[R] regression towards the mean, AS paper November 2007
Rolf Turner
r.turner at auckland.ac.nz
Mon Dec 17 20:24:53 CET 2007
On 18/12/2007, at 7:32 AM, Duncan Murdoch wrote:
> On 12/17/2007 1:21 PM, Troels Ring wrote:
>> Dear friends, regression towards the mean is interesting in medical
>> circles, and a very recent paper (The American Statistician November
>> 2007;61:302-307 by Krause and Pinheiro) treats it at length. An
>> initial
>> example specifies (p 303):
>> "Consider the following example: we draw 100 samples from a bivariate
>> Normal distribution with X0~N(0,1), X1~N(0,1) and cov(X0,X1)=0.7, We
>> then calculate the p value for the null hypothesis that the means
>> of X0
>> and X1 are equal, using a paired Student's t test. The procedure is
>> repeated 1000 times, producing 1000 simulated p values. Because X0
>> and
>> X1 have identical marginal distributions, the simulated p values
>> behave
>> like independent Uniform(0,1) random variables." This I did not
>> understand, and simulating like shown below produced far from uniform
>> (0,1) p values - but I fail to see how it is wrong. I contacted the
>> authors of the paper but they did not answer. So, please, doesn´t the
>> code below specify a bivariate N(0,1) with covariance 0.7? I get p
>> values = 1 all over - not interesting, but how wrong?
>> Best wishes
>> Troels
>>
>> library(MASS)
>> Sigma <- matrix(c(1,0.7,0.7,1),2,2)
>> Sigma
>> res <- NULL
>> for (i in 1:1000){
>> ff <-(mvrnorm(n=100, rep(0, 2), Sigma, empirical = TRUE))
>> res[i] <- t.test(ff[,1],ff[,2],paired=TRUE)$p.value}
>
> Specifying empirical=TRUE means that your sampled values are not
> independent, the means are guaranteed to match exactly, and the mean
> difference is exactly zero. Thus all of the t statistics are exactly
> zero, and the p-values are exactly 1.
>
> Set empirical=FALSE (the default), and you'll see more reasonable
> results.
This has nothing to do really with the question that Troels asked,
but the exposition quoted from the AA paper is unnecessarily confusing.
The phrase ``Because X0 and X1 have identical marginal
distributions ...''
throws the reader off the track. The identical marginal distributions
are irrelevant. All one needs is that the ***means*** of X0 and X1
be the same, and then the null hypothesis tested by a paired t-test
is true and so the p-values are (asymptotically) Uniform[0,1]. With
a sample size of 100, the ``asymptotically'' bit can be safely ignored
for any ``decent'' joint distribution of X0 and X1. If one further
assumes that X0 - X1 is Gaussian (which has nothing to do with X0 and
X1 having identical marginal distributions) then ``asymptotically''
turns into ``exactly''.
cheers,
Rolf Turner
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