Abstract. Introduction

Transcription

1 Derrick, B., White, P., ad Toher, D. A Iverse Normal Trasformatio Solutio for the compariso of two samples that cotai oth paired oservatios ad idepedet oservatios. Astract Iverse ormal trasformatios applied to the partially overlappig samples t-tests y Derrick et.al. (07) are cosidered for their Type I error roustess ad power. The iverse ormal trasformatio solutios proposed i this paper are show to maitai Type I error roustess. For icreasig degrees of skewess they also offer improved power relative to the parametric partially overlappig samples t-tests. The power whe usig iverse ormal trasformatio solutios are comparale to rak ased o-parametric solutios. Itroductio A frequetly asked questio i research is how to compare meas etwee two samples that iclude oth paired oservatios ad upaired oservatios (Derrick, Toher ad White, 07). This is referred to as partially overlappig samples (Derrick et.al., 05). Parametric test statistics for the compariso of meas of two samples that cotai oth paired oservatios ad idepedet oservatios are give y Derrick et.al. (07). These partially overlappig samples t-tests use all of the availale data, ad are Type I error roust ad more powerful tha covetioal methods, whe ay missig data is missig completely at radom. The partially overlappig samples t-tests are ased o the assumptio of ormality (Derrick, 07). To remove the restrictio of the ormality assumptio, the applicatio Page of 4

2 Derrick, B., White, P., ad Toher, D. of a Iverse Normal Trasformatio (INT) is proposed i this paper. This method is compared to the existig partially overlappig samples t-tests, ad a simple rakig ased o-parametric method. The parametric partially overlappig samples test statistic, T ew, is a iterpolatio etwee the paired samples t-test ad the idepedet samples t-test assumig equal variaces, ad is give y Derrick et.al. (07) as: T ew = S p X + X c r where S p = ( ) S + ( ( ) + ( ) S ) The test statistic T ew is refereced agaist the t-distriutio with degrees of freedom: a c v = ) + ( + ) + + ( c a a. + + c where for j = {Group, Group }, X j = mea of Sample j, = umer of upaired oservatios exclusive to Sample, = umer of upaired oservatios exclusive to Sample, c = umer of pairs, j = total umer of oservatios i Sample j, j S = variace of Sample j, ad r = Pearso s correlatio coefficiet for the paired oservatios. a The idepedet samples t-test is ot Type I error roust whe variaces are ot equal ad sample sizes are ot equal. It follows that T ew is also sesitive to deviatios from the equal variaces assumptio. If equal variaces caot e assumed, the alteratively Welch s t-test is Type I error roust uder ormality (Derrick, Toher ad White, 06). The partially overlappig samples t-test ot costraied to equal variaces, T ew, is a iterpolatio etwee the paired samples t-test ad Welch s t-test, is give y Derrick et.al. (07) as: Page of 4

3 Derrick, B., White, P., ad Toher, D. T ew = S S + X X SS r c The test statistic T ew is refereced agaist the t-distriutio with degrees of freedom: c v = ( ) + ( + ) c γ + a + + c a where γ = ( S / ) ( S / ) S S + + For simple rakig ased o-parametric solutios, oservatios are pooled ad assiged rak values i ascedig order. The rak values are sustituted ito the elemets of the calculatio for T ew ad T ew i place of the oserved values. This gives the test statistics T RNK ad T RNK respectively. The degrees of freedom υ ad υ respectively, are calculated usig the pooled rak values. The trasformig of data attempts to overcome violatios of the ormality assumptio, so that the traditioal parametric tests ca e applied. For equal sample sizes, Cohe ad Arthur (99) foud that the idepedet samples t-tests performed o log or squared data, exhiits satisfactory Type I error roustess. However, popular trasformatios such as the Box-Cox trasformatios do ot ecessarily lead to Type I error roust tests whe sample sizes are uequal or variaces are uequal (Zaremka, 974). I additio the Box-Cox trasformatio rarely results i oth ormality ad equal variaces at the same time (Sakia, 99). Eve if researchers are comfortale with the hypothesis of comparig meas of trasformed data, a suitale trasformatio may ot always e foud. A trasformatio that should always give the appearace of ormally distriuted data, is a Iverse Normal Trasformatio (INT), which derives properties from the Normal distriutio. Methods ased o Fisher ad Yates Normal approximatio are the most commoly used INT (Beasley ad Erickso, 009), ad are the most Page 3 of 4

