DIAGNOSTICS AND REMEDIAL MEASURES

Transcription

1 DIAGNOSTICS AND REMEDIAL MEASURES Rajender Parsad and Lal Mohan Bhar I.A.S.R.I., Lbrary Aenue, New Delh-0 0 rajender@asr.res.n; lmbhar@asr.res.n The nterpretaton of data based on analyss of arance (ANOVA) s ald only when the followng assumptons are satsfed:. Addte Effects: Treatment effects and block (enronmental) effects are addte.. Independence of errors: Expermental errors are ndependent. 3. Homogenety of Varances: Errors hae common arance. 4. Normal Dstrbuton: Errors follow a normal dstrbuton. Also the statstcal tests t, F, z, etc. are ald under the assumpton of ndependence of errors and normalty of errors. The departures from these assumptons make the nterpretaton based on these statstcal technques nald. Therefore, t s necessary to detect the deatons and apply the approprate remedal measures. The assumpton of ndependence of errors,.e., error of an obseraton s not related to or depends upon that of another. Ths assumpton s usually assured wth the use of proper randomzaton procedure. Howeer, f there s any systematc pattern n the arrangement of treatments from one replcaton to another, errors may be non-ndependent. Ths may be handled by usng nearest neghbour methods n the analyss of expermental data. The assumpton of addte effects can be defned and detected n the followng manner: Addte Effects The effects of two factors, say, treatment and replcaton, are sad to be addte f the effect of one-factor remans constant oer all the leels of other factors. A hypothetcal set of data from a randomzed complete block (RCB) desgn, wth treatments and replcatons, wth addte effects s gen n Table. Table Treatment Replcaton Replcaton Effect I II I - II A B Treatment Effect (A-B) 0 0 Here, the treatment effect s equal to 0 for both replcatons and replcaton effect s 65 for both treatments. When the effect of one factor s not constant at all the leels of other factor, the effects are sad to be non-addte. A common departure from the assumpton of addtty n bologcal experments s one where the effects are multplcate. Two factors are sad to hae multplcate effects f ther effects are addte only when expressed n terms of percentages. Table llustrates a hypothetcal set of data wth multplcate effects.

2 Treatment Table Replcaton Replcaton Effect I II I - II 00(I - II)/II A (.3003) (.0969) (0.04) B (.04) (.0000) (0.04) Treatment Effect (A-B) 40 (0.0969) 5 (0.0969) 00 (A - B)/B 5 5 In ths case, the treatment effect s not constant oer replcatons and the replcaton effect s not constant oer treatments. Howeer, when both treatment effect and replcaton effect are expressed n terms of percentages, an entrely dfferent pattern emerges. For such olatons of assumptons, Logarthmc transformaton s qute sutable. For llustraton, the Logarthmc transformaton of data n Table s gen n brackets. Ths s, howeer a crude method for testng the addtty. Tukey (949) gae a statstcal test for testng the addtty n a RCB desgn. Ths test s known as one degree of freedom test for non-addtty. In ths test, one degree of freedom s solated from error and ths degree of freedom s called as the degree of freedom for non-addtty. In the sequel, we descrbe the procedure n bref. Suppose that an experment has been conducted to compare treatments usng RCB desgn wth r replcatons. Let y j denote the obsered alue of the response arable for th th treatment n j replcaton; =,,, ; j =,,, r. Arrange the data n a r table as gen below: Treatment K j K r Treatment Total Treatment y y K y y K y j y j K K y r T. y.. Deatons from Grand d C Sum of Cross Product y r T. y. d. C M M M K M K M M M M M y y K y j K y r T. y. d. C M M M K M K M M M M M y y K y j K y r T. y. d. C Replcaton R. R K. R K. j R G (Grand. r Total total) Replcaton y. y K. y K. j y. r G GM = r Deaton d. d K. d K. j from Grand d. r III-

