THE ANALYSIS OF COUNT DATA IN A ONE- WAY LAYOUT

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1 Libraries Cnference n Applied Statistics in Agriculture th Annual Cnference Prceedings THE ANALYSIS OF COUNT DATA IN A ONE- WAY LAYOUT Yuhua Wang Dallas E. Jhnsn Linda J. Yung Fllw this and additinal wrks at: Part f the Agriculture Cmmns, and the Applied Statistics Cmmns This wrk is licensed under a Creative Cmmns Attributin-Nncmmercial-N Derivative Wrks 4.0 License. Recmmended Citatin Wang, Yuhua; Jhnsn, Dallas E.; and Yung, Linda J. (1999). "THE ANALYSIS OF COUNT DATA IN A ONE-WAY LAYOUT," Cnference n Applied Statistics in Agriculture. This is brught t yu fr free and pen access by the Cnferences at. It has been accepted fr inclusin in Cnference n Applied Statistics in Agriculture by an authrized administratr f. Fr mre infrmatin, please cntact cads@k-state.edu.

2 Cnference n Applied Statistics in Agriculture 156 The Analysis f Cunt Data in a One-Way Layut Yuhua Wang Quintiles Inc. P.O. Bx 9708, Kansas City, MO Dallas E. Jhnsn Department f Statistics, Linda J. Yung Department f Bimetry, University f Nebraska-Lincln ABSTRACT An efficient scre statistic fr testing the equality f the means f several grups f cunt data in the presence f a cmmn dispersin parameter is intrduced and a new apprximatin t its distributin is given. The perfrmance f the efficient scre statistic using this apprximatin, the riginal efficient scre statistic apprximated by X 2 (t -1), the likelihd rati statistic and fur mre AN OVA methds based n raw data r transfrmed data are cmpared in terms f size and pwer by using Mnte Carl simulatins. The efficient scre statistic with its new apprximatin is recmmended. An applicatin is given. 1. Intrductin The tw distributins cmmnly used t mdel cunt data, the Pissn and the negative binmial (with knwn dispersin parameter), are members f the expnential family. The generalized linear mdel (GLIM) presents an exciting alternative t the general linear mdel (GLM) in that the frm f the distributin can be incrprated in the mdel. Hwever, unless the ppulatin distributin is nrmal, the tests assciated with the GLIM are asympttic. One cncern is whether we have sufficient data supprt fr the asympttic thery when dealing with standard experiments invlving relatively small sample sizes. Anther cncern with the generalized linear mdel is the rle f the dispersin parameter. Tw pssibilities have been suggested in the literature fr accunting fr ver-dispersin in the mdel. Cx (1983), in the cntext f maximum likelihd estimatin, calls fr the use f the negative binmial distributin as a detailed representatin f ver-dispersin in the Pissn case. In this case, Y j can be cnsidered a Pissn randm variable with mean A j, where AI' A 2,,An are iid gamma with mean Jl and dispersin parameter 1" Then YI, Y 2,, Yn are iid negative binmial with parameters Jl and 1" The use f the negative binmial distributin instead f the Pissn t mdel ver-dispersed cunt data in the generalized linear mdel analysis fllws naturally frm this idea. McCullagh and NeIder (1989) suggest a different apprach fr ver-dispersed data. The data are mdeled based n the anticipated distributin, such as the Pissn, and a ver-dispersin parameter<1> (<1> > 1) is added t accunt fr ver-dispersin. Then the dispersin parameter <1> is

