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# Re: st: Ksmirnov one-sided test interpretation

 From Nick Cox To statalist@hsphsun2.harvard.edu Subject Re: st: Ksmirnov one-sided test interpretation Date Thu, 28 Feb 2013 19:39:46 +0000

```One of the amusing things about statistical science is that every
mildly experienced person has a long personal list of statistical
things that they don't respect, if not as unsound in principle then as
useless or oversold in practice. But the lists don't overlap much!

A more polite list is always that of things that are in one's unhumble
opinion undersold and so deserve more exposure.

Like Joerg, I don't think I've used Kolmogorov-Smirnov for real in any
distributions are different, but then the interesting thing is to see
how and how much they differ (and occasionally change my mind if the
distributions turn out to be practically identical).

I looked at the example for two groups that is bundled with
-ksmirnov-, which is a tiny fake dataset.

To make it more interesting, at least a bit, I suggest

sysuse auto, clear
ksmirnov mpg, by(foreign)

as a canonical example.

Here are five graphs you can draw instead. I leave aside more standard
graphs such as histograms (which here hide too much).

* 1 -stripplot- (SSC)
stripplot mpg, over(foreign) stack box(barw(0.1)) boffset(-0.1) height(.4)

* 2 -qplot- (SJ)
qplot mpg, over(foreign)

* 3 -cquantile- (SSC)
cquantile mpg, by(foreign) gen(mpg1 mpg2)
qqplot mpg?

* 4 -distplot- (SJ)
distplot mpg, over(foreign)

* 5 -devnplot- (SSC)
devnplot mpg foreign

If you don't like even _one_ of these graphs as being more informative
or interesting than the output of -ksmirnov-, then I've failed.

Other specific suggestions of graphs are naturally most welcome.

Nick
On Thu, Feb 28, 2013 at 6:37 PM, Joerg Luedicke
<joerg.luedicke@gmail.com> wrote:
> 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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