Amercan Journal of Mathematcs and Statstcs 07, 7(4): 69-78 DOI: 0.593/j.ajms.070704.05 ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach Samsad Jahan Department of Arts and Scences, Ahsanullah Unversty of Scence and Tehcnology, Dhaka, Bangladesh Abstract Ths paper s an attempt to observe the extent of effect on the power of analyss of varance test to volatons of assumptons.e. normalty assumpton of the error of multple lnear regresson model. The error of the model s consdered as g-and-k dstrbuton because of the fact that t has shown a consderable ablty to ft to data and faclty to use n smulaton studes. The strength of ANOVA s evaluated by observng the power functon of F-test for dfferent combnaton of g (skewness) and k (kurtoss) parameter. From the smulaton results t s observed that the performance of ANOVA s seen to be mmensely affected n presence of excess kurtoss and for small samples (say, n<00). Skewness parameter has not much effect on the power of the test under non-normal stuaton. The effect of sample sze on the exstng test for multple regresson models s also observed here n ths paper under varous non normal stuatons. Keywords The g-and-k dstrbuton, ANOVA-test, Multple lnear regresson model,. Introducton Classcal statstcal procedures are desgned n such a way that they can produce best result when underlyng assumptons on the data s populaton dstrbutons are true. But n practce, we often have to deal wth the stuaton when actual stuatons depart from the deal stuaton descrbed by such assumptons, and t has been proved that the performance of many statstcal technques suffers badly when the real stuaton departs from deal stuaton. The performance of ANOVA test also suffers badly when the valdty of normalty assumpton does not hold. Generally the extent of devaton from normalty s an mportant factor that supervses the strength (weakness) of the ANOVA procedure. The man concern of ths study s to observe the performance of conventonal ANOVA test under varous nonnormal stuatons for multple lnear regresson models. Smultaneous measure of the skewness and kurtoss parameter has been consdered as the measure of extent of non-normalty. The skewness parameter measure the degree of dstorton or devaton from normalty and the kurtoss measures the peakedness or thckness of the tal of the dstrbuton. In ths manuscrpt, the smplest possble multple lnear regresson model.e. three varable multple regresson model wth one dependen t varable and two * Correspondng author: samsad.jahan@gmal.com (Samsad Jahan) Publshed onlne at http://journal.sapub.org/ajms Copyrght 07 Scentfc & Academc Publshng. All Rghts Reserved explanatory varable s consdered and the extent of effect of devaton from normalty s measured by consderng the model error from g-and-k dstrbuton. A number of studes on robustness and tests of normalty shows many contrbutons from the most outstandng theorsts and practtoners of statstcs. The effect of non-normalty on the power of analyss of varance test has been studed by Srvastava (959) by nvestgatng the non-central dstrbuton of the varance rato. Box and Watson (96) demonstrated the overrdng nfluence whch the numercal values of regresson varables have n decdng senstvty to non-normalty and also showed the essental nature of ths dependency. Tku (97) calculated the values of the power of the F test employed n analyss of varance under non-normal stuatons and compared wth normaltheory values of the power. Kanj (976) dscussed about smulaton methods for calculatng power values n the case of non-normal errors. He used Erlangan and contamnated normal dstrbuton as an example of non-normal error dstrbuton. MacGllvray and Balanda (988) studed on skewness and kurtoss, and consdered the concept of ant-skewness to use t as a tool to dscuss the dea of kurtoss n asymmetrc unvarate dstrbutons. Mukhter and Shubhas (996) nvestgated the robustness to nonnormalty of the null dstrbuton of the standard F-tests for regresson coeffcents n lnear regresson models. Assumng the errors to be nonnormal wth fnte moments, the null dstrbuton of the F-statstc s derved. Khan and Rayner (00) made an attempt to study the effects of the strong assumptons requred for ANOVA and also nvestgated the effects of the
