Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

3 placebo 0.1 0.45 -0.350 0.1225
4 anxifree 0.6 0.72 -0.117 0.0136
5 anxifree 0.4 0.72 -0.317 0.1003
BLAH BLAH BLAH
16 joyzepam 1.8 1.48 0.317 0.1003
17 joyzepam 1.3 1.48 -0.183 0.0336
18 joyzepam 1.4 1.48 -0.083 0.0069
If you compare this output to the contents of the table I’ve been constructing by hand, you can see that
R has done exactly the same calculations that I was doing, and much faster too. So, if we want to finish
the calculations of the within-group sum of squares in R, we just ask for the sum() of the squared.devs
variable:
> SSw print( SSw )
[1] 1.39
Obviously, this isn’t the same as what I calculated, because R used all 18 observations. But if I’d typed
sum( squared.devs[1:5] ) instead, it would have given the same answer that I got earlier.
Okay. Now that we’ve calculated the within groups variation, SS w , it’s time to turn our attention
to the between-group sum of squares, SS b . The calculations for this case are very similar. The main
difference is that, instead of calculating the differences between an observation Y ik and a group mean Ȳk
for all of the observations, we calculate the differences between the group means Ȳk and the grand mean
Ȳ (in this case 0.88) for all of the groups...
group group mean grand mean deviation squared deviations
k Ȳ k Ȳ Ȳ k ´ Ȳ pȲk ´ Ȳ q2
placebo 0.45 0.88 -0.43 0.18
anxifree 0.72 0.88 -0.16 0.03
joyzepam 1.48 0.88 0.60 0.36
However, for the between group calculations we need to multiply each of these squared deviations by
N k , the number of observations in the group. We do this because every observation in the group (all N k
of them) is associated with a between group difference. So if there are six people in the placebo group,
and the placebo group mean differs from the grand mean by 0.19, then the total between group variation
associated with these six people is 6 ˆ 0.16 “ 1.14. So we have to extend our little table of calculations...
group ... squared deviations sample size weighted squared dev
k ... pȲk ´ Ȳ q2 N k N k pȲk ´ Ȳ q2
placebo ... 0.18 6 1.11
anxifree ... 0.03 6 0.16
joyzepam ... 0.36 6 2.18
And so now our between group sum of squares is obtained by summing these “weighted squared deviations”
over all three groups in the study:
SS b “ 1.11 ` 0.16 ` 2.18
“ 3.45
As you can see, the between group calculations are a lot shorter, so you probably wouldn’t usually want
to bother using R as your calculator. However, if you did decide to do so, here’s one way you could do it:
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