Test if the slope in simple linear regression equals to a given constant in R

问题

I want to test if the slope in a simple linear regression is equal to a given constant other than zero.

> x <- c(1,2,3,4)
> y <- c(2,5,8,13)
> fit <- lm(y ~ x)
> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
   1    2    3    4 
 0.4 -0.2 -0.8  0.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  -2.0000     0.9487  -2.108  0.16955   
x             3.6000     0.3464  10.392  0.00913 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7746 on 2 degrees of freedom
Multiple R-squared:  0.9818,    Adjusted R-squared:  0.9727 
F-statistic:   108 on 1 and 2 DF,  p-value: 0.009133
> confint(fit)
                2.5 %   97.5 %
(Intercept) -6.081855 2.081855
x            2.109517 5.090483

In this example, I want to test if the slope is equal to 5. I know I won't reject it since 5 is in the 95% CI. But is there a function which can give me the p-value directly?

回答1:

One approach to testing whether a fit is significantly different than a particular coefficient is to construct an "offset" in which that coefficient is used as a factor applied to the x-value. You should think of this as re-setting the "zero" at least the zero-for-the-slope. The Intercept is still "free" to "move around", er, to be estimated.

 fit2 <- lm( y~x +offset(5*x) )
#----------------
 summary(fit2)
#--------
Call:
lm(formula = y ~ x + offset(5 * x))

Residuals:
   1    2    3    4 
 0.4 -0.2 -0.8  0.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  -2.0000     0.9487  -2.108   0.1695  
x            -1.4000     0.3464  -4.041   0.0561 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7746 on 2 degrees of freedom
Multiple R-squared:  0.9818,    Adjusted R-squared:  0.9727 
F-statistic:   108 on 1 and 2 DF,  p-value: 0.009133

Now compare to the results from your fit-object. The coefficients for x differ by exactly 5. The model fit statistics are the same, but as you suspected the p-value for the x-variable is much lower ... er, higher, i.e. less significant.

> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
   1    2    3    4 
 0.4 -0.2 -0.8  0.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  -2.0000     0.9487  -2.108  0.16955   
x             3.6000     0.3464  10.392  0.00913 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7746 on 2 degrees of freedom
Multiple R-squared:  0.9818,    Adjusted R-squared:  0.9727 
F-statistic:   108 on 1 and 2 DF,  p-value: 0.009133

回答2:

You just have to construct the t-statistic for the null hypothesis slope=5:

# Compute Summary with statistics      
sfit<- summary(fit)
# Compute t-student H0: intercept=5. The estimation of coefficients and their s.d. are in sfit$coefficients
tstats <- (5-sfit$coefficients[2,1])/sfit$coefficients[2,2]
# Calculates two tailed probability
pval<- 2 * pt(abs(tstats), df = df.residual(fit), lower.tail = FALSE)
print(pval)

回答3:

My impression is that the linearHypothesis function from the car package provides a standard way to do this.

For example

library(car)

x <- 1:4
y <- c(2, 5, 8, 13)
model <- lm(y ~ x)

linearHypothesis(model, "x = 1")
#> Linear hypothesis test
#> 
#> Hypothesis:
#> x = 1
#> 
#> Model 1: restricted model
#> Model 2: y ~ x
#> 
#>   Res.Df  RSS Df Sum of Sq      F  Pr(>F)  
#> 1      3 35.0                              
#> 2      2  1.2  1      33.8 56.333 0.01729 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

And here the hypothesis tests shows that the restricted model (i.e. the case when the coefficient of x equals 1) is explaining less variance than the full model in a statistically significant way as evaluated by an F statistic.

This is more useful than using offset in a formula as you can test multiple restrictions at once:

model <- lm(y ~ x + I(x^2))
linearHypothesis(model, c("I(x^2) = 1", "x = 1"))
#> Linear hypothesis test
#> 
#> Hypothesis:
#> I(x^2) = 1
#> x = 1
#> 
#> Model 1: restricted model
#> Model 2: y ~ x + I(x^2)
#> 
#>   Res.Df  RSS Df Sum of Sq    F  Pr(>F)  
#> 1      3 30.0                            
#> 2      1  0.2  2      29.8 74.5 0.08165 .
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

来源：https://stackoverflow.com/questions/33060601/test-if-the-slope-in-simple-linear-regression-equals-to-a-given-constant-in-r

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