To describe this issue, we consider the following 2 models:

m1 <- lm(disp ~ drat, mtcars)
m2 <- lm(disp ~ drat + factor(cyl), mtcars)

drat is a continuous variable. cyl is a categorical variable with levels '4', '6', '8', and I treat '4' as the baseline. I want to know if cyl is correlated to disp after controlling drat. I perform an F-test where the null hypothesis is

$$H_0: \beta_{cyl=6} = \beta_{cyl=8} = 0.$$

anova(m1, m2)

# Model 1: disp ~ drat
# Model 2: disp ~ drat + factor(cyl)
#   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
# 1     30 235995                                  
# 2     28  70246  2    165750 33.034 4.286e-08 ***

The p-value ($\approx$ 0) implies that we can reject the $H_0$ statement. Now I wonder which pairs of cyl have significant difference in disp.

I have 2 main questions:

1. How to perform the multiple comparisons on each pair of cyl after controlling drat? In R, pairwise.t.test() doesn't seem to handle this.

2. From the model summary below, the p-value of $\beta_{cyl=6}$ is $0.036 < 0.05$. Could I interpret this value directly and conclude:

   The mean disp with cyl=4 is different from that with cyl=6 after controlling drat.

   Or otherwise, should I adjust the p-value to avoid inflation of the type I error? E.g. multiply it by 3 (with Bonferroni correction) to get p-value $= 0.036 \times 3 = 0.108 > 0.05$, and hence conclude:

   The mean disp with cyl=4 is the same as that with cyl=6 after controlling drat.

   summary(m2)
   
   # Coefficients:
   #              Estimate Std. Error t value Pr(>|t|)    
   # (Intercept)    266.28      97.32   2.736   0.0107 *  
   # drat           -39.58      23.62  -1.676   0.1048    
   # factor(cyl)6    58.97      26.79   2.201   0.0361 *  
   # factor(cyl)8   214.65      28.33   7.578 2.96e-08 ***
   

Thanks for any helps!