3 Applied Statistics in Agriculture Cnference n Applied Statistics in Agriculture 157 estimated frm the data by ne f tw methds. The first apprach is t set <I> equal t the deviance functin divided by the degrees f freedm. The deviance functin has the frm D(y, Il) = 2<1>(l(y; y) -1(Jl; y)), where ley; y) and 1(P,; y) are the likelihds as a functin f the data and estimated parameters, respectively. The secnd methd equates <I> freedm. Pearsn's X 2 statistic is X2 = I (y - Jl)2 / V(Jl), and Pearsn's X2 statistic divided by the degrees f where V(Jl) is the estimated variance functin. Bth prcedures are t liberal when we use them t mdel ver-dispersed cunt data and t test the null hypthesis H : III = 112 =... = Ilt (see Yung, et ai., (1999)). Therefre, we d nt cnsider the apprach f estimating <I> here. The classical statistical setting fr hypthesis testing invlves a sequence f independent randm variables whse distributin depends n a t-dimensinal parameter e = (e l,e2,..., at)' belnging t a sample space e, an pen subset f t-dimensinal Euclidean space 9{t. A null hypthesis H usually invlves restrictins f the parameter, e = (e l, e2,..., e t)', Rie) = 0 fr j = 1,2, "', r, (r:::; t). Nw cnsider tests f the (cmpsite) hyptheses R : R(e) = 0 vs. HI: R(e):;t: 0 (1.1), where R(e) = [R I (e),..., Rr(e)] is a vectr-valued functin R: 9{t ~ 9{r such that the (txr) matrix W(a) ~ (ar/a a ) exists and is cntinuus in a and rank(w(a)) ~ r. Mre specifically, _ suppse e is the MLE f e under the restrictins impsed by the null hypthesis, e is the (unrestricted) MLE f a and n is the sample size. Further, I(a) ~ ( - E[ a' IO:~~~~' a)}] J is Fisher's Infrmatin Matrix. Tests f H against HI have typically invlved ne f the three test statistics: 1. Neyman-Pearsn's likelihd rati statistic given by A= 2 (L(fh - L(e)) where L is the lg likelihd; 2. Ra' s efficient scre statistic ~= S'(e)[ I(e) rl See) where See) is the likelihd scre functin defined by d L(e) / de; and 3. W aid's test statistic A r=nr'(8)( W'(8)(1(8) tw(8) JI R(8). Under R ' each f the three statistics has an asympttic x2(r) distributin. Ra's efficient scre statistic depends nly n the MLE fr the restricted class f parameters under H ' while Wald's

4 Cnference n Applied Statistics in Agriculture 158 statistic depends n the MLE ver the whle parameter space. Because Wald's test is cmplicated and des nt have an explicit frm, it is nt cnsidered here. 2. Testing equality f cunt means with a knwn cmmn dispersin parameter 1: Cnsider the density functin frm fr the negative binmial distributin that was prpsed by Bliss & Owen (1958) in which the randm variable Y fllws a negative binmial distributin with mean Il and dispersin parameter 1:, NB(Il, 1:), if Pr (Y = y) = r(y + 1:- I ) ( ~ JY( _1_ JIlt, y!r(1:-i) 1+1:1l 1+1:1l (2.1) fr y = 0, 1, 2,... ; 1: ~ 0; and Il >. Fr this parameterizatin E(Y) = Il and Var(Y) = Il + Il 1: 2 Nw suppse Yij - ind NB (Ili,1:) fr j = 1, 2,..., n i and i = 1, 2,..., t. (2.2) Cnsider testing H : III = 112 =... = III versus HI : nt all lli's are equal. It fllws frm (2.1) and (2.2) that the lg-likelihd functin f the lli's is L(IlI',IlI)= t[tiog[r(~ij +~~I) J+(t Yij JIOg( 1:lli 2.1. Likelihd rati test 1=1 J=I Yij T(1:) J=I 1 +1:lli 1: J-~IOg(1+1:IlJ]. Bamwal and Paul (1988) btained Neyman-Pearsn's likelihd rati statistic under the assumptin f a cmmn dispersin parameter 1: as A =2(L(~)-L(jI)) (2.3) = 2 t[niyi lg (1:YJ] - 2 t [ni(yi + 1:-1 ) lg (l + 1:YJ] - i=1 i=1 2( t.n; )YIOg(~y) + 2( t.n; }l( Y H-')lg(l HY) 1 (2.4) where Yi is the (unrestricted) maximum likelihd estimatr f Ili under the full mdel ni ~i = Yi = Yi fni = LYij fni (i = 1,..., t). (2.5) j=1 Let Il represent the cmmn value f each Ili under H. Then Y is the maximum likelihd estimatr f each Il under H. That is, we have _ - ~ Yie ~ Il = Y = L.J-' where n = L.Jni. (2.6) i=l n i=l The likelihd rati statistic A has an asympttic distributin that is chi-square with (t -1) degrees f freedm.

