# [R] nested anova not giving expected results

Matthew Norton matthew.norton at umontreal.ca
Mon Apr 15 16:12:48 CEST 2002

```Hello all. This may be a trivially simple question to answer, but I'm a little
bit stumped with respect to the calculation of the F statistics in nested
anovas in R. If I understand correctly, the F statistic for the
among-subgroups but within groups hypothesis is calculated as
MS_subgroups/MS_error, while the F statistic for the factor is calculated as
MS_factor/MS_subgroups (I'm getting this from Sokal & Rohlf's _Biometry_).
However, as I understand the output from R, it calculates the F for the
factor as MS_factor/MS_error, which can significantly change the results.

As an example, I took the values from Sokal & Rohlf's example on mosquitos,
which are as follows:

cage animal length
1     1      a   58.5
2     1      a   59.5
3     1      b   77.8
4     1      b   80.9
5     1      c   84.0
6     1      c   83.6
7     1      d   70.1
8     1      d   68.3
9     2      a   69.8
10    2      a   69.8
11    2      b   56.0
12    2      b   54.5
13    2      c   50.7
14    2      c   49.3
15    2      d   63.8
16    2      d   65.8
17    3      a   56.6
18    3      a   57.5
19    3      b   77.8
20    3      b   79.2
21    3      c   69.9
22    3      c   69.2
23    3      d   62.1
24    3      d   64.5

Using the following R commands, I get this output for a nested anova:

> model<-lm(length~cage/animal)
> anova(model)
Analysis of Variance Table

Response: length
Df  Sum Sq Mean Sq F value    Pr(>F)
cage         2  665.68  332.84  255.70 1.452e-10 ***
cage:animal  9 1720.68  191.19  146.88 6.981e-11 ***
Residuals   12   15.62    1.30
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

According to the book and my understanding of nested anovas, the F statistic
for the cage:animal component is correct, but the F statistic for 'cage'
should be 332.84/191.19, giving a value of 1.741 which is not significant,
and highly different than 255.70.

Perhaps I've misunderstood, but could someone explain to me what R is doing?

In order to guide you, I'm running linux and my R-version is:

R 1.4.1 (2002-01-30).
Copyright (C) 2002 R Development Core Team

--
Matthew Norton
nortonm at magellan.umontreal.ca
Dépt. des Sciences Biologiques, Université de Montréal
C.P. 6128 Succ. centre-ville,  Montréal, Qc H3C 3J7
(514) 343-6111 x1233
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```