Bootstrap and Permutation tests in ANOVA for directional data

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1 strap and utaton tests n ANOVA for drectonal data Adelade Fgueredo Faculty of Economcs of Unversty of Porto and LIAAD-INESC TEC Porto - PORTUGAL Abstract. The problem of testng the null hypothess of a common drecton across several populatons defned on the hypersphere arses frequently when we deal wth drectonal data. We may consder the Analyss of Varance (ANOVA for testng such hypotheses. However, for the Watson dstrbuton, a commonly used dstrbuton for modelng axal data, the ANOVA test s only vald for large concentratons. So we suggest to use alternatve tests, such as bootstrap and permutaton tests n ANOVA. Then, we nvestgate the performance of these tests for data from Watson populatons defned on the hypersphere. Keywords: Hypersphere, Monte Carlo methods, smulaton, Watson dstrbuton. 1 Introducton The statstcal analyss of drectonal data, represented by ponts on the surface of the unt sphere n R q, denoted by S q 1 = {x R q : x x = 1} was wdely developed by Watson (1983, Fsher et al. (1987, Fsher (1993, Marda and Jupp (2000, among other authors. The applcatons of drectonal data are essentally on the crcle (q = 2 and on the sphere (q = 3, but the applcatons on hgher dmensons (q 4 are also relevant. Drectonal data arse n many scentfc areas, such as bology, geology, machne learnng, text mnng, bonformatcs, among others. An mportant problem n drectonal statstcs and shape analyss, as well n other areas of statstcs, s to test the null hypothess of a common mean vector or polar axs across several populatons. Ths problem was already treated for crcular data and sphercal data by several authors, such as Stephens (1969, Underwood and Chapman (1985, Anderson and Wu (1995, Harrson et al. (1986, Jammalamadaka and SenGupta (2001, among others. However, there has been relatvely lttle dscusson of nonparametrc bootstrap approaches to ths problem. strap methods and permutaton tests based on pvotal statstcs were proposed by Amaral et al. (2007 n drectonal 1

2 statstcs and shape analyss. The bootstrap methodology was proposed by Efron (1979 and was used by Fsher and Hall (1989 and Fsher et al. (1996 for constructng bootstrap confdence regons based on pvotal statstcs wth drectonal data. The permutaton tests, wdely used n mult-sample problems were proposed by Wellner (1979 for drectonal data. In ths paper we focus on the ANOVA test for axal data.e. unsgned unt vectors and we consder the bootstrap verson of ths test and the respectve permutaton test. The bootstrap test conssts n resamplng wth replacement from each sample and a permutaton test conssts n resamplng wthout replacement from the whole sample. We evaluate the performance of these tests when data come from Watson populatons defned on the hypersphere. We consder the Watson dstrbuton defned on the hypersphere because t s one of the most used dstrbutons for modelng axal data. For ths dstrbuton, the ANOVA statstc follows an F -dstrbuton that s approprate only for hghly concentrated data (see Stephens, 1992, Gomes and Fgueredo, 1999 and Marda and Jupp, 2000, p Thus, t seems us that the bootstrap test and permutaton test based on the ANOVA statstc may perform better when data are not suffcently hghly concentrated. The artcle s organzed as follows. In Secton 2 we refer the Watson dstrbuton defned on the hypersphere and we present ANOVA test for ths dstrbuton. In Secton 3 we propose the bootstrap approach and the permutaton test to ANOVA test. In Secton 4 we present numercal results about the performance of the tests n the two-sample case and n three-sample case, such as the estmaton of the levels of sgnfcance of the tests and the emprcal power of the tests. In Secton 5 we present an applcaton and fnally, n Secton 6 we conclude the paper wth some remarks. 2 Analyss of Varance for axal data from Watson dstrbutons defned on the hypersphere In ths secton we refer the Watson dstrbuton defned on the hypersphere and the ANOVA test for ths dstrbuton. 2.1 Watson dstrbuton The bpolar Watson dstrbuton defned on the q-dmensonal sphere, denoted by W q (u, κ, has probablty densty functon gven by { ( 1 f (±x = 1F 1 2, q } 1 { 2, κ exp κ ( u x } 2, ± x S q 1, ±u S q 1, κ > 0, (2.1 where 1 F 1 (1/2, q/2, κ s the confluent hypergeometrc functon defned by 1F 1 ( 1 2, q 2, κ = Γ ( q 2 Γ ( ( 1 2 Γ q exp (κt t 0.5 (1 t (q 3 2 dt. (2.2 2