5 Applied Statistics in Agriculture Cnference n Applied Statistics in Agriculture Efficient scre statistic In this sectin, we develp Ra's efficient scre statistic fr testing the hyptheses H : ~I = ~2 =... = ~t vs. HI: nt all ~i 's are equal. Differentiating the lg-likelihd functin f the ~i 's with respect t ~i Next replace ~i yields al(~\,~2""'~t)_ (niyij ni (2.7) a ~i ~Jl + 't~j (1 + 't~j in (2.7) by ji = Y. It fllws frm the definitin f S (ji) that S'(!I)=( Y(~~~Y) - (1+n~y)... y(~~~y) - (1+n~y)} (2.8) The (r, s)th element f Fisher's Infrmatin Matrix is given by irs =_E(a2L(~\' ~2'... '~t)j. (2.9), a~r a~s With the help f the equatin a2l(~\,~2'''''~t) (Yi.) (-1-2't~J 'tni ----''-'---=----=:-----'---'-- = + a~i2 ~;(1+'t~J2 (1+'t~J2' (2.9) becmes ~i (1 + 't~j Then, replacing each ~i if r -:f. s if r = s in (2.10) by Y we btain ( ~ ) = lagna,...,. I -d' I ( n I n t J Y(1+'tY) Y(1+'tY) (2.10) Thus, the efficient scre statistic is S = S' (ji)[ I(jI) ] -I S(jI) t (n.? n J2( n J- I t njy; - y)2 = i=1 Y (1 ~ ~ Y) ~ Y Y (l ~ 't Y = i=1 Y (l + 't Y ) (2.11) Equatin (2.11) is identical t Neyman's C( a) statistic when 't is knwn, as indicated by Bamwal and Paul (1988) F -tests One cmmn way t analyze cunt data in a ne-way layut data is t use analysis f variance and rely n the Central Limit Therem and the rbustness prperties f the F test. If

6 Cnference n Applied Statistics in Agriculture 160 variances are unequal, the impact n the F test can be significant fr the ne-way classificatin. Several variance stabilizing transfrmatins have been suggested t address this cncern. (1) Anscmbe (1949) suggested the transfrmatin, fey) = lg(y + 1), fr the negative binmial distributin. (2) The square rt transfrmatin, fey) =.JY, is cmmnly used when the ppulatins are Pissn distributed. (3) Beall (1942) suggested the fllwing variance-stabilizing transfrmatin fr the negative binmial distributin prvided that 't is knwn: fey) =~ sinh-i ~(y+o.5)'t. Therefre, an analysis f variance culd be perfrmed n the raw data r (1) fl (y) = lg(y + 1), (2) f 2 (y) =.JY, r (3) f3(y) =~ sinh-i ~(y+o.5)'t. If the resulting F statistic is significant then H is rejected. 3. An apprximatin t the distributin f the efficient scre statistic Bamwal and Paul (1988) apprximated the distributin f the efficient scre statistic using X 2 (t -1). This sectin prvides an alternative apprach. Let XI = :t (Yi~ J- Y.~ statistic can be written as and X 2 = Y(1 + 'ty) = Y (1 + ~ y ). Then the efficient scre i=1 ni n n n y XI. ~ (3.1) X 2 In this sectin, we find v such that ~ is apprximately X2 with v degree f freedm. E(S) T accmplish this, we will need t find the mean and variance f Sunder H. Therem 3.1 Under H ' the expected value f the efficient scre statistic is Prf Fr details see Wang (1999). 3.1 An apprximatin t the variance f S E(S) = (t -1) (_n_). (3.2) n+'t In this sectin the variance f S is estimated by using a first-rder Taylr apprximatin (Sectin 5.2.3, Md, Graybill and Bes, 1974). We have

7 Applied Statistics in Agriculture Cnference n Applied Statistics in Agriculture 161 (3.3) (3.4) If n l = n2 =... = n t = m, then (3.4) can be simplified. Prf Fr details see Wang (1999). Crllary 3.1 Under the assumptin f n l = n2 =... = TIt = m, (3.4) becmes Prf var(s) "" 2(t-1)n 2 (n+'tt). (n + 't)3 Under the assumptin f n l = n2 =... = n t = m, we have 1 t 2 n=mt and L-=-' n i n Therefre, the right side f (3.4) can be simplified as 3 n { 2(t-1)(n+'tt)~2 +4't(t-1)(n+'tt)~3 +2't2(t-1)(n+'tt)~4 } (n+'t) (~+'t~ ) 2 n 2 (t -1)(n + 'tt) = (n + 't)3 3.2 A secnd apprximatin t the variance f S In this sectin a better apprximatin t var(s) is btained. This apprximatin is btained by using the fact that (Md, Graybill and Bes, 1974). var(s) = E(var( S I Y )) + var(e( sl Y )) (3.5)

