COMPARISON OF APPROACHES TO TESTING EQUALITY OF EXPECTATIONS AMONG SAMPLES FROM POISSON AND NEGATIVE BINOMIAL DISTRIBUTION

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1 ACTA UNIVERSITATIS AGRICULTURAE ET SILVICULTURAE MENDELIANAE BRUNENSIS Volume 66 0 Number 4, 08 COMPARISON OF APPROACHES TO TESTING EQUALITY OF EXPECTATIONS AMONG SAMPLES FROM POISSON AND NEGATIVE BINOMIAL DISTRIBUTION Mart Tejkal, Zuzaa Hüberová Departmet of Ecoometrcs, Faculty of Mltary Leadershp, Uversty of Defece Bro, Koucova 65, Bro, Czech Republc Departmet of Mathematcs, Faculty of Mechacal Egeerg, Bro Uversty of Techology, Techcká 896 /, Bro, Czech Republc Abstract TEJKAL MARTIN, HÜBNEROVÁ ZUZANA. 08. Comparso of Approaches to Testg Equalty of Expectatos Amog Samples from Posso ad Negatve Bomal Dstrbuto. Acta Uverstats Agrculturae et Slvculturae Medelaae Bruess, 66(4): The paper deals wth testg of the hypothess of equalty of expectatos amog p samples from Posso or egatve bomal dstrbuto. a comparso of two ma approaches s carred out. The frst approach s based o trasformg the samples from ether Posso or egatve bomal dstrbuto order to acheve ormalty or varace stablty, ad the testg the hypothess of equalty of expectatos va the F test. I the secod approach, test statstcs comg from the theory of maxmum lkelhood appearg geeralsed lear models framework, specally desged for testg the hypothess amog samples from the respectve dstrbutos (Posso or egatve bomal), are used. The comparso s doe graphcally, by plottg the smulated power fuctos of the test of the hypothess of equalty of expectatos, whe frst or secod approach was used. Addtoally, the relatoshp betwee the power fuctos obtaed va the respectve approaches ad sample szes s studed by evaluatg the respectve power fuctos as fuctos of a sample sze umercally. Keywords: Posso dstrbuto, egatve bomal dstrbuto, ANOVA, F test statstc, geeralzed lear model, lkelhood rato, score statstc, varace stablzg trasformato, Yeo Johso trasformato, power fucto INTRODUCTION I applcatos, we ofte meet data volvg couts. For modellg of such data Posso dstrbuto, or case of heteroscedastc data, egatve bomal dstrbuto s ofte used. If we wat to test the hypothess of equalty of expectatos amog a gve umber of depedet samples of such data, we caot use the classcal aalyss of varace framework, because the assumptos of the ormalty ad varace stablty of the data are ot met. Hece a dfferet strategy has to be developed. Oe of the possble approaches s to trasform the samples order to meet the assumptos of classcal aalyss of varace (see Scheffé, 999). I the case of Posso or egatve bomal dstrbuto, a logarthmc trasformato s ofte suggested (see Moss ad McPhee, 006). The problem of zero observatos s solved by addg or aother postve costat to the argumet of the logarthm. Aother possble soluto s to use varace stablsg trasformato (see Aděl, 0; Tejkal, 07), or some other trasformato that assures, that the trasformed radom varable s approxmately ormally dstrbuted. Such a trasformato mght be for 05

2 06 Mart Tejkal, Zuzaa Hüberová example of the Yeo Johso famly (see Yeo ad Johso, 000). Aother possble approach s to use wholly dfferet test statstc, specally desged for testg the hypothess amog samples from a gve dstrbuto. Such test statstcs come from the theory of maxmum lkelhood, ad they appear the framework of the geeralsed lear models (GLM). The goal of ths paper s to provde the reader wth a comparso of the above metoed approaches. Ths s doe by comparg the power fuctos of the test statstcs, whe testg the hypothess of equalty of expectatos amog p samples of the same sze from the Posso ad the egatve bomal dstrbuto. Addtoally, the relatoshp betwee power fuctos ad sample sze s studed, ad the comparso of power fuctos for dfferet sample szes s provded. The results of ths comparso may be straghtforwardly appled to expermetal desg to determe sample sze. MATERIALS AND METHODS Assumed models Let Y = (Y,..., Y ) T for =,..., p be the radom samples of a sze from ether Posso dstrbuto Po(θ ) wth expectato parameter θ, or egatve bomal dstrbuto