Oh! That slope! (Of the tangent to the quadratic at x = 0.) Duh!
I was construing ?slope? as just a alternative term for ?coefficient? ? don?t know why. Someday I?ll learn if that if I can?t make sense of something you?ve written, then I?ve construed something wrongly.
When I re-run the analysis, I will use (u/1+u) instead of u. So I assume I would use ln(u/(1+u)), and (ln(u+(1/u)))^2, parallel to ln(u) and (ln(u))^2? (Unless you don?t want me to take logs ? please clarify here.)
3. You wrote:
?For instance, you might get the slope at each data point and then use their literal average, a1 + 2*a2*mean_x. But what if cells with different x-means give the same a1 and a2? Should their "average slope" measures be the same or different? That's the kind of question you have to ask yourself.?
Since I can?t think that far ahead in the abstract (as you can), I will use ?a1 + 2*a2*mean_x? initially, and see if any peculiarities arise of the sort you mention (or others.)
Apart from correcting the general theoretical deficiency which you perceive as arising from uL/uH dichotomization itself, another benefit of doing the re-analysis with the three new regressions is presumably that it will help us improve the two tables I presented at the end of the other thread:
Extent to which Significant 3-way Interactions across Length Intervals are Exhibited at Specific Length Intervals (Data Not Shown for 2RS and 2RC)
3-way u Non-Random Coeff Len Int p Lev 2S 2C % Chg eS ALL .014 L 2.288 6.075 165.5% H 7.993 8.309 4.0%
1 .035 L 1.578 6.545 314.6% H 5.719 4.760 -16.8%
euSe ALL .005 L 2.402 6.154 156.2% H 8.265 7.975 -3.5%
1 .025 L 2.076 6.640 219.8% H 6.002 3.926 -34.6%
4 .011 L 1.250 7.881 530.3% H 10.755 7.658 -28.8%
uS ALL .0003 L 0.277 -0.082 -129.6% H 0.050 -0.848 -1796.0%
5 .040 L -0.287 -0.345 20.0% H 0.289 -0.325 -212.3%
9 .0002 L 0.548 -0.078 -114.2% H 0.436 -1.056 -342.4%
10 .011 L 0.563 -0.006 -101.0% H 0.261 -1.103 -522.9%
11 .032 L 0.617 -0.142 -123.1% H 0.354 -1.178 -432.9%
euSu ALL .036 L 0.240 -0.037 -115.4% H -0.190 -0.760 300.0%
9 .008 L 0.482 -0.067 -113.8% H 0.141 -0.829 -686.3%
11 .012 L 0.415 -0.083 -120.1% H 0.185 -1.253 -777.3%
(A) with respect to the equivalent of Table I that will be constructed from the new 2-ways, I am hoping first that more of the new 2-ways at specific length-intervals will show probabilities < .05, as opposed to the paltry yield of ?good? interval-specific 3-ways in the current table.
(B) with respect to the equivalent of Table II that will be constructed from the p?s in the new Table I, I am hoping that more of these p?s (both cross-interval and interval-specific) will withstand plausible Bonferroni correction.
Is there any reason to assume A PRIORI that (A-B) will not be the case, due to some subtle theoretical reason that I?m too ignorant to see? And more generally, is there a better pair of tables that should be generated for the new 2-ways, instead of tables like (I-II):
Also, just out of curiousity, I?m wondering if it would be legitimate to ?allow? row 3 in Table II as well as rows 1 and 2? (You mentioned once that some manual tinkering with Bonferroni results is allowable.)
Thanks as always for your considered consideration of these matters. It will take me 2-3 days to add code and re-run for c on (u,u^2)with computation of a first-cut ?average? slope, depending on day-job demands.
In the meantime, I?m hoping you?ll continue your exegesis with an explication of c on (e, u, u*e), the second of the three new regressions.