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INFLUENCE OF USING ALTERNATIVE MEANS ON TYPE-I ERROR RATE IN THE COMPARISON OF INDEPENDENT GROUPS.

Byline: H. Mirtagioglu S. Yigit A. Mollaogullari S. Genc and M. Mendes

ABSTRACT

In this study the effect of using trimmed winsorized and modified means instead of arithmetic mean on type-I error rate was investigated when the assumptions of the one-way ANOVA were not satisfied. Therefore random numbers were generated by simulation technique from the populations distributed by Normal (01) Beta (52) and 2 (3) for 3 and 4 groups. The results of 30 000 simulation trials demonstrated that all the means displayed similar type-I error rates when the variances were homogenous regardless of the distribution shape sample size and the number of groups. When homogeneity of variances assumption was not satisfied the most reliable result was obtained by using trimmed mean in terms of keeping the type-I error rate at nominal alpha level and it was followed by modified and winsorized means. Themost biased results were obtained when arithmetic mean was used.

Key words: One-way ANOVA Outlier Non-normality Homogeneity of variance Type-I error.

INTRODUCTION

Outliers in the data set studied may cause serious problems in statistical analysis process (Iglewicz and Hoaglin 1993; Zimmerman 1994; Zimmerman 1998). One of the problems is significant deviation from the assumptions of analysis of variance; namely normality and homogeneity of variance (Jeyaratnam and Othman1985; Sutton 1993; Wilcox 1994; 1996). As a result the statistical differences existing among groups compared cannot be revealed. Analysis of variance (ANOVA) is widely used in practice to compare group means and the arithmetic mean is used for necessary calculations. However outliers have a negative effect on arithmetic mean. There are different strategies to deal with outliers in measured data sets. One of these strategies is to use alternative means instead of arithmetic mean. These are trimmed mean and winsorized mean that are not affected much from outliers (Tukey and McLaughlin 1963; Yuen1974; Huber 1981; Hall and Padmanabhan 1992; Rivest1994; Wilcox 2001). Trimmed mean was introduced to overcome non-normality (especially studying with heavytails) and heterogeneity of variances (Tukey and McLaughlin 1963; Yuen 1974; Wilcox 1994; 1995).In this study the effect of using trimmed mean winsorized mean and a new approach instead of arithmetic mean on type-I error rate in one-way fixed effect ANOVA was investigated.

MATERIALS AND METHODS

The material of this study is random numbers generated from Normal (01) Beta (52) and 2(3) distributions by Monte Carlo simulation technique. RNNOA RNBET and RNCHI functions of IMSL library of Microsoft FORTRAN Developer Studio were used to generate random numbers. Characteristics of these distributions were given in table 1 and figure 1 respectively. We considered k=3 and 4 group cases where each group contained 10 20 30 and 40 observations. Each experimental condition was repeated 30 000 times. It is well known that the trimmed mean is not appropriate for small sample size (nless than 10; Mendes and Yigit 2012). We considered both homogeneity (1:1:1 and 1:1:1:1) and heterogeneity (1:1:4 1:1:10 1:1:1:4 and 1:1:1:10) of the variances on type-I error rates. In order to create outliers in data sets constant numbers were added to some of the random numbers. First type-I error rates for one-way analysis of variance were obtained under these experimental conditions. Type-I error rates regarding ANOVA were calculated by dividing the number of falsely rejected H 0 hypotheses by the total number oftrials (30 000). Then trimmed winsorized and modifiedmeans were used instead of arithmetic mean in ANOVA and type-I error rates were estimated under these circumstances. In the simulation study type-I error rate was determined as 0.05 and the percent of trim was5.00%.