8 162 Cnference n Applied Statistics in Agriculture Under H ' var(e( ~I Y )) = 0, s var(~) = E(var( ~I Y )) Nw expressin (3.6) is used t find the value f var(~). (3.6) Therem 3.3 Fr n i = E.. fr i = 1, 2,..., t, the fllwing is true. t var(~) = E(var( ~I Y )) 2n\t -1)(n + t't) =--~ E( 1 J (n + 't)2 (n + 2't)(n + 3't) (n + 't)2 (n + 2't)(n + 3't) Y (n + 'ty.)2 Prf Fr details see Wang (1999). 't 'E( 1 J (n + 't)(n + 2't)(n + 3't) (n + 'ty.)2. Nte that the secnd and third terms n the right side f (3.7) are very small, s we have the fllwing apprximate expressin var(~)"" 2n\t-1)(n+t't). (3.8) (n + 't)2 (n + 2't)(n + 3't) Our simulatin results shwed that the apprximatins (3.8) and (3.2) are very accurate. T apprximate a critical pint fr the efficient scre statistic we fllw Satterthwaite (1946). We find v such that ~..:. X2 (v) when H is true. E(~) This implies that v must satisfy Var (v%(~))= 2v. Hence, it fllws frm (3.2) and (3.8) that 4. Simulatins (3.7) v = 2[E(~)]2 =...;..(t_-_1.:...:)(...;..n_+_2...;..'t)...;..(n_+_3't...:...) when nj = E.. fr i = 1, 2,..., t (3.9) var (~) n(n + t't) t A Mnte Carl study was perfrmed t cmpare the efficient scre test and its sampling distributin apprximated by a chi-square distributin with degrees f freedm equal t V in (3.9) t several ther previusly intrduced methds f testing fr equal means fr negative binmial distributins. The methds used were: 1. Efficient scre test using the new Satterthwaite apprximatin t the critical pint (RSCR). 2. Efficient scre test using X 2 (t -1) t apprximate the critical pint (SCOR).

9 Applied Statistics in Agriculture Cnference n Applied Statistics in Agriculture Likelihd rati test (LR). 4. F-test (F) n raw data. 5. F-test with square rt transfrmatin (FSQ). 6. F-test with lgarithm (f cunts + I) transfrmatin (FLG). 7. F-test with inverse hyperblic transfrmatin (FSH). Tw majr criteria in evaluating these test methds are the rbustness f 1. The bserved significance level ( &. ) which is the estimated prbability f rejecting the null hypthesis when the ppulatin means are equal. 2. The bserved pwer ( 1- ~ ) which is the estimated prbability f rejecting the null hypthesis when the ppulatin means are nt equal. Empirical significance levels and pwers f the seven tests described abve are derived fr t = 4 means based n 3000 samples frm the negative binmial distributin fr different values f J.LI' J.L2' J.L3 and J.L4 ' and 't. Data were simulated frm negative binmial distributins with means f 0.25,0.5,0.75, 1,2,5, 10, 15, r 20, and values f't f 4,2,4/3, 1 and 0.2. Only balanced designs were cnsidered, with n l = n 2 = n3 = n 4 = 5, 10,25 and 50 replicatins per treatment. Small means, small sample sizes, and large values f't are emphasized because they are mre representative f the situatins mst frequently encuntered in bilgical studies. The estimated and nminal Type I errr rates were cmpared at the 0.01, 0.05, and 0.1 levels fr all tests. Sme f the simulatin results are presented in Tables 1-3. Cnsider the case a = Simulatin results shw that fr large sample sizes ( n I = n 2 = n 3 = n 4 = 50), all f the test statistics hld their significance levels well even fr small means. Fr mderately large sample sizes n l = n 2 = n3 = n 4 = 25, the estimated Type I errr rates are generally clse t the nminal rates fr all the test statistics, but LR tends t be liberal and SCaR and F cnservative. In general, the RSCR, FSQ, and FLG hld their significance levels well. The RSCR perfrms cnsistently well acrss all sample sizes, fr different 't's and different J.L's. Als, it gives the best perfrmance when sample sizes are small 't is large and J.L is small. The simulatin results shw that fr small sample sizes (n I = n 2 = n 3 = n 4 = 5, 10 ) the likelihd rati chi-squared (LR) test is t liberal and gets wrse as J.L decreases and 't increases - that is, when the negative binmial distributin departs frm the Pissn distributin. When 't is large (e.g. 't =4,2,4/3, 1) and sample sizes are small (nl = n 2 = n3 = n4 = 5), the estimated Type I errr rate is smaller than the nminal rate fr LR. In general, the estimated Type I errr rate is smaller than the nminal rate fr F and SCaR. It nly gets better when the sample size is large, 't is small (the negative binmial is clse t the Pissn distributin), and J.L is large. When a = 0.01 and a = 0.1, we btained results similar t thse fr a =

