R version 2.9.2 (2009-08-24)
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ISBN 3-900051-07-0
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>
> supervisor.table <- read.table('http://www-stat.stanford.edu/~jtaylo/courses/stats191/data/supervisor.table', header=T)
> attach(supervisor.table)
>
> # original regression model
>
> supervisor.lm <- lm(Y ~ X1 + X2 + X3 + X4 + X5 + X6)
>
>
> # A function to do "nested" F-tests
>
> f.test.lm <- function(R.lm, F.lm) {
+ SSE.R <- sum(resid(R.lm)^2)
+ SSE.F <- sum(resid(F.lm)^2)
+ df.num <- R.lm$df - F.lm$df
+ df.den <- F.lm$df
+ F <- ((SSE.R - SSE.F) / df.num) / (SSE.F / df.den)
+ p.value <- 1 - pf(F, df.num, df.den)
+ return(data.frame(F, df.num, df.den, p.value))
+ }
>
> # reduced model: only X1 and X3
>
> reduced.lm = lm(Y ~ X1 + X3)
>
> print(anova(reduced.lm, supervisor.lm))
Analysis of Variance Table
Model 1: Y ~ X1 + X3
Model 2: Y ~ X1 + X2 + X3 + X4 + X5 + X6
Res.Df RSS Df Sum of Sq F Pr(>F)
1 27 1254.65
2 23 1149.00 4 105.65 0.5287 0.7158
>
> # enforce constraint of the coefficients for X1 and X3
> # we do this by making a new variable Z=X1+X3
>
> Z = X1 + X3
> equal.lm = lm(Y ~ Z)
>
> print(anova(equal.lm, reduced.lm))
Analysis of Variance Table
Model 1: Y ~ Z
Model 2: Y ~ X1 + X3
Res.Df RSS Df Sum of Sq F Pr(>F)
1 28 1424.59
2 27 1254.65 1 169.95 3.6572 0.06649 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> # another example from the book:
> # test that beta1+beta3=1
>
> Z = X1-X3
> W = Y-X3
> print(f.test.lm(lm(W ~ Z), reduced.lm))
F df.num df.den p.value
1 1.611813 1 27 0.2150713
>
> # this is the same as using the 'offset term' which is
> # subtracted from the response before fitting the model
>
> print(anova(lm(Y ~ Z, offset=X3), reduced.lm))
Analysis of Variance Table
Model 1: Y ~ Z
Model 2: Y ~ X1 + X3
Res.Df RSS Df Sum of Sq F Pr(>F)
1 28 1329.5
2 27 1254.7 1 74.9 1.6118 0.2151
>
> # look at the models themselves
>
> print(summary(lm(Y ~ Z, offset=X3)))
Call:
lm(formula = Y ~ Z, offset = X3)
Residuals:
Min 1Q Median 3Q Max
-9.799 -6.242 0.252 4.911 12.263
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.1665 1.7079 0.683 0.5
Z 0.6938 0.1129 6.147 1.23e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.891 on 28 degrees of freedom
Multiple R-squared: 0.7505, Adjusted R-squared: 0.7415
F-statistic: 84.21 on 1 and 28 DF, p-value: 6.196e-10
> print(summary(lm(W ~ Z)))
Call:
lm(formula = W ~ Z)
Residuals:
Min 1Q Median 3Q Max
-9.799 -6.242 0.252 4.911 12.263
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.1665 1.7079 0.683 0.5
Z 0.6938 0.1129 6.147 1.23e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.891 on 28 degrees of freedom
Multiple R-squared: 0.5744, Adjusted R-squared: 0.5592
F-statistic: 37.79 on 1 and 28 DF, p-value: 1.233e-06
>
>
> proc.time()
user system elapsed
0.644 0.028 0.711
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