1 A Non-prmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University of Perdeniy, Perdeniy, Sri Lnk. ABSTRT A ommon prolem fed in dt nlysis is testing intertion effets when the sle of mesurement is ordinl. Akritl et.l. (997) introdued methodology to study the intertion effets for ordinl dt. owever, their methodology is limited to testing two-wy intertion with fixed levels of ftors. This study suggests the extension of Akrits et l method for three-wy intertion. The test sttistis used hve lose reltion with some reent developments in the symptoti theory for F -test nd -test when the oservtions re independent nd the levels of the ftor re fixed. Also we hve shown how this proedure n get implemented using sttistil softwre pkge (SAS). As n illustrtion, rel dt set is nlyzed nd the results re ompred with the prmetri test, Anlysis of Vrine (ANOVA). The suggested method n e onsidered s nonprmetri lterntive for ANOVA. Key words: Ftoril design, Non-prmetri hypotheses, Rnk, Intertion INTRODUCTION The most importnt reson for onduting ftoril experiment insted of seprte single-ftor experiments is the ility to hek for the presene of intertions mong the ftors. Determining intertion effets in prmetri pproh is well estlished. owever, nonprmetri equivlent is not ville espeilly for higher order intertions. In prmetri pproh, the effets re typilly modeled y deomposing the ell mens into min effets nd intertion effets. Thus for two-ftor design with fixed levels & for two ftors
2 respetively, the k th oservtion from ( i, j) is modeled s ) ; i,..., j,..., Where Y i i j (,, ( ), ( ) nd the error term re i i j j j independently normlly distriuted with zero men nd ommon vrine. The extension of prmetri pproh to testing higher order intertions is well estlished. owever, the nonprmetri lterntives re unommon for testing intertions, espeilly three-wy nd higher orders. The im of this pper is to propose rnk test sttistis for the non-prmetri hypotheses of three-ftor ftoril designs with independent oservtions, fixed numer of levels nd severl independent oservtions (replites) per ell (ftoril omintions) METODOLOGY Model nd Nonprmetri ypothesis th Let A, B, C denote ftors with levels, nd, respetively. The l oservtions in ell ( i, j, k) will e denoted yy l. It is ssumed tht Y l re independent with P( Y x) F ( x) P( Y x) F ( x) l i,..., j,..., k,..., l. ere F is the right ontinuous version nd F is the left ontinuous version of the distriution ofy l. In prtiulr, we do not ssume ontinuous distriutions. For this reson we introdue the nottion F ( x) F ( x) F ( x) (.) This definition of the distriution funtion inludes the se of ties nd moreover, disrete ordinl dt is inluded in this setup. Thus F will e onsidered s the distriution funtion ofy l, nd will e denoted y Y l ~ F (.) Aordingly the generl model does not require tht the distriutions in different ells re relted in ny prmetri wy. Consider the deomposing of F s follows:
3 F x M x Ai x B j x Ck x AB x ik x jk x x Where i i i j i k i A, ( AB ) ( ) ( ) ik jk j B,,, j j k, k k ( AB) ( ) ( ) C ik k, jk,,, ( ), ( ), nd ( ). j (.3) Thus, M F..., Ai Fi M, B j F j M, Ck F k M AB F F F M i j,, Fi k Fi F k M, F F F M F j ik F F k M F F ik F jk jk F i jk j k, will e lled respetively, the men, the non-prmetri min effet of ftor A, min effet of ftor B, min effet of ftor C, intertion effet of A nd B, intertion effet of A nd C, intertion effet of B nd C nd intertion effet of A,B nd C. Therefore, the null hypotheses for intertions n e formulted s follows. AB : AB i..., j... (.4) : ik i..., k... (.5)
