[R] linearHypothesis

[John Fox]

Dear Johan,

It's generally a good idea to keep the conversation on r-help to allow 
list members to follow it, and so I'm cc'ing this response to the list.

I hope that it's clear that car::linearHypothesis() computes the test as 
a Wald test of a linear hypothesis and not as a likelihood-ratio test by 
model comparison. As your example illustrates, however, the two tests 
are the same for a linear model, but this is not true more generally.

As I mentioned, you can find the details in many sources, including in 
Section 5.3.5 of Fox and Weisberg, An R Companion to Applied Regression, 
3rd Edition, the book with which the car package is associated.

Best,
  John

On 2020-09-17 4:03 p.m., Johan Lassen wrote:
> Thank you John - highly appreciated! Yes, you are right, the less 
> complex model may be seen as a restricted model of the starting model. 
> Although the set of variables in the less complex model is not directly 
> a subset of the variables of the starting model. What confused me at 
> first was that I think of a subset model as a model having a direct 
> subset of the set of variables of the starting model. Even though this 
> is not the case in the example, the test still is on a restricted model 
> of the starting model.
> Thanks,
> Johan
> 
> Den tor. 17. sep. 2020 kl. 15.55 skrev John Fox <jfox using mcmaster.ca 
> <mailto:jfox using mcmaster.ca>>:
> 
>     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/
>      >
> 
> 
> 
> -- 
> Johan Lassen
> 
> "In the cities people live in time -
> in the mountains people live in space" (Budistisk munk).

-- 
John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
web: https://socialsciences.mcmaster.ca/jfox/