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# st: Ksmirnov one-sided test interpretation

 From "Tsankova, Teodora" To Subject st: Ksmirnov one-sided test interpretation Date Fri, 1 Mar 2013 09:30:46 -0000

```Thank you Joerg, for your comment. I am using the test not as an
equality of distributions check but as an one-sided (inequality) check.

In my case I want to check whether a parameter is higher than a random
uniform distribution would suggest. So, I basically need to prove that
its values are higher than if they were chosen at random in the range
observed. I am not using a simple ttest because I would like to prove
that not only the mean is higher but that also all the values tend to be
higher than the uniform distribution. Also, it is difficult to deduct
this information from the CDF graphs as I have a limited number of
observations which are sometime above and sometimes below the 45 degree
line which would represent the random uniform distribution.

That being said, most of the interpretation of the KS test are for a
two-sided test and this is why I have trouble making conclusions.

Thank you again,

Teodora

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Joerg
Luedicke
Sent: 28 February 2013 18:38
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Ksmirnov one-sided test interpretation

Yes, why not just looking at your data?

That aside, I am wondering what the point of such a test is? What does
it even mean that one distribution is "lower" than another? Or to quote
the Stata manual, version 11: "We wish to use the two-sample
Kolmogorov-Smirnov test to determine if there are any differences in the
distribution of x for these two groups..." "Any" differences seem to
pick up a mix of differences with regard to the location and shape of
distributions. What is the motivation behind this? If there are
differences in two distributions, why not just looking at what these
differences are? But even if there was a good reason for using this
test, I am wondering what it is telling us. I did not try hard to come
up with the following example:

Let's generate some data for two groups where the distribution in group
one is normal with mean 10 and SD 5, while the distribution in the other
group is a gamma with shape 5 and scale 2:

*---------------
clear
set obs 200
set seed 1234

gen u = runiform()>.5
gen x = rnormal(10,5) if u==0
replace x=rgamma(5,2) if u==1
*---------------

and have a look at the empirical distribution for this data realization:

*---------------
tw kdensity x if u==0 || kdensity x if u==1
*---------------

As expected, these distributions surely look different to me. We can
also have a look at the true functions:

*---------------
tw 	function y = gammaden(5,2,0,x) , range(0 25) || ///
function y = normalden(x,10,5) , range(-5 25) ///
legend(order(1 "Gamma" 2 "Gauss"))
*---------------

Yet, if we run the K-S test:

*---------------
ksmirnov x, by(u) exact
*---------------

we would conclude that we cannot reject the hypothesis that the
distributions are "different"? That does not sound right to me.

So, my bottom line is: a) that I wonder why one would use this test in
the first place, and b) even if there was a good reason, I probably
would not trust it. I may very well be missing something here as I have
never used or studied this test before, so others, please correct me if
I am wrong here with something.

Joerg

On Thu, Feb 28, 2013 at 1:06 PM, Nick Cox <njcoxstata@gmail.com> wrote:
> Why not plot the data to show what is going on?
>
> Nick
>
> On Thu, Feb 28, 2013 at 5:23 PM, Tsankova, Teodora <TsankovT@ebrd.com>
wrote:
>
>> I have a question related to a previous post:
>>
>> http://www.stata.com/statalist/archive/2009-01/msg00525.html
>>
>> The Stata output from this message is as follows:
>>
>> Two-sample Kolmogorov-Smirnov test for equality of distribution
functions:
>>
>> Smaller group       D       P-value  Corrected
>> ----------------------------------------------
>> male:               0.2468    0.002
>> female:             0.0000    1.000
>> Combined K-S:       0.2468    0.005      0.003
>>
>>
>> From the one sided tests (first two lines) on can say which
distribution tends to be lower - for males or for females. However, I am
not sure how to interpret it.
>>
>> Given that the pvalue from the first line is low and that D in the
second line is 0, can we say that this is a proof that the distribution
of male is lower than that of female? To rephrase it - can we claim that
the distribution of male stochastically dominates the one of female
which would imply that the values of the underlying variable tend to be
larger for male than for female?  Or, do we interpret it in the exactly
opposite way - that the values for male tend to be lower than the values
for female?

>
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```