70 Samsad Jahan: ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach departure from the normalty of error on the power functon by usng g-and-k dstrbuton. Khan and Rayner (003) nvestgated the effect of devaton from the normal dstrbuton assumpton by consderng the power of two many sample locaton test procedure: ANOVA (parametrc) and Kruskal-Wals (non-parametrc). Rasch and Gulard (004) presented some results of a systematc research of robustness of statstcal procedures aganst non-normalty. Serln and Harwell (004) observed some more powerful tests of predctor subsets n regresson analyss under non-normalty. A Monte Carlo study of tests of predctor subsets n multple regresson analyss ndcates that varous nonparametrc tests show greater power than the F test for skewed and heavy-taled data. These nonparametrc tests can be computed wth avalable software. (PsycINFO Database Record (c) 0 APA, all rghts reserved) Yanaghara (007) presented the condtons for robustness to non-normalty on three test statstcs for a general multvarate lnear hypothess, whch were proposed under the normal assumpton n a generalzed multvarate analyss of varance (GMANOVA). Mortaza et al. (007) provded a study on partal F-test for multple lnear regresson models. They showed a power comparsons between the partal F tests and new test to assess when the new tests are more or less powerful than the partal F tests. Schmder et al (00) provded emprcal evdence to the robustness of the analyss of varance (ANOVA) concernng volaton of the normalty assumpton s presented by means of Monte Carlo methods. Khan and Hossan (00) suggested a numercal lkelhood rato test for testng the locaton equalty of several populatons under quantle functon dstrbuton approach. Lantz B. (0) nvestgated the relatonshp between populaton non-normalty and sample non-normalty wth respect to the performance of the ANOVA, Brown-Forsythe test, Welch test, and Kruskal-Walls test when used wth dfferent dstrbutons, sample szes, and effect szes. The overall concluson s that the Kruskal-Walls test s consderably less senstve to the degree of sample normalty when populatons are dstnctly non-normal and should therefore be the prmary tool used to compare locatons when t s known that populatons are not at least approxmately normal. Jahan and khan (0) demonstrated the extent of effect of non-normalty on power of the t-test for smple lnear regresson model usng g- andk dstrbuton. It s clear that a wde range of studes have been made on the non-normalty of the model error but so far no studes has been conducted to see what extent of devatons from normalty causes what extent of effect on the sze and power of ANOVA-test for multple lnear regresson model. Ths paper contans a power curve study to examne the extent of effect on sze and power of ANOVA test for multple lnear regresson models wth two explanatory varables on a wde varety of normal and non-normal stuaton and for dfferent sample szes. The power of ANOVA test for multple lnear regresson models s measured numercally and shown graphcally.. Multple Lnear Regresson Model The Three -Varable Model The multple lnear regresson models wth two explanatory varables can be wrtten as follows: Y = β + β X + β X + ε ; 0 =,,..., n (.) Where, Y s the dependent varable, X and X are explanatory varables, ε s the stochastc dsturbance term, and s the th observaton. β s the ntercept term, t 0 gves the mean or average effect on Y of all the varable excluded from the model, although ts mechancal nterpretaton s the average value of Y when X and X are set equal to zero. The coeffcents and β are called partal regresson coeffcents. β measures the change n the mean value of Y, E (Y ), per unt change n X, holdng the value of X constant. Lkewse, measures the change n the mean value of Y per unt change n X, holdng the value of X constant. The coeffcents β and β are called partal regresson coeffcents. β measures the change n the mean value of Y, E (Y ) X constant., per unt change n 3. The g-and-k Dstrbuton X, holdng the value of The g-and-k dstrbuton (MacGlvray and Canon) can be defned n terms of ts quantle functon as: gz e u k QX( u \ A, B, g, k) = A + Bzu( + c )( + zu), gz + e u (3.) Where, A and B >0 are the locaton and scale parameters respectvely, g measures skewness n the dstrbuton, k > measures kurtoss (n general sense of peakness/taledness) n the dstrbuton and z u = ϕ ( u ) s the u th quantle of a standard normal varate, and c s a constant chosen to help produce proper dstrbutons. It can be clearly observed that for g = k = 0, the quantle functon n (3.) s just the quantle functon of a standard normal varate. The sgn of the skewness parameter ndcates the drecton of skewness; g < 0 ndcates the dstrbuton s skewed to the left, and g > 0 ndcates skewness to the rght. Increasng/decreasng the unsgned value of ncreases/ decreases the skewness n the ndcated drecton. When g = 0 the dstrbuton s symmetrc.
