America Joural of Theoretical ad Applied Statistics 07; 6(5-: 5-6 http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.s.0706050.8 ISSN: 36-8999 (Prit; ISSN: 36-9006 (Olie Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio Ukpomwa H. Nosakhare, Ajibade F. Bright Departmet of Geeral Studies, Mathematics ad Computer Sciece Uit, Petroleum Traiig Istitute, Warri, Nigeria Email address: equalright_bright@yahoo.com (U. H. Nosakhare, Nosayk@gmail.com (A. F. Bright To cite this article: Ukpomwa H. Nosakhare, Ajibade F. Bright. Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio. America Joural of Theoretical ad Applied Statistics. Special Issue: Statistical Distributios ad Modelig i Applied Mathematics. Vol. 6, No. 5-, 07, pp. 5-6. doi: 0.648/j.ajtas.s.0706050.8 Received: February 4, 07; Accepted: March, 07; Published: Jue 9, 07 Abstract: This paper examies the sesitivity of ie ormality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D Agostio, Shapiro-Wilk, Shapiro-Fracia, Rya-Joier, Lilliefors ad Aderso Darligs test statistics, with a view to determiig the effectiveess of the techiques to accurately determie whether a set of data is from ormal distributio or ot. Simulated data of sizes 5, 0,, 00 is used for the study ad each test is repeated 00 times for icreased reliability. Data from ormal distributios (N (, ad N (0, ad o-ormal distributios (asymmetric ad symmetric distributios: Weibull, Chi-Square, Cauchy ad t-distributios are simulated ad tested for ormality usig the ie ormality test statistics. To esure uiformity of results, oe statistical software is used i all the data computatios to elimiate variatios due to statistical software. The error rate of each of the test statistic is computed; the error rate for the ormal distributio is the type I error ad that for o-ormal distributio is type II error. Power of test is computed for the o-ormal distributios ad use to determie the stregth of the methods. The rakig of the ie ormality test statistics i order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor s, D Agostio, Rya-Joier, Shapiro-Fracia, Shapiro-Wilk, W/S, Jarque-Bera ad Aderso-Darlig test statistics while for large sample sizes, we have; D Agostio, Rya-Joier, Shapiro-Fracia, Jarque-Bera, Aderso-Darlig, Lilliefor s, Adjusted Jarque-Bera, Shapiro-Wilk ad W/S test statistics. Hece, oly D Agostio test statistic is classified as Uiformly Most Powerful sice it is effective for both small ad large sample sizes. Other methods are Locally Most Powerful. Shapiro-Fracia, a improvemet of Shapiro-Wilk is more sesitive for both small ad large samples, hece should replace Shapiro-Wilk while the Adjsted Jarque-Bera ad the Jarque-Bera should both be retaied for small ad large samples respectively. Keywords: Error Rate, Power-of-Test, Normality, Sesitivity, ad Simulatio. Itroductio Normality tests are used to determie whether a set of data could be modeled by a ormal distributio or to ivestigate whether a set of observatios is ormally distributed. Statistical methods ca be classified ito two; parametric ad oparametric methods. Parametric methods such as Studett test ad Aalysis of Variace require that the set of data to be used is ormally distributed, otherwise, the tests become ivalid. Noparametric tests such as Friedma test, Kruskal- Wallis test, Ma-whitey test, etc require o assumptio o the distributio of the data. Accordig to Noradiah ad Yap (0, ormal distributio is the uderlyig assumptio of may statistical procedures, hece, the eed for ormality test i the field of statistical iferece. I testig for ormality, the primary aim is to ivestigate whether or ot available (sample data was draw from a ormally distributed populatio. I some cases, where real life data do ot satisfy the ecessary parametric assumptios, researchers trasform such data to adjust for ormality ad other basic assumptios of parametric tools such as homogeeity of variace (Seth, 008; Jaso, 00; Jaso, 00; Pitchaya et al, 0. Normality test procedures ca be categorized ito two; graphical ad aalytical method. Graphical methods ivolve the use of charts (or graphs examples iclude quatile-
5 Ukpomwa. H. Nosakhare: Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio quatile plot (Q-Q plot, histogram, box-plot ad stem-adleaf plot. Aalytical test for ormality testig ivolves the use of mathematical formulae, this iclude Shapiro-Wilk test, Jarque-Bera test, Kolmogorov-Smirov test, Rya-Joier test etc. May researchers, especially those with phobia for mathematics, apply or prefer graphical approach because it requires less formula ad ca easily be uderstood, eve by the begier i research. Statistical tests geerally have limits where they ted to be weak ad usuitable for what they are meat for. Normality tests are umerous with sigificat variatios i terms of sesitivity as the powers of the tests vary sigificatly ad eve produce uequal p-value usig the same set of data. This implies the iability of some of the ormality tests to detect the actual distributio of a set of data which could lead to either type I or type II errors. Statistical tests that ca stad the test of time ad applicable i every situatio with miimal error are referred to as Uiformly Most Powerful (UMP ad those that ca oly perform excelletly i a defied eviromet or coditio are referred to as Locally Most Powerful (LMP (Piegorsch ad Bailer, 005. Due to the proliferatio of ormality tests, the sesitivity of tests could vary sigificatly based o some factors such as parameters of the available or sampled data like mea, skewess, kurtosis, sample size, etc. Based o the defiitio give by Piegorsch ad Bailer (005, ormality tests that are cosistet irrespective of sample size are here referred to as UMP while those that are adopted for some specified sample sizes, either small or large, are referred to as LMP. Recetly, may researchers ivestigated the stregth of some ormality techiques ad their adequacy i the detectio of ormality of sets of data. Rya (990 faults his previous research by cocludig that the method proposed i his earlier research ca oly be used perfectly for small sample sizes. Also, Shapiro ad Fracia (976, proposed aother method