10 Cnference n Applied Statistics in Agriculture 164 Using the frmula a ± zooo5~a(1- a)/3000, we were able t identify Type I errr rates that are significantly different frm a at the 0.01 significance level. We cnstructed pwer curves fr all seven statistics (RSCR, SCOR, LR, F, FSQ, FLG, and FSH) with 't = 4, 2, 1,0.2, n1 = n 2 = n3 = n 4 = 5, 10, 25, and sme values f~, with a levels fixed at These pwer curves are pltted as functins f d. Larger values f d crrespnd t greater differences in the ~i 's. See Figures 1-4 fr details. In general, LR and RSCR are mre pwerful than the ANOV A methds. Amng the ANOV A methds, F des nt pssess the rbust prperties with respect t the negative binmial distributin fr small sample sizes, when 't is large, r when ~ is small. The ther three ANOV A methds that are based n the transfrmed data (FSQ, FLG and FSH) hld their significance levels well and their pwer curves are better than thse f F (but nt as gd as fr LR r RSCR). See Figures Example The data in Table 4 (McCaughran & Arnld, 1976) refer t cunts f embrynic deaths in a cntrl grup and tw treatment grups (n1 = n 2 = n3 = 10). Table 4: Cunts f embrynic deaths Number f deaths Cntrl grup Frequency Dse Levell Dse Level Suppse it is knwn frm experience that 't = 0.25, s we might assume that Y jj - ind NB (~j,0.25) fr i = 1, 2, 3 and j = 1, 2,..., 10. T determine if there are differences in the mean cunts f deaths amng the grups we test the hypthesis H : ~1 = ~2 = ~3 versus H1: at least tw differ. Table 5 belw shws the calculated p-values fr the seven test statistics previusly intrduced. Test P-value Table 5: List f P - Values fr Seven Methds (Example) RSCR SCOR LR F FSQ FLG FSH Based n the p-values, all the tests indicate that the grup means d nt differ significantly frm ne anther. This is in agreement with ur simulatin results.

11 Cnference n Applied Statistics in Agriculture Applied Statistics in Agriculture SUMMARY In general, RSCR and LR are mre pwerful tests than the ther tests. Hwever, while RSCR maintains the type I errr rate, LR is t liberal fr small values f Jl and fr small sample sizes (i.e., LR rejects the hypthesis mre than it shuld when the ppulatin means are equal.). Hence it certainly will have greater pwer. All tests have greater pwer fr small 't than fr large 't - that is, fr small departures frm the Pissn assumptin, the tests are mre pwerful. Als all f the tests have greater pwer fr large values f Jl than they d fr small values f Jl. Because LR is smetimes t liberal and the RSCR test is much mre pwerful than all f the ther tests, we recmmend the use f the RSCR test. REFERENCES Anscmbe, F.J. (1949). The analysis f insect cunts based n the negative binmial distributin. Bimetrics, 5, Bliss, C.I. & Owen, A.RG. (1958). Negative binmial distributin with a cmmn k. Bimetrika, 45, Barnwal, R K. and Paul, S. R (1988). Analysis f ne-way layut f cunt data with negative binmial variatin. Bimetrika, 75, Beall, G. (1942), The Transfrmatin frm Entmlgical Field Experiments s that the Analysis f Variance Becmes Applicable. Bimetrika, 32, McCullagh, P. and NeIder, J. A. (1989). Generalized Linear Mdels, 2 nd ed, Lndn: Chapman and Hall. McCaughran, D.A. & Arnld, D.W. (1976) Statistical mdels fr numbers f implantatin sites and embrynic deaths in mice. Txicl.Appl. Pharmacl. 38, Md A. M., Graybill, F. A. and Bes, D. C. (1974). Intrductin t the Thery f Statistics. McGraw-Hill, Inc. Satterthwaite, F. E. (1946). An apprximate distributin f estimates f variance cmpnents. Bimetrics Bulletin 2: Wang, Y. (1999). The analysis f cunt data in a ne-way layut and a new bivariate negative binmial distributin. Ph.D dissertatin, Department f Statistics,. Yung, L. J., Campbell, N. L., and Capuan, G. A. (1999). Analysis f verdispersed cunt data. Jurnal f Agricultural, bilgical, and Envirnmental Statistics, vlume 4, Number 3,

