Re: Problems with DistributionFitTest

*To*: mathgroup at smc.vnet.net*Subject*: [mg122616] Re: Problems with DistributionFitTest*From*: Andy Ross <andyr at wolfram.com>*Date*: Thu, 3 Nov 2011 03:47:16 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111010502.AAA14754@smc.vnet.net> <201111021123.GAA03608@smc.vnet.net> <4EB26A05.813B.006A.0@newcastle.edu.au>

I agree with your clarification completely. However, in this case, Felipe was clearly sampling from the null distribution and was asking why the p-value was unexpectedly small, not how to interpret p-values in general. Here is an empirical result that demonstrates what I was claiming. Set up a sample. In[25]:= data1 = BlockRandom[SeedRandom[94]; RandomVariate[NormalDistribution[], 10000]]; We get a p-value near 0.06 In[26]:= DistributionFitTest[data1, Automatic, "TestData"] Out[26]= {0.121449, 0.0568874} What proportion of test statistics are more extreme? In[27]:= Count[ Table[DistributionFitTest[ RandomVariate[NormalDistribution[], 10000], Automatic, "TestStatistic"], {1000}], x_ /; x >= DistributionFitTest[data1, Automatic, "TestStatistic"]]/1000. Out[27]= 0.06 I will also point out that I claimed that the p-value follows a UniformDistribution[{0,1}]. This too is only true in the context of the problem Felipe posed. In general, if a p-value has a standard uniform distribution, the test size and power are equivalent and so the test is useless. Under a general alternative we would hope for a right-skewed distribution. Andy Ross Wolfram Research On 11/2/2011 6:16 PM, Barrie Stokes wrote: > Hi Felipe > > Can I beg to make a small clarification to Andy's response? > > The whole idea of p values and rejection of the Null Hypothesis continues to be one in which people get tangled up in logical and linguistic knots. > > An observed p value of does *not* allow one to make a *general* claim like "about 3% of the time you can expect to get a test statistic like the one you obtained or one even more extreme". > > Given the context of this p value, it's value being 0.0312946, i.e., less than 0.05, allows a frequentist-classical statistician to say that, *on this occasion*, this observed p value enables me to reject the Null Hypothesis (which is that the data are Gaussian) at the 5% significance level, or some such equivalent phrase. > > The important thing here is that, *by construction*, p values are equal to or less than 0.05 precisely 5% of the time *when the Null Hypothesis holds, i.e., is in fact true*, or "under the Null Hypothesis", as it's usually phrased. > > When one rejects the Null Hypothesis (having obtained a p value<=0.05, one is in fact betting that, in so doing, you will only be wrong in so doing 1 time in 20. > > If anyone doesn't like this explication, please note that I am a Bayesian, s for me to explain a p value is like George Bush explaining the meaning of the French word 'entrepreneur'. :-) > > (Apparently GB once claimed that the trouble with the French is that they don't have a word for 'entrepreneur'. Actually, they do.) > > You may find the following code (built on your original code) helpful - run it as many times as your patience allows. > > numTests = 1000; > resultsList = {}; > Do[ > (data = RandomVariate[NormalDistribution[], 10000]; > AppendTo[ resultsList, DistributionFitTest[data] ]; > ), {numTests} > ] > resultsList // Short > Length[ Select[ resultsList, (s \[Function] s<= 0.05) ] ]/numTests // N > > Cheers > > Barrie > > > >>>> On 02/11/2011 at 10:23 pm, in message<201111021123.GAA03608 at smc.vnet.net>, > Andy Ross<andyr at wolfram.com> wrote: >> This is exactly what you might expect. The p-value from a hypothesis >> test is itself a random variable. Under the null hypothesis the p-value >> should follow a UniformDistribution[{0,1}]. >> >> In your case, the null hypothesis is that the data have been drawn from >> a normal distribution. What that p-value is really saying is that about >> 3% of the time you can expect to get a test statistic like the one you >> obtained or one even more extreme. >> >> Andy Ross >> Wolfram Research >> >> >> On 11/1/2011 12:02 AM, fd wrote: >>> Dear Group >>> >>> I'm not a specialist in statistics, but I spoke to one who found this >>> behaviour dubious. >>> >>> Before using DistributionFitTest I was doing some tests with the >>> normal distribution, like this >>> >>> data = RandomVariate[NormalDistribution[], 10000]; >>> >>> DistributionFitTest[data] >>> >>> 0.0312946 >>> >>> According to the documentation "A small p-value suggests that it is >>> unlikely that the data came from dist", and that the test assumes the >>> data is normally distributed >>> >>> I found this result for the p-value to be really low, if I re-run the >>> code I often get what I would expect (a number greater than 0.5) but >>> it is not at all rare to obtain p values smaller than 0.05 and even >>> smaller. Through multiple re-runs I notice it fluctuates by orders of >>> magnitude. >>> >>> The statistician I consulted with found this weird since the data was >>> drawn from a a normal distribution and the sample size is big, >>> especially because the Pearson X2 test also fluctuates like this: >>> >>> H=DistributionFitTest[data, Automatic, "HypothesisTestData"]; >>> >>> H["TestDataTable", All] >>> >>> Is this a real issue? >>> >>> Any thougths >>> >>> Best regards >>> Felipe >>> >>> >>> >>>

**References**:**Re: Problems with DistributionFitTest***From:*Andy Ross <andyr@wolfram.com>