When performing the Shapiro-Wilk test, we can obtain the critical $W_{alpha}$ statistic from tables, given $alpha$ is 0.05 or 0.01.

If $alpha$ is nonstandard, say 0.07 or 0.02 for example, how we can calculate the value of $W_{alpha}$?

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#### Best Answer

I think you're asking how to calculate a critical value for the Shapiro-Wilk test at significance levels other than the usual tabulated ones (I mention this because it's possible you're really interested in how to compute a p-value, which is a closely related issue, though now Chris and I have edited your question, my doubt looks a bit odd.)

If your desired significance level is between tabulated ones, such as wanting 2% and having 5% and 1%, you could use interpolation, which will at least be approximately suitable. It looks to me like $Phi(1-p)$ is close to linear* in $log(n(1-W))$ for $nge 12$, and even down at $n=5$ it's locally close enough that linear interpolation should work fine if done on those scales.

More generally, computer software is the most obvious choice. Some software offers direct calculation of p-values for the Shapiro-Wilk that may avoid the need to use critical values at all.

Finally, simulation is an option; one can simulate the statistic and hence obtain simulations from the distribution under the null; this allows one to compute estimates of quantiles of the distribution.

* Edit: looking at the p-value code in R, that shift in nearness-to-linearity between n=11 and n=12 is because R is using a different approximation to compute p-values for $n$ below 12; that shouldn't affect the suitability of linear interpolation at say 12, but it does suggest to me that it appearing to be very close to linear is more that the *approximation* is close to linear for $nge 12$; the actual transformed distribution is probably somewhat less linear down that low. [Now I look, the help even gives the information that a different approximation is used below 12.]

R also offers the following references, in case you want to write your own code:

[1] Patrick Royston (1982) An extension of Shapiro and Wilk's W test for normality to large samples. *Applied Statistics*, *31*, 115-124.

[2] Patrick Royston (1982) Algorithm AS 181: The W test for Normality. *Applied Statistics*, *31*, 176-180.

[3] Patrick Royston (1995) Remark AS R94: A remark on Algorithm AS 181: The W test for normality. *Applied Statistics*, *44*, 547-551.