Table 1. Characteristics of populations

###Distributions###Skewness###Kurtosis

###Normal (01)###0.000###0.000

###Beta (52)###-0.592###-0.122

###2 (3)###1.620###3.921

Trimmed mean is one of the robust estimators of the population mean especially when outliers exist in the data set. Therefore this mean is preferred to arithmetic mean in order to reduce the effects of outliers. The trimmed mean is computed after the kth smallest and largest observations are removed from the sample.Test statistics based on trimmed mean arecomputed depending on winsorized variance (Tukey and McLaughlin 1963; Staudte and Sheather 1990). The sample winsorized variance isEquation

RESULTS AND DISCUSSION

Results of the simulations are presented in Tables 2- 4 respectively. Table 2 illustrates represents the empirical type-I error rates of the ANOVA at a=0.05significance level when homogeneity of variance assumption is fulfilled. Table 3 and Table 4 represent type-I error estimates when homogeneity of variance assumption is not met. Bradley (1987) criterion was used to evaluate robustness of these means or approaches. According to Bradley criterion if empirical type-I error estimates of any test fall within the interval [0.5a = a=1.5 a] in this case this test called as robust test. Therefore for 0.05 alpha level the empirical type-I error estimates of a robust test should be between 0.025 and0.075.The type-I error estimates obtained by using arithmetic winsorized trimmed and modified means are generally around 5.00% when homogeneity of varianceswas satisfied regardless of the distribution shapes sample size and the number of groups (Table 2). Thetype-I error rate estimates for arithmetic winsorized trimmed and modified means varied between 4.58-5.27% 4.48-5.21% 4.67-5.51% and 4.48-5.21% respectively. As seen in table 2 all approaches gavesimilar results under these experimental conditions.Therefore it is possible to conclude that all approaches are robust. All approaches except trimmed mean were affected negatively from deviations from homogeneity of variances. This negative effect was more obvious when variance ratio was 1:1:10 for k=3 and 1:1:1:10 for k=4(Table 3 and 4). When there was a small deviation in thehomogeneity of variances (1:1:4 and 1:1:1:4) the type error estimates of all approaches fall within Bradley(1987) criterion. However the most reliable results were obtained when trimmed mean was used. Trimmed mean was followed by the modified mean. When variance ratio increased to 10 (1:1:10 and 1:1:1:10) the type I error rates of all approaches deviated from 5.00% alpha level. However theses deviations were more prominent for arithmetic mean and winsorized mean. Increasing the number of groups to be compared has generally caused higher type I error rates. It is more obvious for winsorized and arithmetic means especially when the variances are 1:1:10 and 1:1:1:10 under Chi-Square distribution with 3 d.f. On the other hand trimmed and modified means were detected as not to be significantlyaffected from the number of groups. Under these conditions most reliable estimations were obtained again by using trimmed and modified means.As can be seen in Table 3 the type I error estimations most approximated for trimmed mean were obtained by using modified mean. The case is more prominent especially when the variance ratios are 1:1:4 for k=3 and 1:1:1:4 for k=4 and the shape of the distribution is Normal or Beta (52). Therefore under these conditions the modified mean can be considered as a good alternative of trimmed mean. At the same time by looking at the steps of calculating trimmed mean and modified mean the modified mean can be considered to be much easier than trimmed mean. This case can be evaluated as modified mean has an important advantage with respect to trimmed mean. Although the most reliable results were obtaining by using trimmed mean if biased results of trimmed mean especially when nless than 10 are taken into consideration it can be interpreted that in Analysis of Variance calculations using modified mean might be considered in case of possessing outliers in data set or the assumptions of the ANOVA test are not satisfied.The present study reflected that the results were similar when the variances were homogenous and all the conditions studied were evaluated together. However the same results were invalid when the variances were not homogenous. When the variances were heterogeneous trimmed mean could be considered regardless of distribution shapes and sample size. The results of this simulation study suggested that the most reliable results were obtained when trimmed mean was used. Trimmed mean was followed by the modified mean. Westfall and Yang (1993) and Wilcox (2003; 2005) reported that testing the hypothesis regarding trimmed mean provides more reliable results in the skewed distributions. The standard error of trimmed mean was determined to be smaller than arithmetic mean in the case of the skewed distributions. Likewise Tukey and McLaughlin (1963) Wilcox (1995) and Luh and Guo (2005) reported that trimmed mean methods could be used efficiently for both non-normality (especially heavy tails) and heterogeneity of variance conditions. On the other hand Wu (2004) reported that a problem occurred when sampling was made from a highly skewed distribution where outliers were more common in the right tail than in the left or vice versa.