called Shapiro-Fracia due to the icompetece of the first proposed method of test of ormality by Shapiro ad Wilk (965 called Shapiro-Wilk. I like maer, Jarque-Bera test was proposed by aother researcher who strictly cosidered kurtosis ad skewess as measures of ormality of a set of data ad later modified the method to form Adjusted Jarque-Bera test. Iadditio, may researchers have ivestigated the competecies of some of the existig methods due to the proliferatio of ormality tests ad there exists sigificat variatios i their coclusios as see i the results of Yap ad Sim (0, Noradiah ad Yap (0, Paagiotis (00 ad Mayette (03. Researchers such as Stephe (974, Douglas ad Edith (00, Siddik (006, Derya et al (006, Frai (006, Nor-Aishah ad Shamsul (007, Riakor ad Kamo (007, Zvi et al (008, Noradiah ad Yap (0, Yap ad Sim (0, ad Shigekazu et al (0 have ivestigated the sesitivity of various ormality techiques usig power of test by simulatig data from desirable distributio. Most of the works failed to categorize the methods based o sample size ad i terms of peakedess ad skewess of the data. Hece, this paper which aims at evaluatig the performace of some existig ormality techiques with regards to a wide rage of sample sizes ad distributios, with a view to classifyig them ito Uiformly Most Powerful (UMP ad Locally Most Powerful (LMP tests.. Review of Normality Techiques Ivestigated.. Shapiro-Wilk Statistic (W Normality Test Accordig to Farrel ad Stewart (006, Shapiro-Wilk statistic is based o order statistics of the sample, the case weights have to be restricted to itegers. The W statistic is defied as: where x( x(... x( ( x ( W = i ( x( x i ( are ordered sample values, is xi the sample size ad x = is the mea. Critical value of W is available i a statistical table ad decisio is based o the compariso of the W statistic value ad the critical value. The ull hypothesis is rejected if the critical value is less tha the statistic W... Shapiro-Fracia Test Accordig to Roysto (983, the Shapiro-Fracia Test statistic (W is computed usig W mx i ( i = mi ( xi Where x i is the ordered observed values arraged i ascedig order ad m i is the vector of the expected values of stadard ormal order statistics. x is the mea of the observed values. Critical value of W is available i a statistical table ad decisio is based o the compariso of statistic ad the critical value. The ull hypothesis is rejected if the critical value is less tha the statistic W..3. Jarque-Bera Normality Test The Jarque-Bera test is a goodess-of-fit test of whether sample data have the skewess ad kurtosis matchig a ormal distributio. The test is amed after Carlos Jarque ad Ail K. Bera (980. The test statistic JB is defied as JB = S + k 3 6 4 ( ( (3
America Joural of Theoretical ad Applied Statistics 07; 6(5-: 5-6 53 where is the umber of observatios (sample size; S is the sample skewess, ad k is the sample kurtosis: ad µ S = = σ µ K = = σ ( xi 3 3 3 ( xi ( xi ( xi 3 4 4 where µ 3 ad µ 4 are the estimates of third ad fourth cetral momets respectively, x is the sample mea, ad σ is the estimate of the secod cetral momet, the variace. The calculated value is compared with the table value usig Chi- Square distributio table at a desirable level of sigificace (usually 0.05. The decisio rule is; accept the ull hypothesis if the table value is greater tha the calculated value, otherwise, reject..4. Modified/Adjusted Jarque-Bera (AJB Test Statistic I the modified Jarque-Bera test statistic, the computatio of both skewess ad kurtosis differ from the previous Jarque-Bera test statistic as modified by Urzua (996. Here, the Skewess (g is; g = ( ( 4 ( xi s ( 3 where is the sample size, s is the stadard deviatio, x i is the i th observatio ad x is the mea (average observatio of the observed data. The kurtosis (g is computed usig; g 3 4 (4 (5 (6 ( + xi 3( = ( ( ( 3 s ( ( 3 (7 The, AJB becomes; ( g ( g AJB = ( g ( g + = + 6 4 6 4 The decisio rule is; accept the ull hypothesis if the table value is greater tha the calculated value, otherwise, reject H 0..5. Rya-Joier Normality Test Accordig to Rya (976, the test assesses ormality by calculatig the correlatio betwee sample data ad its ormal scores. If the correlatio coefficiet is close to, the populatio is said to be ormal. The Rya-Joier statistic assesses the stregth of this correlatio; if it falls below the appropriate critical value, reject the ull hypothesis of (8 populatio ormality. Cosiderig set of observatios; x wherei =,,...,, the z-score of the set of observatios is i xi mx zi = s where m x is the mea of the set of observatios ad sx is the stadard deviatio of the observatios. By defiitio, To test: H 0 : r is isigificat (r = 0 versus H : r is sigificat (r 0 Usig the test statistic: ( x y cov, Correlatio( r = (9 σ σ t calculated t tabulated = r x y r t α, x (0 = ( the ull hypothesis is rejected if the t calculated > t tabulated value, otherwise, it is accepted. Usig p-value, the ull hypothesis is rejected if the p-value is less tha the level of sigificace of the test (0.05. The, for the test of ormality, the hypothesis is accepted if the r is sigificat..6. D Agostio D Test Accordig to D Agostio ad Stephe (986, the test statistic ca be used to detect departure of data set from ormality. It requires that the observatios be ordered i ascedig order ad the mea deviatios of the ordered data, used to compute D calculated which was compared with the rage/iterval of values from table values of D (D tabulated. The ull hypothesis is rejected if the D calculated value falls outside the rage of values from the D Agostio critical values, otherwise, it is accepted. The test statistic is; Where ( x i x SS = T D = ( 3 SS + T = i xi (3, x i is the i th observatio ad = sample size. The critical value of the test is obtaied from the Critical Value for D Agostio D ormality test. The computed value is compared with the tabulated value (D tab ad the ull hypothesis is rejected if the D cal falls outside the rage of critical values, otherwise, it is accepted. Oe of the disadvatages of the test is that the lowest sample size it ca accommodate is 0. Meaig, it caot be used for sample size less tha 0.
