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

[Andrews, Chris]

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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