# [R] Shapiro-Wilk test

Prof Brian Ripley ripley at stats.ox.ac.uk
Mon Jul 2 15:08:05 CEST 2001

```On Mon, 2 Jul 2001, Ralf Goertz wrote:

> does the shapiro wilk test in R-1.3.0 work correctly? Maybe it does, but can
> anybody tell me why the following sample doesn't give "W = 1" and
> "p-value = 1":
>
> R> x<-1:9/10;x
> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
> R> shapiro.test(qnorm(x))
>
>         Shapiro-Wilk normality test
>
> data:  qnorm(x)
> W = 0.9925, p-value = 0.9986
>
> I can't imagine a sample being more "normal" than this. Furthermore, the
> Kolmogorov-Smirnov test gives a p-value of 1.

Well, both I and the Shapiro-Wilk test can.  That sample has too short
tails:

> f <- function(a) shapiro.test(qnorm((a+0:8)/(8+2*a)))
> f(1)

Shapiro-Wilk normality test

data:  qnorm((a + 0:8)/(8 + 2 * a))
W = 0.9925, p-value = 0.9986

> f(0.2)

Shapiro-Wilk normality test

data:  qnorm((a + 0:8)/(8 + 2 * a))
W = 0.9998, p-value = 1

(Values less than 1 are normally used in Q-Q plots. For example

> ppoints(9)
[1] 0.06756757 0.17567568 0.28378378 0.39189189 0.50000000 0.60810811 0.71621622
[8] 0.82432432 0.93243243

corresponds to a = 5/8.)

>
> R> ks.test(qnorm(x),"pnorm",mean=0,sd=1)
>
>         One-sample Kolmogorov-Smirnov test
>
> data:  qnorm(x)
> D = 0.1, p-value = 1
> alternative hypothesis: two.sided

Actually

> ks.test(qnorm(x),"pnorm",mean=0,sd=1)\$p
[1] 0.9999907

> a <- 0.2
> ks.test(qnorm((a + 0:8)/(8 + 2 * a)),"pnorm",mean=0,sd=1)\$p
[1] 0.9999999

> Can someone explain this please,

Tail sensitivity varies between tests, and  K-S is notoriously
insensitive.

>From the FAQ:

If a command does the wrong thing, that is a bug.  But be sure you know
for certain what it ought to have done.  If you aren't familiar with the
command, or don't know for certain how the command is supposed to work,
then it might actually be working right.  Rather than jumping to
conclusions, show the problem to someone who knows for certain.

--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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```