4 Derrick, B., White, P., ad Toher, D. powerful (Bradley, 968). Example INT approaches iclude Blom s method ad Va der Waerde s method, ut it is of little cosequece which of them is selected ecause they are liear trasformatios of oe-aother (Tukey, 96). It should e oted that a INT does ot make a populatio Normal, ut it makes a sample appear Normal. This is ot the same as directly esurig the assumptio of ormally distriuted residuals is true (Servi ad Stephes, 007). I additio, care eeds to e take with the iterpretatio of results. The ull hypothesis of equal meas for the trasformed data, is assumed to e equivalet to a ull hypothesis of equal meas. The Fisher ad Yates INT procedure requires poolig all of the oservatios, sortig ito ascedig order ad rakig each oservatio over the etire sample so that: = Φ yi c Xi where y i is the ordiary rak of oservatio i, N is the total N c + pooled sample size, ad Φ is the stadard Normal quatile fuctio. The most simple INT method is attriuted to Va Der Waerde (95) uses c = 0. Pefield (994) foud that Va Der Waerde trasformatios applied to the idepedet samples t-test, are Type I errors roust across a variety of distriutios. Calculatig the Va Der Waerde scores, of T ew ad ew T, gives distriutio free test statistics X m, ad usig these withi the calculatio T INT ad T INT respectively. The degrees of freedom υ ad υ respectively, are calculated usig the pooled trasformed values. The calculatio of r is Pearso s correlatio coefficiet etwee the trasformed paired oservatios. Methodology For each of the six test statistics defied aove ( T ew, T ew, T RNK, T RNK, T INT, T INT ), the roustess for validity (i.e. Type I errors) ad efficacy (i.e. power) are explored Page 4 of 4

5 Derrick, B., White, P., ad Toher, D. usig simulatio. Values of a,, c {5, 0, 30, 50, 00, 500} are varied i a factorial desig, as well as values of the populatio correlatio coefficiet ρ {-.75,-.50, -.5,.00,.5,.50,.75}. To geerate the idepedet oservatios, firstly the Mersee-Twister algorithm (Matsumoto ad Nishimura, 998) geerates radom U(0,) deviates. These uiform deviates are trasformed ito N(0,) deviates usig the Paley ad Wieer (934) trasformatio. To geerate the paired oservatios, the approach used is equivalet to that used y Derrick ad White (07) ad Derrick et.al. (07). I this approach, additioal Stadard Normal deviates are trasformed to correlated Stadard Normal ivariates, z ji, as follows: z i = + ρ ρ z z i + i ad z i = + ρ ρ z z i i where i = (,,., c ) Each of the test statistics are assessed firstly uder the Stadard Normal distriutio. For the compariso of test statistics uder o-ormality, oservatios are geerated y trasformatio of the Stadard Normal deviates as give i Tale. Tale. Trasformatios applied to Stadard Normal deviates (N) to otai oormally distriuted deviates (X), with the resultig skewess ad kurtosis. Note that Uiform (U) deviates are calculated as the cumulative desity fuctio of N. Distriutio Trasformatio Skewess Kurtosis Normal (N) X Gumel X= -log(-log U) Expoetial X= -log (U) Logormal X= expoetial (N) Page 5 of 4

6 Derrick, B., White, P., ad Toher, D. Followig the trasformatios as per Tale, the calculatio of raks / iverse ormal trasformatios, are performed as appropriate for each statistical test. Each of the tests are performed at the 5% sigificace level, two sided. For each of the parameter comiatios withi the factorial desig, the ull hypothesis rejectio rate (NHRR) is recorded as the proportio of the 0,000 replicates where the ull hypothesis is rejected. The methodology is depicted i Figure. Aalyses uder the alterative hypothesis proceeds similarly, ut with the additio of 0.5 to the oservatios directly followig the trasformatio i Tale. Figure. Overview of the simulatio process. Page 6 of 4

7 Derrick, B., White, P., ad Toher, D. Results Uder the ull hypothesis, 0,000 iteratios are otaied for each of the,5 parameter comiatios. The Type I error rates for each of the test statistics across the simulatio desig are give through Figure to Figure 5. Each parameter comiatio which has a Type I error rate etwee.5% ad 7.5%, is cosidered to e maitaiig reasoale Type I error roustess Figure. Type I error rates for the Stadard Normal distriutio. Figure shows that whe oth samples are take from the Stadard Normal distriutio, all of the test statistics are Type I error roust for all of the parameter comiatios withi the simulatio desig. Page 7 of 4

8 Derrick, B., White, P., ad Toher, D. Figure 3. Type I error rates for the Gumel distriutio. Figure 3 shows that whe oth samples are take from the Gumel distriutio, all of the test statistics are Type I error roust for all of the parameter comiatios withi the simulatio desig. This suggests that all of the test statistics are Type I error roust for distriutios with a relatively small degree of skewess. Page 8 of 4

9 Derrick, B., White, P., ad Toher, D. Figure 4. Type I error rates for the Expoetial distriutio. Figure 4 shows that whe oth samples are take from the Expoetial distriutio, all of the test statistics are Type I error roust whe sample sizes are equal. However the test statistic T ew is ot Type I error roust whe there is a large imalace etwee the size of the two samples. This suggests that all of the test statistics except T ew are Type I error roust for distriutios with a moderate skew. Page 9 of 4