Amercan Journal of Mathematcs and Statstcs 07, 7(4): 69-78 7 The kurtoss parameter k, for the g-and-k dstrbuton, behaves smlarly. Increasng k ncreases the level of kurtoss and vce versa. The value k = 0 corresponds to no extra kurtoss added to the standard normal base dstrbuton. However, ths dstrbuton can represent less kurtoss than the normal dstrbuton, as k > can negatve values. If curves wth more kurtoss requred then base dstrbuton wth less kurtoss than standardzed normal dstrbuton can be used. For these dstrbutons c s the value of overall (MacGlvray). For an arbtrary dstrbuton, theoretcally the overall asymmetry can be as large as one, so t would appear that for c <, data or dstrbuton could occur wth skewness that cannot be matched by these dstrbutons. However for g 0, the larger the value chosen for c, the more restrctons on k are requred to produce a completely proper dstrbuton. Real data seldom produce overall asymmetry values greater than 0.8 (MacGlvray and Canon). The value of c s taken as 0.83 throughout ths paper. To examne extent of the effect of dfferent level of non-normalty on the test of multple lnear regresson models, t s consdered that the random error belongs to the g -and- k dstrbuton. Fgure. Densty curves of g-and-k dstrbuton for dfferent combnaton of g and k
7 Samsad Jahan: ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach 4. The Analyss of Varance Approach for Testng the overall Sgnfcance of an observed Multple Regresson: The F -test Analyss of varance (ANOVA) s a popular and wdely used technque n the feld of statstcs. Besdes beng the approprate procedure for testng the equalty of several Table. ANOVA table for the Three-varable regresson means, the ANOVA has a much wder applcatons. The objectve of the ANOVA procedure le manly n estmatng and testng hypotheses about the treatment effect parameters. The usual tt test cannot be used to test the jont hypothess that the true partal slope coeffcents are zero smultaneously. However ths jont hypothess can be tested by the analyss of varance technque whch can be demonstrated as follows: Sources of varaton (S.V) Degrees of Freedom (D.F) Sum of Square Mean Sum of Square ˆ Due to regresson (ESS) ˆ ˆ β ˆ β YX + β YX YX + β YX Due to resdual (RSS) n 3 Total n Y ˆ ε ε ˆ n 3 Now under the assumpton of normal dstrbuton for ε and for the null hypothess H : β = β = 0 0 H : At least one β s not equal to zero; =,. Then the test statstc ESS / df F = (4.) RSS / df s dstrbuted as the F dstrbuton wth and n 3 df. Therefore, the F value of (4.) provdes a test of null hypothess that the true slope coeffcents are smultaneously zero. The null hypothess H 0 can be rejected f the F value computed from (4.) exceeds the crtcal F value from the F table at α percent level of sgnfcance, H cannot be rejected. otherwse 0 5. Smulaton Study In ths paper, multple lnear regresson models wth two explanatory varables s consdered. As t s known that the error term ε of multple lnear regresson models are normally dstrbuted but here n ths paper, the random error term ε s assumed to follow the g- and -k dstrbuton. The extent of non-normalty on the sze and power of ANOVA test s observed by varyng the skewness and the kurtoss parameter of the g - and -kk dstrbuton. Usng the g and-k dstrbuton allows us to quantfy how much the data depart from normalty n terms of the values chosen for the g (skewness) and k(kurtoss) parameters. For g = k = 0, the quantle functon for g -and- kk dstrbuton s just the quantle functon of a normal varate. To observe the power of the tests, expresson for the power curve s requred. However, n practce, to obtan analytc expressons for these power functons s mpractcal. Instead, a smulaton s conducted to estmate these power functon for varous combnatons of the g and k parameter values for the error dstrbuton from the g -and- kk dstrbuton. Whle smulatng for the test, A s taken to be the locaton whch s the medan n case of g -and- kk dstrbuton but for non-normal stuatons the mean of the dstrbuton moves away from A whch actually s the medan of the dstrbuton. Ths departure vares as the values g and k of vary. The values of g and k are taken as g -: and k -.5, 0,.5, and. At frst, the effect of non-normalty on the sze of the F test s observed. For smulatng the sze of F - test the explanatory varables x and x are generated from unform dstrbuton and the random error ε from g-and-k dstrbuton wth locaton and scale parameters A=0 and B=, respectvely. Usng statstcal software R data are generated for sample sze 0, 30 and 00, and the followng hypothess s tested. H : β = β = 0. 0 Aganst the alternatve H : At least one β s not equal to zero; =,. To determne the sze of the test, data are generated under the null hypothess and the test s repeated 5,000 tmes. The total number of tmes the hypothess s rejected s dvded by 5,000; tests are carred out usng.5 percent level of sgnfcance.