3 r where T. = y j ; y. = T. / r ; R. j = y j ; y. j = R. j / ; d. = y. GM = j= r d. j = y. j GM ; C = yj d. j j= Obtan L = = C d. ; D = d. ; D = d = Sum of squares due to non-addtty (SSNA)= The sum of squares due to treatments, replcatons and total sum of squares are gen by r j=. j D L D T. Sum of squares due to treatments (SST) = r = G r R. Sum of squares due to replcatons (SSR) = Total sum of squares (TSS) = = r j y j G r r j= j G r Sum of squares due to Error (SSE) = TSS SST-SSR-SSNA Then the outlne of ANOVA table s Source df SS MS Treatments - SST MST Replcatons r- SSR MSR Non-addtty SSNA MSNA Error (-)(r-)- SSE MSE Total r- TSS The mean squares (MS) are obtaned by ddng sum of squares (SS) by correspondng degrees of freedom (df). The non-addtty s tested by F-statstc wth and (-)(r-)- MSNA degree of freedom calculated alue of F =. MSE Normalty of Errors The assumptons of homogenety of arances and normalty are generally olated together. To test the aldty of normalty of errors for the character under study, one can take help of Normal Probablty Plot, Anderson-Darlng Test, D'Augstno's Test, Shapro - Wlk's Test, Ryan-Joner test, Kolmogro-Smrno test, etc. In general moderate departures from III-3

4 normalty are of lttle concern n the fxed effects ANOVA as F - test s slghtly affected but n case of random effects, t s more seerely mpacted by non-normalty. The sgnfcant deatons of errors from normalty, makes the nferences nald. So before analyzng the data, t s necessary to conert the data to a scale that t follows a normal dstrbuton. In the data from desgned feld experments, we do not drectly use the orgnal data for testng of normalty or homogenety of obseratons because ths s embedded wth the treatment effects and some of other effects lke block, row, column, etc. So there s need to elmnate these effects from the data before testng the assumptons of normalty and homogenety of arances. For elmnatng the treatment effects and other effects we ft the model correspondng to the desgn adopted and estmate the resduals. These resduals are then used for testng the normalty of the obseratons. In other words, we want to test the null hypothess H 0 : errors are normally dstrbuted aganst alternate hypothess H : errors are not normally dstrbuted. For detals on these tests one may refer to D Agostno and Stephens (986). Most of the standard statstcal packages aalable n the market are capable of testng the normalty of the data. In SAS and SPSS commonly used tests are Shapro-Wlk test and Kolmogro-Smrno test. MINITAB uses three tests z. Anderson-Darlng, Ryan-Joner, Kolmogro-Smrno for testng the normalty of data. Homogenety of Error Varances A crude method for detectng the heterogenety of arances s based on scatter plots of means and arance or range of obseratons or errors, resdual s ftted alues, etc. To be clearer, let Y j be the obseraton pertanng to th treatment ( = () ) n the j th replcaton ( ( ) j = ) r. Compute the mean and arance for each treatment across the replcatons (the range can be used n place of arance) as r r = Y. = Yj ; Varance = S = ( Yj Y. ) r r j= j= Draw the scatter plot of mean s arance (or range). If S. 's ( = () ) are equal (constant) or nearly equal, then the arances are homogeneous. Based on these scatter plots, the heterogenety of arances can be classfed nto two types:. Where the arance s functonally related to mean.. Where there s no functonal relatonshp between the arance and the mean. For llustraton some scatter - dagrams of mean and arances (or range) are gen as: Varance Varance (a) Homogeneous arance (b) Heterogeneous arance where arance s proportonal to mean III-4

5 Varance (c) Heterogeneous arance wthout any functonal relatonshp between arance and mean The frst knd of arance heterogenety (fgure b) s usually assocated wth the data whose dstrbuton s non-normal z., negate bnomal, Posson, bnomal, etc. The second knd of arance heterogenety usually occurs n experments, where, due to the nature of treatments tested, some treatments hae errors that are substantally hgher (lower) than others. For example, n aretal trals, where arous types of breedng materal are beng compared, the sze of arance between plots of a partcular arety wll depend on the degree of genetc homogenety of materal beng tested. The arance of F generaton, for example, can be expected to be hgher than that of F generaton because genetc arablty n F s much hgher than that n F. The arances of aretes that are hghly tolerant of or hghly susceptble to, the stress beng tested are expected to be smaller than those of hang moderate degree of tolerance. Also n testng yeld response to a chemcal treatment, such as, fertlzer, nsectcde or herbcde, the non-unform applcaton of chemcal treatments may result n a hgher arablty n the treated plots than that n the untreated plots. The scatter-dagram of means and arances of obseratons for each treatment across the replcatons ges only a prelmnary dea about homogenety of error arances. Statstcally the homogenety of error arances s tested usng Bartlett's test for normally dstrbuted errors and Leene test for non-normal errors. These tests are descrbed n the sequel. Bartlett's Test for Homogenety of Varances Let there are - ndependent samples drawn from same populaton and th sample s of sze r and ( r + r r ) = N. In the present case, the ndependent samples are the resduals of the obseratons pertanng to treatments and th sample sze s the number of replcatons of the treatment. One wants to test the null hypothess H 0 σ = σ =... = σ aganst the alternate hypothess H : at least two of the σ ' s arance for treatment. : are not equal, where σ s the error Let e j denotes the resdual pertanng to the obseraton of treatment from replcaton j, then t can easly be shown that the sum of resduals pertanng to a gen treatment s zero. In III-5