10 Derrick, B., White, P., ad Toher, D. Figure 5. Type I error rates for the Logormal distriutio. Figure 5 shows that whe oth samples are take from the Logormal distriutio, all of the test statistics are Type I error roust with the exceptio of T ew which ca e occasioally coservative whe sample sizes are equal, ad ca e lieral whe sample sizes are ot equal. This suggests that all of the test statistics except T ew maitai Type I error roustess for heavily skewed distriutios. For the simulatios uder the alterative hypothesis Tale gives the average power across the simulatio desig of sample size ad correlatio coefficiet for each of the distriutios. Power is oly recorded for scearios where the test statistic maitais Type I error roustess. Page 0 of 4

11 Derrick, B., White, P., ad Toher, D. Tale. Power for α =.05, µ =.5, two sided, over all values of c. µ Distriutio Sample size ρ Tew Tew TRNK T RNK TINT TINT Normal Gumel Expoetial Logormal a = a a = a a = a a = a > < > < > < > < > < > < > < > < Whe populatio variaces are equal, Tale shows that the test statistics ot assumig equal variaces, T ew, T RNK ad T INT, perform similarly to their couterparts where equal variaces are assumed T ew, T RNK ad T INT respectively. From Tale it ca e see that for ormally distriuted data, the parametric statistics T ew ad T ew are margially more powerful tha the other test statistics cosidered, ut ot to ay meaigful extet. Page of 4

12 Derrick, B., White, P., ad Toher, D. The relative power advatage of T INT ad T INT over T ew ad T ew icreases as the degree of skewess icreases. However, the proposed statistics usig iverse ormal trasformatios, T INT ad T INT, yield very similar results to T RNK ad T RNK. Coclusio Test statistics makig use of all of the availale data i a partially overlappig samples desig are compared usig simulatio. Assumig ormality the partially overlappig samples t-test proposed y Derrick et.al. (07) for equal variaces, T ew, is more powerful tha o-parametric equivalets ad Iverse Normal trasformatios. The test statistics makig use of Iverse Normal Trasformatios offer o sustatial improvemet over the o-parametric tests. Due to its Type I error roustess, power properties ad relative simplicity, T RNK is recommeded over T INT as the est solutio for comparig partially overlappig samples from o-ormal distriutios. Refereces Beasley, T. M., Erickso, S. & Alliso, D. B. (009). Rak-ased iverse ormal trasformatios are icreasigly used, ut are they merited?. Behavior geetics. 39(5), Bradley, J. V. (968). Distriutio-free statistical tests. Pretice-Hall; New York. Page of 4

13 Derrick, B., White, P., ad Toher, D. Cohe, M. & Arthur, J. (99). Radomizatio aalysis of detal data characterized y skew ad variace heterogeeity. Commuity detistry ad oral epidemiology. 9(4), Derrick, B. (07) Statistics: New t-tests for the compariso of two partially overlappig samples. I: Faculty of Eviromet ad Techology Degree Show, UWE, Frechay Campus, UWE, Jue 07. Availale from: Derrick, B., Doso-McKittrick, A., Toher, D. & White P. (05). Test statistics for comparig two proportios with partially overlappig samples. Joural of Applied Quatitative Methods. 0(3), -4. Derrick, B., Russ, B., Toher, D. & White P. (07). Test statistics for the compariso of meas for two samples which iclude oth paired oservatios ad idepedet oservatios. Joural of Moder Applied Statistical Methods. 6(), Derrick, B., Toher, D. & White, P. (06). Why Welch s test is Type I error roust. The Quatitative Methods for Psychology. (), Derrick, B., Toher, D. & White, P. (07). How to compare the meas of two samples that iclude paired oservatios ad idepedet oservatios: A compaio to Derrick, Russ, Toher ad White (07). The Quatitative Methods for Psychology. 3(), 0-6. Derrick, B. ad White, P. (07) Comparig two samples from a idividual Likert questio. Iteratioal Joural of Mathematics ad Statistics. 8(3), -3. Page 3 of 4

14 Derrick, B., White, P., ad Toher, D. Paley, R. E. A. C. & Wieer, N. (934). Fourier trasforms i the complex domai. America Mathematical Society. 9 Sakia, R. M. (99). The Box-Cox trasformatio techique: a review. The Statisticia Servi, B. & Stephes, M. (007). Imputatio-ased aalysis of associatio studies: cadidate regios ad quatitative traits. PLoS Geet, 3(7), e4. Tukey, J. W. (96). The future of data aalysis. The Aals of Mathematical Statistics, 33(), -67. Va der Waerde, B. L. (95). Order tests for the two-sample prolem ad their power. Idagatioes Mathematicae, 4(53), Zaremka, P. (97). Trasformatio of variales i ecoometrics. Trasformatio of variales i ecoometrics. I Ecoometrics, Page 4 of 4