[R] ANOVA
Prof Brian Ripley
ripley at stats.ox.ac.uk
Thu Jun 29 14:59:24 CEST 2000
> Date: Thu, 29 Jun 2000 14:22:24 +0000
> From: Lilla Di Scala <lilla at dimat.unipv.it>
> I have a problem regarding the anova() output. When I apply it to a
> single regression model, I do not understand how the values
> corresponding to the F statistics are obtained by the software. I
> believe that they are computed using differences between residual sums
> of squares of sequential models obtained from the imputed one (removing
> one independent variable at a time). I have tried to do the computations
> by hand, but my figures do not match R's.
That's essentially correct, but only if you mean removing one variable
at a time and not replacing it. Here is an example:
> data(stackloss)
> fm <- lm(stack.loss ~ ., data=stackloss)
> anova(fm)
Analysis of Variance Table
Response: stack.loss
Df Sum Sq Mean Sq F value Pr(>F)
Air.Flow 1 1750.12 1750.12 166.3707 3.309e-10
Water.Temp 1 130.32 130.32 12.3886 0.002629
Acid.Conc. 1 9.97 9.97 0.9473 0.344046
Residuals 17 178.83 10.52
> fm1 <- update(fm, . ~ . - Acid.Conc.)
> anova(fm1, fm)
Analysis of Variance Table
Model 1: stack.loss ~ Air.Flow + Water.Temp
Model 2: stack.loss ~ Air.Flow + Water.Temp + Acid.Conc.
Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)
1 18 188.795
2 17 178.830 1 9.965 0.9473 0.3440
> fm2 <- update(fm1, . ~ . - Water.Temp)
> fm3 <- update(fm2, . ~ . - Air.Flow)
> anova(fm2, fm1)
Analysis of Variance Table
Model 1: stack.loss ~ Air.Flow
Model 2: stack.loss ~ Air.Flow + Water.Temp
Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)
1 19 319.12
2 18 188.80 1 130.32 12.425 0.002419
> anova(fm3, fm2)
Analysis of Variance Table
Model 1: stack.loss ~ 1
Model 2: stack.loss ~ Air.Flow
Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)
1 20 2069.24
2 19 319.12 1 1750.12 104.20 3.774e-09
Note that the SSqs are all the same, but the sequential table compares
them to the residual MSq, not to the next-larger model (as there is
no test that terms which have already been dropped are not significant).
--
Brian D. Ripley, ripley at stats.ox.ac.uk
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272860 (secr)
Oxford OX1 3TG, UK Fax: +44 1865 272595
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