Organization: SYRACUSE UNIV RESEARCH DATA CENTER
X-Acknowledge-To: <METRMDHM@SUVM.SYR.EDU>
Newsgroups: bit.listserv.stat-l
Date:         Fri, 8 Jan 1993 01:42:12 LCL
From:         "Mark D. H. Miller" <METRMDHM@SUVM.SYR.EDU>
Subject:      Re: 2 x 3 contingency table

On Wed, 6 Jan 1993 19:11:29 GMT David Nichols <nichols@spss.com> said:
>Another way to look at the problem of figuring out where a significant
>chi-square statistic comes from is to examine residuals, or more
>specifically, standardized and adjusted residuals. The standardized
>residuals give a picture of the contribution of the misfit of observed
>vs. expected in each cell of the table, and the adjusted residuals are
>asymptotically normally distributed and can thus in large samples be
>used to test for cells that are mispredicted under the independence
>model. See pp. 224-on in Agresti's _Categorical Data Analysis_ for more
>information, or some of Haberman's stuff. These residuals are provided
>in SPSS in CROSSTABS.
>

I agree with David Nichols and will go even further in urging that
Haberman's "standardized residuals" or "adjusted standardized
residuals" really should be examined.  This is, in fact, just an
an extension of the tack being taken in the initial posting.  If
we take seriously Hamming's dictum that "the purpose of computing
is insight not numbers" then these calculations can be extremely
useful in studying a table, sometimes highlighting patterns which
might otherwise go unnoticed.  The chi-square test provides no
insight as to the source of significant deviations nor a real
alternative hypothesis.  Since it only takes one skewed cell to
produce a significant chi-squared statistic, it can be quite
informative to examine the adjusted (or standardized) residuals;
even graphically since the adjusted residuals lend themselves
quite nicely to half-normal plots thus extending the utility of
this common diagnostic tool.  Shelby Haberman's original paper
"The Analysis of Residuals in Cross-classified Tables" BIOMETRICS
29(1973):205-220 is still worth a read; he uses half-normal plots
as illustrations.

Procedures for computing these statistics have been available in
BMDP for years (Morton B.  Brown, BMDP-77) and they are available
in the more recent versions of SAS. I'm sure they're available
in other packages as well.

As suggested by other comments, there are many other ways to slice
and dice contingency tables < perhaps billions :-) >.  Another
simple plotting method for smaller tables (now possibly available
with correspondence anlaysis routines) was written up years ago by
Ronald Snee, "Graphical Display of Two-way Contingency Tables",
THE AMERICAN STATISTICIAN 28(Feb 74): 9-12.

(IMHO The overall chi-square test is is pitifully uninformative since
it's just a go/no-go gauge of a "hypothesis" (independence) which
is often untranslatable into any scientifically interesting statement
about the phenomena under investigation.)

What with log-linear models (in a dizzying variety of forms) and
correspondence analysis (and a plethora of disguises), it's a
mystery to me why omnibus measures of association or fit like the
standard chi-squared still seem to hold center stage in analysis
of cross-tabs when several alternatives are more versatile and
interesting as well as more consistent with mainline statistical
modeling.  Go figure ??  !

  *------------------------------------------------*
    Mark D. H. Miller (metrmdhm@suvm.syr.edu)
    Syracuse University, Faculty Computing Services
    Research Data Center   { +1 315 443-1144 }
  *------------------------------------------------*
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU