Re: Shapiro-Wilks Statistic
Posted by
Jon Peck on
URL: http://spssx-discussion.165.s1.nabble.com/Shapiro-Wilks-Statistic-tp5739693p5739713.html
And the SPSSINC HETCOR extension command calculates Pearson, polychoric,and polyserial correlations based on he measurement levels.
Thanks for providing that detail, Harley. For the benefit of readers who may
not be terribly familiar with polychoric correlation, here is the definition
from the relevant Wikipedia page:
"In statistics, polychoric correlation[1] is a technique for estimating the
correlation between two theorised normally distributed continuous latent
variables, from two observed ordinal variables."
And this paragraph from Rigdon & Ferguson (1991) goes even further in saying
that the underlying variables are assumed to have a bivariate normal
distribution:
"The polychoric correlation coefficient, introduced by Pearson, is an
alternative to the Pearson r specifically for situations in which the
variables of interest are continuous but the measurement instruments yield
data that may only be ordinal (Pearson and Pearson 1922; Ritchie-Scott
1918). Olsson (1979b) developed two maximum likelihood procedures for
estimating the polychoric, both based on the assumption that the unseen
underlying variables are continuous and have a bivariate normal
distribution."
Latent variables are a bit outside of my statistical wheelhouse, so maybe
this is a naive question, but I'll ask it anyway! If the underlying
variables are (approximately) normal or bivariate normal, how can the
distributions of the observed ordinal variables by U-shaped? I was about to
ask if there is any literature on how non-normality of the latent variables
affects things, but then I found an article by Jin & Yang-Wallentin (2017)
which tells me there has been some work on that question. I don't have time
to read it right now, but a quick skim suggests that it does not discuss
U-shaped distributions.
Bruce
PS- I am well aware of what George Box (1976) said about normality and
linearity in the real world, and understand that when we work with real
world data, approximate normality is the best we can hope for.
References
Box, G. E. (1976). Science and statistics. Journal of the American
Statistical Association, 71(356), 791-799.
http://mkweb.bcgsc.ca/pointsofsignificance/img/Boxonmaths.pdf
Jin, S., & Yang-Wallentin, F. (2017). Asymptotic robustness study of the
polychoric correlation estimation. psychometrika, 82(1), 67-85.
Rigdon, E. E., & Ferguson Jr, C. E. (1991). The performance of the
polychoric correlation coefficient and selected fitting functions in
confirmatory factor analysis with ordinal data. Journal of marketing
research, 28(4), 491-497.
Baker, Harley wrote
> 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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Bruce Weaver
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