Winsorized Modified One Step M-estimator in Alexander-Govern Test

This research centres on independent group test of comparing two or more means by using the parametric method, namely the Alexander-Govern test. The Alexander-Govern (AG) test uses mean as a measure of its central tendency. It is a better alternative to the Welch test, James test and the ANOVA, because it has a good control of Type I error rates and produces a high power efficient for a normal data under variance heterogeneity, but not for non-normal data. As a result, trimmed mean was applied on the test under non-normal data for two group condition, but as the number of groups increased above two, the test fails to be robust. Due to this, when the MOM estimator was applied on the test, it was not influenced by the number of groups, but failed to give a good control of Type I error rates under skewed heavy tailed distribution. In this research, the Winsorized MOM estimator was applied in AG test as a measure of its central tendency. 5,000 data sets were simulated and analysed using Statistical Analysis Software (SAS). The result shows that with the pairing of unbalanced sample size with unequal variance of (1:36) and the combination of both balanced and unbalanced sample sizes with both equal and unequal variances, under six group condition, for g = 0.5 and h = 0.5, for both positive and negative pairing condition, the test gives a remarkable control of Type I error rates. In overall, the AGWMOM test has the best control of Type I error rates, across the distributions and across the groups, compared to the AG test, the AGMOM test and the ANOVA.


Introduction
This research focuses on comparing the performance of the Type I error rates and power of the AG test, the AGMOM test, the AGWMOM test, the t-test and the ANOVA, for two, four and six group conditions.In order to see of the five tests, which one of the tests will give good control of Type I error rates and also produce high power efficient, under skewed heavy tailed distribution.The independent group tests such as the analysis of variance (ANOVA) methods have been employed in many areas, for example, in medicine, economics, sociology and agriculture, as discussed by Pardo, Pardo, Vincente and Esteban (1997).Several assumptions have to be satisfied before the method can perform effectively, which are (i) homogeneity of variances, (ii) normal distribution of the data and (iii) independent observations.The ANOVA is a classical method of analysis that is used for comparing the differences between three or more means.It is used for testing the equality of the measure of the central tendency of a distribution and is robust to small deviations from normality, especially when the sample size is large enough to guarantee normality as stated by Wilcox (1997Wilcox ( , 2003)).Yusof, Abdullah, Yahaya and Othman (2011) observed that the two major problems confronting the analysis of variance are the presence of non-normality and variance heterogeneity in a data distribution.As a result of this, Type I error rates is increased and there is a reduction in the power efficiency of the test.When the data distribution is seen to be heavy tailed, the standard error of the mean can be badly increased (Wilcox & Keselman, 2002).This makes the standard error of the ANOVA to be larger than it ought to be and the power of the test would be reduced.To obtain a good test, Type I error rates should be controlled and likewise the power of the test.This implies that neither should Type I error rates be increased nor should there be a lost in the power of the test.
The ANOVA is very sensitive to the assumptions of homogeneity of variance such that when there is a violation, the outcome of the analysis could be unreliable: then the p-value becomes too conservative or may be large.Therefore, it is very important to test for the homogeneity of the variance and to check for the equality of the variance assumptions by using the correct test, so as to increase the genuineness of the results (Brown & Forsythe, 1974;Wilcox, Charlin & Thompson, 1986).The problem of heterogeneity of variance has been discussed by few researchers and some alternatives were proposed.Welch (1951) introduced the Welch test that is used to test for the hypothesis of two populations having equal means.It was mentioned in different literatures as an alternative to the ANOVA (Algina, Oshima & Lin, 1994;Keselman, 1982;Lix, Keselman, & Keselman, 1996;Wilcox et al., 1986).James (1951) also introduced a better solution for ANOVA under heterogeneity of variance, namely the James test.This test is used for weighing sample means and is recommended by different researchers (Lix et al., 1996;Oshima & Algina, 1992;Wilcox, 1988).When the sample size is small, and the data distribution is non-normal, the James test fails to give a good control of Type I error rates.Both the Welch test and the James test are used for analysing a data distribution that is non-normal with unequal variance (Brunner, Dette, & Munk, 1997;Kohr & Games, 1974;Krishnamorthy, Lu, & Mathew, 2007;Wilcox & Keselman, 2003).
The Alexander-Govern test was introduced in (1994) as to deal with heterogeneity of variance under the condition of normality, but is a test that is not robust to non-normal data.The Alexander-Govern test has been compared with the Welch test and the James test and it was agreed by Schneider and Penfield (1997) and Myers (1998) that the Alexander-Govern test and the James test were better under most conditions being studied.The performance of the Alexander-Govern test was comparable to the James test, and the calculation of the Alexander-Govern test was simpler than that of the James test.
This finding was agreed by Myers (1998) suggesting that the Alexander-Govern test provides a good solution to the problem of variance heterogeneity.Although, the Alexander-Govern test is a good alternative to the ANOVA when it comes to variance heterogeneity, this test still has some drawbacks.The main weakness of the test is that it cannot handle a deviation from normality, as proven by Myers (1998).This method can acceptably put under control Type I error rates when there is heterogeneity in the variances and when the distribution of the data is normal.Lix and Keselman (1998) proposed an alternative approach by substituting the common mean with trimmed mean in a few robust test statistics which improved the performance of the tests under non-normal data.The trimmed mean and Winsorized variance are widely used as alternative to the common mean and variance respectively, due to some good properties, such as having a remarkable control over Type I error rates and the power of the test, when there is a violation under the assumptions of homogeneity of the variance and when the distribution is normal (Wilcox, 1995).
Trimmed mean is calculated by averaging only the centred data after discarding a particular amount of the percentage of the largest and the smallest data value, while its variance is estimated by Winsorized variance.In applying trimmed mean in a data distribution, it possesses some disadvantages, which are firstly: the percentage of trimming is placed at prior, resulting in the elimination process performed regardless of the shape of the distribution.Secondly in trimming process, it should be done carefully, to minimize loss of information.Thirdly, it cannot handle large size of extreme values (Yahaya, Othman & Keselman, 2006).
One of the suggested estimator as a replacement for the trimmed mean is a better alternative referred to as the modified one-step M-estimator (MOM), which is able to identify the presence of outliers in a data distribution (Yusuf, Abdullah, Yahaya & Othman, 2011).It empirically trims only extreme data (Othman, Keselman, Padmanabhan, Wilcox & Fradette, 2004).The main disadvantage of using the MOM estimator as a measure of the central tendency, for example in Alexander-Govern test, is that it cannot control the error rates in the test when the level of skewness and kurtosis is at peak.In this research, the Winsorized MOM was applied in Alexander-Govern test under variance heterogeneity for non-normal data, under a skewed heavy tailed distribution, and it gave the test a remarkable control of Type I error rates and produced a high power efficient for test.