Amercan Journal of Mathematcs and Statstcs 07, 7(4): 69-78 73 To compute the power of the F test, the explanatory varables x and x are consdered from unform dstrbuton and the random error ε from g-and-k dstrbuton wth locaton parameter A = 0 and scale parameter B =. The value of c s consdered as 0.83. To smulate power, the followng hypothess s tested H : β = β = 0. 0 Aganst the alternatve H : At least one β s not equal to zero; =,. Data are generated usng (-,-.5,-,-.5,0,.5,,.5,) and β (-,-.5,-,-.5,0,.5,,.5,) and the test procedures are repeated 5000 tmes for each par of (, ) (-,-),(-,-.5), (,.5),(,). Frstly the number of rejectons of the test out of the 5000 tmes s determned for each par of ( β, β ) n the mentoned set and the total number of rejectons are dvded by 5000, wth the level of sgnfcance α = 5. 6. Sze of F-test Frst, the effect of non-normalty on the sze of the F test s consdered. For smulatng the sze of F - test the explanatory varables x and x s generated from the unform dstrbuton and the random error ε from g-and-k dstrbuton wth locaton and scale parameters A = 0 and B =, respectvely. Data are generated for sample sze 0, 30 and 00 usng statstcal software R and the followng hypothess s tested: H : β = β = 0. 0 Aganst the alternatve H : At least one β s not equal to zero; =,. Table. Sze of ANOVA for dfferent combnatons of (g, k) wth varyng sample szes To determne the sze of the test, data are generated under the null hypothess and repeat the test 5,000 tmes and dvde the total number of tmes the hypothess s rejected by 5,000; tests are carred out usng.5 percent level of sgnfcance. The sze of ANOVA for dfferent combnatons of (g, k) are presented n Table. In table, some smulaton results are presented to see the effect of dfferent level of non-normalty on the sze of F-test. F-test s sze robust under normal stuaton, but under non-normal stuaton there s a lttle effect on the sze of the test. For sample sze 0 and 30, t s seen that skewness parameter has a very lttle effect and the kurtoss parameter has moderate effect on the sze of F-test. For sample sze 00, even n the case of non-normal stuaton, F -test s almost sze robust. 7. of F-test To compute the power of the F test, frstly the explanatory varables x and x are generated from the unform dstrbuton and the random error ε s consdered from g-and-k dstrbuton wth locaton parameter A = 0 and scale parameter B =. The value of c s taken to be 0.83 throughout the paper. To smulate power, the followng hypothess s consdered H : β = β = 0. 0 H : β s not equal to zero; =,. To see how the power dffers as the values of g and k change, the power for specfed values of g and k s plotted to get the power curve for ANOVA test wth sample szes n= 0, 30 and 00. To get smooth power curve, many ponts for dfferent combnatons of g and k are used. For each combnaton we get power. The process s repeated where for each pont 5,000 smulatons are run. Fgure () through (7) shows the power curves for dfferent combnaton of (g,k) for sample sze n = 0, 30 and 00. g 0.5.0.5.0 0.5.0 k 0.3 0.5 0.8.0-0. -0.3-0.5 0.5.0 Sample sze 0 5 3 9 86 38 58 56 6 04 5 Sample sze 30 4 8 84 08 7 68 6 3 6 4 86 94 6 Sample sze 00 6 38 46 34 3 38 6 3 38 94 4 4 8. Dscusson of Results From the fgure () to fgure (7) t s seen that the powers of the test s badly affected by the sample sze and kurtoss parameter. The smulaton results can be summarzed n the followng ways: ) In fgure, the skewness parameter s consdered to be fxed at g=0 but the kurtoss parameter s vared from k=0 to k=. It s apparent that as the kurtoss parameter ncreases n postve drecton power of the test s vastly decreased than that of normal data. The effect of sample sze on the power of the test s also observed n ths paper. It s found that the rate of decreasng power n presence of excess kurtoss for small sample (n=0, 30) s hgher than that of larger sample sze say n=00.