6 ths test S r r = ( ej e. ) = ej r j= r j= s taken as unbased estmate of σ. The procedure noles computng a statstc whose samplng dstrbuton s closely approxmated by the χ dstrbuton wth - degrees of freedom. The test statstc s χ q 0 =.306 c and null hypothess s rejected when χ 0 > χ α,, where χ α, s the upper α percentage pont of χ dstrbuton wth - degrees of freedom. To compute χ 0, follow the steps: Step : Compute mean and arance of all -samples. Step : Obtan pooled arance S p = = ( r ) N Step 3: Compute q = ( N ) log S ( r ) Step 4: Compute Step 5: Compute χ 0. c = + ( ) 3 = S 0 p log0 = ( r ) ( N ) Bartlett's χ test for homogenety of arances s a modfcaton of the normal-theory lkelhood rato test. Whle Bartlett's test has accurate Type I error rates and optmal power when the underlyng dstrbuton of the data s normal, t can be ery naccurate f that dstrbuton s een slghtly non-normal (Box 953). Therefore, Bartlett's test s not recommended for routne use. An approach that leads to tests that are much more robust to the underlyng dstrbuton s to transform the orgnal alues of the dependent arable to dere a dsperson arable and then to perform analyss of arance on ths arable. The sgnfcance leel for the test of homogenety of arance s the p-alue for the ANOVA F-test on the dsperson arable. Commonly used test for testng the homogenety of arance usng a dsperson arable s Leene Test gen by Leene (960). The procedure s descrbed n the sequel. Leene Test for homogenety of Varances The test s based on the arablty of the resduals. The larger the error arance, the larger the arablty of the resduals wll tend to be. To conduct the Leene test, we dde the data nto dfferent groups based on the number of treatments f the error arance s ether ncreasng or S III-6

7 decreasng wth the treatments, the resduals n the one treatment wll tend to be more arable than those n others treatments. The Leene test than conssts smply F statstc based on one way ANOVA used to determne whether the mean of absolute/ Square root deaton from mean are sgnfcantly dfferent or not. The resduals are obtaned from the usual analyss of arance. The test statstc s gen as ( r ) r ( d. d.. ) = = F = F(( ), r ( r ) = ( d d ) where d j j = j= = e e ; j. r dj j= d. = ; r r dj = j= d.. = and r j r j= e s the mean of the resduals of the th treatment. e s the j th resdual for the th plot, Ths test was modfed by Brown and Forsythe (974). In the modfed test, the absolute deaton s taken from the medan nstead of mean n order to make the test more robust. In the present nestgaton, the Bartlett's χ -test has been used for testng the homogenety of error arances when the dstrbuton of errors s normal and Leene test for non-normal errors. Remark : In a block desgn, t can easly be shown that the sum of resduals wthn a gen block s zero. Therefore, the resduals n a block of sze wll be same wth ther sgn reerse n order. As a consequence, q n Bartlett s test and numerator n Leene test statstc becomes zero for the data generated from experments conducted to compare only two treatments n a RCB desgn. Hence, the tests for homogenety of error arances cannot be used for the experments conducted to compare only two treatments n a RCB desgn. Inferences from such experments may be drawn usng Fsher-Behren t-test. Further, Bartlett s test cannot be used for the expermental stuatons where some of the treatments are sngly replcated. Remark : In a RCB desgn, t can easly be shown that the sum of resduals from a partcular treatment s zero. As a consequence, the denomnator of Leene test statstc s zero for the data generated from RCB desgns wth two replcatons. Therefore, Leene test cannot be used for testng the homogenety of error arances for the data generated from RCB desgns wth two replcatons. Remedal Measures In ths secton, we shall dscuss the remedal measures for non-normal and/or heterogeneous data n greater detals. III-7

8 Data transformaton s the most approprate remedal measure, n the stuaton where the arances are heterogeneous and are some functons of means. Wth ths technque, the orgnal data are conerted to a new scale resultng nto a new data set that s expected to satsfy the homogenety of arances. Because a common transformaton scale s appled to all obseratons, the comparate alues between treatments are not altered and comparson between them remans ald. Error parttonng s the remedal measure of heterogenety that usually occurs n experments, where, due to the nature of treatments tested some treatments hae errors that are substantally hgher (lower) than others. Here, we shall concentrate on those stuatons where character under study s non-normal and arances are heterogeneous. Dependng upon the functonal relatonshp between arances and means, sutable transformaton s adopted. The transformed arate should satsfy the followng:. The arances of the transformed arate should be unaffected by changes n the means. Ths s also called the arance stablzng transformaton.. It should be normally dstrbuted. 3. It should be one for whch effects are lnear and addte. 4. The transformed scale should be such for whch an arthmetc aerage from the sample s an effcent estmate of true mean. The followng are the three transformatons, whch are beng used most commonly, n bologcal research. a) Logarthmc Transformaton b) Square root Transformaton c) Arc Sne or Angular Transformaton a) Logarthmc Transformaton Ths transformaton s sutable for the data where the arance s proportonal to square of the mean or the coeffcent of araton (S.D./mean) s constant or where effects are multplcate. These condtons are generally found n the data that are whole numbers and coer a wde range of alues. Ths s usually the case when analyzng growth measurements such as trunk grth, length of extenson growth, weght of tree or number of nsects per plot, number of eggmass per plant or per unt area etc. For such stuatons, t s approprate to analyze log X nstead of actual data, X. When data set 3 noles small alues or zeros, log (X+), log( X + ) or log X + should be used 8 nstead of log X. Ths transformaton would make errors normal, when obseratons follow negate bnomal dstrbuton lke n the case of nsect counts. b) Square-Root Transformaton Ths transformaton s approprate for the data sets where the arance s proportonal to the mean. Here, the data conssts of small whole numbers, for example, data obtaned n countng rare eents, such as the number of nfested plants n a plot, the number of nsects caught n III-8

9 traps, number of weeds per plot, parthenocarpy n some aretes of mango. Ths data set generally follows the Posson dstrbuton and square root transformaton approxmates Posson to normal dstrbuton. For these stuatons, t s better to analyze X than that of X, the actual data. If X s 3 confrmed to small whole numbers then, X + or X + should be used nstead of X. 8 Ths transformaton s also approprate for the percentage data, where, the range s between 0 to 30% or between 70 and 00%. c) Arc Sne Transformaton Ths transformaton s approprate for the data on proportons,.e., data obtaned from a count and the data expressed as decmal fractons and percentages. The dstrbuton of percentages s bnomal and ths transformaton makes the dstrbuton normal. Snce the role of ths transformaton s not properly understood, there s a tendency to transform any percentage usng arc sne transformaton. But only that percentage data that are dered from count data, such as % barren tllers (whch s dered from the rato of the number of non-bearng tllers to the total number of tllers) should be transformed and not the percentage data such as % proten or % carbohydrates, %ntrogen, etc. whch are not dered from count data. For these stuatons, t s better to analyze sn ( X ) than that of X, the actual data. If the alue of X s 0%, t should be substtuted by and the alue of 00% by 4n number of unts upon whch the percentage data s based. 00, where n s the 4n It s nterestng to note here that not all percentage data need to be transformed and een f they do, arc sne transformaton s not the only transformaton possble. The followng rules may be useful n choosng the proper transformaton scale for percentage data dered from count data. Rule : The percentage data lyng wthn the range 30 to 70% s homogeneous and no transformaton s needed. Rule : For percentage data lyng wthn the range of ether 0 to 30% or 70 to 00%, but not both, the square root transformaton should be used. Rule 3: For percentage that do not follow the ranges specfed n Rule or Rule, the Arc Sne transformaton should be used. The other transformatons used are recprocal square root [, when arance s X proportonal to cube of mean], recprocal [ X, when arance s proportonal to fourth power of mean] and tangent hyperbolc transformaton. The transformaton dscussed aboe are a partcular case of the general famly of transformatons known as Box-Cox transformaton. III-9