The Alexander-Govern Test
The Alexander-Govern test is a test introduced by Alexander-Govern (1994).This test uses mean as a measure of its central tendency.It produces a good control of Type I error rates and high power efficient under variance heterogeneity, but it is not robust for a non-normal data.As a result, the test fails to give a good control of Type I error rates for a non-normal data.This test is used for comparing two or more groups.The test statistic for the Alexander-Govern test is obtained by using the following procedures.

The Alexander-Govern Test Statistic
The procedure to calculate the test statistic for the Alexander-Govern test starts by first ordering the data set, with population j (j = 1, …, J).Then for each of the data set, the mean is calculated using: (1) where represents the observed random samples and denote the sample sizes of the observation.The mean is used as the central tendency measure in the Alexander-Govern (1994) procedure.After obtaining the mean, the estimate of the usual unbiased variance is calculated using: (2) where is used for estimating for the population j.The standard error of the mean is calculated for each group using: (3) The weight ( for the group sizes with j population of the ordered sample data is defined such that should be equal to 1. So, the weight ( for each of the group is calculated using the formula below: (4) The null hypothesis for the Alexander-Govern test (1994) method for the equality of the mean, under variance heterogeneity is expressed as: (5) The alternative hypothesis contradicts the claim or statement made by the null hypothesis.The variance weighted estimate of the total mean for all the groups in the data distribution is calculated using: (6) where denotes the weight for each group in the data set and denotes the mean of each of the groups in the ordered sample.The statistic for each group is calculated by using: (7) where is the mean for each group.
is the grand mean for all the groups under analysis and is the standard error of the mean for each of the group with population j.
The t statistic is distributed as a t variable, having degrees of freedom for Where, is the degree of freedom for each of the groups in the order sample data.The t statistic calculated for each group is converted to a standard normal deviates by using the Hill's (1970) normalization approximation in the Alexander-Govern (1994) procedure.The formula is expressed as: (8) where ( 9) where (10) The test statistic for the Alexander-Govern procedure is expressed as: (11) After obtaining the test statistic for the Alexander-Govern test, we select a significance level of with J -1 chi-square degrees of freedom.The p-value is also obtained for the Alexander-Govern method.The standard chi-square distribution is used to calculate the critical value of the test.After which, we compare our chi-square value with the test statistic value of the Alexander-Govern test.If A calculated is greater than the chi-square value from the chi-square distribution table, then we reject and conclude that the means of the independent groups are different from each other, otherwise, we do not.

