This example can also be applied to a GLM but I choose to use a LM because the coefficients are more easily interpreted.

set.seed(3) dat <- data.frame(var1 = as.factor(rep(c("a","b"), each = 8)), var2 = as.factor(rep(c("c","d"), 8)), response = round(rnorm(16, 10, 4), 0)) mod <- lm(response ~ var1 * var2, data = dat) summary(mod) interaction.plot(dat$var2, dat$var1, dat$response) # the intercept in the first line of summary(mod) tests var1a_var2c == 0 # in the second line of summary(mod) you test var1a_var2c - var1b_var2c == 0, # which is the effect of var1 within var2c # in the third line you test var1a_var2c - var1a_var2d == 0, # which is the effect of var2 within var1a # in the fourth line you test the interaction, which is: # (var1a_var2c - var1a_var2d) - (var1b_var2c - var1b_var2d) == 0 # relevel var2, then the intercept is var1a_var2d # and the second line tests var1a_var2d - var1b_var2d == 0, # which is the effect of var1 within var2d dat1 <- dat dat1$var2 <- relevel(dat$var2, ref = "d") mod1 <- lm(response ~ var1 * var2, data = dat1) summary(mod1) # test this with calling contrast(): require(contrast) var2_in_var1 <- contrast(mod, list(var2 = levels(dat$var2), var1 = "b"), list(var2 = levels(dat$var2), var1 = "a")) print(var2_in_var1, X = TRUE)

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