54 Ukpomwa. H. Nosakhare: Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio.7. W/S Test The test uses sample stadard deviatio ad the data rage which is the differece betwee maximum observatio ad the miimum observatio. The test statistic is expressed as; w q = (4 s where w is the rage of available set of data ad s is the sample stadard deviatio. s = ( xi (5 The critical value is read from W/S ormality test table which has boudary values a ad b, where a is the lower boudary or the left-sided critical values ad b is the upper boudary or the right-sided critical value. The ull hypothesis is rejected if the calculated q value falls outside the boudary of the table values, otherwise, it is accepted. Oe of the advatages of the test is that it ca be used for eve samples size less tha 5 as the table has the least value as low as 3..8. Modified Kolmogorov-Smirov Normality Test (Lilliefor s Test Modified Kolmogorov-Smirov test is a o-parametric test for equality of cotiuous probability distributios which ca be used to compare a sample with referece probability distributio or to compare two samples (Wikipedia, 03. Kolmogorov-Smirov statistic quatifies the distace betwee the empirical distributio fuctio of the sample ad the cumulative distributio fuctio of the referece distributio, or betwee the empirical distributio fuctios of two samples. If the observed differece is sufficietly large, the test will reject the ull hypothesis of the populatio ormality. The test statistic is give as; [ ] L = Max l( zi N( zi, l ( zi N( zi = Max D0, D (6 Where xi mx zi = s x s x is the stadard deviatio of the set of observatios, mx is the mea ad xi is the i th observatio. l ( z i is the proportio of scores smaller tha Z i ad N( z i is the cumulative probability associated with Z i for the stadard ormal distributio. Therefore, L is the maximum absolute value of the biggest differece betwee the probability associated to z i whe z i is ormally distributed ad the observed frequecy ( l ( z i. The ull hypothesis of ormality is rejected whe the L value is greater tha or equal to the critical value (Moli ad Abdi, 007..9. Aderso Darlig Test Accordig to D'Agostio ad Stephe (986, the test compares the Empirical Cumulative Distributio (E.C.D fuctio of sample data with the distributio expected. If this observed differece is sufficietly large, the test will reject the ull hypothesis of populatio ormality. The test statistic is a squared distace that is weighted more heavily i the tails of the distributio. The Aderso-Darlig ormality test statistic is give as: i AD = F x + F x * AD AD ( l ( i l ( ( + i 0.75.5 = + + (7 (8 Where F is the cumulative distributio fuctio of the ormal distributio ad x i is the ordered observatios. 3. Yardstick for Classificatio of Normality Techiques ito UMP ad LMP The study ivolves both small ad large sample sizes. Normality testig techiques that ca detect ormality of a set of data irrespective of the sample size, either small or large will be referred to as UMP but those that are better for either of small or large sample size will be categorized as LMP. This implies that ay ormality testig techique classified as UMP ca be used by researchers for test of ormality, irrespective of sample size of the data. 4. Source ad Sample Size of the Data Simulatio was used i the geeratio of the required data usig defied statistical distributios with kow parameters. Symmetric distributios such as ormal distributio; N(, ad N(0,, Studet-t distributio (8, Cauchy distributio (7, 3, ad asymmetric distributios such as Chi-square distributio ( χ (4, ad Weibull distributio [7,3] were used i the simulatio of data. Sice the sample space of the parameter values are ulimited, the values were chose radomly sice from the literature, the methods have similar behaviour irrespective of the parameters of the distributios cosidered. (Siddik, 006; Normadiah ad Yap, 0; Yap ad Sim, 0; Marmolejo-Ramos ad Goza Lez-Burgos, 0. Also from literature, it is cofirmed that at sample size of 00 ad above, the ormality techiques have the same behaviour ad sesitivity to both ormally distributed ad o-ormally distributed data (Rya ad Joier, 976; Derya et al, 006; Normadiah ad Yap, 0, Nor-Aisha et al., 007 ad Mayette ad Emily, 03; Abbas, 03, therefore, maximum sample size cosidered i this study is 00. 5. Algorithm for Mote Carlo Simulatio Accordig to Schaffer (00, Mote Carlo (MC