NB(θ, κ) wth expectato parameter θ ad shape parameter κ. For the sake of coveece, the parameter of the Posso dstrbuto ad the frst parameter of the egatve bomal dstrbuto are deoted by the same symbol. Furthermore, assume that for both Posso ad egatve bomal case we have EY j = θ for all =,..., p ad j =,...,. For completeess, we preset the probablty desty fucto of the egatve bomal dstrbuto uder ths parametrsato: ( x ) ( ) x κ Γ + κ θ κ p( x; θκ, ) =, x! Γ κ κ + θ κ + θ 0 otherwse. for x 0, For detals about ths parametrsato of egatve bomal dstrbuto see Tejkal (07). Assume that the p radom samples are mutually depedet. We wat to test the hypothess H 0 : θ =... = θ p () of equalty of expectatos amog the p samples agast the alteratve H :, k {,..., p}, k, such that θ θ k (3) Approach va Trasformatos For the samples from Posso dstrbuto we cosder the followg trasformatos, the logarthmc trasformato () Z = l (Y + ), (4) the square root trasformato Z= Y + k, (5) where the value of the costat k s chose to be optmal accordg to Ascombe (948),.e. k = 3. 8 Fally, we cosder the so called Yeo Johso trasformato gve by ( X) λ + for X 0, λ 0 λ l( + X) for X 0, λ = 0 Z = λ (( + X) ) for X < 0, λ λ l( X) for X < 0, λ =, where he value of the parameter λ ca be estmated by maxmum lkelhood estmato (see Yeo ad Johso, 000). For the samples from egatve bomal dstrbuto we cosder aga the logarthmc trasformato (4), the Yeo Johso trasformato (6), ad addtoally also the argumet of hyperbolc se trasformato Y + c Z = sh κ +, d where 3 c = 6κ( 6κ 7κ + 4κ ) 6κ + 9κ κ 3 6 ad d = c are the optmal values of the costats c ad κ accordg to Ascombe (948). The formulas (5) ad (7) are geeralsed versos of varace stablsg trasformatos for the respectve dstrbutos (see Aděl, 0). Full dervatos of the optmal values of the costats ca be foud Ascombe (948) or Tejkal (07). It s assumed, that after applyg oe of the troduced trasformatos to the tal model, the classcal oe way aalyss of varace settg s obtaed. I. e. for each =,..., p s Z = (Z,..., Z ) T the radom sample of a sze from N(μ,σ ) ad, furthermore, the p radom samples are mutually depedet. The extet of how much ths was satsfed the respectve cases of dfferet trasformatos was checked by computg the estmato of the varace ad the estmato of skewess of the trasformed sample. The followg estmators were used s = (Z j Z ), (8) b j= for estmatg varace, ad (6) (7) 3 ( Z Z j j ), (9) 3 ( Z Z j j )

3 Comparso of Approaches to Testg Equalty of Expectatos Amog Samples from Posso ad Negatve for estmatg skewess, where = Z j= Z j s the arthmetc mea of the th sample. The F test statstc used to test the hypothess of the equalty of expectatos of the trasformed samples s gve by F p p p ( Z Z) ( Z Z ), (0) p j j p where Z = p = j= Zj. I what follows, we wll deote Q d ( p ) a quatle of a dstrbuto wth d degrees of freedom. The F test s carred out by comparg the value of the test statstcs wth the respectve quatle Q ( α ) F p, p ( for the selected ) level of sgfcace α. Approach va Tests from GLM Framework I the case of Posso dstrbuted samples, the followg two test statstcs wll be used to test the hypothess of the equalty of expectatos. The lkelhood rato test statstc (see Hrdlčková, 00), whch s gve by LR Po p j Y Yj l, () Y ad the score test statstc (see Hrdlčková, 006), whch s gve by Sc Po p ( Y Y), Y () p where Y = j= Yj ad Y = p = j= Yj. I the case of egatve bomally dstrbuted samples, the followg two test statstcs wll be used. The lkelhood rato test statstc (see Hüberová ad Doudová, 0), that s gve by p Y Y LRNB Y l Y l Y Y, (3) ad the score test statstc, that for the egatve bomal case (see Hüberová ad Doudová, 0) s gve by Sc NB Y Y p Y ( Y ). (4) For all the test statstcs (), (), (3), ad (4) the test s carred out by comparg the value of the test statstc wth the respectve quatle Q ( α ) χ p for the selected level of sgfcace α. Comparg Power Fuctos Let Ω be the parametrc space, let θ be a elemet of Ω. Let us deote β(θ) the codtoal probablty of rejectg the ull hypothess, gve that the alteratve, charactersed by the value of