Posted by
Rich Ulrich on
URL: http://spssx-discussion.165.s1.nabble.com/Shapiro-Wilks-Statistic-tp5739693p5739712.html
Thanks -
I agree that univariate normality should not be at issue, and I
wonder anew at what your reviewer had in mind. Showing the
extreme kurtosis should be enough (if there were any question).
From Bruce's citation - I have doubts that the assumptions of
underlying bivariate normality are met, when it comes to polychoric.
If the polychoric r's are larger, then they also have larger standard
errors than the Pearson r's. I think I had one set of data that I ever
played with, testing polychoric for factoring, and I ended up using
Pearson's.
What also came to mind was Item (or Latent) Response Theory, which
uses logistic transformations. I know IRT from reading, too, rather than
hands-on experience. I don't know if they have examples with U-shapes.
(a) Underlying Logistic is shaped like Normal with slightly fatter tails, so
it might not meet assumptions any better for U shaped data. (b) If both
tails are always fat, the correlations won't change by much. (c) The
people who perform IRT have some interesting ancillary statistics.
--
Rich Ulrich
I have a number of religious identity labels that participants rate in terms of how well they fit. There are five response categories from ‘not at all’ to ‘very well.’ As is the case in most research like this, the distribution of responses is basically
U-shaped. (Basically, a label either fits or it does not, very few in-between responses.) Mardia’s statistic shows a huge violation of multivariate normality, as one would expect. As such, univariate normality is not an issue, which is what I should have said
originally. Pearson correlations underestimate the degree of linear relationship in these situations, and we often turn to polychoric correlations instead. (I am loathe to use nonparametric correlations when perfectly good parametric methods exist. When categorical
data are well-behaved, I also prefer to use parametric models. I agree with one of my mentors, Fred Lord: the data don’t know where they came from. So, if they act parametric, treat them that way. If not, don’t.) Polychoric correlations provide a more accurate
estimate of the true degree of linear association under these conditions. The matrix was then submitted to an EFA to see what patterns of self-labeling emerge, if any.
Hope this clarifies the situation a bit better.
Harley
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I'm a little confused. You say,
"The data are ordinal in nature,
so the issue of normality doesn't even
apply to my situation."
"... the issue doesn't even apply ... "
That implies to me that you are doing analyses after rank-transform
("nonparametric") without considering ordinary ANOVA -- and the
reviewer asks you to justify abandoning the raw scores.
If I were reviewer, I might ask the same. I hope I would put the
request more clearly than what you seem to have received.
Bruce writes, unclear about the situation -
If it is to justify use of a parametric test, my advice would be, "Don't
bother!" Like me, Bruce is biased against non-parametric testing for
decently-behaved ordinal data.
--
Rich Ulrich
Hi Bruce,
An editorial reviewer is requiring me to do so. Otherwise, I would not. The data are ordinal in nature, so the issue of normality doesn't
even apply to my situation. Regardless, I am complying . . .
Harley
Dr. Harley Baker
Professor Emeritus of Psychology
California State University Channel Islands
One University Drive
Camarillo, CA 93012
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