America Joural of Theoretical ad Applied Statistics 07; 6(5-: 5-6 55 simulatio is used to determie the performace of a estimator or test statistic uder various scearios. The structure of a typical Mote Carlo exercise is as follows:. Specify the Data Geeratio Process.. Choose a sample size N for the MC simulatio. 3. Choose the umber of times to repeat the MC simulatio. 4. Geerate a radom sample of size N based o the Data Geeratio Process. 5. Usig radom sample geerated i 4 above, calculate the statistic(s. 6. Go back to (4 ad repeat (4 ad (5 util desirable replicate is achieved. 7. Examie parameter estimates, test statistics, etc. 6. Determiatio of Sesitivity of Normality Techiques The P-value was used i the study of sesitivity of ormality testig techiques cosidered. I statistical aalysis, the p-value defies the poit at which the test starts beig sigificat. The decisio rule is to reject the ull hypothesis if the p-value is less tha the level of sigificace. I this study, a p-value greater tha level of sigificace implies the set of data is ormally distributed ad the higher the value, the more sesitive the statistic used amog the rest provided the data is simulated usig ormal distributio.. Accordig to Noradiah ad Yap (0, power of a statistical test is the probability that the test will reject the ull hypothesis whe the alterative hypothesis is true (i.e. the probability of ot committig a Type II error. The power is i geeral a fuctio of the possible distributios, ofte determied by a parameter, uder the alterative hypothesis. As the power of test icreases, the chace of a Type II error occurrig decreases. The probability of a Type II error occurrig is referred to as the false egative rate (β. Therefore power of test is equal to β, which is also kow as the sesitivity (Gupta, 0. 7. Data Aalysis This sectio deals with aalysis of simulated data from symmetric ad asymmetric distributios usig ie selected ormality test methods (uivariate methods. The data aalysis ivolves followig steps;. Simulate data of sizes =5, 0,..., 00 from the six distributios; N(,, N(0,, Weibull (7,3, χ (4, t(8, ad Cauchy (7,3.. For each sample size,, replicated 00 times, i Calculate the ormality test statistics for each of the ie uivariate ormality tests ii Reject/Accept the ull hypothesis; H 0 : the data are ormally distributed. H : the data are ot ormally distributed. iii Calculate the error rate = umber of times wrog decisio is recorded umber of replicate (00 iv Calculate the type I error or power of test = p( typeii error See Tables to 9 for the error rates obtaied for the ie ormality tests methods. 8. Error Rates Table. Error Rate of W/S Test Statistic. From Table 3, it ca be see vividly that the test is highly icosistet as the error rate is cosiderably high irrespective of the sample sizes ad for all the distributios except Cauchy (7, 3. However, the behaviour of the test is differet for t- distributio which belogs to the same category as Cauchy distributio highlightig the icosistecy of W/S test. Table. Error Rate of JB Test Statistic. N[,] N[0,] 5 0.0 0.03 0.9 0.86 0.9 0.9 0 0. 0.08 0.88 0.97 0.89 0.7 5 0.07 0.09 0.9 0.9 0.80 0.48 0 0.07 0.07 0.8 0.94 0.79 0.5 5 0.08 0.06 0.88 0.8 0.76 0.7 30 0.07 0.08 0.87 0.85 0.85 0.4 35 0.3 0.06 0.9 0.85 0.77 0.08 40 0.08 0.09 0.9 0.8 0.78 0.04 45 0.0 0.05 0.89 0.90 0.65 0.04 50 0. 0.05 0.88 0.78 0.76 0.0 55 0. 0.06 0.86 0.78 0.79 0.0 60 0. 0.06 0.76 0.79 0.70 0.0 65 0.0 0.07 0.78 0.8 0.7 0.0 70 0.3 0.08 0.84 0.83 0.73 0.0 75 0.08 0.08 0.75 0.76 0.66 0.00 80 0. 0.08 0.79 0.77 0.69 0.0 85 0. 0.07 0.73 0.76 0.65 0.00 90 0.06 0.09 0.77 0.78 0.64 0.00 95 0.3 0.07 0.78 0.79 0.60 0.00 00 0.06 0.08 0.74 0.8 0.6 0.00 N[,] N[0,] 5 0.00 0.00 0 0.00 0.00 0.98 0.9 0.6 5 0.0 0.03 0.97 0.83 0.9 0.9 0 0.0 0.0 0.95 0.73 0.90 0. 5 0.0 0.0 0.96 0.68 0.85 0.5 30 0.0 0.03 0.97 0.5 0.89 0.0 35 0.05 0.05 0.93 0.4 0.80 0.0 40 0.04 0.03 0.89 0.9 0.8 0.0 45 0.07 0.03 0.8 0.33 0.7 0.0 50 0.0 0.0 0.8 0.7 0.76 0.00 55 0.08 0.0 0.8 0. 0.80 0.00 60 0.07 0.0 0.84 0.4 0.7 0.0 65 0.05 0.0 0.76 0. 0.7 0.0 70 0.03 0.04 0.83 0.08 0.7 0.0 75 0.0 0.03 0.83 0. 0.68 0.00 80 0.04 0.0 0.85 0.07 0.67 0.00 85 0.0 0.0 0.85 0.04 0.70 0.00 90 0.00 0.0 0.76 0.03 0.6 0.00
56 Ukpomwa. H. Nosakhare: Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio From Table the error rate of Jarque-Bera test decreases as the sample size icreases, a idicatio that the method is more active or efficiet for large sample size especially for the o-ormal distributios. Table 3. Error Rate of AJB Test Statistic. For the o-ormal distributio, error rate icreases as sample size icreases except for the Cauchy (7, 3. For the ormal distributio, there is o impact of sample size o the error rate. It was observed that the error rate is sigificatly reduced for small sample sizes for the o-ormal distributios for the ADJ test statistic, a idicatio that the adjusted Jarque-Bera Test is a improvemet o the Jarque-Bera test for small sample sizes. Table 4. Error Rate of D Agostio Test Statistic. From Table 4, the stregth of the test statistic icreases as sample size icreases, a idicatio that the test is good or more sesitive for large sample sizes. Table 5. Error Rate of Lilliefor s Test Statistic. Lilliefors test statistic has lower sesitivity, a idicatio that the test statistic is weak i test of ormality except Cauchy (7, 3. Table 6. Error Rates of Shapiro-Wilk Test Statistic. N[,] N[0,] 95 0.0 0.03 0.70 0.00 0.63 0.00 00 0.0 0.03 0.70 0.00 0.58 0.00 N[,] N[0,] 5 0.03 0.05 0.03 0.04 0.07 0.7 0 0.0 0.03 0.05 0.7 0.05 0.4 5 0.04 0.05 0.07 0.4 0. 0.8 0 0.04 0.05 0.06 0.35 0.3 0.6 5 0.05 0.0 0.06 0.37 0.8 0.0 30 0.05 0.05 0.04 0.53 0.4 0.09 35 0.08 0.08 0. 0.6 0.3 0.0 40 0.04 0.03 0.4 0.73 0.8 0.00 45 0.08 0.05 0.3 0.75 0.70 0.0 50 0.0 0.0 0.0 0.73 0.7 0.00 55 0.09 0.05 0. 0.89 0.68 0.00 60 0.0 0.05 0.7 0.90 0.68 0.0 65 0.05 0.04 0.8 0.89 0.69 0.0 70 0.03 0.05 0.0 0.93 0.67 0.0 75 0.03 0.04 0.0 0.88 0.63 0.00 80 0.05 0.04 0.8 0.94 0.65 0.00 85 0.0 0.0 0.6 0.96 0.65 0.00 90 0.0 0.0 0.6 0.97 0.60 0.00 95 0.0 0.03 0.3.00 0.6 0.00 00 0.03 0.03 0.3.00 0.55 0.00 N[,] N[0,] 5 - - - - - - 0 0.05 0.04 0.53 0.64 0.73 0.5 5 0.04 0.04 0.53 0.59 0.7 0.8 0 0.06 0.05 0.5 0.7 0.74 0.5 5 0.04 0.05 0.5 0.68 0.73 0.3 30 0.04 0.06 0.5 0.5 0.78 0. 35 0.04 0.06 0.4 0.34 0.74 0.09 40 0.03 0.05 0.43 0.4 0.6 0.06 45 0.03 0.06 0.3 0.36 0.64 0.05 50 0.04 0.05 0.3 0.4 0.63 0.0 55 0.03 0.05 0.3 0.36 0.6 0.0 60 0.03 0.04 0.36 0.47 0.65 0.0 65 0.0 0.0 0. 0.3 0.6 0.0 70 0.0 0.0 0.3 0.6 0.6 0.0 75 0.0 0.03 0. 0. 0.64 0.0 80 0.0 0.0 0. 0.3 0.66 0.00 N[,] N[0,] 85 0.0 0.0 0.9 0.6 0.6 0.0 90 0.0 0.0 0. 0.8 0.65 0.00 95 0.0 0.0 0. 0. 0.6 0.00 00 0.0 0.0 0.0 0.8 0.6 0.00 N[,] N[0,] 5 0.08 0.06 0.96 0.94 0.93 0.67 0 0.06 0.06 0.95 0.83 0.46 5 0.04 0.05 0.97 0.76 0.90 0.0 0 0.0 0.0 0.9 0.69 0.96 0.8 5 0.03 0.0 0.89 0.67 0.89 0. 30 0.0 0.04 0.9 0.5 0.86 0.08 35 0.0 0.04 0.89 0.46 0.94 0.0 40 0.03 0.04 0.93 0.4 0.94 0.0 45 0.03 0.03 0.89 0.44 0.88 0.0 50 0.04 0.03 0.85 0.3 0.88 0.00 55 0.0 0.0 0.8 0.5 0.9 0.00 60 0.0 0.03 0.8 0.4 0.83 0.0 65 0.0 0.03 0.79 0.5 0.93 0.0 70 0.0 0.04 0.80 0.0 0.86 0.0 75 0.0 0.03 0.86 0.0 0.88 0.00 80 0.0 0.05 0.85 0.08 0.90 0.00 85 0.0 0.04 0.93 0.0 0.87 0.00 90 0.0 0.04 0.75 0.03 0.85 0.00 95 0.0 0.04 0.8 0.03 0.83 0.00 00 0.0 0.03 0.79 0.04 0.8 0.00 N[,] N[0,] 5 0.05 0.06 0.97 0.95 0.88 0.7 0 0.06 0.07 0.9 0.8 0.99 0.44 5 0.06 0.06 0.96 0.6 0.87 0. 0 0.0 0.04 0.89 0.57 0.89 0.3 5 0.0 0.03 0.90 0.4 0.87 0.0 30 0.03 0.0 0.80 0.33 0.86 0.05 35 0.04 0.0 0.85 0. 0.85 0.00 40 0.03 0.04 0.86 0.09 0.90 0.00 45 0.0 0.04 0.74 0.07 0.77 0.0 50 0.03 0.03 0.76 0.07 0.78 0.00 55 0.03 0.03 0.73 0.04 0.80 0.00 60 0.0 0.0 0.79 0.00 0.8 0.0 65 0.0 0.04 0.67 0.0 0.76 0.0 70 0.0 0.04 0.68 0.00 0.74 0.0 75 0.03 0.04 0.74 0.00 0.70 0.00 80 0.0 0.03 0.80 0.0 0.73 0.00 85 0.0 0.03 0.8 0.0 0.78 0.00 90 0.0 0.03 0.63 0.00 0.67 0.00 95 0.0 0.0 0.66 0.00 0.65 0.00 00 0.0 0.0 0.67 0.00 0.65 0.00
America Joural of Theoretical ad Applied Statistics 07; 6(5-: 5-6 57 From Table 6, the error rate decreases as sample size icreases, a idicatio of higher sesitivity for large sample sizes. Table 7. Error Rate of Shapiro-Fracia Test Statistic. N[,] N[0,] 5 0.03 0.0 0.87 0.84 0.85 0.6 0 0.04 0.03 0.84 0.8 0.87 03 5 0.04 0.03 0.85 0.74 0.85 0.8 0 0.0 0.03 0.89 0.43 0.86 0.0 5 0.0 0.03 0.87 0.4 0.84 0.0 30 0.0 0.0 0.84 0.43 0.83 0.04 35 0.0 0.0 0.8 0. 0.8 0.0 40 0.0 0.03 0.8 0.05 0.87 0.0 45 0.0 0.03 0.78 0.06 0.73 0.00 50 0.03 0.03 0.78 0.04 0.75 0.00 55 0.0 0.03 0.7 0.03 0.76 0.00 60 0.0 0.0 0.7 0.0 0.78 0.0 65 0.0 0.03 0.66 0.0 0.7 0.0 70 0.0 0.04 0.6 0.00 0.73 0.0 75 0.0 0.03 0.7 0.00 0.7 0.0 80 0.0 0.03 0.76 0.0 0.7 0.00 85 0.0 0.0 0.76 0.0 0.75 0.00 90 0.0 0.0 0.6 0.00 0.6 0.00 95 0.0 0.03 0.6 0.00 0.6 0.00 00 0.0 0.03 0.6 0.00 0.6 0.00 The Shapiro-Fracia has decreasig error rates as sample size icreases. Whe compared with Shapiro-Wilk, it also has lower error rates, which makes it better tha Shapiro- Wilk test statistic. Table 8. Error Rate of Rya-Joier Test Statistic. From Table 8, Rya-Joier test statistic is good for small sample sizes but poor for large sample sizes except Cauchy ad ormal distributios. Table 9. Error Rate of Aderso Darlig Test Statistic. N[,] N[0,] 5 0.06 0.05 0.06 0.09 0.9 0.75 0 0.03 0.05 0.05 0.8 0.9 0.65 5 0.03 0.03 0.06 0.8 0.8 0.43 0 0.04 0.05 0.06 0.4 0. 0.3 5 0.0 0.05 0.06 0.3 0.7 0. 30 0.05 0.05 0.06 0.47 0.9 0. 35 0.05 0.03 0.08 0.76 0.09 0.08 40 0.04 0.0 0. 0.73 0.7 0.04 45 0.05 0.03 0.3 0.75 0.70 0.0 50 0.0 0.03 0.0 0.73 0.8 0.00 55 0.05 0.04 0. 0.8 0.78 0.0 60 0.08 0.04 0.7 0.84 0.78 0.0 65 0.03 0.04 0.8 0.83 0.69 0.0 N[,] N[0,] 70 0.03 0.03 0.0 0.83 0.57 0.0 75 0.03 0.03 0.0 0.83 0.43 0.00 80 0.04 0.0 0.8 0.9 0.55 0.00 85 0.0 0.0 0.6 0.9 0.55 0.00 90 0.0 0.03 0.6 0.9 0.40 0.00 95 0.0 0.0 0. 0.93 0.5 0.00 00 0.0 0.0 0. 0.94 0.35 0.00 N[,] N[0,] 5 0.00 0.00 0.99 0 0.00 0.00 0.98 0.98 0.9 0.77 5 0.0 0.03 0.98 0.97 0.9 0.3 0 0.0 0.0 0.9 0.87 0.90 0. 5 0.0 0.0 0.9 0.8 0.88 0.4 30 0.0 0.03 0.96 0.73 0.76 0. 35 0.05 0.04 0.93 0.44 0.79 0.05 40 0.04 0.03 0.85 0.3 0.80 0.04 45 0.04 0.03 0.83 0.30 0.73 0.05 50 0.0 0.0 0.83 0.3 0.7 0.0 55 0.05 0.0 0.8 0.9 0.76 0.0 60 0.05 0.0 0.83 0. 0.7 0.0 65 0.05 0.0 0.75 0.0 0.7 0.0 70 0.03 0.0 0.76 0.09 0.70 0.0 75 0.0 0.03 0.7 0.0 0.69 0.0 80 0.03 0.0 0.78 0.06 0.69 0.00 85 0.0 0.0 0.77 0.03 0.70 0.00 90 0.00 0.0 0.76 0.03 0.6 0.00 95 0.00 0.0 0.73 0.0 0.6 0.00 00 0.0 0.0 0.7 0.00 0.4 0.00 The error rates show that Aderso Darlig s test statistic is good for small sample sizes but poor for large sample sizes except for Cauchy ad ormal distributios. See Table 0 for power of the ie methods cosidered. 9. Power of Test Table 0. Power of Nie Selected Uivariate Normality Tests Uder Evaluatio. 5 0 5 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 Weibull (7,3 0.09 0. 0.09 0.8 0. 0.3 0.08 0.09 0. 0. 0.4 0.4 0. 0.6 0.5 0. 0.7 0.3 0. 0.6 W/S Chi-Sqr (4 0.4 0.03 0.09 0.06 0.9 0.5 0.5 0.8 0. 0. 0. 0. 0.9 0.7 0.4 0.3 0.4 0. 0. 0.8 t(8 0.09 0. 0. 0. 0.4 0.5 0.3 0. 0.35 0.4 0. 0.3 0.9 0.7 0.34 0.3 0.35 0.36 0.4 0.38 Cauchy (7,3 0.08 0.9 0.5 0.75 0.83 0.86 0.9 0.96 0.96 0.99 0.99 0.99 0.99 0.98 0.99 Weibull (7,3 0 0.0 0.03 0.05 0.04 0.03 0.07 0. 0.9 0.8 0.8 0.6 0.4 0.7 0.7 0.5 0.5 0.4 0.3 0.3 JB Chi-Sqr (4 0 0.08 0.7 0.7 0.3 0.48 0.58 0.7 0.67 0.73 0.88 0.86 0.88 0.9 0.88 0.93 0.96 0.97 t(8 0 0 0.09 0. 0.5 0. 0. 0.8 0.8 0.4 0. 0.8 0.8 0.9 0.3 0.33 0.3 0.38 0.37 0.4 Cauchy (7,3 0 0.39 0.7 0.79 0.85 0.9 0.99 0.99 0.99 0.99 0.99 0.99
