ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN

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1 Colloquum Bometrcum 4 ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka 3, -95 Lubl e-mals: zofa.hausz@up.lubl.pl, joaa.tarasska@up.lubl.pl Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, -787 Warszawa e-mal: wojtek.zelsk@statystyka.fo Summary Shapro-Wlk W test s wdely used for checkg ormalty of data. The paper cosders ts modfcato to the case of ormalty wth kow mea. The table wth crtcal values of modfed test for dfferet sample szes ad several sgfcace levels s gve. A applcato for resduals two-way ANOVA model s preseted. Keywords ad phrases: Shapro-Wlk W test, ormalty, two-way expermetal layout Classfcato AMS : 6G. Itroducto Shapro ad Wlk (965) troduced the W test for ormalty based o statstc W = ( ) ( X X ) a X,

2 where X ( ) X () X ( ) are the ordered values of the sample ( X, X,, X ) ad a are tabulated coeffcets. The W test s cosdered as very powerful for the hypothess that a radom varable X s ormally dstrbuted wth ukow parameters µ ad However, frequetly we are terested testg ull hypothess that dstrbuto of X s ormal wth kow expectato µ. Adaptato of the Shapro-Wlk W test to the case of kow mea s descrbed Secto. I Secto 3 we gve two examples llustratg applcatos of the Shapro-Wlk W test ad ts modfcato. Some cocludg remarks are preseted Secto 4. σ.. Descrpto of W statstc Let us cosder the ull hypothess of the form: H : X s ormally dstrbuted wth a kow expectato µ. To test the H hypothess we propose modfcato of the Shapro-Wlk W statstc the followg form W =. ( X µ ) a X ( ) The hypothess H s rejected at a sgfcace level α f W s less the the crtcal value W ( α ;). The crtcal values of W ca be evaluated smulato study. For each sample sze of = 3, 4,, 5 ; =,, N pseudoradom samples from (, ) for each sample the value W was calculated, so the sample w,, N were geerated ad w W N of values of the statstc were obtaed. The crtcal value W ( α ;) was take as the α-th quatle of w,,. All calculatos were doe depedetly Mathematca ad R program. I w N program R we used the procedure shapro.test whch Roysto s procedure s applcated (Roysto 99; Hausz, Tarasska ). The results are gve Table. Table. Crtcal values of W statstc for sample szes ad sgfcace level α α =. α =. 5 α =. α =. α =. 5 α =

3 The statstc W has smlar propertes to the W statstc, amely, W s scale varat ad the maxmum of W s oe. However, the mmum of W s zero, whereas the mmum of W s ε = a (Shapro ad Wlk, 965). It s suffcet to cosder the maxmzato of ( x µ ) subject to x = x a ad ote that ( µ ) 3. Applcato of W test may be arbtrarly large. The advatage of usg of the W test we llustrate wth umercal examples. Let us cosder a two-way expermetal layout whch volves two treatmet factors A ad B. Let us assume that the factor A has a levels A,, B, B b. For each possble value of (,,a) B, observato of X ( k,, ) = affected by levels A ad, A, Aa, ad the factor B has b levels = ad j ( j,,b) B j. Let us assume that observatos x jk fulfll the followg model wth j j ( ) j jk j jk =, let x jk be a kth x = µ + e (3.) µ = α + β + αβ, where α deotes a effect of th level of A, ) j β j deotes a effect of jth level of B, ( αβ deotes a teracto betwee th level of A ad jth level of B ( =,, a, j =,, b, k =,, ). We assume that jk e s are depedet N (, σ ) 3

4 varables. If the model (3.) s adequate to the expermetal data the for each combato (, j) the resduals eˆ jk = xjk xj, where x = j x jk k=, should be dstrbuted as (, σ ) N. Whe the teracto model (3.) s eglected, the the followg model s cosdered: x = α + β + e. (3.) jk j jk I model (3.), resduals are equal to e ˆ jk = xjk x x j + xj, where b = x x j b j= ad x a j = x j a. Example. Let us cosder a expermet wth two levels of a factor A ( a = ) ad three levels of a factor B ( = 3) b. Oe set of data wth = replcatos was geerated accordg to model (3.) wth µ =, µ =, µ = 3 3, µ =, µ = 4, µ 4 ad σ =. The results of aalyss of varace are gve Table. Table. The results of aalyss of varace for model (3.) 3 = Sum of Mea Source d.f. F Test p-value Squares Square A B AxB Error Thus, for our data the teracto betwee factors tured out to be sgfcat. I spte of the fact that wth gve µ j s teracto was volved the model. If we cosder model (3.) the we get the results gve Table 3. Table 3. The results of aalyss of varace for model (3.) Source d.f. Sum of Mea Squares Square F Test p-value A B Error Now, we wll focus o checkg whether for each combato (, j) the resduals both models (3.) ad (3.) are ormally dstrbuted wth ull mea.e. the hypothess that resduals are (, σ ) N should be verfed. Results, rouded to the thrd decmal place, are gve Table 4. The values of W are the same (3.) ad (3.) models. I model (3.) they are also the same as W values. 4

5 Table 4. Results of checkg ormalty for resduals ANOVA. Cell Resduals model (3.) W W A B A B A B A B A B A B Resduals model (3.) A B A B A B A B A B A B For = ad α =. 5, crtcal value of the W test s equal to.585 (see Table ) ad of the Shapro-Wlk W test s equal to.8449 (Hausz, Tarasska, ). I the case of model (3.), the hypothess of ormalty wth ull mea was rejected for observatos from the cell A B (bold umber), whle the case of model (3.) was ever rejected. Let us otce that the Shapro-Wlk W test ever rejected the ormalty of resduals both models. Example. I ths example we cosder smlar model as Example, just takg µ = 5 stead of µ = 4. A set of data wth = replcatos was geerated. The results of aalyss of varace for model (3.) are preseted Table 5. Table 5. The results of aalyss of varace for model (3.) Sum of Mea Source d.f. F Test p-value Squares Square A B AxB Error I ths example, the teracto betwee factors s sgfcat. The results of testg ormalty of the resduals for each combato (, j) for models (3.) ad (3.) are preseted Table 6. Table 6. Results of checkg ormalty for resduals ANOVA. Cell Resduals model (3.) W W A B A B A B A B A B A B Resduals model (3.) A B A B A B

6 A B A B A B For the model (3.) both tests W ad W dd ot reject the ull hypothess about ormalty. However, whe we cosder ot adequate model (3.), the W test rejected ormalty of resduals four cells (bold umbers), whle the Shapro-Wlk W test ever rejected the ormalty of resduals. 4. Coclusos I the paper some modfcato of the Shapro-Wlk W test for testg ormalty s proposed. Ths test should be appled whe the mea of radom varable s kow. I geeral, suggested test should be recommeded for testg ormalty of resduals, whe observato are affected by some factors. I the paper we show by the examples that the W test ca reject ormalty whe data does ot fulfll the theoretcal model, cotrary to the Shapro-Wlk W test whch does ot reject ormalty such a stuato. Refereces Hausz Z., Tarasska J. (). Tables for Shapro-Wlk W statstc accordg to Roysto approxmato, Colloquum Bometrcum 4, -9. Roysto P. (99). Approxmatg the Shapro-Wlk W-test for o-ormalty. Statstcs ad Computg,, 7-9. R Developmet Core Team (8). R: A laguage ad evromet for statstcal computg. R Foudato for Statstcal Computg. Vea, Austra. ISBN , URL Shapro S.S., Wlk M.B. (965). A aalyss of varace test for ormalty (complete samples). Bometrka 5,