[R] How to read ANOVA output

[(Ted Harding)]

On 18-Aug-10 07:42:23, Stephen Liu wrote:
> Hi folks,
> Where can I find document re "how to read anova output"?
> Google found many of them.  But seemingly non of them can
> explain to me following output:-
> 
>> tabA = c(5.67, 5.67, 5.55, 5.57)
>> tabB = c(5.75, 5.47, 5.43, 5.45)
>> tabC = c(4.74, 4.45, 4.65, 4.94)
>> tabs = data.frame(tabA, tabB, tabC)
> 
>> tablets = stack(tabs)
> 
>> anova(lm(values ~ ind, data = tablets))
> Analysis of Variance Table
> Response: values
>          Df    Sum Sq  Mean Sq  F value      Pr(>F)
> ind       2   2.05787  1.02893   45.239   2.015e-05 ***
> Residuals 9   0.20470  0.02274
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The above is the basic standard format for the results of a
1-way analysis of variance (differences between means of groups
compared with within-group differences between observations
and group means). You need to understand how that works (basic
statistical theory) before even thinking of looking at the
Tukey thing (omitted in this reply).

The following is an explanation of your 1-way ANOVA written
entirely in R (preceded by a duplicate of your ANOVA output):

## anova(lm(values ~ ind, data = tablets))
## Analysis of Variance Table
## Response: values
##          Df    Sum Sq   Mean Sq   F value      Pr(>F)
## ind       2   2.05787   1.02893    45.239   2.015e-05 ***
## Residuals 9   0.20470   0.02274
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

tabA = c(5.67, 5.67, 5.55, 5.57)
tabB = c(5.75, 5.47, 5.43, 5.45)
tabC = c(4.74, 4.45, 4.65, 4.94)

nA <- length(tabA) ; nB <- length(tabB) ; nC <- length(tabC)
nG <- nA + nB + nC
mG <- mean(c(tabA,tabB,tabC))
mA <- mean(tabA) ; mB <- mean(tabB) ; mC <- mean(tabC)
SSres <- sum((tabA-mA)^2) + sum((tabB-mB)^2) + sum((tabC-mC)^2)
SSres # = 0.2047

SSeff <- nA*(mA-mG)^2 + nB*(mB-mG)^2 + nC*(mC-mG)^2
SSeff # = 2.057867

## Number of groups = 3 hence df.groups = (3-1) = 2
df.groups <- 2
meanSSeff <- SSeff/df.groups
meanSSeff # = 1.028933

## df for residuals in each group = (n.group - 1):
df.res <- (nA-1) + (nB-1) + (nC-1)  ## = 3 + 3 + 3 = 9
meanSSres <- SSres/df.res
meanSSres # = 0.02274444

## Fisher's F-ratio statistic = meanSSeff/meanSSres:
F <- meanSSeff/meanSSres
F         # = 45.23889

## P-value for F as test of difference between group means
## relative to within-group residuals (upper tail):
Pval <- pf(F, df.groups, df.res, lower.tail=FALSE)
Pval      # = 2.015227e-05


Ted.

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E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
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Date: 18-Aug-10                                       Time: 09:41:08
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