74 Samsad Jahan: ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach ) In fgure 3, the kurtoss parameter s fxed at k=0 but the skewness parameter s vared from g=0 to g=. It s examned that power of the test has not much effect when the skewness parameter s vared n postve drecton. As the sample sze ncreases power of the test seems to almost robust although the skewness parameter s ncreased. ) In fgure 4 and 5, the combnaton of (g=0, k=0), (g=, k=0), (g=0, k=), (g=, k=) for sample sze 0 and 30 s consdered and t s notced that varyng the kurtoss parameter n postve drecton has more effect n decreasng the power than that of varyng the skewness parameter. v) In fgure 6 and 7, the effect of ncreasng the kurtoss parameter n negatve drecton s observed. The combnaton of (g=0, k=0), (g=0, k=-.3), (g=0, k=-.5) for sample sze 0 and 30 s shown. At frst glance t may seems that varyng the kurtoss parameter n negatve drecton gves better power but f a close attenton s gven at the sze of the test t s clearly seen that the sze of the test s ncreased. v) From fgure to 7, t s apparent that the power of ANOVA test s decreased more for small sample sze (n=0, 30) than that of large sample sze (n=00) under non-normal stuaton. g=0, k=0 g=0, k=0.5 g=0, k=0.8 g=0,k=.0 a) n=0 b) n=30 c) n=00 Fgure. curve of ANOVA for fxed value of g and varyng Kurtoss parameter for (a) sample sze n=0, (b) sample sze n=30, (c) sample sze n=00
Amercan Journal of Mathematcs and Statstcs 07, 7(4): 69-78 75 g=0, k=0 g=,k=0 g=.5, k=0 g=, k=0 a) n=0 b) n=30 c) n=00 Fgure 3. curve of ANOVA for fxed value of kurtoss and varyng skewness parameter for (a) sample sze n=0, (b) sample sze n=30, (c) sample sze n=00
76 Samsad Jahan: ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach a) g=0, k=0 b) g=, k=0 c) g=0, k= d) g=, k= Fgure 4. curves of ANOVA for a) g=0, k=0, b) g=,k=0, c)g=0,k=, d) g=, k= for sample sze 0 a) g=0, k=0 b) g=, k=0 c) g=0, k= d) g=, k= Fgure 5. curves of ANOVA for a) g=0, k=0, b) g=,k=0, c)g=0,k=, d) g=, k= for sample sze 30 a) g=0, k=0 b) g=0, k=-0.3 c) g=0, k=-0.5 Fgure 6. curves of ANOVA for a) g=0, k=0,b) g=0, k=-0.3, c) g=0, k=-0.5 for sample sze 0
Amercan Journal of Mathematcs and Statstcs 07, 7(4): 69-78 77 a) g=0, k=0 b) g=0, k=-0.3 c) g=0, k=-0.5 Fgure 7. curves of ANOVA for a) g=0, k=0, b) g=0, k=-0.3, c) g=0, k=-0.5 for sample sze 30 9. A Real Lfe Example (Wolf Rver Polluton data) In real lfe applcatons, sometmes t may happen that the data do not follow the normal dstrbuton. The examner needs to dentfy the amount of devaton from normalty and take necessary acton to mnmze the nonstandard condtons. Khan and Hossan (00) examned the wolf Rver Polluton data to nvestgate how the ANOVA and Kruskal Walls test perform. Ther focus was on hexachlorobenzene (HCB) concentraton data (n nanograms per lter) that came out wth some features of non-normalty. The ANOVA test was carred out for testng the equalty of average HCB concentraton for dfferent depth although the assumptons were not fully satsfed. The ANOVA test dd not provde any strong evdence for the hypothess that the average HCB concentraton for dfferent depths are dfferent, producng a p-value of 0.65. The Kruskal Walls test produces almost smlar p-value of 64 lke ANOVA. Khan and Hossan (00) also ftted the data wth g-and-k dstrbuton to test whether the data lack normalty. MLE was used to estmate the dstrbuton parameters A,B,g, and k to dentfy the amount of devaton from normalty n terms of skewness and kurtoss. The estmated value of g ˆ = 0.40 and k ˆ = 0. 0668 presented the data to be slghtly suffered from asymmetry and lght taledness. 