[R] Shapiro-Welch W value interpretation

[P Ehlers]

Omar Baqueiro wrote:
> Hello,
> 
> I have tested a distribution for normality using the Shapiro-Welch
> statistic. The result of this is the following:
> 
> 
>         Shapiro-Wilk normality test
> 
> data: mydata
> W = 0.9989, p-value = 0.8791
> 
> 
> I know that the p-value > 0.05 (for my purposes) means that the data
> IS normally distributed but what I am not sure is with the W value,
> what values tell me that the data is normally distributed.   I know
> that my data is normally distributed, but what I want to know if how
> to interpret the W value, I have read that "if W is very small then
> the distribution is probably not normally distributed", but how
> "small"  is "very small", and also, what happens is, say W = 0.000001
> but the p-value is > my significance level (0.05)? is the hypothesis
> rejected?
> 

There is some confusion in your query.
First, how do you know that your data are indeed normally distributed?
That's *not* what the p-value of the test says.
Consider the following result of the Shapiro-Wilk test applied to
a vector x:

data: x
W = 0.9856, p-value = 0.988

Here x was not sampled from a normal distribution (code at end).

Second, the point of a p-value is to formalize decision-making
so that critical regions of tests are converted to p-value intervals.
Thus, your emphasis on the value of W is misplaced. It's
not how small W is but how small it is for the given sample size,
and the p-value takes care of the significance. (This is not to
say, of course, that the distribution of W is not of interest.)

Finally, what exactly, in your view, is "the hypothesis"?

I hope this doesn't sound too critical. I'm trying to be helpful.

Peter Ehlers

> thank you!
> 
> Omar
> 
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> 

set.seed(34); shapiro.test(rexp(10))