Newsgroups: bit.listserv.stat-l
Date:         Tue, 17 Nov 1992 09:10:00 EST
From:         "Philip Gallagher,(919)966-1065" <UPHILG@UNC.BITNET>
Subject:      Do assumptions violations mean anything?

    The recent discussion of assumption-violations has reinforced
    my long standing (not very popular) observation that statisticians
    and others attempting to practice statistics often get so
    comfortable with our (valid) jargon and (again valid) knee-jerk
    practices that we gloss over, perhaps even in our own minds, any
    real meaning that our practices may be accessing.  I illustrate
    with a one-way ANOVA.  So we do some kind of test for homo-
    scedasticity;  it fails.  What does that mean?  Not "The data
    violate the assumptions", not really.  What it means is that we
    have one or more cells in which the distribution of the data differs
    from the distribution of the data in other cells.  If that were
    true, what real meaning would an analysis that certifies the cell
    means are not equal have?  Suppose the distribution in one cell
    were highly skewed to the right and another cell to the left,
    but with equal means and equal variances?  It would almost
    certainly be a very unusual set of scientific data indeed where
    the failure of an ANOVA to detect differences in means would
    be the result the scientist wanted to have called to his attention.
    It strikes me that the testing of assumptions is most easily taught
    by showing the students that violation of the assumptions means
    that the distributions differ, but not necessarily in the way that
    the specific procedure (say, ANOVA) is directed at (equality of
    means).  Once the student sees clearly the underlying phenomon
    that leads to an assumption violation it becomes very hard to
    prevent the student from looking for those violations (and in a
    very perceptive way, too).

    I have had amazingly good experiences in the last few years by
    encouraging students to "look for systematic characteristics in
    those persons for whom the model does not predict well" rather
    than "examine the residuals".  My first real success along this
    line came after having begged, cajoled, and demanded that a
    osteoporosis student examine the residuals in the model for three
    months and gotten nowhere;  when I said "Well, forget about looking
    at the residuals, just figure out which groups the model doesn't
    fit well" I got the best part of the answer in (gasp!) two hours.
    Two months later over a celebratory drink I flabbergasted the
    student by explaining that she had actually done an analysis of
    the residuals.  (There is success in this world sometimes - this
    person is now faculty at another school, and last month one of her
    students was complaining to me about being forced to examine
    residuals!  Hallelujah!)

    The gist of this not extensively edited comment is that we often
    become so entranced by the mathematical aspects of what we are
    doing that we fail to remember that the mathematics (at least
    for statistical analyses) is usually a reflection of some aspect
    of the data that one need not be a statistician to understand.

    I conclude with my favorite way of examing the similarity of two
    distributions, attributable to Dana Quade.  One plots the empirical
    distribution of distn1 on the Y-axis against the empirical distribu-
    tion of distn2 on the X-axis.  (When not looking for differences
    in location I center both distributions at zero first).  If the
    distributions are similar, the result will be a straight line (at
    45 degrees if you scale the axes cleverly).  Large differences
    in dispersion result in S-shaped pictures;  the graph is very
    informative, both to the statistician and to the scientist.
    Sometimes this picture makes the differences in the distributions
    so clear that everyone gladly abandons the original intention of
    testing means.  Thank goodness.

                              Phil Gallagher
                              uphilg@unc
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU