The POSDdecDiag.lng Model

The educational testing problem, illustrating @POSD

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We are given a covariance matrix, thus it is positive definite.

We want to

maximize the sum of decrements to the diagonal of the matrix

such that the adjusted matrix is still positive semi-definite.

Application:

A comprehensive test consists of N subtests on different topics.
The overall score of a student who takes the test is the sum of the scores
on the N subtests.

If a large number of students take a test,
one measure of the goodness of the test is
how large the covariances among different subtests are relative
to the variances (the diagonal) in scores on a test.
The scores on the subtests of a well prepared test taker
taking a well prepared test should be positively correlated.
On the otherhand, if none of the test takers can understand
the questions, then the variances in scores will be large,
but the covariances between subtest scores will be close to zero.
Thus, a measure of interest is how large the variances are
relative to the covariances. One measure of this is to ask
how much we can decrease the diagonal terms (the variances on each test)
so the matrix remains positive semi-definite. For a good test,
the diagonal (variances) should be small relative to the off-diagonal (covariances);

Make sure to turn on

Solver --> Options --> Nonlinear Solver --> Quadratic Recognition;

Ref.

Fletcher, R.(1981), "A nonlinear programming problem in
statistics(Educational testing)", SIAM J. of Scientific and
Statistical Computing, vol. 2, no. 3, pp. 257-267.;

Keywords:

Statistics | Positive Definite | SDP | Covariance | Test Evaluation | POSD |