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Re: Some preliminary numbers and three preliminary simple questions ...
Posted:
Aug 18, 2012 1:22 AM
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On Aug 17, 6:07 am, djh <halitsky.d@att.net> wrote:
> Thanks as always for your willingness to contine this discussion.
>
> Your summary is remarkably accurate ? the only clarification I would
> add is to modify this statement:
>
> ?Then you create a 12 x 12 matrix, say M12, by comparing the
> hyperboloids from the column 1 regressions to those from the column 2
> regressions; and a similar matrix, say M13, by comparing the
> hyperboloids from the column 1 regressions to those from the column 3
> regressions.?
>
> to this:
>
> ?Then you create a 12 x 12 matrix, say M12, by comparing pairs of
> hyperboloids from the column 1 regressions to pairs from the column 2
> regressions; and a similar matrix, say M13, by comparing pairs of
> hyperboloids from the column 1 regressions to pairs from the column 3
> regressions.?
>
> Turning now to the ?data frames?, I?m sorry I used this term unclearly
> (I thought I had used it before in our discussions.)
>
> My previous statement was in error. There are actually 240 non-random
> data frames (not 180), and 240 random data frames (not 180.)
>
> In particular, since we?re working with 6 folds f1,...,f6, we have 15
> unique pairs of folds:
>
> f1f2
> f1f3
> f1f4
> f1f5
> f1f6
>
> f2f3
> f2f4
> f2f5
> f2f6
>
> f3f4
> f3f5
> f3f6
>
> f4f5
> f4f6
>
> f5f6
>
> and from these we can form the following 20 pairs of pairs of folds:
>
> f1f2 f1f3
> f1f2 f1f4
> f1f2 f1f5
> f1f2 f1f6
> f1f3 f1f4
> f1f3 f1f5
> f1f3 f1f6
> f1f4 f1f5
> f1f4 f1f6
> f1f5 f1f6
>
> f2f3 f2f4
> f2f3 f2f5
> f2f3 f2f6
> f2f4 f2f5
> f2f4 f2f6
> f2f5 f2f6
>
> f3f4 f3f5
> f3f4 f3f6
> f3f5 f3f6
>
> f4f5 f4f6
>
> So if we adopt your notion of a 12 x 3 table of regressions, each of
> the above 20 pairs of pairs will yield one of these tables, eg.
> f1f2,f1f3 will yield T123, etc.
>
> But we can generate the 20 tables Tijk for data derived by choosing:
>
> 1) one of 6 non-random dicodon sets
> 2) restricting u to uL or uH
>
> and so we will have 20*6*2 = 240 tables Tijk computed from data
> obtained using non-random dicodon sets.
>
> And similarly, we can also generate the 20 tables Rijk for data
> derived by choosing:
>
> 3) one of 6 random dicodon sets
> 4) restricting u to uL or uH
>
> and so we will again have 20*6*2 = 240 matrices Rijk computed from
> data obtained using non-random dicodon sets.
>
> Each of the 240 tables Tijk will yield a pair of matrices Mij,Mik for
> which the scalars Vij and Vik can be computed using your scalar
> function V.
>
> And each of the 240 tables Rijk will yield a pair of matrices rMij,
> rMik for which the scalars rVij and rVik can be computed using your
> scalar function V.
In the analysis of a triple {i,j,k}, j & k are interchangeable with
one another but not with i, which plays a different role. Is there a
reason that different folds get to be 'i' different numbers of times?
Instead of the 20 triples
{1,2,3},{1,2,4},{1,2,5},{1,2,6},{1,3,4},{1,3,5},{1,3,6},{1,4,5},
{1,4,6},{1,5,6},
{2,3,4},{2,3,5},{2,3,6},{2,4,5},{2,4,6},{2,5,6},
{3,4,5},{3,4,6},{3,5,6},
{4,5,6}}
do you really want the 60 triples
{1,2,3},{1,2,4},{1,2,5},{1,2,6},{1,3,4},{1,3,5},{1,3,6},{1,4,5},
{1,4,6},{1,5,6}, {2,1,3},{2,1,4},{2,1,5},{2,1,6},{2,3,4},{2,3,5},
{2,3,6},{2,4,5},
{2,4,6},{2,5,6},
{3,1,2},{3,1,4},{3,1,5},{3,1,6},{3,2,4},{3,2,5},{3,2,6},{3,4,5},
{3,4,6},{3,5,6},
{4,1,2},{4,1,3},{4,1,5},{4,1,6},{4,2,3},{4,2,5},{4,2,6},{4,3,5},
{4,3,6},{4,5,6},
{5,1,2},{5,1,3},{5,1,4},{5,1,6},{5,2,3},{5,2,4},{5,2,6},{5,3,4},
{5,3,6},{5,4,6},
{6,1,2},{6,1,3},{6,1,4},{6,1,5},{6,2,3},{6,2,4},{6,2,5},{6,3,4},
{6,3,5},{6,4,5}
in which each fold gets to be 'i' the same number of times?
>
> So we will actually have:
>
> 5) a set S of 480 values of the function V derived from data obtained
> using non-random dicodon sets, and a distribution D of these values of
> V.
>
> 6) a set Sr of 480 values of the function V derived from data obtained
> using random dicodon sets, and a distribution Dr of these values of V.
>
> And therefore, my first question (Q1) is whether there is some
> legitimate way to determine whether D differs significantly from
> Dr.
No, at least not with your current resources. The problem is that the
480 (or whatever) V-values share data, both within and between pairs.
You can easily estimate the difference between the means of the V and
Vr distributions, but the complex dependence structure of the data
means that there is no easy estimate of the standard error of the
difference.
>
> Furthermore, suppose for the sake of discussion that the answer to
> this question is ?yes?, and that in fact, there is a significant
> difference between D and Dr.
>
> Then my second question (Q2) would be whether this entitles us to
> treat the values of V in the set S as characterizing some ?real?
> properties of the data associated with at least SOME non-random
> dicodon sets (namely our 6 non-random dicodon sets.)
>
> But of course, if your answer to Q1 is ?no?, then my second question
> would be whether the nature of the distributions D and Dr will tell
> you what you need to know in order to figure how to bootstrap in order
> to decide whether the distribution of values of V differs in data
> obtained using non-random dicodon sets versus data obtained using
> random dicodon sets.
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