R help - Sep 2010 - Depletion of small p values upon iterative testing of identical normal distributions

Dear all,

I'm performing a t-test on two normal distributions with identical mean
&
standard deviation, and repeating this tests a very large number of times to
describe an representative p value distribution in a null case. As a part of
this, the program bins these values in 10 evenly distributed bins between 0
and 1 and reports the number of observations in each bin. What I have
noticed is that even after 500,000 replications the number in my lowest bin
is consistently ~5% smaller than the number in all the other bins, which are
similar within about 1% of each other. Is there any reason, perhaps to do
with random number generation in R or the nature of the normal distribution
simulated by the rnorm function that could explain this depletion?

Here are two key parts of my code to show what functions I'm working with:

#Calculating the p values
while(i<numtests){
Group1<-rnorm(6,-0.0065,0.0837)
Group2<-rnorm(6,-0.0065,0.0837)
PV<-t.test(Group1,Group2)$p.value
pscoresvector<-c(PV,pscoresvector)
i<-i+1
}

#Binning the results
freqtbl1<-binning(pscoresvector,breaks=bins)

Thanks in advance for any insights,

Andrew

	[[alternative HTML version deleted]]

On 20/09/2010 9:54 AM, A wrote:
> Dear all,
>
> I'm performing a t-test on two normal distributions with identical
mean&
> standard deviation, and repeating this tests a very large number of times
to
> describe an representative p value distribution in a null case. As a part
of
> this, the program bins these values in 10 evenly distributed bins between 0
> and 1 and reports the number of observations in each bin. What I have
> noticed is that even after 500,000 replications the number in my lowest bin
> is consistently ~5% smaller than the number in all the other bins, which
are
> similar within about 1% of each other. Is there any reason, perhaps to do
> with random number generation in R or the nature of the normal distribution
> simulated by the rnorm function that could explain this depletion?
No, equal sized bins should expect equal numbers of entries.  But your 
code may have errors in it.

This is a very slow, and slightly dangerous way to program
R:
> Here are two key parts of my code to show what functions I'm working
with:
>
> #Calculating the p values
> while(i<numtests){
> Group1<-rnorm(6,-0.0065,0.0837)
> Group2<-rnorm(6,-0.0065,0.0837)
> PV<-t.test(Group1,Group2)$p.value
> pscoresvector<-c(PV,pscoresvector)
> i<-i+1
> }
The slowness comes because pscoresvector is growing by one entry every 
iteration.  The danger comes because the initialization of pscoresvector 
is not shown.  If it wasn't initialized, you'll be binning whatever junk
was there before, as well as the new values.

I'd suggest initializing it as

pscoresvector <- numeric(numtests)

and updating using

pscoresvector[i] <- PV

to avoid both problems.

Duncan Murdoch
> #Binning the results
> freqtbl1<-binning(pscoresvector,breaks=bins)
>
> Thanks in advance for any insights,
>
> Andrew
>
> 	[[alternative HTML version deleted]]
>
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A <petlyakov <at> gmail.com> writes:
> 
> Dear all,
 [snip]
> Here are two key parts of my code to show what functions I'm working
with:
> 
> #Calculating the p values
> while(i<numtests){
> Group1<-rnorm(6,-0.0065,0.0837)
> Group2<-rnorm(6,-0.0065,0.0837)
> PV<-t.test(Group1,Group2)$p.value
> pscoresvector<-c(PV,pscoresvector)
> i<-i+1
> }
> 
> #Binning the results
> freqtbl1<-binning(pscoresvector,breaks=bins)
Also consider
> f <- function() {
+   Group1 <- rnorm(6,-0.0065,0.0837)
+   Group2 <- rnorm(6,-0.0065,0.0837)
+   t.test(Group1,Group2)$p.value
+ }
> pscoresvector <- replicate(10000,f())
> table(cut(pscoresvector,breaks=...))
  Is your lowest bin perhaps narrower than your other bins
(although admittedly I can't see an easy reason it should be
only 5% smaller -- 50% would be a more typical result of a binning
mistake)?

On 20-Sep-10 13:54:56, A wrote:
> Dear all,
> I'm performing a t-test on two normal distributions with identical
> mean & standard deviation, and repeating this tests a very large
> number of times to describe an representative p value distribution
> in a null case. As a part of this, the program bins these values
> in 10 evenly distributed bins between 0 and 1 and reports the number
> of observations in each bin. What I have noticed is that even after
> 500,000 replications the number in my lowest bin is consistently ~5%
> smaller than the number in all the other bins, which are similar
> within about 1% of each other. Is there any reason, perhaps to do
> with random number generation in R or the nature of the normal
> distribution simulated by the rnorm function that could explain
> this depletion?
> 
> Here are two key parts of my code to show what functions I'm
> working with:
> 
>#Calculating the p values
> while(i<numtests){
> Group1<-rnorm(6,-0.0065,0.0837)
> Group2<-rnorm(6,-0.0065,0.0837)
> PV<-t.test(Group1,Group2)$p.value
> pscoresvector<-c(PV,pscoresvector)
> i<-i+1
> }
> 
>#Binning the results
> freqtbl1<-binning(pscoresvector,breaks=bins)
> 
> Thanks in advance for any insights,
> Andrew
The issue lies in the t-test, not in the random number generation.

Look at '?t.test':
  t.test(x, y = NULL,
         alternative = c("two.sided", "less",
"greater"),
         mu = 0, paired = FALSE, var.equal = FALSE,
         conf.level = 0.95, ...)

and note "var.equal = FALSE" as the default, Then:

  var.equal: a logical variable indicating whether to treat the two
        variances as being equal. If ?TRUE? then the pooled variance
        is used to estimate the variance otherwise the Welch (or
        Satterthwaite) approximation to the degrees of freedom is
        used.

So the default t-test is not the t-test you were perhaps expecting!
I used a variant of your code (to be absolutely sure of what I was
doing):

  numtests<-100000 ; PVals <- numeric(numtests)
  for(i in (1:numtests)){
    Group1<-rnorm(6,-0.0065,0.0837)
    Group2<-rnorm(6,-0.0065,0.0837)
    PVals[i] <- t.test(Group1,Group2)$p.value
  }
  hist(PVals,breaks=0.1*(0:10))

and observed similar behaviour to what you report. But, with
"var.equal=TRUE":

  numtests<-100000 ; PVals <- numeric(numtests)
  for(i in (1:numtests)){
    Group1<-rnorm(6,-0.0065,0.0837)
    Group2<-rnorm(6,-0.0065,0.0837)
    PVals[i] <- t.test(Group1,Group2,var.equal=TRUE)$p.value
  }
  hist(PVals,breaks=0.1*(0:10))

all the bin-values were similar.

So what heppans here is that the Welch/Satterthwaite approxumation
does not produce uniformly distributed P-values when the Null
Hyptohesis is true (at any rate for sample sizes s as small as the
6 you are using).

Hoping this helps,
Ted.

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Date: 20-Sep-10                                       Time: 16:08:41
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