linearHypothesis - R-help | R Mailing Lists

John Fox

Thu, Sep 17, 2020 2:20 PM #

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 at mcmaster.ca 
<mailto:jfox at 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/