BOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS. M. Krishna Reddy, B. Naveen Kumar and Y. Ramu

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1 BOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS M. Krshna Reddy, B. Naveen Kumar and Y. Ramu Department of Statstcs, Osmana Unversty, Hyderabad , Inda. Abstract: In ths paper, a graphcal procedure usng bootstrap method s developed as an alternatve to the ANOVA to test the hypothess on equalty of several means. An example s gven to demonstrate the advantage of bootstrap graphcal procedure over the ANOVA n decson makng pont of vew. Keywords: Mean, Analyss of varance, Bootstrap method.. Introducton The analyss of varance frequently referred to by the contracton ANOVA s a statstcal technque specally desgned to test whether the means of more than two quanttatve populatons are equal. Bascally, ths method conssts of classfyng and crossclassfyng statstcal results and testng whether the means of specfed classfcaton dffer sgnfcantly. In ths study, applcablty of bootstrap method for testng of several means s dscussed and ths bootstrap method can be treated as an alternatve method to ANOVA when the samples are very small. Statstcal nference based on data resamplng has drawn a great deal of attenton n recent years. The man dea about these resamplng methods s not to assume much about the underlyng populaton dstrbuton and nstead tres to get the nformaton about the populaton from the data tself varous types of resamplng methods. Bootstrap method (Efron, 979) use the relatonshp between the sample and resamples drawn from the sample,

2 to approxmate the relatonshp between the populaton and samples drawn from t. Wth the bootstrap method, the basc sample s treated as the populaton and a Monte Carlo style procedure s conducted on t. Ths s done by randomly drawng a large number of resamples of sze n from ths orgnal sample wth replacement [] []. Both bootstrap and tradtonal parametrc nference seek to acheve the same goal usng lmted nformaton to estmate the samplng dstrbuton of the chosen estmatorθˆ. The estmate wll be used to make nferences about a populaton parameterθ. The key dfference between these nferental approaches s how they obtan ths samplng dstrbuton where as tradtonal parametrc nference utlzes a pror assumptons about the shape of θˆ s dstrbuton, the non-parametrc bootstrap s dstrbuton free, whch means that t s not dependent on a partcular class of dstrbutons. Wth the bootstrap method, the entre samplng dstrbuton θˆ s estmated by relyng on the fact that the sample s dstrbuton s good estmate of the populaton dstrbuton [] [].. Bootstrap method for testng of equalty of several means Let { X,,,..., k, j =,,..., } = represent k ndependent random samples of j n szes n,, n,... nk respectvely and we assume that X ~ (, j N σ ) µ for =,,..., k. Here, we are nterested n testng the null hypothess. H : 0 µ = µ = K = µ k = µ (Unknown) aganst the alternatve hypothess H : µ µ K µ. ANOVA s used for testng k H 0 n the lterature. Ths test demonstrates only the statstcal sgnfcance of the equalty of means beng compared. The Bootstrap graphcal procedure for testng H 0 s gven n the followng steps.

3 k Let the avalable combned sample s{ Z j, j =,, LN} of sze N = n =,. Draw the B- bootstrap samples of sze N wth replacement from the combned sample { j,, LN} Z j, = and the b th bootstrap sample of sze N s gven by { y b, =,, K, N and b =,, K, B} (.). Compute the mean for the b th bootstrap sample. y N b = y b N =, b =,, L, B. (.) 3. Obtan the samplng dstrbuton of mean usng B-bootstrap estmates and compute the mean and standard error of mean. y B = y b B b= B. and s = ( y b y ) B b= (.3) 4. The lower decson lne (LDL) and the upper decson lne (UDL) for the comparson of each of the x are gven by LDL = y UDL = y Z + Z α / α / s s (.4) where Zα / s the crtcal value at α level and Z 0.05 =. 96at 5% level. 5. Plot x aganst the decson lnes. If any one of the ponts plotted les outsde the respectve decson lnes, H 0 s rejected at α level and conclude that the means are not homogenous. 3

4 The proposed method s very useful n handlng of small samples of sze less than 30. Ths method not only tests the sgnfcant dfference among the means but also dentfy the source of heterogenety of means. 3. Numercal Example The lfetmes (n hours) of samples from three dfferent brands of batteres were recorded wth the followng results [3]. Table. Lfetmes (n hours) of the batteres Brand X X X We wsh to test whether the three brands have same average lfetmes or not. We wll assume that the three samples come from normal populatons wth common (unknown) standard devaton. From the data, we have n =, n = 4, n 6, x = 40, x = 55 and x3 = = One-way ANOVA s carred out to test the hypothess H : µ = µ = µ 0 3 and the results gven below. Table. ANOVA Source of Varaton SS df MS F P-value F crt. Between Brands Wthn Brands Total

5 Snce the sgnfcant probablty value s 0.040, whch s less than the level of sgnfcance α = 0. 05, thus we reject the null hypothess and we may conclude that the mean lfetmes of three brands are not homogenous. We apply bootstrap method to test the same hypothess for the gven example. The procedure s explaned n the followng steps.. Draw B (=000) bootstrap sample of sze N=5 from the combned sample of sze N=5.. Compute the mean for each bootstrap sample usng (.) and form the samplng dstrbuton. 3. Compute the mean and standard devaton of the B-bootstrap estmates of mean usng (.3) and t s observed that y = and s = The decson lnes are computed usng (.4) and are obtaned at α = s as follows. LDL= and UDL= Prepare a chart as n Fgure wth the above decson lnes and plot the ponts x (=,, 3). From Fg., we observe that x and x3 le outsde the decson lnes. Hence, H 0 may be rejected and we may conclude that the three populaton means are not same. 4. Conclusons From the above study, we observe that the concluson s same, that s Ho s rejected by both ANOVA and bootstrap method and bootstrap method not only tests the sgnfcant dfference among the means but also dentfes the source of heterogenety of means. 5

6 Snce x 3 les above the UDL, therefore the customer s advsed to purchase the brand X 3 batteres and try to avod the brand X batteres. References. Efron, B (979), Bootstrap methods: Another look at the Jackknfe, Ann. Statst., 7, pp -6.. Efron, B, Tbshran, R.J (993), An ntroducton to the Bootstrap, Chapman & Hall, New York. 3. Rohatg, V.K. (976), An Introducton to probablty theory and mathematcal statstcs, Wley Eastern Ltd., pp Fgure. Chart of decson lnes x 3 UDL x LDL x 6