Hi
EACH person's deviation from the grand mean is partitioned into a within-group
component and a between-group component. That is, we start with the following
equality.
y - Mg = (y - Mj) + (Mj - Mg)
Squaring and summing these deviations across individuals and groups leads to
the ANOVA:
SStotal = SSwithin + SSbetween
Because SSbetween is summed over individuals within-groups and over groups,
note that there will be nj (# subjects per group) deviations of Mj from Mg
within each group. Instead of summing the constant (Mj - Mg) nj times for each
group, we simply multiply by nj.
If you teach regression before ANOVA (as I do in an upper level course), one
can easily demonstrate this by running the regression and saving the predicted
and residual values. The predicted values will be Mj, the group means.
Compute Mj-Mg, square, and sum, and one has SSbetween.
This can also be done in GLM in SPSS, because GLM allows you to save predicted
and residual scores. Here's a simple example to illustrate what is happening.
Note the bet column has Mj-Mg replicated nj times. (Change to fixed font ...
courier? ... if formatting messed up).
data list free / group score.
begin data
1 1 1 2 1 3 2 2 2 3 2 4 3 6 3 7 3 8
end data
GLM score /SAVE PRED(mg).
...
COMPUTE tot = score - mg.
COMPUTE sstot = tot**2.
GLM score BY group /SAVE PRED(mj) RESI(err).
Source Type III Sum of df Mean Square F Sig.
Squares
Corrected Model 42.000(a) 2 21.000 21.000 .002
Intercept 144.000 1 144.000 144.000 .000
group 42.000 2 21.000 21.000 .002
Error 6.000 6 1.000
Total 192.000 9
Corrected Total 48.000 8
COMPUTE sswit = err**2.
COMPUTE bet = mj - mg.
COMPUTE ssbet = bet**2.
DESCR sstot sswit ssbet /STAT = SUM.
N Sum
sstot 9 48.0000
sswit 9 6.0000
ssbet 9 42.0000
LIST.
group score mg tot sstot mj err sswit bet ssbet
1.0000 1.0000 4.0000 -3.000 9.0000 2.0000 -1.000 1.0000 -2.000 4.0000
1.0000 2.0000 4.0000 -2.000 4.0000 2.0000 .0000 .0000 -2.000 4.0000
1.0000 3.0000 4.0000 -1.000 1.0000 2.0000 1.0000 1.0000 -2.000 4.0000
2.0000 2.0000 4.0000 -2.000 4.0000 3.0000 -1.000 1.0000 -1.000 1.0000
2.0000 3.0000 4.0000 -1.000 1.0000 3.0000 .0000 .0000 -1.000 1.0000
2.0000 4.0000 4.0000 .0000 .0000 3.0000 1.0000 1.0000 -1.000 1.0000
3.0000 6.0000 4.0000 2.0000 4.0000 7.0000 -1.000 1.0000 3.0000 9.0000
3.0000 7.0000 4.0000 3.0000 9.0000 7.0000 .0000 .0000 3.0000 9.0000
3.0000 8.0000 4.0000 4.0000 16.000 7.0000 1.0000 1.0000 3.0000 9.0000
If you use SPSS, GLM has some nice features for teaching anova concepts.
Take care
Jim
James M. Clark
Professor of Psychology
204-786-9757
204-774-4134 Fax
[email protected]
>>> Marc Carter <[email protected]> 22-Feb-12 7:57 AM >>>
Hi, all --
I fear this to be a really stupid question, but...
In ANOVA, why are the squared deviations in the SS between groups scaled
(multiplied) by _n_? I was once told it was to weight them, but that somehow
doesn't seem right.
Can someone help?
TIA,
marc
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