Solved – two-sample KS test: data becomes significantly different after normalization

I'm currently working on a data set with two sets of samples. The csv file of the data could be found here. 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 look the same even after normalization. Anything wrong with my usage of the R function?

Since the data contains ties, I also tried ks.boot, 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.)

I hope to replace this with a full answer once we have sorted out what's going on.

In trying to show you what's going on with the dependence in large vs small values, I see another problem:

enter image description here

The y-values have been monotonically transformed (log(y-2.e6)) for clarity.

The green box shows how the large values always occur together.

But they're also astonishingly regular. It looks like the large ones are every 20th value.

The red ovals show another problem. Notice patches of almost pure-black followed by almost pure-blue in the red ovals? There's something weird going on. Why would the middling-size values alternate, with a patch from column 1 then a patch from column 2?

You have neither independence within columns nor within pairs, but I am not sure what is causing that particular alternation in the red ovals.

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