linearHypothesis
Dear Johan,
On 2020-09-17 9:07 a.m., Johan Lassen wrote:
Dear R-users, I am using the R-function "linearHypothesis" to test if the sum of all parameters, but the intercept, in a multiple linear regression is different from zero. I wonder if it is statistically valid to use the linearHypothesis-function for this?
Yes, assuming of course that the hypothesis makes sense.
Below is a reproducible example in R. A multiple regression: y = beta0*t0+beta1*t1+beta2*t2+beta3*t3+beta4*t4 It seems to me that the linearHypothesis function does the calculation as an F-test on the extra residuals when going from the starting model to a 'subset' model, although all variables in the 'subset' model differ from the variables in the starting model. I normally think of a subset model as a model built on the same input data as the starting model but one variable. Hence, is this a valid calculation?
First, linearHypothesis() doesn't literally fit alternative models, but rather tests the linear hypothesis directly from the coefficient estimates and their covariance matrix. The test is standard -- look at the references in ?linearHypothesis or most texts on linear models. Second, formulating the hypothesis using alternative models is also legitimate, since the second model is a restricted version of the first.
Thanks in advance,Johan # R-code: y <- c(101133190,96663050,106866486,97678429,83212348,75719714,77861937,74018478,82181104,68667176,64599495,62414401,63534709,58571865,65222727,60139788, 63355011,57790610,55214971,55535484,55759192,49450719,48834699,51383864,51250871,50629835,52154608,54636478,54942637) data <- data.frame(y,"t0"=1,"t1"=1990:2018,"t2"=c(rep(0,12),1:17),"t3"=c(rep(0,17),1:12),"t4"=c(rep(0,23),1:6)) model <- lm(y~t0+t1+t2+t3+t4+0,data=data)
You need not supply the constant regressor t0 explicitly and suppress the intercept -- you'd get the same test from linearHypothesis() for lm(y~t1+t2+t3+t4,data=data).
linearHypothesis(model,"t1+t2+t3+t4=0",test=c("F"))
test = "F" is the default.
# Reproduce the result from linearHypothesis: # beta1+beta2+beta3+beta4=0 -> beta4=-(beta1+beta2+beta3) -> # y=beta0+beta1*t1+beta2*t2+beta3*t3-(beta1+beta2+beta3)*t4 # y = beta0'+beta1'*(t1-t4)+beta2'*(t2-t4)+beta3'*(t3-t4) data$t1 <- data$t1-data$t4 data$t2 <- data$t2-data$t4 data$t3 <- data$t3-data$t4 model_reduced <- lm(y~t0+t1+t2+t3+0,data=data) anova(model_reduced,model)
Yes, this is equivalent to the test performed by linearHypothesis() using the coefficients and their covariances from the original model. I hope this helps, John
John Fox, Professor Emeritus McMaster University Hamilton, Ontario, Canada web: https://socialsciences.mcmaster.ca/jfox/ >