3 Ths dstrbuton has two parameters: a drectonal parameter ±u and a concentraton parameter κ, whch measures the concentraton around ±u. It s rotatonally symmetrc about the prncpal axs ±u. The Watson dstrbuton W q (u, κ has the followng property gven n Marda and Jupp (2000 p. 236: For ±x S q 1 from a bpolar Watson populaton, we have for large κ (κ { 2κ 1 ( u x } 2. χ 2 q 1. (2.3 Let X = [±x 1 ± x 2... ± x n ] be a random sample of sze n from the bpolar Watson dstrbuton W q (u, κ. The maxmum lkelhood estmators of the parameters, gven for nstance, n Marda and Jupp (2000, p. 202 and Watson (1983, p are defned by The maxmum lkelhood estmator û of u s the egenvector of the orentaton matrx XX assocated wth the largest egenvalue ŵ. The maxmum lkelhood estmator κ of κ s the soluton of Y ( κ = ŵ n, where Y (κ s defned by Y (κ = d ( 1 1F 1 2, q 2, κ. (2.4 dκ 2.2 ANOVA test for Watson dstrbuton Let X = [±x 1 ± x 2... ± x n ], = 1,..., k be k ndependent random samples of szes n 1,...,n k from Watson dstrbutons W q (±u, κ wth polar axs ±u and concentraton parameter κ around ±u, = 1,..., k and let n = n n k be the total sample sze. Suppose that we wsh to test H 0 : ±u 1 = ±u 2 =... = ±u k = ±u, (2.5 aganst the alternatve that at least one of the equaltes s not satsfed. Next we consder κ known. We note that when the concentraton parameters κ are unknown, we may replace them by ther maxmum lkelhood estmates. The maxmum lkelhood estmate κ for = 1,..., k s the soluton of the equaton Y ( κ = ŵ n, where Y (. s defned n (2.4 and ŵ s the largest egenvalue assocated wth X X. Consder the followng dentty k n { κ 1 ( û } 2 x j = j=1 k n { κ 1 ( û 2 x j }+ j=1 k n j=1 κ { (û x j 2 (û x j 2 }, (2.6 3

4 where û s the egenvector assocated wth the largest egenvalue λ of the matrx κ X X,.e., ( κ X X û = λ û, (2.7 and û s the egenvector assocated wth the largest egenvalue λ of the matrx.e., k κ X X, ( k κ X X û = λû. (2.8 The dentty (2.6 s the decomposton of the total varablty nto the sum of the wthn- -groups varablty and the between-groups varablty, and t may be wrtten as ( k k ( κ n λ k k = κ n λ + λ λ. (2.9 The test statstc s defned by ( k λ λ (k 1 (q 1 F = k (κ n λ, (2.10 (n k (q 1 and t may be wrtten as: ( k û (κ X X û û F = ( k k κ X X û (k 1 (q 1. (2.11 (n k (q 1 (κ n û (κ X X û In the partcular case of all concentraton parameters equal to κ (known or unknown, the statstc gven by (2.10 reduces to the followng statstc ( k ŵ ŵ (k 1 (q 1 F = ( n k, (2.12 ŵ (n k (q 1 where ŵ s the largest egenvalue of k X X and ŵ s the largest egenvalue of X X. The test statstc F has under the null hypothess, approxmately F (k 1(q 1,(n k(q 1 dstrbuton, for known and large concentraton parameters κ (κ, = 1,..., k. 4