10 d) Box-Cox Transformaton By now we know that f the relaton between the arance of obseratons and the mean s known then ths nformaton can be utlzed n selectng the form of the transformaton. We now elaborate on ths pont and show how t s possble to estmate the form of the requred transformaton from the data. The transformaton suggested by Box and Cox (964) s a power transformaton of the orgnal data. Let y ut be the obseraton pertanng to the u th plot; then the power transformaton mples that we use y ut s as * λ y ut = y ut. * λ The transformaton parameter λ n y ut = y ut may be estmated smultaneously wth the other model parameters (oerall mean and treatment effects) usng the method of maxmum lkelhood. The procedure conssts of performng, for the arous alues of λ, a standard analyss of arance on λ yut λ 0 (A) λy& ut λ ( λ) yut = y& ut ln yut λ = 0 N nu where y& ut = ln (/ n) ln yut. u= t= y& ut s the geometrc mean of the obseratons. The maxmum lkelhood estmate of λ s the alue for whch the error sum of squares, say SSE (λ), s mnmum. Notce that we cannot select the alue of λ by drectly comparng the error sum of squares from analyss of arance λ on y because for each alue of λ the error sum of squares s measured on a dfferent scale. Equaton (A) rescales the responses so that the error sums of squares are drectly comparable. Ths s a ery general transformaton and the commonly used transformatons follow as partcular cases. The partcular cases for dfferent alues of λ are gen below. λ Transformaton No Transformaton ½ Square Root 0 Log -/ Recprocal Square Root - Recprocal Remark 3: If any one of the obseratons s zero then the geometrc mean s undefned. In the expresson (A), geometrc mean s n denomnator so t s not possble to compute that expresson. For solng ths problem, we add a small quantty to each of the obseratons. III-0

11 Note: It should be emphaszed that transformaton, f needed, must take place rght at the begnnng of the analyss, all fttng of mssng plot alues, all adjustments by coarance etc. beng done wth the transformed arate and not wth the orgnal data. At the end, when the conclusons hae been reached, t s permssble to 're-transform' the results so as to present them n the orgnal unts of measurement, but ths s done only to render them more ntellgble. As a result of ths transformaton followed by back transformaton, the means wll rather be dfferent from those that would hae been obtaned from the orgnal data. A smple example s that wthout transformaton, the mean of the numbers, 4, 9, 6 and 5 s. Suppose a square root transformaton s used to ge,, 3, 4 and 5, the mean s now 3, whch after back- transformaton ges 9. Usually the dfference wll not be so great because data do not usually ary as much as those gen, but logarthmc and square root transformaton always lead to a reducton of the mean, just as angles of equal formaton usually lead to ts mong away from the central alue of 50%. Howeer, n practce, computng treatment means from orgnal data s more frequently used because of ts smplcty, but ths may change the order of rankng of conerted means for comparson. Although transformatons make possble a ald analyss, they can be ery awkward. For example, although a sgnfcant dfference can be worked out n the usual way for means of the transformed data, none can be worked out for the treatment means after back transformaton. Non-parametrc tests n the Analyss of Expermental Data When the data remans non-normal and/or heterogeneous een after transformaton, a recourse s made to non-parametrc test procedures. A lot of attenton s beng pad to deelop non-parametrc tests for analyss of expermental data. Most of these non-parametrc test procedures are based on rank statstc. The rank statstc has been used n deelopment of these tests as the statstc based on ranks s. dstrbuton free. easy to calculate and 3. smple to explan and understand. Another reason for use of rank statstc s due to the well known result that the aerage rank approaches normalty quckly as n (number of obseratons) ncreases, under the rather general condtons, whle the same mght not be true for the orgnal data {see e.g. Conoer and Iman (976, 98)}. The non-parametrc test procedures aalable n lterature coer completely randomzed desgns, randomzed complete block desgns, balanced ncomplete block desgns, desgn for boassays, splt plot desgns, cross-oer desgns and so on. For an excellent and elaborate dscussons on non-parametrc tests n the analyss of expermental data, one may refer to Segel and Castellan Jr. (988), Deshpande, Gore and Shanubhogue (995), Sen (996), and Hollander and Wolfe (999). Kruskal-Walls Test can be used for the analyss of data from completely randomzed desgns. Skllngs and Mack Test helps n analyzng the data from a general block desgn. Fredman Test and Durbn Test are partcular cases of ths test. Fredman Test s used for the analyss of data from randomzed complete block desgns and Durbn test for the analyss of data from balanced ncomplete block desgns. III-

