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Can the two 12x12 M's in this post be "t-tested"? If so, how?
Posted:
Aug 11, 2012 12:19 AM
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At Ray Koopman's suggestion, I am posting this question concerning
whether the following two 12x12 matrices can be "t-tested" to see if
they differ significantly. (If this post does not appear in a fixed-
font such as Courier, copy/paste the two matrices into a plain text
file.)
Matrix M13
1 1 1
1 2 3 4 5 6 7 8 9 0 1 2
1 0 0 0 1 1 0 0 0 1 1 0
2 0 0 1 1 2 1 0 1 0 0
3 0 1 1 1 0 0 1 1 0
4 0 0 0 1 1 0 1 0
5 0 0 1 1 1 1 0
6 0 1 1 0 1 1
7 0 0 2 0 0
8 0 0 0 2
9 0 0 1
10 0 1
11 0
12
Matrix M11
1 1 1
1 2 3 4 5 6 7 8 9 0 1 2
1 0 0 0 1 1 0 2 0 1 1 2
2 0 0 1 1 0 0 2 2 1 0
3 0 1 1 0 0 1 1 1 0
4 0 0 1 2 1 2 2 0
5 0 0 3 1 0 0 1
6 0 1 1 0 0 0
7 0 0 0 0 0
8 0 0 0 0
9 0 0 0
10 0 0
11 0
12
(Note that in both matrices, no entries will exist for cells with i <
j, except the duplicatory entries obtained by interchanging i and j.)
Here is how these matrices were derived, in case this information is
needed to answer the question:
Derivation of the matrix M13:
1 1 1
1 2 3 4 5 6 7 8 9 0 1 2
1 0 0 0 1 1 0 0 0 1 1 0
2 0 0 1 1 2 1 0 1 0 0
3 0 1 1 1 0 0 1 1 0
4 0 0 0 1 1 0 1 0
5 0 0 1 1 1 1 0
6 0 1 1 0 1 1
7 0 0 2 0 0
8 0 0 0 2
9 0 0 1
10 0 1
11 0
12
Step 1.
Given the set of 12 regression line slopes and intercepts:
i slope(i) intcpt (i)
1 0.919189333 3.272106625
2 0.970203428 3.472702989
3 0.954830994 3.596580505
4 0.953796551 3.74717413
5 0.939894987 3.879741157
6 0.985961826 4.047887396
7 0.964293811 4.166318333
8 0.970053118 4.289706924
9 1.016286753 4.471448044
10 1.011206594 4.596713064
11 1.028200331 4.741754866
12 0.979762299 4.804540122
construct the 66 "n choose 2" pairs ((sl(i),in(i)),(sl(j),in(j))) such
that i < j.
Step 2.
Use each of these 66 pairs to determine a unique hyperboloid H(i,j) of
one sheet, and rotate each H(i,j) so as to obtain its canonical
equation (no cross-terms). For example, using the slope and intercept
pairs from rows 3 and 7, generate the equation of H(3,7) with these co-
efficients for x^2, y^2, and x^2 respectively:
+ 0.045532445463041852339
- 2.6228375259947364878
+ 2.2795579677139341259
Step 3.
Repeat Steps 1-2 to obtain a second set of 66 hyperboloids, using this
set of 12 slopes and intercepts:
i slope(i) intcpt (i)
1 0.835833663 3.268537562
2 0.930007862 3.511291774
3 0.921386712 3.644709513
4 0.921623772 3.776648594
5 0.975267321 3.975225513
6 0.985528515 4.118473417
7 0.980759597 4.246615012
8 0.955637928 4.335339283
9 1.050954764 4.570948528
10 1.028807082 4.673430858
11 0.944424362 4.701392744
12 0.962875894 4.847733313
For ease of reference, denote each of these 66 H's by H(2,i,j) and
denote each of the 66 original H's obtained in Steps 1-3 by H(1,i,j).
Step 4.
Compute the distance between any two hyperboloids H(1,u,v) and
H(2,g,h) as follows:
"Each hyperb has one negative and two positive coeffs. Say that gives
a[1], a[2], a[3] and b[1], b[2], b[3]. For each i = 1, 3 find the
larger of a[i]/b[i], b[i]/a[i]. Add up those three ratios. Identical
hyperbs produce a norm of 3. Of course it would be trivial to use
some different norm."
(Bob Lewis (Fordham) came up with this norm; as you can see, he's
perfectly willing to change it):
Step 5.
Find all pairs of hyperboloids H(1,u,v) and H(2,g,h) for which:
i) the distance between H(1,u,v) and H(2,g,h) is less than 3.1
(according to Bob's norm);
ii) u,g differ by no more than 1 and v,h differ by no more than 1;
iii) if u <> g then v = h, and vice-versa.
Step 6.
Initialize a 12 x 12 matrix "M13" to 0 in each cell, and for each pair
of H's found in Step 5, add 1 to cells (u,v) and (g,h) of M13.
Derivation of the Matrix M11:
M11
1 1 1
1 2 3 4 5 6 7 8 9 0 1 2
1 0 0 0 1 1 0 2 0 1 1 2
2 0 0 1 1 0 0 2 2 1 0
3 0 1 1 0 0 1 1 1 0
4 0 0 1 2 1 2 2 0
5 0 0 3 1 0 0 1
6 0 1 1 0 0 0
7 0 0 0 0 0
8 0 0 0 0
9 0 0 0
10 0 0
11 0
12
Repeat steps 1-6 using these two sets of slopes and intercepts:
0.919189333 3.272106625
0.970203428 3.472702989
0.954830994 3.596580505
0.953796551 3.74717413
0.939894987 3.879741157
0.985961826 4.047887396
0.964293811 4.166318333
0.970053118 4.289706924
1.016286753 4.471448044
1.011206594 4.596713064
1.028200331 4.741754866
0.979762299 4.804540122
0.905986553 3.27364294
0.885904618 3.395887597
0.963660407 3.631431109
0.914517315 3.719920424
0.932219739 3.881191654
0.936877024 4.018221025
0.961521047 4.177249199
0.959417369 4.294850847
0.962399241 4.423458466
0.958904534 4.551600259
0.975759512 4.6940141
0.967654684 4.807156534
(Note that the first of these two sets is identical to the set used in
Step 1 above, but the second is not identical to the set used in Step
3 above.)
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