5 3 strap procedure and permutaton test We consder the null hypothess of a common polar axs, H 0 : ±u 1 = ±u 2 =... = ±u k = ±u for k populatons wth polar axs ±u and concentraton parameter κ around ±u. We propose the bootstrap and permutaton versons of the ANOVA statstc defned by (2.11. The algorthms for performng the bootstrap and permutaton tests are based on Monte Carlo samplng n both algorthms. Amaral et al. (2007 refer that a key pont n bootstrap hypothess testng s make a prelmnary transformaton of the data before performng resamplng under the null hypothess. Ths s because typcally the data do not satsfy the null hypothess exactly. These authors refer a method to move û to û, whch wll be descrbed next. Gven two unt vectors a and b n R q, the rotaton matrx to move b to a along the geodesc path on the unt sphere n R q that connects b to a s gven by Q = I p + (sn α A + {(cos α 1} ( aa +cc, (3.1 where α = cos 1 (a b (0, π and A = ac +ca, wth c = b a(a b b a(a b, where. denotes the Eucldean norm on R q. Then, Qb = a, n our case b = û and a = û. The theoretcal accuracy of the bootstrap procedure was analyzed n Amaral et al. (2007. The algorthm for the bootstrap test can be mplemented n the followng steps: 1. For each sample of sze n, calculate the estmate of u defned by (2.7, û and the correspondng egenvalue λ, = 1,..., k. 2. Determne the estmate of the common polar axs û, defned by (2.8, and the correspondng egenvalue λ. Then, calculate the statstc value F obs defned n ( Transform each sample usng the rotaton matrx (3.1 to move û to û ( = 1,..., k. 4. For each bootstrap cycle b, b = 1,..., B do as follows. For = 1,..., k draw a re-sample of sze n randomly wth replacement, from the sample, and calculate the egenvalue λ (b λ (b usng (2.7 for known concentraton parameters κ and calculate the egenvalue and κ (b for unknown concentraton parameters. Then, determne the bootstrap statstc F (b defned by and F (b = F (b = ( k k λ (b ( κ n ( k k ( κ (b λ (b λ (k 1 (q 1 n (b λ (n k (q 1 for known and unknown κ, respectvely.. (3.2 λ (k 1 (q 1, (3.3 (b λ (n k (q 1 5

6 5. Determne the bootstrap p-value by 1 + B b=1 p = B + 1 I {F (b F obs} where the ndcator functon s defned by I A =, (3.4 { 1 f A occurs 0 otherwse The algorthm for mplementng the permutaton test can be descrbed n the followng four steps. Let [±x 1 ± x 2... ± x n ] be the -sample of unt vectors. 1. For each sample = 1,..., k, calculate the egenvalue λ defned by (2.7, and then the egenvalue λ defned by ( Determne the statstc value F obs gven n ( At each permutaton cycle c, c = 1,..., C do as follows. Sample randomly, wthout replacement, from the pooled set of observatons [±x 1 ± x 2... ± x n ], = 1,..., k, j = 1,..., n to form k subsamples of szes n 1,..., n k and for each, calculate the egenvalue λ (c usng (2.7 for known concentraton parameters κ and for unknown concentraton parameters, calculate λ (c and κ (c. Next, determne the permutaton verson of the statstc F (c defned by ( k λ (c F (c λ (k 1 (q 1 = k ( (c κ n λ (n k (q 1 and F (c = ( k k ( κ (c λ (c n for known and unknown κ, respectvely. 4. Determne the permutaton p-value by 1 + C c=1 p = C + 1 I {F (c F obs}, (3.5 λ (k 1 (q 1, (3.6 (c λ (n k (q 1 where I(. s the ndcator functon prevously defned., (3.7 The permutaton tests n k-samples problems are n general vald f under the null hypothess the k sets of observatons are exchangeable,.e., the k populatons are dentcal wth the same parameters (see Wellner, 1979, Romano, 1990, Good, 2004 and Amaral et al., 6

7 2007. In our expermental analyss n next secton, we wll consder not only the case of dentcal populatons wth the same parameters under the null hypothess, but also the case of populatons wth the same polar axs and dfferent concentraton parameters under the null hypothess. 4 Performance of the tests for data from Watson populatons In ths secton, we present the results of the performance of the tests obtaned wth a smulaton study. We note that n ths study as we can not present all the cases studed, we selected only some cases that seemed relevant to us. For the smulaton of the Watson dstrbuton defned on the hypersphere, we used the acceptance-rejecton method gven n L and Wong (1993. Frst, we suppose two Watson populatons wth known concentratons n subsecton 4.1. Second, we consder two Watson populatons wth estmated concentratons n subsecton 4.2. Thrd, we suppose three Watson populatons wth a common and known concentraton parameter n subsecton Two Watson populatons wth known concentratons (equal or dfferent We consdered two Watson populatons W q (u 1, κ 1 and W q (u 2, κ 2, where the concentraton parameters κ 1 and κ 2 are known. An extensve smulaton study was undertaken and we present the results for the dmensons of the sphere q = 2, 3, 4, 5 to test H 0 : ±u 1 = ±u 2 = ±u. Ths study was carred out for nvestgatng the performance of the three tests for the ANOVA statstc gven by (2.10, the tabular test, based drectly on the null asymptotc dstrbuton of the statstc, the bootstrap test and the permutaton test. We note that for equal concentraton parameters, the ANOVA statstc reduces to the statstc (2.12, whch does not depend on the common concentraton parameter. Frst we estmated the sgnfcance level of the three tests and second, we determned the emprcal power of the tests Estmated sgnfcance levels We consdered wthout loss of generalty, that under H 0 : ±u 1 = ±u 2 = ±e q, where e q = (0,..., 0, 1. We generated two samples of szes n 1 and n 2 of the populatons W q (e q, κ 1 and W q (e q, κ 2, supposng samples of equal sze and also samples of dfferent szes. The estmated levels of sgnfcance obtaned for a nomnal sgnfcance level of 5% are ndcated 7