parameter θ, holds. The fucto β(θ) wth values θ Ω s called the power fucto of a test. I ths paper, the ma focus wll be o provdg a comparso of power fuctos obtaed va smulatos. a more theoretcal approach s developed Tejkal (07). Its use s, however, lmted oly for obtag power fuctos of the F test after ether trasformato (4) or (5) s appled the Posso case or trasformato (4) or (7) s appled the egatve bomal case. Addtoaly, the p samples have to be of the same sze. The method used for obtag the results of the theoretcal approach wll be explaed brefly the followg subsecto. For detals see Tejkal (07). The process of computg power fuctos for Posso case by ay of the approaches does ot dffer from the egatve bomal case, therefore, whe provdg the descrpto, t wll ot be dfferetated betwee the two dstrbutos. Recall that the expectato parameters of both dstrbutos are deoted θ. Theoretcal Power Fuctos The theoretcal approach to obtag the approxmatos of power fuctos of the F test cases descrbed above s based o the followg. The power fucto of the F test ca be approxmated as descrbed the Proposto. Proposto. The power of the F test β α (θ) o the level of sgfcace α may be wrtte as follows ( ) β α (θ) = Q α, (5) Fp p p p,,, ( ) ( ) δ F ( ) s the dstrbuto fucto of ocetral where p, ( p ), δ F dstrbuto wth degrees of freedom d = p, d = p ( ) ad a ocetralty parameter p ( ), where θ = p j = θ p j. Proof. Straghtforward use of propertes of codtoal probablty. Furthermore, t s assumed, that the expectato parameters of the dstrbuto of the trasformed radom samples are kow. I practcal computatos approxmatos of the umercal characterstcs are used. Wth these assumptos, the value of the power fucto of the F test gve by formula (5) ca be computed at ay pot θ. Approxmatos of the power fuctos of the tests GLM ca be foud Hrdlčková (008). Smulated Power Fuctos The smulated power fuctos are computed the followg way. A tal value of expectato parameter θ are chose. Addtoally, the value of the parameter κ s selected for the egatve bomal case. The sgfcace level α = 0.05 s cosdered hereafter. The p radom samples Y for =,..., p of a sze from ether Posso or egatve bomal dstrbuto are geerated. The values θ,..., θ p are obtaed from θ by addg ad subtractg multples of a umber h j accordg to the formula

4 08 Mart Tejkal, Zuzaa Hüberová θ + h j, for =, 4,..., p, θ = θ hj, for = 3, 5,..., p, (6) where j s the umber of the ru of the algorthm. Deote s a fxed step, ad set h = s for j =. For j > the value of h j s gve by the followg formula h j + = h j + s (7) that characterses θ β(θ). Whe usg the approach va trasformatos, the radom samples are trasformed usg (4), (5), ad (6) for the Posso case ad (4), (7), ad (6) for the egatve bomal case, obtag trasformed radom samples Z for =,..., p. The optmal value of the parameter λ of trasformato (6) s determed va maxmum lkelhood method (see Yeo ad Johso, 000). The F statstc s computed usg formula (0). The trasformed samples Z = (Z,..., Z ) T for =,..., p stacked oe above each other are used as the put vector Z p = (Z,..., Z,......, Z p,..., Z p ) T. The value of the F statstc s the compared wth quatle Q ( α ) F p, p ( to decde ) about the result of the test. I case of the approach va test statstcs of GLM framework, the orgal samples Y for =,..., p are used as put. The values of the test statstcs () ad () the Posso case ad (3) ad (4) egatve bomal case are computed ad compared wth quatle Q ( α ). Ths process s repeated k tmes for the same settg of parameters order to compute the relatve frequecy of rejectg hypothess H 0 (see ()). The value of h j creases wth every ru of the algorthm (see (7)). Hece, by re rug the algorthm the values of the smulated power fucto are obtaed. The practcal computato was doe for the values of the parameters collected Tab. I, from whch the most terestg cases wll be preseted the Results secto of ths paper. The umber of repettos of the algorthm used the practcal computato was k = 000. Notce that for p = 3 the value h j each ru of the program s the dfferece betwee the fxed expectato parameter θ ad the expectato parameters θ, θ 3 of the other two dstrbutos, whch crease ad decrease respectvely. I. e. h = ǀθ θ ǀ = ǀθ θ 3 ǀ. χ p Comparso of Power Fuctos for Dfferet Sample Szes Lastly, the relato betwee the power fuctos ad the sze of the samples was studed va smulatos. The method s based o the computatos descrbed the secto Smulated Power Fuctos ad was carred out for values of parameters gve by the Tab. I. By dog these computatos for varyg sample sze, a set of pots of the smulated power fucto for chose approach for dfferet sample szes was obtaed. I. e. the set of values descrbg the relato β = β(h j, ). By fxg the value h j, meag that the dfferece betwee the expectatos θ, θ, ad θ 3 s fxed, a set of pots descrbg the relato β = β () for gve h j s obtaed. The descrbed procedure s doe for each method of the two approaches, ad the graphcal comparso of the results s preseted for each method for selected choces of parameters. The fxed values of h j were chose to be / 4, / ad 3 / 4 of the terval, o whch the smulated power fuctos were computed. The value of h j the quarter of the computatoal terval was chose to observe what happes for varyg sample szes, whe the dffereces betwee the expectatos θ, θ, ad θ 3 are relatvely small. The value of h j the half of the computatoal terval was pcked to observe the chages, that happe for creasg dffereces betwee the expectatos. Fally, the value the three quarters of the computatoal terval was selected to observe what happes, whe the dffereces betwee the expectatos are relatvely bg. Noparametrc regresso methods were appled to ft the obtaed data. RESULTS Posso Case We wll start wth a comparso of the estmatos of varace ad skewess of the trasformed radom sample, whe trasformatos (4), (5), ad (6) were appled. I the left graph of Fg. oe ca see a comparso of sample varace estmates for θ [0, 00]. Notce, that oly the trasformato (5) has truly the varace stablsg property. The comparso of sample skewess s represeted graphcally by the rght graph of Fg.. Here oe ca observe that both trasformato (5) ad (6) perform very well. The skewess of the samples trasformed va (5) ad (6) s very close to zero, ad therefore t ca be assumed, that I: Values of the parameters Dstrbuto Number of samples Sample sze Expectato parameter Shape parameter Po(θ) p = 3 = 00 θ = 5, 0, 0, 50 NB(θ, κ) p = 3 = 00 θ = 5, 30, 50, 00 κ = 3, 5, 0

5 Comparso of Approaches to Testg Equalty of Expectatos Amog Samples from Posso ad Negatve the trasformed sample s approxmately ormally dstrbuted. Observe that whe the trasformato (4) s appled, however, for small values of expectato parameter, the departure from ormalty may be sgfcat. We wll cotue by presetg the graphcal comparso of the power fuctos for the Posso case. The smooth les Fg. represet the power fuctos of trasformatos (4) ad (5) obtaed va the theoretcal approach. The pots represet power fuctos obtaed va smulatos. Observe that the trasformato (4) performs the worst, (5) ad (6) both perform smlarly, better tha (4). The approach va lkelhood rato test statstc () gves slghtly better results tha the approach va trasformatos. The score statstc smulated power fucto () attas the hghest values out of all the power fuctos the pots, where the umercal computato was carred out. However, t does ot atta the value β = 0.05 whe the expectato parameters amog the samples are equal. We coclude that the score test teds to have hgher smulated test sze tha the chose level of sgfcace α = The smulato results for trasformatos (4) ad (5) are accordace wth the theoretcal power fuctos. For the creasg value of the parameter θ, the dffereces betwee the smulated power fuctos become eglgble. We assume that the weak performace of the trasformato (4) s caused by the fact that the skewess of the trasformed sample s dfferet from zero (see Fg. ) ad hece, the departure from the ormalty of the trasformed sample mght be sgfcat. Furthermore, the trasformato (4) lacks the varace stablsg property (see Fg. ). We wll coclude the Posso case by provdg fgures of the comparso of the power fuctos for dfferet