58 Ukpomwa. H. Nosakhare: Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio 5 0 5 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 Weibull (7,3 0.97 0.95 0.93 0.94 0.94 0.96 0.88 0.86 0.77 0.8 0.78 0.83 0.7 0.8 0.8 0.8 0.84 0.74 0.69 0.69 AJB Chi-Sqr (4 0.96 0.83 0.76 0.65 0.63 0.47 0.39 0.7 0.5 0.7 0. 0. 0. 0.07 0. 0.06 0.04 0.03 0 0 t(8 0.93 0.95 0.89 0.87 0.8 0.86 0.77 0.8 0.3 0.9 0.3 0.3 0.3 0.33 0.37 0.35 0.35 0.4 0.38 0.45 Cauchy (7,3 0.9 0.59 0.8 0.84 0.9 0.9 0.99 0.99 0.99 0.99 0.99 Weibull (7,3-0.47 0.47 0.48 0.48 0.49 0.58 0.57 0.68 0.68 0.68 0.64 0.78 0.77 0.79 0.79 0.8 0.79 0.78 0.8 D'Agostio Chi-Sqr (4-0.36 0.4 0.9 0.3 0.49 0.66 0.58 0.64 0.59 0.64 0.53 0.69 0.74 0.79 0.77 0.74 0.8 0.79 0.8 t(8-0.7 0.8 0.6 0.7 0. 0.6 0.38 0.36 0.37 0.38 0.35 0.38 0.39 0.36 0.34 0.38 0.35 0.38 0.38 Cauchy (7,3-0.85 0.8 0.85 0.87 0.89 0.9 0.94 0.95 0.98 0.98 0.99 0.99 0.99 0.99 0.99 Weibull (7,3 0.04 0.05 0.03 0.09 0. 0.08 0. 0.07 0. 0.5 0.8 0.9 0. 0. 0.4 0.5 0.07 0.5 0.8 0. Lilliefor's Chi-Sqr (4 0.06 0.7 0.4 0.3 0.33 0.48 0.54 0.59 0.56 0.69 0.75 0.76 0.75 0.9 0.9 0.9 0.9 0.97 0.97 0.96 t(8 0.07 0 0. 0.04 0. 0.4 0.06 0.06 0. 0. 0.09 0.7 0.07 0.4 0. 0. 0.3 0.5 0.7 0.9 Cauchy (7,3 0.33 0.54 0.8 0.8 0.89 0.9 0.98 0.99 0.99 0.99 0.99 0.99 Weibull (7,3 0.03 0.09 0.04 0. 0. 0. 0.5 0.4 0.6 0.4 0.7 0. 0.33 0.3 0.6 0. 0.8 0.37 0.34 0.33 SW Chi-Sqr (4 0.05 0.8 0.38 0.43 0.58 0.67 0.79 0.9 0.93 0.93 0.96 0.99 0.99 0.99 t(8 0. 0.0 0.3 0. 0.3 0.4 0.5 0. 0.3 0. 0. 0.8 0.4 0.6 0.3 0.7 0. 0.33 0.35 0.35 Cauchy (7,3 0.9 0.56 0.78 0.87 0.9 0.95 0.99 0.99 0.99 0.99 Weibull (7,3 0.3 0.6 0.5 0. 0.3 0.6 0.9 0.8 0. 0. 0.9 0.8 0.34 0.38 0.9 0.4 0.4 0.39 0.38 0.38 SF Chi-Sqr (4 0.6 0.9 0.6 0.57 0.59 0.57 0.79 0.95 0.94 0.96 0.97 0.99 0.99 0.99 0.99 t(8 0.5 0.3 0.5 0.4 0.6 0.7 0.9 0.3 0.7 0.5 0.4 0. 0.9 0.7 0.9 0.8 0.5 0.38 0.39 0.39 Cauchy (7,3 0.39 0.68 0.8 0.9 0.9 0.96 0.99 0.99 0.99 0.99 0.99 0.99 Weibull (7,3 0.94 0.95 0.94 0.94 0.94 0.94 0.9 0.88 0.87 0.9 0.88 0.83 0.8 0.9 0.9 0.8 0.84 0.84 0.79 0.79 RJ Chi-Sqr (4 0.9 0.8 0.7 0.76 0.69 0.53 0.4 0.7 0.5 0.7 0.9 0.6 0.7 0.7 0.7 0.09 0.08 0.09 0.07 0.06 t(8 0.8 0.8 0.8 0.89 0.83 0.8 0.9 0.8 0.3 0.9 0. 0. 0.3 0.43 0.57 0.45 0.45 0.6 0.48 0.65 Cauchy (7,3 0.5 0.35 0.57 0.68 0.79 0.89 0.9 0.96 0.98 0.99 0.99 0.99 0.99 Weibull (7,3 0.0 0.0 0.0 0. 0. 0.04 0.07 0.5 0.7 0.7 0.8 0.7 0.5 0.4 0.8 0. 0.3 0.4 0.7 0.9 AD Chi-Sqr (4 0.00 0.0 0.03 0.3 0.8 0.7 0.56 0.69 0.7 0.69 0.7 0.79 0.8 0.9 0.9 0.94 0.97 0.97 0.99 t(8 Cauchy (7,3 0.00 0.00 0.08 0.9 0.08 0.69 0.0 0.79 0. 0.86 0.4 0.89 0. 0.95 0.0 0.96 A test statistic with higher power is better, hece, the higher the power of a method, the better is the method. From Table 3., it ca be see that as the sample size icreases, the power of ormality tests icrease, collaboratig the fidigs of researchers like; Stephe (974, Douglas ad Edith (00, Siddik (006, Derya et al (006, Frai (006, Nor-Aishah ad Shamsul (007, Riakor ad Kamo (007, Zvi et al (008, Noradiah ad Yap (0, Yap ad Sim (0, ad Shigekazu et al (0. The superiority of the method is easier visualised from graph of the power of the tests agaist sample sizes, see Figures ad. Compariso of Power of the Test Statistics Usig Lie Chart 0.7 0.95 0.8 0.98 0.4 0.99 0.8 0.99 0.8 0.99 0.30 0.99 0.3 0.99 0.3 0.30 0.38 0.38 0.58 Figure. Power of Tests usig Symmetric distributio (Cauchy Distributio.