The Modified Alexander-Govern Test
In this research, we modified the mean as a central tendency measure in the Alexander-Govern test by replacing it with the Winsorized modified one step M-estimator (WMOM) as a measure of its central tendency.The Winsorized MOM estimator is applied on the data distribution where the outlier detected value is replaced or exchanged with a preceding value closest to the position where the outlier is located.The Winsorized MOM estimator is obtained by using the formula: where, is the mean of the Winsorized data distribution, and n is the sample size of the Winsorized data distribution.
The WMOM estimator becomes a replacement for the common mean as a measure of the central tendency in Alexander-Govern test, for the following reasons: i.
To remove the presence of outliers from the data distribution ii.
To make the Alexander-Govern test to be robust to non-normal data.
The Winsorized sample variance is expressed as: The standard error of the WMOM is obtained by using the bootstrapping methods in order to estimate the standard error and the bootstrapping algorithm for estimating the standard errors is defined below: Firstly, we choose B independent bootstrap samples defined as: where each of these random samples comprises of n data values selected with replacement from x defined below: The indication of the star shows that is not the real data set of x, but it refers to a randomized or resampled version of x.
In estimating the standard error of the bootstrap samples, the number of B falls within the range of .In this research, we made use of 50 amount of the bootstrap samples is sufficient to give reasonable estimate of the standard error of the MOM estimator (Efron & Tibshirani, 1998).
Secondly, we evaluate the bootstrap replications equating to each of the bootstrap samples defined below: The weight for the Winsorized data distribution for each group is expressed below: where is the sum of the inverse of the square of the standard error for all the groups in the ordered data distribution, from the real data distribution.
The variance weighted estimate of the total mean for the Winsorized data distribution for all the groups is expressed as: (19) The t statistic for the Winsorized data distribution for each of the groups is obtained using the formula below: where is the Winsorized sample standard error from the Winsorized data distribution for each of the independent group of In the Alexander-Govern (1994) method, the value is converted to standard normal by using the Hill's (1970) normalization approximation and the hypothesis testing of the Winsorized data distribution, where is the usual estimate of the Winsorized sample variance of the WMOM estimator for is expressed as: Thus, the normalization approximation formula for the Alexander-Govern method, using the Winsorized data distribution is expressed as AGWMOM.The test statistic of the AGWMOM for all the groups in the ordered data sample is expressed as: ( The test statistic for the AGWMOM test follows a chi-square distribution at level of significance, having J -1 chi-square degrees of freedom.The p-value can be determined using a chi-square distribution table.

The Variables Used in this Research
There are five different types of variables that were used in this research.They are balanced and unbalanced sample size, equal and unequal variance, group sizes, types of distribution and nature of pairing.All these variables mentioned above, were manipulated to show the strength and the weakness of the original AG test, the AGMOM test, the AGWMOM test, the t-test and the ANOVA.

The Research Design
The Alexander-Govern test is a test that is not robust for non-normal data under variance heterogeneity.For the design of this research, both balanced and unbalanced sample size were paired with equal and unequal variance for two groups (J = 2), for four groups, (J = 4), and for six groups (J = 6), positively and negatively with each of the g-and h-distribution.
For each of the test namely: the AG test, the AGMOM test, the AGWMOM test, the t-test and the ANOVA, 5,000 data sets were simulated in the research design.To obtain the pseudo random variates, SAS generator RANNOR (SAS, Institute, 1999) was used with a nominal level of for the analysis of the tests in this research.The robustness of the Winsorized modified one step M-estimator was obtained by manipulating the five listed ... : ... , ( 1,..., ).
variables as mentioned previously with regards to the Type I error rates and power of the test.

