Marken spreadsheet II

[From Bruce Abbott (971204.1605 EST)]

Rick Marken (971204.1100) --

Bruce Abbott (971204.1210 EST)

If you want, I can run the ANOVA for you. Then you only have to
provide the table of data. Fair enough?

OK. I had to try to recreate it in a rush here. I don't know
if this will show up as significant (.01). While playing with
the data I realized that it's quite easy to get a significant
result when all the subjects have _very_ different functions
relating IV to DV. It all depends on the range of IV values
you use. For some IV values, you only pick up the increasing
part of the range for a non-monotonic subject. But I figure
you are intersted in seeing if it is possible to get a
significant group effect when the individual subject effects
are visibly different within the range of the IV used. So
here's some data that _might_ be statistically significant

IV 0.00 1.00 2.00

S1 1 43 46
S2 11 20 11
S3 -4 0 1
S4 2 12 22
S5 3 53 74
S6 0 14 57
S7 0 8 -11
S8 1 6 41
S9 0 - 5 -10
S10 10 9 8

XBar 2.4 16.0 24.9

S 4.65 18.4 29.3
Var 21.6 338.2 855.7
SEM 1.47 5.82 9.25

I have multiplied all scores by 10 to eliminate the decimal; this is just a
rescaling and does not affect the analysis.

The repeated measures ANOVA was significant, F(2,18)=4.456, p=.0268

However, this ANOVA requires that the variances in the populations be equal
(homogeniety of variance assumption) and (essentially) that the correlations
of scores between treatments be equal in the populations (homogeniety of
covariance). The latter will be tend to be violated if the effect of the
independent variable is inconsistent across subjects.

A rough test of homogeniety of variance is to examine the ratio of the
largest to smallest variance for each pair of treatments. The assumption is
suspect if the ratio is 3 to 1 or higher. Level 0 vs Level 1 = 15.7; Level
0 vs Level 2 = 39.6; Level 1 vs Level 2 is 2.5. Homogeniety of variance is
violated. A rough test of homogeniety of covariance is to examine the
correlations; these should be in the same ballpark. The correlations are
Level 0 vs Level 1 = +.21, Level 0 vs Level 3 = -.02, Level 2 vs Level 3 =
.75. Homogeniety of covariance is violated. The ANOVA results are
therefore untrustworthy.

A plot of scores from individual subjects reveals that as the IV increased,
some scores went sharply up, some went up less sharply, some went up and
then down, some stayed approximately level, some went up a bit and then down
sharply, and some declined a small amount across levels.

The standard errors of the mean increased from 1.47 to 9.25 across levels of
the IV, indicating increasingly divergent values as the IV went up.

Pairwise comparison of treatments (using t-tests) revealed significant
effects for 0 vs 1 (p=.0408) and 0 vs 2 (p=.0479) but not for 0 vs 2
p=.0251). For both significant t-tests were invalid because of a
significant violation of the homogeniety of variance assumption (Hartley's

To me it looks easy to spot what is going on in these data, both overall and
for individual subjects. Anyone who plots the data and connects the dots
for same subject (row) can see what I mean.