Recall that covariates are variables that may be related to the outcome but are unaffected by treatment assignment. In a randomized experiment with one or more observed covariates, an analysis of covariance (ANCOVA) addresses this question: How would the mean outcome in each treatment group change if all groups were equal with respect to the covariate? The goal is to remove any variability in the outcome associated with the covariate from the unexplained variability used to determine statistical significance.
- How to do a one-way analysis of variance (ANOVA)
- How to compare two nested linear models
- How to conduct a mixed designs ANOVA
- How to conduct a repeated measures ANOVA
The solution below uses an example dataset about car design and fuel consumption from a 1974 Motor Trend magazine. (See how to quickly load some sample data.)
1 2 df <- mtcars df$vs <- as.factor(df$vs)
Let’s use ANCOVA to check the effect of the engine type (0 = V-shaped, 1 = straight, in the variable
vs) on the miles per gallon when considering the weight of the car as a covariate. We will use the
ancova function from the
pingouin package to conduct the test.
1 2 cov.model <- lm(mpg ~ wt + vs, data = df) anova(cov.model)
1 2 3 4 Df Sum Sq Mean Sq F value Pr(>F) wt 1 847.72525 847.725250 109.704168 2.284396e-11 vs 1 54.22806 54.228061 7.017656 1.292580e-02 Residuals 29 224.09388 7.727375 NA NA
The $p$-value for each variable can be found in the final column of the output, called
The $p$-value for the
wt variable tests the null hypothesis, “The quantities
mpg are not related.” Since it is below 0.05, we reject the null hypothesis, and conclude that
wt is significant in predicting
The $p$-value for the
vs variable tests the null hypothesis, “The quantities
mpg are not related if we hold
wt constant.” Since it is below 0.05, we reject the null hypothesis, and conclude that
vs is significant in predicting
mpg even among cars with equal weight (
If we wish to create a 2-factor ANCOVA model, we can test to see if the engine type (0 = V-shaped, 1 = straight) and transmission type (0 = automatic, 1 = manual) have an effect on the Miles/gallon per car when considering the weight of the car as a covariate.
1 2 cov.model.2 <- lm(mpg ~ wt + vs + am, data = df) anova(cov.model.2)
1 2 3 4 5 Df Sum Sq Mean Sq F value Pr(>F) wt 1 847.725250 847.725250 109.729918 3.420018e-11 vs 1 54.228061 54.228061 7.019303 1.310627e-02 am 1 7.778149 7.778149 1.006807 3.242621e-01 Residuals 28 216.315728 7.725562 NA NA
The $p$-values are again in the final column of output. They show that at the 5% significance level, we would conclude that engine type (
vs) significantly impacts the Miles/gallon per car while accounting for the weight of the car (
wt) but the transmission type (
am) does not.
Content last modified on 24 July 2023.
Contributed by Krtin Juneja (KJUNEJA@falcon.bentley.edu)