Re: Q: dichotomous variables

On 11 Apr 2008 07:08:58 GMT, Erkki.Komulainen@xxxxxxxxxxxxxxxxxxx
wrote:

Hi!

Two questions about dichotomous variables:

1) Some elementary books present (exploratory) factor analysis as a
parametric method where the variables have to be continuous, at least in
interval scale, and with normal distribution (which view is rather
dogmatic). However dichotomies have been used in factor analysis with
certain things in mind. Gorsuch (Factor Analysis 2nd Ed. , 1983) and
Rummel (Applied Factor Analysis, 1970) give such examples and present
some difficulties but do not deny the use of dichotomous variables
categorically. I know that LISREL, Mplus and other programs offer
sophisticated solutions to the problem. However PCA or PFA (setting
communality iteration value low) give quite consistent results. Would
you happen to have other EFA books or other sources where this question
is dealt with or enlightened opinios?

This has been discussed before. You might
Google groups for < factor-analysis binary groups:sci.stat.* >

Consistency is not the immediate problem. Correlations
between dichotomies are restricted in range by the
relationship of their skewness. Because of this, the "mean-level"
introduces an excess number of factors, as an artifact of
the method.

Also, because dichotomies carry less information than
continuous variables, correlations tend to be small and thus
the Ns required are larger than some may expect.


2) Long ago it was quite clear to me, that methods like one-way ANOVA
and cross-tabulation with khii-square gave very similar results (when
testing the differences in proportions). Recently I calculated from a
real material the p-values with three methods: khii-square, logistic

- chi-square, in the English literature.

regression, and one-way ANOVA. The tables were 8*2. Highest frequency in
8 categories was 55 and lowest 21 and the proportions were regularly
lower than .5 in the dichotomies. I did 38 real data calculations. The
obtained p-values between khii-square and one-way ANOVA correlated (to
my surprise) .99996.

If you examine the formulas for t-tests and chi-squared,
you will see that both are based on product moment
formulations; the difference you observe, I think, is only
the difference between using a t-test or a z-test for the
resulting statistic.

Range of p-values was from .885 to .005 thus the
values were practically identical. Number of cases was 350. Correlation
with logistic was .871 thus similar but not the same. I understand that

- I'm not sure what you are testing with your 38 models.
Logistic regression does not use an additive model of
raw means of proportions, so that will account for differences
when the model is at all complicated.

with larger data sets the above mentioned khii-square and one-way anova
are identical. What bothers me is that basic texts present the anova
requirements in a very categorical way so that the reader is ready to
adopt the view "anova is basically wrong and gives definitely wrong
results" in testing differences in proportions (percentages). Comments
and possible references to this second problem are wellcome as well.

I think I would not recommend any textbook that
tried to hide the similarities and relationships between
normal-z and chi-squared, chi-squared and F-tests, and
so on. On the other hand, I'm not reading the texts you
have on hand, and I have sympathy with the quandry of
writing to various audiences. the authors should encourage
the use of the *right* tests, even while recognizing that
the other tests have similar computations and give generally
similar results. I've known psychologists whose statistical
education seemed to leave them with that weak understanding
of proportions (among other things). I don't know how much
that was from their textbooks, or from their teachers.


--
Rich Ulrich
http://www.pitt.edu/~wpilib/index.html
.