C84
This research design was used to determine the robustness of the modified AG test.By using this research design, the best procedure was obtained.As stated by Lix and Keselman (1998) the empirical rate of Type I error must fall within the interval of to judge the robustness of a given test at level of significance.The range of the value selected for this research gave a strict condition for the robustness of the tests within this interval, with the aim of producing minimum error rate with deviation from model assumptions.
According to Abdullah, Yahaya and Othman (2007) in their research used the range of 0.042 and 0.058 to evaluate the robustness of the tests for their analysis.The interval selected by these researchers, reveal that a test is said to be robust if its Type I error rates falls within the stringent criterion of robustness, otherwise if the test falls outside this interval to Judge the robustness of the test, then the Type I error rates is out of control.According to Bradley's (1978) the lenient criteria of robustness must fall within the range of (0.025 -0.075).
This interval of robustness is also selected in this research, to see those tests that can give remarkable control over the probability of Type I error rates.
In the Table for the Type I error rates for two, four and six group condition, the bolded values represent those values that fall within the stringent criteria of robustness.The un-bolded values represent those values that fall within the lenient criteria of robustness.The red-coloured values represent those values that are not robust at all.They neither fall within the stringent criteria of robustness nor within the lenient criteria of robustness.In Table 5, for g = 0 and h = 0, in all the conditions of pairing, the AG test, the AGMOM test and the AGWMOM test have equal number of Type I error rates that are considered robust compared to the t-test.The t-test have two of its Type I error rates fall within the stringent criteria of robustness and only one of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with equal variance and the pairing of unbalanced sample size with both equal and unequal variance.The remaining two of its Type I error rates are regarded as not robust.
have their Type I error rates fall within the lenient criteria of robustness and the three tests are said to be robust between the interval of 0.025 and 0.075.While the t-test has just one of its Type I error rate, fall within the stringent criteria of robustness, two of its Type I error rates fall within the lenient criteria of robustness and the remaining two of its Type I error rates are considered not robust.In Table 7, for g = 0.5 and h = 0, in all the conditions of pairing, the AG test, the AGMOM test and the AGWMOM test are more robust compared to the t-test.The AG test and the AGMOM test have all of their Type I error rates fall within the stringent criteria of robustness.While the AGWMOM test have two of its Type I error rates fall within the stringent criteria of robustness and the remaining three of its Type I error rates fall within the lenient criteria of robustness.The t-test have two of its Type I error rates fall within the stringent of robustness, only one of its Type I error rates fall within the lenient criteria of robustness and the remaining two of its Type I error rates are considered not to be robust.In table 8, for g = 0.5 and h = 0.5, in all the conditions of pairing, the AG, the AGMOM and the AGWMOM test have equal number of Type I error rates that are regarded as robust, compared to the t-test.The t-test only has one of its Type I error rate fall within the stringent criteria of robustness and the other one of its Type I error rates fall within the lenient criteria of robustness.The remaining three of its Type I error rates are considered not robust.In table 9, for g = 0 and h = 0, in all the conditions of pairing, the AG test, the AGMOM test and the AGWMOM have equal number of Type I error rates that are said to be robust, compared to the ANOVA.The ANOVA has two of its Type I error rates fall within the stringent criteria of robustness, the other two of its Type I error rates fall within the lenient criteria of robustness.The remaining four of its Type I error rates are said not to be robust.In Table 10, for g = 0 and h = 0.5, the AGWMOM test is more robust compared to the AG test, the AGMOM test and the ANOVA.The AGWMOM test have five of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with unequal variance and the pairing of unbalanced sample size with equal variance and the pairing of unbalanced sample size with unequal variance, positively and negatively.For the AG test, three of its Type I error rates fall within the lenient criteria of robustness and are said to be robust.The AGMOM test have only two of its Type I error rates fall within the lenient criteria of robustness, the remaining six of its Type I error rates are considered not robust.The ANOVA have only one of its Type I error rate fall within the stringent criteria of robustness, the other two of its Type I error rates fall within the lenient criteria of robustness.While the remaining five of its Type I error rates are considered not robust.
Table 11.Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA under a skewed normal tailed distribution for four group condition.For the AGMOM and the AGWMOM test, all their Type I error rates fall within the stringent criteria of robustness, in all the conditions of pairing.The AG test has five of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with both equal variance and unequal variance.Also, with the pairing of unbalanced sample with equal variance and unbalanced sample size with unequal variance for negative pairing condition only.For the ANOVA, three of its Type I error rates fall within the stringent criteria of robustness and the remaining five of its Type I error rates are regarded as not robust.