sample szes. From ths graphcal comparso oe ca observe, what the ecessary sample sze s to dstgush the dfferece h j wth a requred probablty, for a gve approach. Recall that the value of h j s the dfferece betwee the fxed θ ad the computed θ ad θ 3 respectvely. The sample sze eeded to dstgush the dfferece h j = 0.53 wth the probablty β = 0.8 by score test statstc () s = 8. By lkelhood rato test statstc () t s = 89. By the worst performg method usg logarthmc trasformato the requred sample sze s = 99 (see Fg. 3). : Posso case varace (left) ad skewess (rght) estmates for trasformatos (3) (red), (4) (blue) ad (6) (black) : Posso case power fuctos comparso for θ = 5, θ = θ + h, θ 3 = θ h,

6 030 Mart Tejkal, Zuzaa Hüberová 3: Posso case power fuctos as fuctos of a sample sze for hj = 0.53 (approxmately a quarter of the computato terval), θ = 5, θ = θ + h = 5.53, θ3 = θ h = Negatve Bomal Case As before, we wll start by comparg the varace ad skewess estmates of the trasformed samples whe trasformatos (4), (7), ad (6) were appled. I Fg. 4 we ca see the comparso of sample varaces for creasg values of κ. Notce that both trasformatos (4) ad (7) have the varace stablsg property, whle (6) clearly does ot. The graphcal comparso of sample skewess for creasg values of s provded Fg. 5. Notce that the sample skewess of the samples trasformed va (6) s always close to 0. For the samples obtaed by applyg the other two trasformatos ths s ot true. We may, however, observe, that wth creasg value of κ the value of skewess of these two samples gets closer to 0. Addtoally, for smaller 4: Negatve bomal case varace estmato va sample varace for trasformatos (4) (red), (7) (blue) ad (6) (black) for values of parameter κ = 3, 5, 0 from left to rght 5: Fg. 5: Negatve bomal case skewess estmato va sample skewess for trasformatos (4) (red), (7) (blue) ad (6) (black) for values of parameter κ = 3, 5, 0 from left to rght

7 Comparso of Approaches to Testg Equalty of Expectatos Amog Samples from Posso ad Negatve values of θ the trasformato (7) outperforms (4) the terms of skewess. I geeral we may say that for the samples trasformed va (4) or (7) the departure from ormalty may be sgfcat especally for small values of κ. For greater values of κ the performace of (7) slghtly mproves. We wll cotue by presetg the graphcal comparso of smulated power fuctos. Out of the multtude of computatoal results for varous choces of parameters, we wll preset the cases for the followg pars of parameters θ = 5, κ = 3; θ = 5, κ = 0; θ = 00, κ = 0; ad θ = 50, κ = 5. The reaso for the choce of the frst par s to observe, how the methods of the dfferet approaches tackle wth the worst case scearo a relatvely small expectato parameter ad a small shape parameter. The secod par represets the case of a relatvely small value of the expectato parameter but relatvely bg value of the shape parameter. The thrd par represets the choce of bg values of both parameters ad fally, the last par represets the choce of moderately bg expectato ad shape parameters. Addtoally, for the case ad the comparso s erched by the theoretcal power fuctos for samples trasformed va (4) ad (7) (see Fg. 9, red ad blue les respectvely). For the other two cases that are covered the graphcal comparso, the theoretcal power fuctos are ot preseted because the approxmatos of umercal characterstcs used the computatos do ot behave well (for more detal see Tejkal, 07). From the Fgures 6, 7, ad 9 we ca see that the worst performace regardless of the settg of the parameters has the trasformato (4). I case of small shape parameter κ (see Fg. 6), the dfferece betwee (4) ad (7) s ot very sgfcat. However, case of bg value of the shape parameter κ (see Fg. 7) (7) performs better tha (4). Ths ceases to be true for bg values of the expectato parameter θ (see Fg. 8). The best performace out of the approach va trasformatos has trasformato (6). 6: Negatve bomal case Power fuctos comparso for κ = 3, θ = 5, θ = θ + h, θ3 = θ h 7: Negatve bomal case power fuctos comparso for κ = 0, θ = 5, θ = θ + h, θ3 = θ h

