[R] two-sample KS test: data becomes significantly different after normalization

[Monnand]

I know this must be a wrong method, but I cannot help to ask: Can I only
use the p-value from KS test, saying if p-value is greater than \beta, then
two samples are from the same distribution. If the definition of p-value is
the probability that the null hypothesis is true, then why there's little
people uses p-value as a "true" probability. e.g. normally, people will not
multiply or add p-values to get the probability that two independent null
hypothesis are both true or one of them is true. I had this question for
very long time.

-Monnand

On Tue Jan 13 2015 at 2:47:30 PM Andrews, Chris <chrisaa at med.umich.edu>
wrote:

> This sounds more like quality control than hypothesis testing.  Rather
> than statistical significance, you want to determine what is an acceptable
> difference (an 'equivalence margin', if you will).  And that is a question
> about the application, not a statistical one.
> ________________________________________
> From: Monnand [monnand at gmail.com]
> Sent: Monday, January 12, 2015 10:14 PM
> To: Andrews, Chris
> Cc: r-help at r-project.org
> Subject: Re: [R] two-sample KS test: data becomes significantly different
> after normalization
>
> Thank you, Chris!
>
> I think it is exactly the problem you mentioned. I did consider
> 1000-point data is a large one at first.
>
> I down-sampled the data from 1000 points to 100 points and ran KS test
> again. It worked as expected. Is there any typical method to compare
> two large samples? I also tried KL diverge, but it only gives me some
> number but does not tell me how large the distance is should be
> considered as significantly different.
>
> Regards,
> -Monnand
>
> On Mon, Jan 12, 2015 at 9:32 AM, Andrews, Chris <chrisaa at med.umich.edu>
> wrote:
> >
> > The main issue is that the original distributions are the same, you
> shift the two samples *by different amounts* (about 0.01 SD), and you have
> a large (n=1000) sample size.  Thus the new distributions are not the same.
> >
> > This is a problem with testing for equality of distributions.  With
> large samples, even a small deviation is significant.
> >
> > Chris
> >
> > -----Original Message-----
> > From: Monnand [mailto:monnand at gmail.com]
> > Sent: Sunday, January 11, 2015 10:13 PM
> > To: r-help at r-project.org
> > Subject: [R] two-sample KS test: data becomes significantly different
> after normalization
> >
> > Hi all,
> >
> > This question is sort of related to R (I'm not sure if I used an R
> function
> > correctly), but also related to stats in general. I'm sorry if this is
> > considered as off-topic.
> >
> > I'm currently working on a data set with two sets of samples. The csv
> file
> > of the data could be found here: http://pastebin.com/200v10py
> >
> > I would like to use KS test to see if these two sets of samples are from
> > different distributions.
> >
> > I ran the following R script:
> >
> > # read data from the file
> >> data = read.csv('data.csv')
> >> ks.test(data[[1]], data[[2]])
> >     Two-sample Kolmogorov-Smirnov test
> >
> > data:  data[[1]] and data[[2]]
> > D = 0.025, p-value = 0.9132
> > alternative hypothesis: two-sided
> > The KS test shows that these two samples are very similar. (In fact, they
> > should come from same distribution.)
> >
> > However, due to some reasons, instead of the raw values, the actual data
> > that I will get will be normalized (zero mean, unit variance). So I tried
> > to normalize the raw data I have and run the KS test again:
> >
> >> ks.test(scale(data[[1]]), scale(data[[2]]))
> >     Two-sample Kolmogorov-Smirnov test
> >
> > data:  scale(data[[1]]) and scale(data[[2]])
> > D = 0.3273, p-value < 2.2e-16
> > alternative hypothesis: two-sided
> > The p-value becomes almost zero after normalization indicating these two
> > samples are significantly different (from different distributions).
> >
> > My question is: How the normalization could make two similar samples
> > becomes different from each other? I can see that if two samples are
> > different, then normalization could make them similar. However, if two
> sets
> > of data are similar, then intuitively, applying same operation onto them
> > should make them still similar, at least not different from each other
> too
> > much.
> >
> > I did some further analysis about the data. I also tried to normalize the
> > data into [0,1] range (using the formula (x-min(x))/(max(x)-min(x))), but
> > same thing happened. At first, I thought it might be outliers caused this
> > problem (I can see that an outlier may cause this problem if I normalize
> > the data into [0,1] range.) I deleted all data whose abs value is larger
> > than 4 standard deviation. But it still didn't help.
> >
> > Plus, I even plotted the eCDFs, they *really* look the same to me even
> > after normalization. Anything wrong with my usage of the R function?
> >
> > Since the data contains ties, I also tried ks.boot (
> > http://sekhon.berkeley.edu/matching/ks.boot.html ), but I got the same
> > result.
> >
> > Could anyone help me to explain why it happened? Also, any suggestion
> about
> > the hypothesis testing on normalized data? (The data I have right now is
> > simulated data. In real world, I cannot get raw data, but only normalized
> > one.)
> >
> > Regards,
> > -Monnand
> >
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> >
> >
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