Balanced and
Table 12.Comparison of the Type I error for the AG, the AGMOM, the AGWMOM and the ANOVA under skewed heavy tailed distribution for four group condition.In Table 12, for g = 0.5 and h = 0.5, the AG test is more robust compared to the AGMOM test, the AGWMOM test and the ANOVA.The AG test have all its Type I error rates fall within the lenient criteria of robustness, in all the conditions of pairing.The AGMOM test have five of its Type I error rates fall within the lenient criteria of robustness and the remaining three of its Type I error rates are regarded as not robust.The AGWMOM test have six of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with unequal variance and the pairing of unbalanced sample size with equal variance.Also, with the pairing of unbalanced sample size with unequal variance, positively and negatively.The ANOVA has only one of its' Type I error rates fall within the stringent criteria of robustness, the other one of its Type I error rates fall within the lenient criteria of robustness.The remaining six of its Type I error rates are considered not robust.In Table 13, for g = 0 and h = 0, the four different tests have equal number of Type I error rates that are said to be robust.With the combination of balanced sample size with both equal and unequal variance, the AG test and AGMOM test have three of their Type I error rates fall within the stringent criteria of robustness.While the remaining five of their Type I error rates are said not to be robust.The AGWMOM test has only one of its Type I error rate fall within the stringent criteria of robustness, the other two of its Type I error rates fall within the lenient of robustness.The remaining five of its Type I error rates are considered not robust.The ANOVA have two of its Type I error rates fall within the stringent criteria of robustness, the other one of its Type I error rates fall within the lenient criteria of robustness.The remaining five of its Type I error rates are regarded as not robust.

Balanced and
Table 14.The Comparison of the Type I error rates, for the AG test, AGMOM test, the AGWMOM test and the ANOVA, under symmetric heavy tailed distribution, for six group condition.In Table 14, for g = 0 and h = 0.5, the AG test is more robust compared to the AGMOM test, the AGWMOM test and the ANOVA.The AG test have four of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with both equal and unequal variance and the pairing of unbalanced sample size with unequal variance, for positive pairing only.The AGMOM test has only one of its Type I error rate fall within the stringent criteria of robustness.The remaining seven of its Type I error rates are considered not robust.For the AGWMOM test, only one of its Type I error rates fall within the stringent criteria of robustness, the other two of its Type I error rates fall within lenient criteria of robustness.The remaining five of its Type I error rates are regarded as not robust.The ANOVA have three of its Type I error rates fall within the lenient criteria of robustness, the remaining five of its Type I error rates are said not to be robust.In Table 15, for g = 0.5 and h = 0, the AGMOM test, the AGWMOM test and the ANOVA have equal number of Type I error rates that are said to be robust, compared to the AG test.For both the AGMOM and the AGWMOM test, with the combination of balanced sample size with both equal and unequal variance, their Type I error rates fall within the stringent criteria of robustness.The AG test have two of its Type I error fall within the lenient criteria of robustness and the remaining six of its Type I error rates are regarded as not robust.

