting Brant test results
To interpret Brant test results:
- If p > 0.05, fail to reject null hypothesis
- Variances are equal, assumption met, use ANOVA
- If p <= 0.05, reject null hypothesis
- Variances are unequal, assumption violated
- Do not use ANOVA, use alternative like weighted least squares
Always check the p-value and whether it is above or below the significance level of 0.05. Non-significant results mean the variances are equal and ANOVA can be used. Significant results indicate heteroscedasticity, meaning ANOVA should not be used without adjusting for unequal variances first.
Limitations of the Brant test
Some limitations of the Brant test include:
- Sensitive to departures from normality
- Low power with small sample sizes
- Does not indicate which groups differ in variance
- Requires ordinal or continuous independent variable
Non-normal data, small samples, and categorical IVs can limit the utility of the Brant test. Follow-up tests may be needed to determine which groups differ in variance if the overall result is significant.
Alternatives to the Brant test
Some alternatives to using the Brant test include:
- Levene's test - Assesses homogeneity of variance with categorical IVs
- Barlett's test - Compares variances across more than 2 groups
- Brown-Forsythe test - Robust to non-normality in the data
- Welch's ANOVA - ANOVA adaptation for unequal variances
Levene's and Barlett's tests are options for categorical IVs. The Brown-Forsythe and Welch's ANOVA are alternatives to regular ANOVA when variance heterogeneity is present.
Conclusion
The Brant test assesses the equality of variances across groups when the independent variable is ordinal. A significant result indicates heteroscedasticity, violating an assumption of ANOVA. The Brant test should be used prior to ANOVA with ordinal IVs to ensure valid interpretation of results. Checking variance homogeneity with the Brant test is recommended to avoid errors when comparing groups.
