[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

Fmax).

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.

Regards,

Bruce