AG AGMOM AGWMOM ANOVA
The ANOVA has only one of its Type I error rate fall within stringent criteria of robustness and the other two of its Type I error rates fall within the lenient criteria of robustness.While the remaining five of the Type I error rates of the ANOVA are considered not robust.In Table 16, for g = 0.5 and h = 0.5, the AGWMOM test is more robust compared to the AG test, the AGMOM test and the ANOVA.The AGWMOM test have three of its Type I error rates fall within the lenient criteria of robustness and only one of its Type I error rate fall within the stringent criteria of robustness, with the combination of balanced sample size with both equal and unequal variance and the pairing of unbalanced sample size with unequal variance, for positive pairing only.The AG test have three of its Type I error rates fall within the lenient criteria of robustness and the remaining five of its Type I error rates are regarded as not robust.The AGMOM test has only one of its Type I error rate fall within the stringent criteria of robustness and the remaining seven of its Type I error rates are considered not robust.The ANOVA has only one of its Type I error rate fall within the stringent criteria of robustness and the other one of its Type I error rate fall within the lenient criteria of robustness.The remaining six of its Type I error rates are regarded as not robust.In Table 17, SR represents the number of each test that fall within the stringent criteria of robustness, LR represents the number of each test that fall within the lenient criteria of robustness and NR represents the number of each test that are considered not robust.
Note.We have eighty-four conditions of pairing from the research design i.e C1 -C84 to determine the Type I error rates of each of the tests in this research.

Discussion and Conclusion
In this research, the stringent criteria of robustness to give a strict control of Type I error rates must fall within the interval of 0.042 -0.058, as stated by Lix and Keselman (1998) and the lenient criteria of robustness must fall within the interval of 0.025 -0.075, as discussed by Bradley's (1978).These two conditions were selected in this research to give a strict control of the Type I error rates for the test.
In Table 17, the overall results of the Type I error rates shows 56 out of 84 of the Type I error rates of the AG test are said to be robust, where 21 of its Type I error rates fall within the stringent criteria of robustness, 35 of its Type I error rates fall within the lenient criteria of robustness and 28 of its Type I error rates are regarded as not robust.The AGMOM test has 51 out 84 of its Type I error rates that are said to be robust, where 32 of its Type I error rates fall within the stringent criteria of robustness, 19 of its Type I error rates fall within the lenient criteria of robustness and 33 of its Type I error rates are considered not robust.The AGWMOM test has 60 out of 84 of its Type I error rates are considered to be robust, where 26 of its Type I error rates fall within the stringent criteria of robustness, 34 of its Type I error rates fall within the lenient criteria of robustness and 24 of its Type I error rates are regarded as not robust.The ANOVA has 34 out of 84 of its Type I error rates that are said to be robust, where 17 of its Type I error rates fall within the stringent criteria of robustness, 17 of its Type I error rates fall within the lenient criteria of robustness and the remaining 50 of its Type I error rates are regarded as not robust.
In all, the AGWMOM test has the highest number of Type I error rates that are regarded as robust.Hence, the AGWMOM test is the best test compared to the other four tests in the control of Type I error rates.Also, for g = 0.5 and h = 0.5, under six group condition, the AGWMOM test has more of its Type I error rates that are said to be robust compared to the AG test, the AGMOM test and the ANOVA.
sample variance for the Winsorized data distribution, is the observed values and is the Winsorized MOM estimator for the Winsorized data distribution.

Table 1 .
The characteristics of the g-and h-distribution

Table 2 .
Research Design for Two Groups

Table 5 .
Comparison of the Type I error rates for the AG test, AGMOM test, AGWMOM test and t-test under normal distribution for two group condition.

Table 6 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the t-test under a symmetric heavy tailed distribution for two group condition.
In Table6, for g = 0 and h = 0.5, in all the conditions of pairing, the AG, AGMOM and the AGWMOM test all

Table 7 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the t-test under a skewed normal tailed distribution for two group condition.

Table 8 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the t-test under a skewed heavy tailed distribution for two group condition.

Table 9 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA under a normal distribution for four group condition.

Table 10 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA under a symmetric heavy tailed distribution, for four group condition.

Unbalanced Sample Size Equal and Unequal Variance AG AGMOM AGWMOM ANOVA
In Table11, both the AGMOM test and the AGWMOM test are more robust compared to the AG test and the ANOVA.

Table 13 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA, for six group condition, under a normal distribution

Table 15 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA under a skewed normal tailed distribution for six group condition

Table 16 .
Comparison of the Type I error rates for the AG test, the AGMOM test, the AGWMOM test and the ANOVA for six group condition, under a skewed heavy tailed distribution

Table 17 .
Overall Summary of the Type I error rates for the five tests, for all the distributions and group sizes: