Normality identifier in Shapiro-Wilk test

When using the Shapiro-Wilk test, should I look at the p-values or the W values in order to find out the "most" normal value among my different samples (e.g. iq, age, weight -- if I'm running Shapiro-Wilk test on iq, age, and weight and want to find out which is the most normal value, should I look at the P-value or the W? And what number should it be close to?)

• Do you have the same amount of data in each variable? Why would you want to know this? None of these variables will ultimately be perfectly normal. What kind of deviation from normality do you care about (eg, skew or kurtosis or something else)? – gung - Reinstate Monica Oct 6 '15 at 19:12
• What's the purpose of this exercise? (i.e. why do you want to identify one that's 'most normal' across very different variables?) – Glen_b Oct 7 '15 at 0:28

2 Answers

You're asking for something like an effect size (A "how big?" type question).

P-values don't measure that; at a given value of W, the p-value tends to go down as n goes up.

The Shapiro-Wilk statistic, W, is in some sense a measure of "closeness to what you'd expect to see with normality", akin to a squared correlation (if I recall correctly, the closely related Shapiro-Francia test is actually a squared correlation between the data and the normal scores, while the Shapiro Wilk tends to be slightly larger; I seem to recall that it takes into account correlations between order statistics).

Specifically values closer to 1 indicate "closer to what you'd expect if the distribution the data were drawn from is normal".

However, keep in mind it's a random variable; samples can exhibit random fluctuations that don't represent their populations, and summary statistics will follow suit.

It's not immediately clear that it necessarily makes sense to compare Shapiro-Wilk statistics across data-sets in order to declare one set "more normal" than another; even less so with very different variables and different sample sizes.

Further, choosing the one closest to 1 among a collection of samples may actually be choosing something other than values randomly selected from a normal distribution, for a variety of reasons. For example, goodness of fit tests generally tend to be biased tests; what makes their criterion "closest" isn't necessarily the thing the test is actually designed to pick up. (I don't know what sorts of small-sample biases the Shapiro-Wilk specifically may have, however.)

Finally, I don't see any useful point to such an exercise. What possible value can there be in such a procedure?

You can't find out the "most" Normal value: that's not what Shapiro-Wilk is for. You can't find the "least" Normal value, either. All you can do is set a wrong-rejection threshold, e.g. 5%, then use S-W to reject (with that error rate) your sample for non-Normality. It's the $p$-value that you need to look at: too small (e.g. $<0.05$) and your data can be rejected as non-Normal.