Kolmogorov-Smirnov Test for Goodness of Fit in an ordered sequence

Transcription

1 Biostatistics 430 Kolmogorov-Smirnov Test ORIGIN Model: Assumptions: Kolmogorov-Smirnov Test for Goodness of Fit in an ordered sequence The Kolmogorov-Smirnov Test is designed to test whether observed counts, frequencies, or continuous numerical values collected along an ordered sequence (Y) conform to that expected for a known probability distribution such as equal probability or Normal distribution, or the like. The test proceeds by considering each value in an ordered sequence Y to represent bins (as in a histogram) for which Observed and Expected Cumulative probabilites (Y) are calculated. (I use Y instead of X here only to conform to Table 7.4 in Sokal & Rohlf 995). Let Expected Cumulative Probabilities (Y) be specified according to some model over ordered values in Y. Frequency or numerical data collected for each value of X are independent of the others. Hypotheses: H 0 : P j are distributed according to the model H : P j differ from the model < Two sided test Sokal & Rohlf Table 7.4 based on data in Box 5.2 n 2 i n Y sort SR 3 i Y stdy ExpF F0.5 g0.5 F0 g0 F g Construct Cumulative Frequencies: 2.05 mean sd max δ max δ max SR READPRN ("c:/data/biostatistics/sr5.2r.txt" ) < Ordered values which will be tested against some probability distribution. stdy Y mean( Y) Var( Y) mean( Y) Var( Y) < standardizing Y ExpF pnorm( stdy0 ) < Expected (Y) - In this instance, we will be testing a Normal distribution ~N(0,) for the standardized data stdy.

2 Biostatistics 430 Kolmogorov-Smirnov Test 2 i Construct Statistics g 0.5, g 0 & g : Y stdy ^ standardized Y ExpF ^ (Y) for ~N(0,) g ObsF 0.5 HKL corrected ObsF 0.5 g 0 - ObsF 0 Khamis corrected ObsF 0.5 where = 0 g - ObsF Khamis corrected ObsF 0.5 where = ( i 0.5) F 0.5i n g 0.5i ExpF F i 0.5i < Harter, Khamis & Lamb correction SR p. 708 < Exp-Obs 0.5 F g i F 0i n 2 g 0i ExpF F i 0i < Khamis SR p. 7 < 0 correction < Exp-Obs 0 for = 0 F g

3 Biostatistics 430 Kolmogorov-Smirnov Test 3 < Khamis SR p. 7 i F i n 2 < correction g i ExpF F i i < Exp-Obs for = Test Statistic d max : F maxg maxg max g d max maxg 0.5 d max n Sampling Distribution: If Assumptions hold and H 0 is true, then D max ~D (,n) g Critical Value of the Test: 0.05 <Type I error must be set ^ Kolmogorov-Smirnov distribution for continuous data in specialized tables (e.g., Zar 200 Appendix B.9). Sokal & Rohlf offer two different tables depending on whether distribution ExpF is derived from internal estimates or external parameters. C Decision Rule: < from Zar 200 Appendix B.9 n 2 IF d max > C THEN REJECT H 0 d max IF P < THEN REJECT H 0 Probability Value: The Kolmogorov-Smirnov distribution is not available for calculating this, so until I find one, we'll have to depend on R's function ks.test() to calculate this for us. Prototype in R: #KOLMOGOROV SMIRNOV SINGLE SAMPLE TEST #FOR CONTINUOUS DATA #SOKAL & ROHLF TABLE 7.4 & BOX 5.2 SR=read.table("c:/DATA/Biostascs/SR5.2R.txt") SR aach(sr) Y=sort(bodywt) stdy=(y mean(y))/sqrt(var(y)) stdy One sample Kolmogorov Smirnov test ks.test(stdy,pnorm) data: stdy D = 0.225, p value = alternave hypothesis: two sided > SR gillwt bodywt ^ Note: in this test, I standardized Y (making stdy) in order to compare stdy with the Normal distribution ~N(0,)

4 Biostatistics 430 Kolmogorov-Smirnov Test 4 Kolmogorov-Smirnov Test for Goodness of Fit for Two Samples: A similar Kolmogorov-Smirnov approach may be used to compare the distribution of one sample with another. According to Sokal & Rohlf, this test is not as sensitive as the Mann-Whitney Test for assessing differences in median. However, it also looks for differences in shapes of the the two distributions. Construct Chart: Sokal & Rohlf Biometry 3rd Edition 995 Example Box 3.9 Testing differences in distribution between Sample A & B: length sample F F2 F/n F2/n2 d 00 B A B B B B A B A A A B A A A B A B A B A A A A A A n n2 max d F & F2 are cumulative counts respectively for samples A & B. n & n2 are total counts for each sample. d is the absolute difference: F/n - F2/n2 Test Statistic: d max

5 Biostatistics 430 Kolmogorov-Smirnov Test 5 Sampling Distribution: If Assumptions hold and H 0 is true, then D max ~D () 0.05 Probability Value: Decision Rule: Prototype in R: <Type I error must be set IF P < THEN REJECT H 0 ^ Sokal & Rohlf Table W The Kolmogorov-Smirnov distribution is not available for calculating this, so until I find one, we'll have to depend on R's function ks.test() to calculate this for us. d max #KOLMOGOROV SMIRNOV TWO SAMPLE TEST #FOR CONTINUOUS DATA #SOKAL & ROHLF EXAMPLE 3.7 SR=read.table("c:/DATA/Biostascs/SREX3.7R.txt") SR aach(sr) X=length[sample=="A"] Y=length[sample=="B"] ks.test(x,y) Two sample Kolmogorov Smirnov test data: X and Y D = 0.475, p value = alternave hypothesis: two sided Warning message: In ks.test(x, Y) : cannot compute correct p values with es ^ Here Calcuulation of max(d) match hand calculation above and in Sokal & Rohlf (995). This test is supposed to cover continuous distributions, so there should be no ties. As a result, R gives a warning message. Systat Output: Categorical values encountered during processing are: SAMPLE$ (2 levels) A, B > SR length sample 04 A 2 09 A 3 2 A 4 4 A 5 6 A 6 8 A 7 8 A 8 9 A 9 2 A 0 23 A 25 A 2 26 A 3 26 A 4 28 A 5 28 A 6 28 A 7 00 B 8 05 B 9 07 B B 2 08 B 22 B 23 6 B B 25 2 B B Kolmogorov-Smirnov Two Sample Test results Maximum differences for pairs of groups A B A B < test statistic matches Two-sided probabilities A B A. B < probability doesn't exactly match!