4 : jk j..., k... (.6) : i..., j..., k... (.7) The hypotheses ( AB ), ( ), ( ), respetively re the hypothesis of no intertion (AB), () () nd () effet. Test sttistis Let F F, F,...., F, denote the x olumn vetor onsisting of the F, nd set CAB M M distriution funtion C C C M M M M M M M (.8) M is defined s M I d d Where for ny integer d, d d, where d denotes d olumn vetor of s, nd I d is the d-dimensionl identity mtrix. The nonprmetri hypotheses (.4), (.5), (.6) nd (.7) re equivlently written s AB : ( M M ) F CABF : ( M M ) F C F :( M M ) F C F :( M M M ) F C F (.9) Rnk sttistis for these hypotheses re derived y onsidering estimtes for the quntities (Bkeerthn, 3), Tˆ ˆ AB AB d ˆ Fˆ Tˆ ˆ d ˆ Fˆ Tˆ ˆ d ˆ Fˆ Tˆ ˆ d ˆ Fˆ Where ˆ df ˆ is the x vetor Whose omponents re ˆ df ˆ nd x i j k F x n, where lim N nd N
5 N n i j k. Define Fˆ x Fˆ x Fˆ x (.) to e the empiril distriution funtion from the oservtion in ell ( i, j, k) n Where Fˆ x n IY l x l n distriution is funtion nd F x n IY l x of the empiril distriution funtion. Set ˆ x ˆ Fˆ x i j k n Where ˆ. N Then, the rnk of oservtion the right ontinuous version of the empiril ˆ is the left ontinuous version l Y l mong ll N oservtions is given y Rl Nˆ Y l (.) If there re no ties, then this redues to the usul rnk ofy l. In the presene of ties, R l gives the verge rnk ofy l. Let e n ritrry full-rnk ontrst mtrix. The proposed sttistis for testing non-prmetri hypothesis of the form F re sed on the symptoti distriutions of ˆ d ˆ Fˆ (.) T Fˆ ˆ ˆ ˆ. Where F, F,..., F Note tht euse of eqution (.), ˆ T N R,..., R. (.3) Thus Tˆ d ˆ Fˆ is vetor of liner rnk sttistis. Let V dig with Vr,....., nd Vˆ denote the mtrix V with Y l for ll l,,...,n repled y
6 ˆ n n Rl R N nd l Akrits et l (997), the following results n e estlished. (i) Are onsistent estimtor of. ˆ (ii) Under the hypothesis F, the test sttisti Q NTˆ Vˆ Tˆ repled y ˆ. Thus, ording to (.4) hs, s N, r distriution, where r denotes hi-squred distriution with r degrees of freedom nd r =rnk ( ). In prtiulr for testing ( AB ), ( ), nd ( ), the following test sttistis respetively n e used. Q AB NTˆ AB ABV ˆ AB Tˆ AB (.5) hs, s N, hi-squred distriution with ( )( ) degrees of freedom. Q NTˆ V ˆ Tˆ hs, s N, hi-squred distriution with ( )( ) degree of freedom. NTˆ V ˆ Tˆ Q (.7) s N, hi-squred distriution with ( )( ) degrees of freedom. NTˆ V ˆ Tˆ Q (.8) hs s N, hi-squred distriution with ( )( )( ) degrees of freedom. owever the omputtion will e esier if null hypotheses re defined using projetion mtries (Akrits et l, 997, Brunner et l, 997) s desried elow. Let d denote the d olumn vetor, J d d d nd I d dig,...,. The projetion mtries re defined s P I J, P I J, nd P I J.
7 Using the projetion mtries the nonprmetri hypotheses ( AB ), ( ), nd ( ) my e written respetively of the form F s P P J with the degrees of freedom ( )( ) h ( )( ) P J P with the degrees of freedom h ( )( ) J P P with the degrees of freedom h nd ( )( )( ) P P P with the degrees of freedom h. The qudrti form Q ( ) n get omputed y using PROC MIXED of SAS (SAS, 99) with option CISQ, pplied to the rnked dt. In ft, PROC MIXED utilizes hypotheses defined using projetion mtries in omputtion. The following SAS sttements n e used to ompute to Q ( ) sttistis. DATA TRE_WAY; INPUT A B C Y; CARDS; ; PROC RANK DATA= TRE_WAY OUT=NEW; VAR Y; RANKS RY; PROC PRINT; RUN; PROC MIXED DATA =NEW; CLASS A B C; MODEL RY=A B C / CISQ; REPEATED/TYPE=UN() GROUP=A*B*C; RUN; Smll-smple pproximtions Consider the null hypothesis : F. Let denotes orthogonl projetion ( ) tht is e n ritrry projetion mtrix. Then F if nd only if F. Thus under : F, the sttisti N P ˆ symptotilly hs
8 multivrite norml distriution with men nd ovrine mtrix V (Akrits et l., 997), where ˆ ˆ ˆ V dig ˆ... ˆ This suggests the pplition of the pproximtion proedure to the qudrti form Q N NPˆ Pˆ, where Pˆ d ˆ Fˆ. Then under : F, it is resonle to pproximte the distriution of F N. ( V) Q( ) ( ) N h tr (.9) To follow entrl F distriution with degrees of freedom ˆf nd ˆf, where ˆ h tr( Vˆ) f, ˆ tr( Vˆ) f, tr( VV ˆ ˆ) ( ˆ tr V D) ( D dig n )...