12 Cnference n Applied Statistics in Agriculture 166 Table 1: Observed Type I Errr Rates based n 3000 replicatins Nminallevel: a = 0.05 n]=n2 n 3 =n 4 1: Jl RSCR SCOR LR F FSQ FLG FSH * 0.006* 0.012* 0.021* 0.027* 0.026* 0.027* * 0.035* 0.028* * * * * 0.060* 0.030* * 0.082* 0.030* * 0.090* 0.028* 0.039* * 0.089* 0.032* 0.040* * 0.086* 0.032* * 0.078* 0.028* 0.036* * 0.028* 0.025* 0.034* 0.032* 0.034* * * * * * * * * * * * * * * * 0.023* * * 0.032* 0.034* 0.027* 0.038* 0.036* 0.036* * 0.067* * 0.079* 0.038* * 0.072* * * * 0.060* * * 0.039* * * * * * 0.038* * 0.031* 0.030* 0.030* * 0.040* * * * * * *Indicates empirically significantly different frm the nminal level a = The 99% cnfidence interval is (0.04, 0.06)

13 Cnference n Applied Statistics in Agriculture Applied Statistics in Agriculture 167 Table 2: Observed Type I Errr Rates based n 3000 replicatins Nminallevel: ex = 0.05 n 1 =n 2 n 3 =n 4 't 11 RSCR SCOR LR F FSQ FLG FSH * 0.020* * 0.036* 0.036* 0.035* * 0.075* 0.038* * 0.078* 0.033* * 0.022* 0.066* 0.025* 0.040* 0.039* * 0.070* 0.030* * 0.068* 0.034* * 0.062* 0.037* * 0.062* 0.036* * * * 0.064* 0.033* * 0.038* * 0.067* 0.038* * 0.066* 0.038* * 0.063* 0.037* * * * * * 0.060* 0.038* * * * 0.084* 0.036* 0.040* 0.039* 0.039* * 0.039* * * * * * * * * * *Indicates empirically significantly different frm the nminal level ex = The 99% cnfidence interval is (0.04, 0.06)

14 Cnference n Applied Statistics in Agriculture 168 Table 3: Observed Type I Errr Rates based n 3000 replicatins Nminallevel: ex = 0.05 n]=n2 n 3 =n 4 l' Il RSCR SCOR LR F FSQ FLG FSH * * * * * * * * * * * * * * * * *Indicates empirically significantly different frm the nminal level ex = The 99% cnfidence interval is (0.04, 0.06)

15 Applied Statistics in Agriculture Cnference n Applied Statistics in Agriculture 169 Empirical pwer curve crrespnding t nminal level = 0.05 c RSCORE ~~ORE F-TEST FSQ FLG ~""'"."" ",,"==="""'''"''''= '" d Figure replicatins; sample sizes n 1 =n2 =n3 = n4 =5; l' = 2; f.ll = f.l2 = f.l3 = 0.75; f.l4 = f.ll (1 + d), d > 0 Empirical pwer curve crrespnding t nminal level = 0.05 c RSCORE SCORE :_.",,---:",.-=-~-_---"::.--::::".-:::"=.-::::,,.-::::::,,.. LR '_ ;....,l.,w F-TEST.~:"": - - FSQ FLG FSH... '- <0 ~ 0 Q) 8. '<t '" d Figure replicatins; sample sizes n 1 = n 2 = n 3 = n 4 = 10 ; l' = 2; f.ll = f.l2 = f.l3 = 5; f.l4 = f.ll (1 + d), d > 0

16 170 Cnference n Applied Statistics in Agriculture Empirical pwer curve crrespnding t nminal level = 0.05 (;; 5: 0 C CO! c 0 < RSCORE SCORE LR F-TEST FSQ FLG FSH.'.'. ',., N d Figure replicatins; sample sizes n 1 = n 2 = n 3 = n 4 = 10; 't = 1; Jll = 5, Jl2 = Jll (1 + d), Jl3 = Jll (1 + 2d), Jl4 = Jll (1 + 3d), d > 0 Empirical pwer curve crrespnding t nminal level = 0.05 c RSCORE SCORE.'.' '.'." LR F-TEST FSQ "... """."". FLG -- "" /,/ />;;>;:>;.~:-~-~ N //"~<;?~~ /1,;4/..,...-"--- ///,...-" ii~... / ~, d Figure replicatins; sample sizes n 1 = n 2 = n 3 = n 4 = 5 ; 't = 2; Jll = Jl2 = 0.75; Jl3 = Jl4 = Jll (1 + d), d > 0