8 n les 1-2 for known and equal concentraton parameters (κ 1 = κ 2 = κ and known and dfferent concentraton parameters (κ 1 κ 2, respectvely. In these tables we hghlghted n bold the levels of sgnfcance between 4.5% and 5.5%, that may be consdered close to the nomnal level 5%. Each estmated sgnfcance level,.e., the proporton of tmes that H 0 s ncorrectly rejected, was obtaned through a smulaton study wth Monte Carlo smulatons n the tabular test and 5000 Monte Carlo smulatons n the bootstrap and permutaton tests. The number of bootstrap re-samples, B, n each Monte Carlo smulaton was B = 200 and the number of permutaton samples was C = 200. For obtanng the sgnfcance levels, we used the 0.95-percentle of an F -dstrbuton n the tabular test and the 0.95-percentle of the dstrbuton of values of the bootstrap and permutaton statstcs n the bootstrap and permutaton tests, respectvely. The estmated sgnfcance levels obtaned for equal and dfferent concentraton parameters enable us to draw smlar conclusons. Frst, we note that n the tabular test, although we used the crtcal pont of an F -dstrbuton for a sgnfcance level of 5%, the estmated sgnfcance levels obtaned are not exactly equal to the nomnal sgnfcance level of 5%. The estmated sgnfcance levels n the tabular test are n general more dstant from the nomnal sgnfcance level for small concentraton parameters. We note that n these tables we dd not consder very large values of the concentraton parameters, for whch the sgnfcance levels obtaned n the tabular test are the closest to the nomnal level of sgnfcance. Ths would be expected snce the F -dstrbuton of the test statstc s only an asymptotc dstrbuton, vald for large concentratons. Thus, t s necessary to consder other versons of the ANOVA statstc such as the bootstrap and permutaton versons. Second, the sgnfcance levels obtaned n the permutaton test n the case of equal or dfferent concentraton parameters are very close to the nomnal sgnfcance level (5% n almost all cases. Consequently, the permutaton test s generally very relable n what concerns the type I error. Thrd, from the estmated sgnfcance levels obtaned n the bootstrap test, we conclude that the bootstrap statstc s very relable n most part of the consdered cases. Addtonally, n general the bootstrap test has smlar accuracy to the permutaton test, essentally n the case of equal concentraton parameters and has generally smlar accuracy to the tabular test n the case of large concentraton parameters. Fnally, we note that the estmated levels of sgnfcance of the tests for a nomnal level of sgnfcance of 1% led us to smlar conclusons Emprcal power of the tests Second, we determned the emprcal power of the tabular, bootstrap and permutaton tests for a nomnal sgnfcance level of 5%. We supposed the same null hypothess as before and n the alternatve hypothess H 1, two drectonal parameters ±u 1 and ±u 2 whch form an angle θ between them, wth θ = 18, 36, 54, 72, 90. Thus, under ths alternatve 8

9 le 1: Estmated sgnfcance levels (n % of the tabular, bootstrap and permutaton tests, for a common and known concentraton parameter κ and several sample szes n 1, n 2. n 1, n 2 κ q = 2 q = 3 q = 4 q = , , , , , hypothess and wthout loss of generalty, we generated one( sample from W q (e q, κ 1 and the other sample from W q (u, κ 2, where u s defned by u = 0,..., 0, (1 cos 2 θ 1/2, cos θ. We note that f the angle θ s equal to 0, we obtan the sgnfcance level of the tests. As n the estmaton of the sgnfcance levels, to determne the emprcal power of the tests, we used the 0.95-percentle of an F -dstrbuton n the tabular test and the 0.95-percentle 9

