[S] QQ plot x-axis values ['Watchdog': checked]

[Paul Matthias Diderichsen]

Dear S-plus users again.

Of course, there's still one issue: There seems to be a discrepancy
between what wikipedia says and what S-plus does.

Apparently, S-plus selects the midpoint of n intervals between 0 and 1 for
it's horizontal values in qqmath. Of course, for n->infinity the sets used
by S-plus and wikipedia converge. But which one is "most right" for
finite/small n?

Thanks again, PMD.

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[Li, Timothy - BLS]

S-plus appears to be using the formula (k-0.375)/(n+0.25) that is
advocated elsewhere on the internet. See
http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap55/sect44.htm

and

http://www.okstate.edu/sas/v8/saspdf/qc/chap10.pdf#search=%22constructio
n%20of%20q-q%20plot%22

Tim

________________________________

From: s-news-***@lists.biostat.wustl.edu
[mailto:s-news-***@lists.biostat.wustl.edu] On Behalf Of Paul Matthias
Diderichsen
Sent: Tuesday, October 10, 2006 9:57 AM
To: Paul Matthias Diderichsen
Cc: s-***@lists.biostat.wustl.edu; s-news-***@lists.biostat.wustl.edu
Subject: Re: [S] QQ plot x-axis values ['Watchdog': checked]



Dear S-plus users again.

Of course, there's still one issue: There seems to be a discrepancy
between what wikipedia says and what S-plus does.

Apparently, S-plus selects the midpoint of n intervals between 0 and 1
for it's horizontal values in qqmath. Of course, for n->infinity the
sets used by S-plus and wikipedia converge. But which one is "most
right" for finite/small n?

Thanks again, PMD.
========================================================================
====================================================
The information contained in this communication is confidential, may be
subject to legal privileges, may constitute inside information, and is
intended only for the use of the addressee. It is the property of Abbott
Laboratories or its relevant affiliate. Unauthorized use, disclosure or
copying of this communication or any part thereof is strictly prohibited
and may be unlawful. If you have received this communication in error,
please notify Abbott Laboratories immediately by return e-mail and
destroy this communication and all copies thereof, including all
attachments.
========================================================================
=====================================================

[Stephen Kaluzny]

Dear S-plus users.
I learned more or less all I know about Q-Q plots by reading
http://en.wikipedia.org/wiki/Qq_plot and related ressources on the web. I
think that I now understand more or less how these beasts work.
Of course, there's still one issue: There seems to be a discrepancy
between what wikipedia says and what S-plus does.

a<-data.frame(NORM=rnorm(3))

# a$NORM could be any 3 values since I'm only interested in x below...

qqmath(~NORM,data=a,panel=function(x,y)print(sort(x)))

[1] -0.9674216 0.0000000 0.9674216

qnorm(c(1/6,3/6,5/6))

[1] -0.9674216 0.0000000 0.9674216
versus: "k/(n + 1)-quantiles of the comparison distributon (e.g. the
normal distribution) on the horizontal axis (for k = 1, ..., n)"
In the example above, I create a qqplot with three points. As I understand
wikipedia, these should be (on the x-axis) at the 1/4, 2/4, and 3/4
quantiles of the normal distribution. However, instead they are at the
1/6, 3/6, and 5/6 quantiles.
I'd appreciate it a lot, if anybody can explain to me what is "right". (Of
course: Gurus - feel free to correct/add to the wikipedia entry in case
it's not accurate!)
Apparently, S-plus selects the midpoint of n intervals between 0 and 1 for
it's horizontal values in qqmath. Of course, for n->infinity the sets used
by S-plus and wikipedia converge. But which one is "most right" for
finite/small n?
S-plus appears to be using the formula (k-0.375)/(n+0.25) that is
advocated elsewhere on the internet. See
http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap55/sect44.htm
and
http://www.okstate.edu/sas/v8/saspdf/qc/chap10.pdf#search=%22construction%20of%20q-q%20plot%22

...

By default, S-PLUS uses the f(i) = (i-0.5)/n formula. This is described
in Cleveland, W. (1993) Visualizing Data where it is noted that the
precise form of f(i) is not important for a quantile plot (it may make
a difference when n = 3 but you probably should not be looking at
a quantile plot with only three data points).

The probability values used in S-PLUS for qqmath and qqnorm are computed
from the ppoints function. The function has an optional argument a that
controls the precise values of f(i). The values returned are: (seq(n) -
a)/(n + 1 - 2*a). The default value of a=0.5 leads to the (i-0.5)/n
values. Setting a=0.375 leads to the formula noted by Timothy Li and
setting a=0 gives the results noted in Wikipedia.

The probability values used in the qqmath function are specified in the
f.value argument to qqmath. The default for f.value is ppoints. To
create a qqmath call using the Wikipedia probability values you
could do:

x <- rnorm(12)
myppoints <- function(n) ppoints(n, a=0)
qqmath(~ x, f.value=myppoints)

To use different probability values in qqnorm you need to define your
own ppoints with the appropriate default value of a that you want
since qqnorm.default calls ppoints using the default value of a.

-Stephen Kaluzny
Insightful Corp.
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