0. Conclusons From the above dscusson, the followng concludng remarks can be made: ) As the kurtoss parameter ncreases n postve drecton of ANOVA test for multple lnear regresson model decreases mmensely than that of the normal data. ) The skewness parameter seems to have not much effect on the power of ANOVA under non normal stuaton. ) Kurtoss Parameter has more effect n decreasng power of ANOVA test than that of the skewness parameter under non normal stuaton. v) Small sample szes have more effect n reducng power than that of large sample szes under non-normal stuaton. v) Negatve kurtoss gves better power and ncreases the sze of the test. REFERENCES [] Srvastava, A. B. L. (959). Effect of Non-normalty on the of the Analyss of Varance Test. Bometrka, 46, No./ 4-. [] Box, G.E.P. and Watson, G.S. (96). Robustness to non-normalty of regresson Tests. Bometrka, 49, Issue /, 93-06. DOI: 0.093/bomet/49.-.93. [3] Tku, M.L. (97). Functon of the F-Test under Non-Normal Stuatons. Journal of Amercan Statstcal Assocaton, 66(336), 93-96. [4] Kanj, G.K. (976). Effect of non-normalty on the power n analyss of varance: A smulaton study. Internatonal Journal of Mathematcal Educaton n Scence and Technology, 7(),55-60. DOI: 0.080/0073976007004. [5] Mukhtar, M. Al and Subhash, C. Sharma, Volume 7, Issues, Robustness to nonnormalty of regresson F-tests, Journal of Econometrcs, March Aprl 996, Pages 75 05. [6] MAcGIllvray, H.L and BAlanda K.P. (988), The relatonshp between skewness and Kurtoss, Australan Jounal of Statstcs, 30:39-337. [7] Khan, A. and Rayner, G.D. (00). ANOVA Procedures wth Quantle-functon Error Dstrbutons. Journal of Appled Mathematcs and Decson Scences, 5(), -9. [8] Khan, A. and Rayner, G.D. (003a). Robustness to non-normalty of common tests for many sample locaton problem. Journal of Appled Mathematcs and Decson Scences, 7(4), 87-07. [9] Rasch, D. and Gulard, V. (004). The robustness of parametrc Statstcal Methods. Psychology Scence, 46(), 75-08. [0] Serln, Ronald C.; Harwell, Mchael R. (004). More ful Tests of Predctor Subsets n Regresson Analyss
78 Samsad Jahan: ANOVA Procedures for Multple Lnear Regresson Model wth Non-normal Error Dstrbuton: A Quantle Functon Dstrbuton Approach under Nonnormalty. Psychologcal Methods, Vol 9(4), Dec 004, 49-509. do: 0.037/08-989X.9.4.49. [] Yanaghara, H. (007). Condtons for robustness to Nonnormalty on Test Statstcs n a GMANOVA Model. J.Japan. Statst., Soc.37 (), 35-55. [] Mortaza Jamshdan, Robert I. Jennrch and We Lu (007). A study of partal F tests for multple lnear regresson models. Computatonal Statstcs & Data Analyss 5 (007) 669 684. [3] Schmder, Emanuel; Zegler, Matthas; Danay, Erk; Beyer, Luz; Bühner, Markus (00), Is t really robust? Renvestgatng the robustness of ANOVA aganst volatons of the normal dstrbuton assumpton. Methodology: European Journal of Research Methods for the Behavoral and Socal Scences, Vol 6(4), 47-5. do: 0.07/64-4/a00006. [4] Khan, A. and Hossan, S.S. (00), Many Sample locaton Test wth Quantle Functon Error Dstrbutons: An almost robust Test, J. Stat. & Appl. Vol.5, No., 39-60. [5] Jahan, S. and Khan, A. (0), of t-test for smple Lnear regresson Model wth Nonnormal Error Dstrbuton: A Quantle Functon Dstrbuton Approach, Journal of Scentfc Research, Volume 4 No. 3, Page 609-6. [6] MacGllvray, H. L. and Cannon, W. H. (Preprnt, 00). Generalzatons of the g-and-h dstrbutons and ther uses. [7] Neter, John, Wsserman, W. and Kutner, Mchael H. (983). Appled Lnear Regresson Models, Publsher: Rchard D. Irwn, INC. [8] Smyth, G. (00). [Webdocument], http://www.statsc.org/d ata/general/wolfrve.html, [Accessed 04/0/07].