12 Some examples of testng the assumptons of normalty and homogenety of errors and remedal measures are dscussed n the Appendx. Some Useful References Anderson,V.L. and McLean,R.A.(974). Desgn of Experments: A realstc approach. Marcel Dekker Inc., New York. Bartlett,M.S.(947). The use of transformaton. Bometrcs, 3, Brown, M. B. and Forsythe, A.B. (974). Robust tests for the equalty of arances. Journal of the Amercan Statstcal Assocaton, 69, Box, G.E.P., (953). Non-normalty and tests on arances. Bometrka, 40 ( & ), Box,G.E.P. and Cox,D.R.(964). An analyss of transformaton. J. Roy. Stastst. Soc. B, 6, -5. Conoer, W.J. and Iman, R.L. (976). In Some Alternate Procedure Usng Ranks for the Analyss of Expermental Desgns. Comm. Statst.: Theory & Methods, A5(4), Conoer,W.J. and Iman,R.L.(98). Rank transfomatons as a brdge between parametrc and non-parametrc statstcs (wth dscusson). Amercan Statstcan, 35, D Agostno, R.B. and Stephens, M. A.(986). Goodness-of-ft Technques. Marcel Dekkar, Inc., New York. Deshpande, J.V., Gore, A.P. and Shanubhogue, A. (995). Statstcal Analyss of Non-normal Data. Wley Eastern Lmted, New Delh. Dean, A and Voss, D (999). Desgn and Analyss of Experments. Sprnger, New York. Dolby,J.L.(963). A quck method for choosng a transformaton. Technometrcs, 5, Draper,N.R. and Hunter,W.G.(969). Transformatons: Some examples rested. Technometrcs,, Hollander, M. and Wolfe, D.A. (999). Nonparametrc Statstcal Methods, nd Ed, John Wley, New York. Leene, H., (960). Robust Tests for Equalty of Varances. I Olkn, ed., Contrbutons to Probablty and Statstcs, Palo Alto, Calf.: Stanford Unersty Press, Royston, P. (99), Approxmatng the Shapro-Wlk W-Test for non-normalty. Statstcs and Computng,, 7-9. Shapro, S. S. and Wlk, M. B.. (965). An analyss of arances test for normalty (Complete Samples). Bometrka, 5, 59-6 Segel, S. and Catellan, N.J. Jr. (988). Non parametrc statstcs for the Behaoural Scences, nd Ed., McGraw Hll, New York. Tukey, J.W.(949). One degree of freedom for non-addtty. Bometrcs, 5, 3-4. III-

13 Appendx Example : Suppose an entomologst s nterested n determnng whether four dfferent knds of traps caught equalent nsects when appled to same feld. Each of the traps s used sx tmes on the feld and resultng data (number of nsects per hour) are as shown below alongwth mean, arance and range. Treatment Replcaton Varance Range I II III IV V VI Y A B C D A scatter plot of mean and arance and mean ersus range are gen as follows: S 80 Range Varance Both plots ndcate that arances are heterogeneous and arance s proportonal to mean. Obtan the resduals for testng the normalty and homogenety of error terms. The resduals obtaned are gen below: Treatment Replcaton Varance I II III IV V VI S A B C D III-3

