(I) Further suggestive CI plot evidence for a subset effect.
I?m glad you wrote:
?so it certainly looks like there's a subset effect.?
because the postulation of such an effect is supported by additional CI plot evidence that I?ll get to in a moment.
But first, I want to clarify what CI?s and slopes we?re talking about here.
Denote the first of the three new reqressions by Ruq:
Ruq = c on (u?, (u?)^2), where u?= u/(1+u)
with average slope Auq = first derivative of the quadratic (as you defined it) and SE of average slope also as you defined it: sqrt[ var(a1) + 4*var(a2)*(mean_x)^2 + 4*cov(a1,a2)*mean_x ], with df = n-3. (Note that Ruq is always executed per length interval per fold per dicodon subset per dicodon set per ?method?, where ?method? is your term for N=nonrandom vs R=random.)
Now define R?uq as the simple linear regression:
R?uq = Auq on the index values 1-12 of our length intervals.
where R?uq is always performed per fold, dicodon set, dicodon subset, and method
Then inasmuch as the plot you just examined graphically exhibits the slopes and CI?s for the executions of R?uq in the twelve cells defined by (fold=each, dicodon set = 1, dicodon subset = each, method = N), we can refer to this plot as the CI plot of R?uq for 1N.
And if you look at the CI plots at the end of this post for 2N, 3N, 1R, 2R, and 3R, you?ll see that:
a) the plot for 2N is almost as good as the plot for 1N (with just one inversion of S and C slightly ?below the middle?;);
b) the plot for 3N is a jumble ? S?s and C?s are interspersed throughout;
c) the plots for 1R,2R,3R are all also jumbles with interspersed S?s and C?s.
So, these six plots alone tell us that we should expect our new set of ?two-ways? for Auq ITSELF to be better for dicodon sets 1 and 2 than for 3. (By ?our new set of two-ways? I mean subset/method two-ways, as we discussed previously, i.e. the two-ways that result from the disappearance of u-Level (L,H)from the design.)
And in the next post I?m going to make following this one, you?ll see that this prediction is borne out quite nicely. In particular, the replacement of c on u with c on (u,u^2), together with the replacement of u by u?=u/(1+u), sharpens the Bonferroni tables very nicely (so as usual, kudos to your intuitions born of experience.) (Note also these two-ways for Auq were done with the same custom t-tests we used for the 3-ways, less the final mechanics for the 3-way from each pair of two-ways; I mention this because of your cautionary note not to use Excel for the tests.)
(II) Definition of slopes and their SE?s for the second new regression.
For the second of the new regressions:
c on (e, u?, u?*e), u=1/(1+u)
would you take a moment to define the relevant slopes (or ?average slopes?, as the case may be), and their associated SE?s? I?m anxious to add these to the overall computation.
(III) CI plots of R?uq for 2N, 3N, 1R, 2R, and 3R.
(Again, if the following wrap, paste them into wide fixed font document.)