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From |
"Tsankova, Teodora" <TsankovT@ebrd.com> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution |

Date |
Sat, 9 Mar 2013 13:49:57 -0000 |

Dear David, Thank you for the suggestion. What I mean is that I create a uniform distribution between 0 and 1 with 15 observation. Given that every value should have the same probability under a uniform distribution I divide 1 by 14 and create those equally spaces 15 values. Plotting the CDF of those values would result in a straight diagonal line which is ultimately what the ksmirnov test would test against as well. The output from the ksmirnov test is as follows: ksmirnov mean_random_BTWGr_Fx=uniform() One-sample Kolmogorov-Smirnov test against theoretical distribution uniform() Smaller group D P-value Corrected ---------------------------------------------- mean_ra~r_Fx: 0.8221 0.000 Cumulative: -0.8983 0.000 Combined K-S: 0.8983 0.000 0.000 So, it seems that although I can reject the inequality of the two distributions, I cannot say anything about which one tends to have larger values. In Stata the -porder- option of the ranksum command gives the probability that a random draw from the first sample is larger than a random draw from the second sample. I like this as it seems very intuitive. I use those constructed values to perform this test. My results are as follows: ranksum mean_random_BTWGr_Fx, by( ObservedORUniform) porder Two-sample Wilcoxon rank-sum (Mann-Whitney) test ObservedOR~m | obs rank sum expected -------------+--------------------------------- Observed | 15 259 232.5 Uniform | 15 206 232.5 -------------+--------------------------------- combined | 30 465 465 unadjusted variance 581.25 adjustment for ties 0.00 ---------- adjusted variance 581.25 Ho: mea~r_Fx(Observ~m==Observed) = mea~r_Fx(Observ~m==Uniform) z = 1.099 Prob > |z| = 0.2717 P{mea~r_Fx(Observ~m==Observed) > mea~r_Fx(Observ~m==Uniform)} = 0.618 Those results, although not very strong, seem much easier to interprpet. Thank you again, Teodora -----Original Message----- From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of David Hoaglin Sent: 09 March 2013 13:34 To: statalist@hsphsun2.harvard.edu Subject: Re: st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution Teodora, It seems odd to use a two-sample test when you actually have only one sample. What was the basis for the advice to use the Wilcoxon-Mann-Whitney test? A one-sided KS test would be all right. Some people might be more comfortable making the test two-sided, unless you would not have any interest in a situation where the data departed from the null hypothesis in the other direction. I don't know what the literature says about whether any other test has greater power for the type of alternative that you are interested in. With only 15 observations, the departure would have to be substantial to reject the uniform null hypothesis. I have an offbeat suggestion. Transform the sample to normal deviates by applying the inverse of the standard normal cumulative distribution function to each observation, and test whether the transformed sample departs from the standard normal distribution. You can also make a normal probability plot of the transformed sample. What do you mean by "a constant markup of 1/14"? David Hoaglin On Thu, Mar 7, 2013 at 3:11 PM, Tsankova, Teodora <TsankovT@ebrd.com> wrote: > Some time ago I posted on statlist with a question regarding the use of a one-sided KS test and I was advised that for my purpose I can use the Wilcoxon-Mann-Whitney test (ranksum command in Stata). > > I basically have 15 observations that go from 0 to 1 and constitute my empirical distribution and I want to prove that those take higher values than a uniform distribution would suggest. I have three questions related to the test: > > 1) I generate myself 15 more observation which take values from 0 to 1 with a constant markup of 1/14 (I simulate a uniform distribution of 15 variables in the same interval). Has anyone else used this method for creating uniform distribution and do you see any problems with it? > > 2) I use the ponder option to compute the p-value for the one sided test and I get the following output: > > Two-sample Wilcoxon rank-sum (Mann-Whitney) test > > ObservedOr~m | obs rank sum expected > -------------+--------------------------------- > Observed | 15 236 232.5 > Uniform | 15 229 232.5 > -------------+--------------------------------- > combined | 30 465 465 > > unadjusted variance 581.25 > adjustment for ties 0.00 > ---------- > adjusted variance 581.25 > > Ho: ktaub_~m(Observ~m==Observed) = ktaub_~m(Observ~m==Uniform) > z = 0.145 > Prob > |z| = 0.8846 > > P{ktaub_~m(Observ~m==Observed) > ktaub_~m(Observ~m==Uniform)} = 0.516 > 999996 > (15 real changes made) > (0 real changes made) > (0 real changes made) > > I would interpret it in the following way: In 51.6% of the cases you would draw a random number from Observed that would be higher than a random draw from Uniform. Is this the correct interpretation? > > 3) My last question is related to the fact that Wilcoxon Mann-Whitney test is used to analyse ordinal data. My data has an ordinal meaning in the sense higher values represent more homogenous group lending villages in my case. However, the values the variable takes are not interval but continuous ones. Can I still use this test? > > Thank you, > > Teodora * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/faqs/resources/statalist-faq/ * http://www.ats.ucla.edu/stat/stata/ EBRD SECURITY NOTICE This email has been virus scanned ______________________________________________________________ This message may contain privileged information. If you have received this message by mistake, please keep it confidential and return it to the sender. 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**Follow-Ups**:**Re: st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution***From:*David Hoaglin <dchoaglin@gmail.com>

**Re: st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution***From:*"JVerkuilen (Gmail)" <jvverkuilen@gmail.com>

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