Suppose we have fitted a standard linear model with a categorical variable which has 3 levels A, B and C.
x <- factor(sample(c("A", "B", "C"), 200, replace = TRUE))
y <- rnorm(200)
fit <- lm(y ~ x)
How can we use this model to determine whether the coefficient for level $B$ = coefficient for level $C$?
You can perform two regression, one with separate values for $B$ and $C$, and another in which just observe $x$ being either $B$ or $C$
# individual variables for B as well as C
reg1 = lm(y ~ I(x == "B") + I(x == "C"))
# either B or C
reg2 = lm(y ~ I(x %in% c("B", "C")))
Then you can do an ANOVA analysis (F-Test or chi square) to check whether the p-value suggests to separate $B$ and $C$
# F-Test
anova(reg1, reg2, test = "F")
In your example, the results look as follows
set.seed(1)
x <- factor(sample(c("A", "B", "C"), 200, replace = TRUE))
y <- rnorm(200)
# individual variables for B as well as C
reg1 = lm(y ~ I(x == "B") + I(x == "C"))
# either B or C
reg2 = lm(y ~ I(x %in% c("B", "C")))
# F-Test
anova(reg1, reg2, test = "F")
# Results
Analysis of Variance Table
Model 1: y ~ I(x == "B") + I(x == "C")
Model 2: y ~ I(x %in% c("B", "C"))
Res.Df RSS Df Sum of Sq F Pr(>F)
1 197 200.82
2 198 200.87 -1 -0.051686 0.0507 0.8221
The p-value is far away from any conventional significance levels, suggesting that you can consider $B=C$ (but be aware that conclusions based on p-values are controversial).
