[R] F-test where the coefficients in the H_0 is nonzero

Achim Zeileis Ach|m@Ze||e|@ @end|ng |rom u|bk@@c@@t
Thu Aug 9 11:32:36 CEST 2018

```On Thu, 9 Aug 2018, John wrote:

> Hi,
>
>   I try to run the same f-test by lm (with summary) and the function
> "linearHypothesis" in car package. Why are the results (p-values for the
> f-test) different?

The standard F test in the summary output tests the hypothesis that all
coefficients _except the intercept_ are zero. Thus, all of these are the
same:

summary(lm1)
## ...
## F-statistic: 0.3333 on 1 and 1 DF,  p-value: 0.6667

linearHypothesis(lm1, "x = 0")
## ...
##   Res.Df RSS Df Sum of Sq      F Pr(>F)
## 1      2 2.0
## 2      1 1.5  1       0.5 0.3333 0.6667

lm0 <- lm(y ~ 1, data = df1)
anova(lm0, lm1)
## ...
##   Res.Df RSS Df Sum of Sq      F Pr(>F)
## 1      2 2.0
## 2      1 1.5  1       0.5 0.3333 0.6667

>
>> df1<-data.frame(x=c(2,3,4), y=c(7,6,8))
>> lm1<-lm(y~x, df1)
>> lm1
>
> Call:
> lm(formula = y ~ x, data = df1)
>
> Coefficients:
> (Intercept)            x
>        5.5          0.5
>
>> summary(lm1)
>
> Call:
> lm(formula = y ~ x, data = df1)
>
> Residuals:
>   1    2    3
> 0.5 -1.0  0.5
>
> Coefficients:
>            Estimate Std. Error t value Pr(>|t|)
> (Intercept)    5.500      2.693   2.043    0.290
> x              0.500      0.866   0.577    0.667
>
> Residual standard error: 1.225 on 1 degrees of freedom
> Multiple R-squared:   0.25, Adjusted R-squared:   -0.5
> F-statistic: 0.3333 on 1 and 1 DF,  p-value: 0.6667
>
>> linearHypothesis(lm1, c("(Intercept)=0", "x=0"))
> Linear hypothesis test
>
> Hypothesis:
> (Intercept) = 0
> x = 0
>
> Model 1: restricted model
> Model 2: y ~ x
>
>  Res.Df   RSS Df Sum of Sq      F Pr(>F)
> 1      3 149.0
> 2      1   1.5  2     147.5 49.167 0.1003
>
> 2018-08-03 13:54 GMT+08:00 Annaert Jan <jan.annaert using uantwerpen.be>:
>
>> You can easily test linear restrictions using the function
>> linearHypothesis() from the car package.
>> There are several ways to set up the null hypothesis, but a
>> straightforward one here is:
>>
>>> library(car)
>>> x <- rnorm(10)
>>> y <- x+rnorm(10)
>>> linearHypothesis(lm(y~x), c("(Intercept)=0", "x=1"))
>> Linear hypothesis test
>>
>> Hypothesis:
>> (Intercept) = 0
>> x = 1
>>
>> Model 1: restricted model
>> Model 2: y ~ x
>>
>>   Res.Df     RSS Df Sum of Sq      F Pr(>F)
>> 1     10 10.6218
>> 2      8  9.0001  2    1.6217 0.7207 0.5155
>>
>>
>> Jan
>>
>> From: R-help <r-help-bounces using r-project.org> on behalf of John <
>> miaojpm using gmail.com>
>> Date: Thursday, 2 August 2018 at 10:44
>> To: r-help <r-help using r-project.org>
>> Subject: [R] F-test where the coefficients in the H_0 is nonzero
>>
>> Hi,
>>
>>    I try to run the regression
>>    y = beta_0 + beta_1 x
>>    and test H_0: (beta_0, beta_1) =(0,1) against H_1: H_0 is false
>>    I believe I can run the regression
>>    (y-x) = beta_0 +beta_1‘ x
>>    and do the regular F-test (using lm functio) where the hypothesized
>> coefficients are all zero.
>>
>>    Is there any function in R that deal with the case where the
>> coefficients are nonzero?
>>
>> John
>>
>>         [[alternative HTML version deleted]]
>>
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>> and provide commented, minimal, self-contained, reproducible code.
>>
>>
>>
>
> 	[[alternative HTML version deleted]]
>
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