( n ), tr () = tre of the squre mtrix nd h= identil digonl element of projetion mtrix (Akrits et l. 997, Brunner et l,997). With the null hypotheses defined using projetion mtries the qudrti form F n e omputed y using pkge PROC MIXED or PROC GLM of SAS pplied to the rnked dt. owever, for unequl smple sizes F nnot e otined using PROC GLM. Lrge smple se EXAMPLE To show the ppliility of the proedures derived, we give n exmple where the dt onsists of humidity reords (dy nd night) t Anurdhpur, Colomo, mntot nd Nuwr Eliy of Septemer to Deemer for the period The dt re ville on monthly sis. owever, for the nlysis the dt from months August to Deemer were used. Aordingly, three ftors were defined s Sttion with 4 levels (Anurdhpur, Colomo, mntot nd Nuwr Eliy), Month with 5 levels (August to Deemer) nd Time with levels, (dy nd night). The yers 99 to 995 were tken s replites. The outputs of using PROC MIXED of SAS with the option CISQ re given in Tle.. In the output, NDF refers to degrees of freedom for eh ftor; DDF refers to error degrees of freedom. In the Tle. P-vlues orresponding to ll the min effets nd intertions re smll. Thus it ould e onluded tht ll effets inluding 3- wy intertion to e present. Vlidtion of the suggested method s vlidtion of the suggested method prmetri ANOVA ws performed using the sme dt
9 nd ompred the results. The output from ANOVA using PROC ANOVA of SAS is given in Tle.. In terms of power of the test, for most of the effets, s expeted, ANOVA hs provided higher power of the test. Tle.: SAS output from testing Fixed Effets (Lrge smple se) Soure NDF DDF Pr >ChiSq Pr > F STATION 3.. MONT 4.. STATION*MONT.. TIME.. STATION*TIME 3.. MONT*TIME STATION*MONT*TIME.5. Tle.: ANOVA tle (Lrge smple se) Soure DF F Vlue Pr>F STATION MONT STATION*MONT 5.8. TIME STATION*TIME MONT*TIME STATION*MONT*TIME Smll Smple Cse The dt used here is n extrt from ove exmple nd onsidered ftors nd levels similrly. The extrted dt set onsidered from Sttions for Months
10 August nd Septemer nd for the yers 99 nd 99. Sine the numer of replites were the sme (equl smple size) the F sttisti using PROC MIXED is sme s F from PROC GLM. The results of the tests using PROC MIXED of SAS re given in Tle.3. In the output, P-vlues orresponding to ll the min effets nd intertions re smll exept the min effet Month nd intertion effet Sttion nd Time. Thus it ould e onluded tht 3-wy intertion is not present. Similrly, other effets lso n e evluted. Vlidtion of the suggested method As vlidtion of the suggested method prmetri ANOVA ws performed using sme dt nd the results were ompred. The output from ANOVA using PROC ANOVA of SAS is given in Tle.4. Both tles led to the sme onlusions nd thus the suggested method n e onsidered vlid method. With respet to power of the test, results re onsistent s in the lrge smple se. DISCUSSION In this study, we hve suggested sttistil proedure for testing intertion in thee-ftor ftoril design with independent oservtions nd fixed effet to e useful nonprmetri lterntive for ANOVA. This method n get implemented using sttistil softwre pkge SAS for lrge smple se s well smll smple se. For smll smple se with equl smple sizes, F n e omputed y using PROC MIXED s well s PROC GLM pplied to the rnked dt. owever, the sttistis F nnot e otined from PROC GLM of SAS under unequl smple sizes. Nevertheless, Q n e omputed y SAS proedure PROC MIXED with option CISQ pplied to the rnked dt in oth ses. Tle.3: SAS output from testing Fixed Effets (Smll smple se) Soure NDF DDF Pr > ChiSq Pr > F STATION MONT STATION*MONT 8.. TIME STATION*TIME MONT*TIME STATION*MONT*TIME
11 Tle.4: ANOVA tle (Smll smple se) Soure DF F Vlue Pr>F STATION 68.. MONT 59.. STATION*MONT TIME.8.7 STATION*TIME.8.65 MONT*TIME.8.7 STATION*MONT*TIME REFERENCES Akrits, M.G., Arnold S.F. nd Brunner E. (997). Nonprmetri ypothesis nd Rnk Sttistis for Unlned Ftoril Designs, Journl of the Amerin Sttistil Assoition, 9, Bkeerthn G. (3). A onept of nonprmetri proedure for testing homogeneity of vrine nd non-prmetri pproh in testing higher order intertions. Brunner, E., Dette. nd Munk A. (997). Box-type pproximtions in nonprmetri ftoril design, Journl of the Amerin Sttistil Assoition,9, Brunner E. nd Puri M.L. (996). ndook of Sttistis 3.