10 le 2: Estmated sgnfcance levels (n % of the tabular, bootstrap and permutaton tests for dfferent and known concentraton parameters κ 1 and κ 2 and several sample szes n 1, n 2. n 1, n 2 κ 1, κ 2 q = 3 q = 4 q = , 5 1, , , , , 10 1, , , , , 5 1, , , , , 10 1, , , , , 20 1, , , of the bootstrap or permutaton dstrbuton n the bootstrap or permutaton tests. In the tabular test, the emprcal power was obtaned from replcates of the test statstc under the alternatve hypothess. In the bootstrap or permutaton test, the emprcal power was obtaned from 5000 Monte Carlo smulatons, where n each smulaton, two samples were generated under H 1 and 200 bootstrap or permutaton re-samples were consdered. We ndcate the emprcal power of the tests for equal and known concentraton parameters n le 3 for q = 2, n le 4 for q = 3 and n le 5 for q = 4, 5. In these tables we hghlghted the values of the power, n whch the bootstrap test s more powerful than the tabular and permutaton tests. Addtonally, we show n Fgure 1 the emprcal power of the three tests for equal and known concentraton parameters when q = 4, 5 and n 1 = 5, n 2 =

11 le 3: Emprcal power (n % of tabular, bootstrap and permutaton tests, for q = 2, common and known concentraton κ, angle θ( and sample szes n 1, n 2. ular strap utaton n 1, n 2 θ\κ , , , , We ndcate the emprcal power of the tests for dfferent and known concentraton parameters n le 6 for q = 3 and n le 7 for q = 4. As before, n these tables we hghlghted the values of the power, n whch the bootstrap test s more powerful than the tabular and permutaton tests. Fgure 2 shows the emprcal power of the tests for dfferent and known concentraton parameters when q = 5. The results obtaned wth a common concentraton parameter and wth dfferent concentraton parameters are smlar. For both cases of equal and dfferent concentraton parameters and for each dmenson of the sphere q, we conclude from the three tests, that the permutaton test s the one that s least powerful. For each q, despte the sgnfcance level of the permutaton test be very close to the nomnal level of sgnfcance n many cases, 11

12 le 4: Emprcal power (n % of tabular, bootstrap and permutaton tests, for q = 3, common and known concentraton κ, angle θ( and sample szes n 1, n 2. ular strap utaton n 1, n 2 θ\κ , , , , , the emprcal power of ths test remans close to the sgnfcance level for low values of the concentraton parameters (equal or not or when the sample szes are dfferent. For large values of the concentraton parameters (equal or not and for each q, the permutaton test has better performance for equal-szed samples than for samples of dfferent szes, snce the emprcal power ncreases as the angle ncreases for equal-szed samples, whle t remans 12

13 le 5: Emprcal power (n % of tabular, bootstrap and permutaton tests, for q = 4, 5, common and known concentraton κ, angle θ( and sample szes n 1, n 2. ular strap utaton n 1, n 2 q θ\κ l , , ,

14 q=4, n1=5,n2=10 (conc. parameter= 1 q=4,n1=5,n2=10 (conc. parameter= 2 q=4,n1=5,n2=10 (conc. parameter= 5 q=5, n1=5,n2=10 (conc. parameter= 1 q=5,n1=5,n2=10 (conc. parameter= 2 q=5,n1=5,n2=10 (conc. parameter= 5 Fgure 1: Emprcal power of the tests for a common and known concentraton approxmately equal to the nomnal level of sgnfcance for dfferent-szed samples. For each q, we also observed that for samples of equal sze, the emprcal power of the permutaton test ncreases as the concentraton parameters (equal or not ncrease. For each q and large values of the concentraton parameters (equal or not, the emprcal power of the permutaton test ncreases n general, when the common sample szes ncreases. For each dmenson of the sphere q, the emprcal power of the tabular and bootstrap tests ncreases rapdly and n some cases tends quckly to 1 when the angle between drectonal parameters θ ncreases for equal or dfferent concentraton parameters and samples of equal or dfferent szes. For each q, the emprcal power of the tabular and bootstrap tests ncreases n general when the concentraton parameters (equal or not ncrease. For equal or dfferent concentraton parameters and for each q, the emprcal power of the tabular and bootstrap tests ncreases n general as the sample szes (equal or not ncrease. In both cases of equal and dfferent concentraton parameters, for each dmenson of the sphere q and samples of szes equal or not, the bootstrap test s generally more powerful than the tabular test for small concentraton parameters or small angle θ between the drectonal parameters. The superorty of the bootstrap test compared to the tabular test s more pronounced for small samples than for large samples. For nstance, s greater for n = 5 than for n = 10, and for n 1 = 3, n 2 = 5 than for n 1 = 5 and n 2 = 10. In fact, for 14