America Joural of Theoretical ad Applied Statistics 07; 6(5-: 5-6 59 Figure. Power of Test usig Weibull Distributio (Asymmetric Distributio. From Figure, D Agostio test statistic has highest power for a sample size of 0 while Shapiro-Fracia test statistic has highest power for sample size 5 to 30 (except whe = 0. For sample size 35 ad above, Shapiro-Fracia has equal power as Shapiro-Wilk test statistic. It ca also be see that the best method for small sample size is ot the best method for large sample size ad that as the sample size icreases, the power of the tests became uilateral/uiform. From Figure, it ca be deduced that Adjusted Jarque- Bera, Rya-Joier ad D Agostio tests have the highest power for asymmetric distributios irrespective of sample size. Other techiques have cosiderably ad cosistetly low power for both small ad large sample sizes. Usig both type I error ad power of test, the rakig of the cosidered ormality tests is as show i Table 3.3; Table. Normality Tests i Order of Superiority by Sample Sizes usig Power of Test ad Type I Error. Sample Size Order of Superiority of ormality tests st d 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 5 AJB Lf RJ SW SF W/S AD JB DA 0 AJB Lf DA RJ SF SW W/S JB AD 5 AJB Lf DA RJ SF SW JB W/S AD 0 AJB Lf DA RJ SF SW AD JB W/S 5 AJB Lf DA RJ SF SW W/S JB AD 30 AJB Lf DA SF RJ SW JB AD W/S 35 AJB Lf DA RJ SF JB AD SW W/S 40 DA Lf AJB RJ SF AD JB SW W/S 45 DA RJ SF Lf AJB JB AD SW W/S 50 DA RJ SF AJB Lf JB AD SW W/S 55 DA RJ SF Lf JB AJB AD SW W/S 60 DA RJ SF JB AJB AD Lf SW W/S Sample Size Order of Superiority of ormality tests st d 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 65 DA RJ SF JB AD Lf AJB SW W/S 70 DA RJ SF Lf JB AD SW AJB W/S 75 DA Lf RJ SF AD JB AJB SW W/S 80 DA RJ AD SF JB Lf AJB SW W/S 85 DA AD RJ JB SF Lf AJB SW W/S 90 DA RJ SF JB AD Lf SW AJB W/S 95 DA RJ SF JB AD Lf SW AJB W/S 00 DA AD RJ JB SF Lf SW AJB W/S 0. Coclusio From all the results obtaied from this study, we coclude as follows;. For small sample data, 30, the best three ormality techiques i order of superiority are; (i Adjusted Jaque- Bera test (ii Lilliefor s test ad (iii D Agostio test.. For large sample data, > 30, the best three ormality techiques i order of superiority are; (i D Agostio test (ii Rya-Joier test ad (iii Shapiro-Fracia test. Thus, D Agostio test is oe of the best methods for small sample sizes, as well as large sample sizes, thus, it ca be referred to as Uiformly Most Powerful amogst of all the tests evaluated. 3. The Adjusted Jarque-Bera (AJB is cosistetly better tha Jarque-Bera (JB for sample data of size 50 while for > 50, JB is better, thus, the AJB should be used for small sample while JB should be used for large sample. This fidig validates the purpose of modificatio of JB (Urzua, 996. 4. The Shapiro-Fracia (SF is however better tha
60 Ukpomwa. H. Nosakhare: Evaluatio of Techiques for Uivariate Normality Test Usig Mote Carlo Simulatio Shapiro-Wilk (SW for small ad large sample data, hece, the SW should be replaced by SF i ormality test. This cotradicts the claims of researchers such as Yap ad Sim (0, Nor-Aisha et al (0, Marmolejo-Ramos ad Goza Lez-Burgos (0, ad Mayette (03 that Shapiro-Wilk test statistic is the best method without cosideratio of SF. 5. W/S test is the worst amogst all the ormality testig procedures; hece, it should be removed from the list of ormality tests or adjusted for improved efficiecy. We recommed as follows; D Agostio test statistic ca be referred to as Uiformly Most Powerful amog cosidered methods. Shapiro-Fracia is a true improved method over SW for all sample sizes, hece, SF should replace SW. Proper modificatio of W/S to improve its sesitivity is very ecessary, otherwise, W/S should be removed from the list of ormality test statistics. 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