F test for β1=β2 in R-CodePudding

If my model looks like this, Y=β0 β1X1 β2X2 β3X3 β4X4, and I want to perform an F test (5%) in R for β1=β2, how do I do it? The only tutorials I can find online deal with β1=β2=0, but that's not what I'm looking for here.

CodePudding user response：

Here's an example in R testing whether the coefficient for vs is the same as the coefficient for am:

data(mtcars)
mod <- lm(mpg ~ hp   disp   vs   am, data=mtcars)
library(car)
linearHypothesis(mod, "vs=am")
# Linear hypothesis test
# 
# Hypothesis:
#   vs - am = 0
# 
# Model 1: restricted model
# Model 2: mpg ~ hp   disp   vs   am
# 
# Res.Df    RSS Df Sum of Sq      F Pr(>F)
# 1     28 227.07                           
# 2     27 213.52  1    13.547 1.7131 0.2016

CodePudding user response：

The glht function from multcomp package can do this (among others). For example, if your model is

mod1 <-lm( y ~ x1   x2   x3   x4)

then you can use:

summary(multcomp::glht(mod1, "x1-x2=0"))

CodePudding user response：

Run the model with and without the constraint and them use anova to compare them. No packages are used.

mod1 <- lm(mpg ~ cyl   disp   hp   drat, mtcars)
mod2 <- lm(mpg ~ I(cyl   disp)   hp   drat, mtcars) # constraint imposed
anova(mod2, mod1)

giving:

Analysis of Variance Table

Model 1: mpg ~ I(cyl   disp)   hp   drat
Model 2: mpg ~ cyl   disp   hp   drat
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     28 252.95                           
2     27 244.90  1    8.0513 0.8876 0.3545

The underlying calculation is the following. It gives the same result as above.

L <- matrix(c(0, 1, -1, 0, 0), 1)  # hypothesis is L %*% beta == 0
q <- nrow(L) # 1
co <- coef(mod1)
resdf <- df.residual(mod1) # = nobs(mod1) - length(co) = 32 - 5 = 27
SSH <- t(L %*% co) %*% solve(L %*% vcov(mod1) %*% t(L)) %*% L %*% co
SSH/q # F value
##           [,1]
## [1,] 0.8876363
pf(SSH/q, q, resdf, lower.tail = FALSE) # p value
##           [,1]
## [1,] 0.3544728