15 le 6: Emprcal power (n % of tabular, bootstrap and permutaton tests, for q = 3, dfferent and known concentratons κ 1, κ 2, angle θ( and sample szes n 1, n 2. ular strap utaton n 1, n 2 θ\κ 1, κ 2 1,2 3,5 5,10 1,2 3,5 5,10 1,2 3,5 5, , , , , , small samples (of equal sze or not, the bootstrap test s better than the tabular test for small values of the concentraton parameters (equal or not and also for large concentraton parameters and small angle between the drectonal parameters. Thus, the results of the power ndcate that the bootstrap test may be a good alternatve to the tabular test for small concentraton parameters (equal or not and small samples (of equal sze or not or 15

16 le 7: Emprcal power (n % of tabular, bootstrap and permutaton tests, for q = 4, dfferent and known concentratons κ 1, κ 2, angle θ( and sample szes n 1, n 2. ular strap utaton n 1, n 2 θ\κ 1, κ 2 1,2 3,5 5,10 1,2 3,5 5,10 1,2 3,5 5, , , , , when the alternatve hypothess s not far away from the null hypothess. 4.2 Two Watson populatons wth estmated concentratons Next, we study the effect n the performance of the tests of the estmaton of the concentraton parameters based on ANOVA statstc defned by (2.10. Frst we determned the estmated levels of sgnfcance of the tests and second, the emprcal power of the tests. We consdered the same number of Monte Carlo smulatons n the tests as before. The number of bootstrap or permutaton samples was also the same as before. The estmated levels of sgnfcance are ndcated n le 8 for q = 3, 4 n the case of equal or dfferent concentraton parameters. The emprcal power of the tests obtaned wth concentraton 16

17 q=5,n1=3,n2=5, (conc. parameters= 1,2 q=5,n1=3,n2=5, (conc. parameters= 3,5 q=5,n1=3,n2=5, (conc. parameters= 5,10 q=5,n1=n2=5, (conc. parameters= 1,2 q=5,n1=n2=5, (conc. parameters= 3,5 q=5,n1=n2=5, (conc. parameters= 5,10 Fgure 2: Emprcal power of the tests for dfferent and known concentratons estmates for q = 3, 4, wth a nomnal sgnfcance level of 5% s ndcated n le 9 for equal concentraton parameters and n le 10 for dfferent concentraton parameters. In these tables we hghlght the values of the power, n whch the bootstrap test s the most powerful. The emprcal power obtaned for q = 3, 4 and n 1 = 5, n 2 = 10 when we estmate the concentraton parameters can be seen n Fgures 3 and 4 for equal and dfferent concentraton parameters, respectvely. As we may observe from these tables and fgures, the results obtaned for the estmated level of sgnfcance and the emprcal power are not substantally affected by the estmaton of the concentraton parameters through the maxmum lkelhood method. When the concentratons are estmated, the behavor of the tests s smlar to the case when the concentratons are known. Consequently, we have the same conclusons for both cases of known and estmated concentraton parameters. 4.3 Three Watson populatons wth a common and known concentraton parameter We consdered three Watson populatons W q (u 1, κ 1,W q (u 2, κ 2 and W q (u 3, κ 3, and we wsh to test H 0 : ±u 1 = ±u 2 ± u 3 = ±u, usng the tabular, bootstrap and permutaton 17

18 le 8: Estmated sgnfcance levels (n % of the tabular, bootstrap and permutaton tests, for estmated concentraton parameters, equal (κ 1 = κ 2 = κ or not (κ 1 κ 2 and several sample szes n 1, n 2. n 1, n 2 κ q = 3 q = , , , n 1, n 2 κ 1, κ 2 q = 3 q = , 5 1, , , , , 5 1, , , , , 10 1, , , ,

19 le 9: Emprcal power (n % of the tests for p = 3, 4, estmated concentratons κ 1, κ 2 (κ 1 = κ 2 = κ, angle θ ( o and sample szes n 1, n 2. ular strap utaton n 1, n 2 p θ/κ , , ,

