From: feh@duke.edu (Frank Harrell)
Newsgroups: bit.listserv.stat-l
Subject: Re: qualitative principal components
Summary: Discussion of qualitative principal components etc.
Keywords: scaling
Date: 2 Jan 93 01:33:03 GMT


Steve Blinkhorn has a good idea in opening discussion on this and
related scaling methods. Let me kick it off with an example.

For this example, SAS's PROC PRINQUAL would not converge, but a simple
approach resulted in excellent data reduction (to 1 d.f. for regression)
and a powerful predictor. In a 4000 patient dataset, I have data on
12 physiologic variables (heart rate, blood pressure, etc.). Many of
these variables have a normal range, so their relationship with
patient risk is very non-monotonic. There are also several binary,
and ordinal variables I wish to consider.

The first principal component (PC) of all the physiologic measures
had a chi-square of 450 in predicting time until death using a Cox model.
I fit a least squares regression for each physiologic var., predicting
the first PC. To not assume a shape for the regression, I used
restricted cubic spline functions with 5 knots. The prediction of the
first PC 1 yielded a first approximation to the transformation for
each variable. Once all were transformed, these new transformed variables
were used to derive a new PC 1. This new PC 1 had a chi-square of 600
for predicting death. More importantly, the derived transformations looked
amazingly like the variables' risk relationships if I peeked at the
dependent variable. I iterated this procedure 4 times, getting the
first PC to have a chi-square of about 700, and getting better
transformations. [We normally would not be looking at these chi-squares
since we want to do data reduction without examining Y, but it's
informative to do so for this example.]

If I considered polytomous variables, I would estimate transformations
by getting cell means of PC 1. For ordinal predictors, I would estimate
their re-scalings by isotonic regression on PC 1.

This dataset is ideal for such data reduction because even though
many transformations are U-shaped, the very first PC 1 derived
captured important patient severity of illness information that
could be used to scale the variables.

I would be interested in this procedure has already been described
in the literature, and if anyone has a better method. I would
also be interested in readers' reaction to including dummy variables
when deriving PC 1, using the correlation matrix to derive the PCs.
The theory isn't there, but it seems to work well most of the time.
SAS PRINQUAL's approach, I think, is more general, because it
tries to predict each variable from the best linear combination
of transformations of all the others. But I think this procedure
is more prone to non-convergence and convergence to silly scores.
In one example, age was transformed sensibly except that the lowest
age in the dataset (one year lower than the next lowest) was transformed
to a value way off the scale of the rest of the ages. This MAY have
been because I was using non-linear-tail restricted cubic splines,
and ordinary cubic splines have some tail difficulties.

Comments to stat-l welcomed.

-- 
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Frank E Harrell Jr			feh@biostat.mc.duke.edu
Associate Professor of Biostatistics
Division of Biometry                    Duke University Medical Center
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