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Re: st: Ksmirnov one-sided test interpretation


From   Joerg Luedicke <joerg.luedicke@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Ksmirnov one-sided test interpretation
Date   Thu, 28 Feb 2013 13:37:30 -0500

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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