# [R] linearHypothesis

Johan Lassen joh@n|@@@en @end|ng |rom gm@||@com
Thu Sep 17 15:07:46 CEST 2020

```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?
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?

# 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)

linearHypothesis(model,"t1+t2+t3+t4=0",test=c("F"))

# 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)

--
Johan Lassen

"In the cities people live in time -
in the mountains people live in space" (Budistisk munk).

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