20 q=3, n1=5,n2=10 (conc. parameter= 1 q=3,n1=5,n2=10 (conc. parameter= 2 q=3,n1=5,n2=10 (conc. parameter= 5 q=4, n1=5,n2=10 (conc. parameter= 1 q=4,n1=5,n2=10 (conc. parameter= 2 q=4,n1=5,n2=10 (conc. parameter= 5 Fgure 3: Emprcal power of the tests for estmated equal concentratons versons for the ANOVA test. We carred out a smulaton study to estmate the level of sgnfcance and to determne the emprcal power of the tests, consderng the dmensons of the sphere q = 3, 4 and a common and known concentraton parameter for the populatons κ 1 = κ 2 = κ 3 = κ = 1, 2, 5, 10. We also consdered equal samples sze n 1 = n 2 = n 3 = n = 5, 10. We supposed, wthout loss of generalty, that under H 0 : ±u 1 = ±u 2 = ±u 3 = ±e q, where e q = (0,..., 0, 1. The estmated levels of sgnfcance were obtaned for a nomnal level of sgnfcance of 5% under H 0. We determned the emprcal power of the tests, for ths nomnal level of sgnfcance, supposng three types of alternatve hypothess. Let θ 1 be the angle between u 1 and u 2, θ 2 be the angle between u 2 and u 3 and θ 3 has the same defnton as θ 2. H (1 1 : u 1 = e p, u 2 = We ( supposed, wthout loss of generalty, ( n the alternatve hypothess: 0,..., 0, ( /2, 0.95, u 3 = 0,..., 0, ( /2, 0.59,.e, ( 0,..., 0, ( /2, 0.95, u 3 = e 1, θ 1 = 18, θ 2 = 54, θ 3 = 36, H (2 1 : u 1 = e q, u 2 =.e, θ 1 = 18, θ 2 = θ 3 = 90 and H (3 1 : u 1 = e q, u 2 = e q 1, u 3 = e 1,.e, θ 1 = θ 2 = θ 3 = 90. The number of replcates n the tests and the number of bootstrap or permutaton samples consdered to determne the levels of sgnfcance and the emprcal power were the same as n the prevous smulaton study done for two populatons. The estmated level of sgnfcance, obtaned when θ 1 = θ 2 = θ 3 = 0 and the emprcal power for the three types of alternatve hypothess are ndcated n le 11. In ths table we hghlght the values of the power, n 20

21 le 10: Emprcal power (n % of the tests, for q = 3, 4, estmated concentratons κ 1, κ 2 (κ 1 κ 2, angle θ( and several sample szes n 1, n 2. ular strap utaton n 1, n 2 q θ\κ 1, κ 2 1,2 3,5 5,10 1,2 3,5 5,10 1,2 3,5 5, , , ,

22 q=3,n1=5,n2=10, (conc. parameters= 1,2 q=3,n1=5,n2=10, (conc. parameters= 3,5 q=3,n1=5,n2=10, (conc. parameters= 5,10 q=4,n1=5,n2=10, (conc. parameters= 1,2 q=4,n1=5,n2=10, (conc. parameters= 3,5 q=4,n1=5,n2=10, (conc. parameters= 5,10 Fgure 4: Emprcal power of the tests for estmated dfferent concentratons whch the bootstrap test s the most powerful. The conclusons are smlar to those obtaned for two Watson populatons, despte of the estmated levels of sgnfcance seem to be a bt worse. The estmated levels of sgnfcance n the bootstrap test are smlar to the values for the permutaton test, although n ths latter test they are slghtly better. In what concerns to the estmated level of sgnfcance for the tabular test, as ths test s vald only for large concentratons, t would be expected that the estmated level of sgnfcance s not good for small concentratons. Smlarly, for each dmenson of the sphere, the emprcal power ncreases n general, as the separaton between populatons ncreases or the common concentraton parameter ncreases. We concluded that the bootstrap test s a good alternatve to the tabular test for small concentraton parameter or small samples or poor separaton between the Watson populatons. Among the three tests, the permutaton test s the one that s least powerful, although t s the test that has n general the best estmated level of sgnfcance. 22

23 le 11: Estmated sgnfcance level and emprcal power (n % of the tabular, bootstrap and permutaton tests, for three Watson populatons, wth q = 3 and q = 4, common and known concentraton parameter κ, common samples sze n and angles between drectonal parameters θ 1, θ 2 and θ 3 (n. q = 3 q = 4 Test n θ 1 θ 2 θ 3 \κ ular strap utaton Applcaton We used the vectorcardogram data of Downs et al. (1971 obtaned wth two systems (Frank system and McFee lead system. From these data we took the unt sphercal vector 23