8 03 Mart Tejkal, Zuzaa Hüberová 8: Negatve bomal case power fuctos comparso for κ = 0, θ = 00, θ = θ + h, θ3 = θ h 9: Negatve bomal case power fuctos comparso for κ = 5, θ = 50, θ = θ + h, θ3 = θ h 0: Negatve bomal case power fuctos as fuctos of a sample sze at hj = 7.98 (approxmately a half of the computato terval) for the settg of parameters κ = 3 ad θ = 30, θ = θ + h = 37.98, θ3 = θ h =.0

9 Comparso of Approaches to Testg Equalty of Expectatos Amog Samples from Posso ad Negatve Notce also, that for a good performace of a trasformato t seems to be more mportat, that the skewess of the trasformed sample s close to 0, tha that the varaces amog the samples are equal. Observe that (6). that performs the best out of the cosdered trasformatos, does ot stablse varace the partcular studed case (see Fg. 4), however, the skewess of samples trasformed va (6) s close to 0 (see Fg. 5). Usg test statstc comg from the theory of maxmum lkelhood seems to be the best approach sce the power fucto of the test whe ether (3) or (4) s used attas hgher values tha ay power fucto of the approach va trasformatos. The smulated power fucto of the test statstc (4) teds to atta slghtly hgher values tha the oe of (3). We may further observe, that for creasg values of θ ad κ the dffereces betwee the power fuctos become smaller (cf. Fg. 6 wth 9, ad 6 wth 7). Lastly, we wll preset the graphcal comparso of the power fuctos for dfferet sample szes for all the studed approaches. Recall that the value of h j s the dfferece betwee the fxed θ ad the computed θ ad θ 3 respectvely. From Fg. 0 we ca observe, that for the best performg method (lkelhood rato test statstc (3)), to dstgush the dfferece h j = 7.98 wth probablty β = 0.8, samples of a sze = 3 are eeded. For the worst performg par of methods (logarthmc trasformato (4) ad argumet of hyperbolc se trasformato (7)), to dstgush the dfferece h j = 7.98 wth probablty β =0,8, samples of a sze = 8 are eeded. DISCUSSION The early tred tacklg the o ormal heteroscedastc data volvg couts lear regresso ad the respectve tests was to apply varous trasformatos order to acheve ormalty ad homoscedastcty. To mprove the desred effect of the trasformatos of the radom varables wth Posso ad egatve bomal dstrbuto, Ascombe (948) troduced geeralsatos of the varace stablsg trasformatos. For data volvg couts t s ofte coveet to use the logarthmc scale, therefore the logarthmc trasformato comes, perhaps, as a atural choce. The problem of zero observatos s usually solved by addg a costat (usually oe) to the argumet of the logarthm. Some authors eve suggest dfferet costats. Ascombe (948) derves the logarthmc trasformato for egatve bomally dstrbuted radom varable as a approxmato of the more complcated argumet of hyperbolc se trasformato ad fds a optmal value of the costat added wth the argumet to be a fucto of the shape parameter κ of the dstrbuto. Yamamura (999) suggests usg trasformato Y = l(x + 0.5) stead of Y = l(x + ). Numercal aalyss carred out whe collectg data for ths paper, however, dd ot suggest, that there s a sgfcat mprovemet the power of the test whe Ascombe s or Yamamura s choce of the optmal costat was used stead of addg oe wth the argumet of the logarthm. O the other had, the famly of trasformatos developed by Yeo ad Johso (000) performed our comparso sgfcatly better tha both the logarthmc trasformato wth oe added wth the argumet or the trasformatos proposed by Ascombe. More recet developmet (see O Hara ad Kotze, 00) has show that for modellg data volvg dscrete couts, models based o Posso or egatve bomal dstrbuto atta better results tha the approach va trasformatos. The results of ths paper geeral support ths statemet. The GLM based tests outperformed the approach va trasformatos all studed cases. However, for some cases, the dfferece betwee the GLMs ad the approach va Yeo Johso trasformato, the best performg oe out of the studed trasformatos, was rather small. CONCLUSION The preseted aalyss provdes a comparso of dfferet methods of testg the hypothess of equalty of expectatos for data volvg couts. Posso ad