14 Normalty of error terms n SPSS: Analyze Descrpte Statstcs Explore (Enter the arable whose normalty s to be tested n Dependent Lst) Clck on the opton Plots Check the opton Normalty Plots wth tests Shapro-Wlk Test Kolmogro-Smrno Test Statstc p-alue Statstc p-alue (SW) (KS) The errors were found to be normally dstrbuted. Therefore, homogenety of error arances was tested usng Bartlett's test. It s descrbed n the sequel. 5 Pooled Varance ( ) ( ) S p = 0 = [ log log log 650.0] q = 0 log log = c = + = χ = Snce χ 0.05,3 = 7. 8, therefore, we reject the null hypothess and conclude that the arances are unequal. The S Y. are 5.77, 5.3, 3.43 and 5.43, ndcatng that arance s proportonal to mean. Therefore, square root transformaton should be used. After applcaton of square root transformaton, the resduals are Treatment Replcaton Varance I II III IV V VI S A B C D Normalty of error terms on the transformed data: Shapro-Wlk Test Kolmogro-Smrno Test Statstc p-alue Statstc p-alue (SW) (KS) III-4

15 The errors reman normally dstrbuted after transformaton. The results of homogenety of error arances usng Bartlett's test are Bartlett's Test (normal dstrbuton): Test statstc = 0.89, p-alue = 0.88 Hence, we conclude that the errors are normally dstrbuted and hae a constant arance after transformaton. The results of analyss of arance wth orgnal and transformed data are gen n the sequel. ANOVA: Orgnal Data Source DF Seq SS Adj. SS Square F (F-calc) p(pr>f) Replcaton Treatment Error Total R-Square Root MSE 9.80% 9.37 Tukey Smultaneous Tests for All Parwse Treatment Comparsons ANOVA: Transformed Data Source DF Seq SS Adj. SS Square F (F-calc) p(pr>f) Replcaton Treatment Error Total R-Square Root MSE 93.99%.89 Tukey Smultaneous Tests for All Parwse Treatment Comparsons Wth transformed data treatments and are sgnfcantly dfferent whereas wth orgnal data, they were not. III-5

16 Example : A aretal tral on Rapeseed-Mustard was conducted at Fazabad wth aretes usng a randomzed complete block desgn wth 3 replcatons. The expermental data (Yeld n kg/ha ) obtaned from the aboe experment s Treatments Replcatons R R R3 MCN MCN MCN MCN MCN MCN MCN MCN MCN MCN MCN The analyss of arance of the orgnal data s gen as ANOVA: Orgnal Data Sources DF SS MS F Prob. >F Replcaton Treatment Error Total R-Square CV RMSE Yld Normalty of error terms was tested, the results are gen as Shapro-Wlk Test Kolmogro-Smrno Test Statstc p-alue Statstc p-alue (SW) (KS) >0.500 Snce the data s normal, therefore, Bartlett s test s used for testng the homogenety of error arances. The results are gen as Bartlett s Test Test Statstc : 0.77 P-Value : The errors were found to be heterogeneous. Therefore, we can conclude that the data s heterogeneous and normal. Therefore, Box-Cox transformaton was used as a remedal measure. In the sequel we descrbe the results of the Box-Cox transformaton. III-6

17 For ths we transform the data by aryng λ from -0 to +0 wth an ncrement of 0.0. The error sum of squares are computed for each alue of λ. The alue of λ wth mnmum error sum of squares s used for transformaton gen n (A). The mnmum alue SSE s obtaned for λ =.38. Therefore, recprocal transformaton was used. The assumptons of normalty and homogenety of errors are agan tested usng the transformed data. Normalty of error terms was tested, the results are gen as Shapro-Wlk Test Kolmogro-Smrno Test Statstc p-alue Statstc p-alue (SW) (KS) >0.500 Snce the data s normal, therefore, Bartlett s Test s used for testng the homogenety of error arances. The results are gen as Bartlett's Test (normal dstrbuton) Test Statstc : 5.75 P-Value : The transformed obseratons were found to be normal and homogeneous Therefore, ANOVA was performed on the transformed data. The results obtaned are: ANOVA: Transformed Data Sources DF SS MS F Prob. >F Replcaton E E Treatment E E Error E E3 Total E5 R-Square CV RMSE Transformed Yld We can see that there s no change n the results of sgnfcance of treatment and replcaton effects. Howeer, the transformed data satsfed the assumptons of ANOVA. III-7