24 assocated wth each vectorcardogram, whch represents the spatal drecton of the vector of the QRS loop havng the greatest magntude. Then, we consdered the axes assocated to these drectons. We selected data for eght chldren from each of the eght combnatons of the categores (sex-age and type of system. Data, n radans, are n the le 12. le 12: Sphercal vectorcardogram data (n radans Boy aged 2-10 Boy aged Grl aged 2-10 Grl aged Frank system McFee lead system

25 We are nterested n nvestgatng whether for each sex-age category, the type of system (Frank system or McFee lead system affects the result of the vectorcardogram. In ths applcaton we supposed the ANOVA statstc for dfferent concentraton parameters (general statstc and also the ANOVA statstc for equal concentraton parameters for each sex-age category. Then for each category sex-age, we determned the values of the ANOVA le 13: Largest egenvalues and estmates of the concentraton parameters of the groups, and statstc values and p-values of the tests for each sex-age category Sex-Age Group System Concen- Statstc p-value (% Frank McFee lead traton value... j 1 2 parameters Boy aged ŵ j Dfferent κ j Equal Boy aged ŵ j Dfferent κ j Equal Grl aged ŵ j Dfferent κ j Equal Grl aged ŵ j Dfferent κ j Equal statstcs gven by (2.10 and (2.12, whch are ndcated n le 13, as well as the p-values obtaned for the tabular method, the bootstrap and permutaton versons of the ANOVA statstc. The p-values of the bootstrap and permutaton tests were obtaned wth B = 1000 bootstrap re-samples and C = 1000 permutaton samples. Frst, the dfference between the p-values of the tests for both statstcs s very small. Second, on one hand, the three tests led to the same concluson for chldren aged and boys aged More precsely, we can conclude that there s no sgnfcant dfference between the systems for boys aged 2-10 whle there s dfference for chldren aged On the other hand, for grls aged 2-10 there s no evdence to conclude that the systems dffer usng the tabular and bootstrap tests. Based on the permutaton test, we can not conclude that the systems dffer at a level of sgnfcance 1%, but we conclude that there s dfference between the systems at a level of 5%. The code for applyng these tests s avalable n the web page: pag_d= &pct_parametros=p_codgo=205276&pct_grupo=23660#

26 6 Concludng remarks We have concluded that the bootstrap and permutaton versons of the ANOVA statstc for testng a common mean polar axs across several Watson populatons defned on the hypersphere gave relable estmates of the sgnfcance level, n most part of the smulated cases, and n partcular, for small concentratons and small samples. Addtonally, from the three tests, the bootstrap test s n general the most powerful test n the case of small samples for small concentratons or bad separaton between the Watson populatons. So, n these cases, the bootstrap and permutaton tests based on ANOVA statstc may consttute useful alternatves to the ANOVA statstc, that has an asymptotc dstrbuton, vald only for large concentratons. Acknowledgements The author s grateful to the Professors Mchael Stephens and Rchard Lockhart for ther suggestons to ths paper. The author also thanks the helpful comments gven by the referees of ths journal, that helped to mprove ths paper. Ths work s fnanced by the FCT - Fundação para a Cênca e a Tecnologa (Portuguese Foundaton for Scence and Technology wthn project UID/EEA/50014/2013. References [1] Amaral, G. J. A., Dryden, I. L. and Wood, A. T. A. (2007. Pvotal bootstrap methods for k-sample problems n drectonal statstcs and shape analyss. Journal of the Amercan Statstcal Assocaton, 102:478, [2] Anderson, C. M. and Wu, C. F. J. (1995. Measurng locaton effects from factoral experments wth a drectonal response. Internatonal Statstcal Revew, 63, [3] Downs, T., Lebman, J. and Mackay, W. (1971. Statstcal methods for vectorcardogram orentatons. In: Hoffman, R. I., Glassman, E. (eds. Vectorcardography 2: proceedngs of XI th nternatonal symposum on vectorcardography. North-Holland, Amsterdam, [4] Efron, B. (1979. strap methods: another look at the jacknfe. The Annals of Statstcs, 7:1, [5] Fsher, N. I. (1993. Statstcal analyss of crcular data, Cambrdge Unversty Press, Cambrdge, Great Brtan. 26

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