egatve bomal dstrbutos were used to model the data. Two ma approaches were used, the approach va ormalsg ad varace stablsg trasformatos, followed by applyg the classcal ANOVA towards the trasformed data, ad the approach va GLM test statstcs. I the Posso case, the GLM test statstcs performed better tha the trasformatoal approach. By far the hghest values were attaed by the smulated power fucto of the score test statstc () followed by the power fucto of the lkelhood rato test statstc (). The lkelhood rato test statstc, tur, performed slghtly better tha ay of the trasformatos. It should be oted, however, that the score test teds to have greater smulated test sze tha the chose level of sgfcace α = 0.05, sce the respectve power fucto dd ot atta the value 0.05 for h = 0. The square root (5) ad the Yeo Joso trasformato (6) both performed smlarly ad both better tha the logarthmc (4) trasformato, whch performed the worst. Wth creasg values of the expectato parameter θ of the Posso dstrbuto, the dffereces betwee the power fuctos of the dfferet approaches became eglgble. I the egatve bomal case, the results were depedet both o the choce of the tal value of the expectato parameter θ ad the shape parameter κ. I all settgs, better result was obtaed whe

10 034 Mart Tejkal, Zuzaa Hüberová the GLM test statstcs were used. The smulated power fucto of the score statstc (4) attaed hgher values tha the oe of the lkelhood rato statstc (3). All of the trasformatos performed worse tha the GLM test statstcs. The Yeo Johso trasformato (6) outperformed the argumet of hyperbolc se trasformato (7) ad the logarthmc trasformato (4). The argumet of hyperbolc se acheved better results tha the logarthm for small values of θ ad bg values of κ. The dfferece betwee these two trasformatos for small values of θ ad κ ad for bg values of θ depedetly of κ were ot sgfcat. For creasg values of the parameters θ ad κ the dffereces betwee the power fuctos of the dfferet approaches became eglgble. Fally, by comparg the power fuctos for dfferet sample szes we saw, that by choosg the rght approach, we ca obta a gve power of the test wth sgfcatly smaller sample, tha the case of pckg a less sutable approach. Namely, the dfferece betwee the worst ad best performg method the comparso provded ths paper was 0 samples the Posso case ad 5 samples the egatve bomal case. REFERENCES ANDĚL, J. 0. Bascs of mathematcal statstcs [ Czech: Základy matematcké statstky]. Praha: Matfyzpress. ANSCOMBE, F. J The trasformato of Posso, bomal ad egatve bomal data. Bometrka, 35(3 4): HRDLIČKOVÁ, Z Comparso of the power of the tests oe-way ANOVA type model wth Posso dstrbuted varables. Evrometrcs, 7(3): HRDLIČKOVÁ, Z Approxmato of powers of some tests oe-way MANOVA type multvarate geeralzed lear model. Computatoal Statstcs & Data Aalyss, 5(8): HRDLIČKOVÁ, Z. 00. Log-lear models wth Posso radom varables [ Czech: Log-leárí modely s Possoovským proměým]. Dploma Thess. Masaryk Uversty Bro, Faculty of Scece. Supervsor: doc. RNDr. Jaroslav Mchálek, CSc. HÜBNEROVÁ, Z. ad DOUDOVÁ, L. 0. Ifluece of estmate of shape parameter o tests models wth egatve bomal dstrbuto. I: Bometrc Methods ad Models Curret Scece ad Research. Bro: UKZUZ, pp O HARA, R. B. ad KOTZE, D. J. 00. Do ot log-trasform cout data. Methods Ecology ad Evoluto, (): 8. MOSS, D. ad MCPHEE, D. P The mpacts of recreatoal four-wheel drvg o the abudace of the ghost crab (Ocypode Cordmaus) o sub-tropcal sady beaches SE Queeslad. Coastal Maagemet, 34: SCHEFFÉ, H The aalyss of varace. Wley classcs lbrary ed. New York: Wley-Iterscece Publcato. TEJKAL, M. 07. Selected radom varables trasformatos used classcal lear regresso. Dploma thess. Bro Uversty of Techology, Faculty of Mechacal Egeerg. Supervsor: Doc. Mgr. Zuzaa Hüberová, Ph. D. YAMAMURA, K Trasformato usg (x + 0.5) to stablze the varace of populatos. Researches o Populato Ecology, 4(3): YEO, I., ad JOHNSON R. A A ew famly of power trasformatos to mprove ormalty or symmetry. Bometrka, 87(4): Mart Tejkal: mart.tejkal@uob.cz Zuzaa Hüberová: huberova@fme.vutbr.cz Cotact formato