Table 2. Type-I error rates (%) under homogeneity of variances.

###k=3###k=4

n###Mean###N (01)###(52)###2 (3)###N (01)###(52)###2 (3)

###X###4.70###4.78###4.58###5.24###4.97###4.83

###X wr###4.69###4.72###4.48###5.21###4.84###4.58

10

###X###5.26###5.22###4.95###5.70###5.22###4.92

###tr

###X new###4.83###4.76###4.65###5.21###4.91###4.76

###X###4.87###5.00###4.62###4.86###5.27###4.61

###X wr###4.88###4.93###4.48###4.81###4.92###4.51

20

###X###5.13###5.11###4.67###5.06###5.04###4.75

###tr

###X new###4.98###4.93###4.48###4.89###5.10###4.54

###X###5.13###4.95###4.77###4.91###4.87###4.68

###X wr###5.19###4.88###4.68###4.93###4.80###4.54

30

###X###5.51###5.09###4.71###5.27###4.97###4.79

###tr

###X new###5.15###4.95###4.70###4.91###4.88###4.56

###X###5.04###4.98###5.17###4.98###4.84###4.84

###X wr###5.03###4.98###5.13###5.03###4.84###4.73

40

###X###5.33###5.15###5.03###5.22###4.85###4.87

###tr

###X new###5.00###4.90###5.05###5.02###4.85###4.77

Table 3. Type-I error rates (%) for k=3 when variances are not homogeneous

###Groups###N(01)###(52)###2 (3)

N###Mean###1:1:4###1:1:10###1:1:4###1:1:10###1:1:4###1:1:10

###X###6.55###8.06###6.86###8.30###6.98###10.09

###X wr###6.42###8.07###6.40###7.85###6.41###8.73

10###X###5.17###5.49###5.87###6.13###6.25###7.68

###tr

###X new###6.13###7.36###6.12###7.49###6.32###8.26

###X###6.17###7.29###6.24###7.84###6.68###8.88

###X wr###6.14###7.26###6.09###7.56###6.14###8.24

20###X###4.69###4.74###5.45###5.56###5.96###7.33

###tr

###X new###5.98###7.02###5.97###7.12###6.21###7.55

###X###5.99###7.40###6.55###7.54###6.46###8.36

###X wr###6.04###7.44###6.38###7.34###6.20###7.93

30

###X

###tr

###5.10###4.82###5.35###5.51###5.91###6.57

###X new###5.80###6.57###5.89###7.05###6.01###7.06

###X###6.16###7.38###6.24###7.38###6.36###7.99

###X wr###6.16###7.36###6.15###7.34###6.09###7.53

40###X###4.75###4.57###4.47###4.40###4.43###5.86

###tr

###X new###5.77###6.12###5.67###6.39###5.89###6.97

Table 4. Type-I error rates (%) for k=4 when variances are not homogeneous

###Groups###N(01)###(52)###2 (3)

N###Mean###1:1:1:4###1:1:1:10###1:1:1:4###1:1:1:10###1:1:1:4###1:1:1:10

###X###6.79###9.03###7.13###8.75###7.29###11.06

###X wr###6.70###9.14###6.86###9.25###6.45###10.08

10

###X###5.49###5.90###5.87###6.25###6.70###7.94

###tr

###X new###6.42###7.67###6.45###8.01###6.59###9.20

###X###6.63###8.41###6.58###9.03###6.81###10.51

###X wr###6.59###9.11###6.84###8.77###6.27###9.39

20

###X tr###4.94###4.53###5.71###6.21###6.19###7.93

###X new###6.19###7.44###6.30###7.89###6.51###8.43

###X###6.62###8.48###6.54###8.36###6.69###9.29

###X wr###6.56###8.65###6.44###8.51###6.62###8.97

30###X###5.23###4.96###5.30###5.43###5.94###6.66

###tr

###X new###6.16###6.39###6.02###7.10###6.04###6.96

###X###6.42###8.18###6.77###8.71###6.71###9.10

###X wr###6.44###8.49###6.65###8.54###6.49###8.95

40###X###4.68###4.57###4.78###4.60###4.53###4.77

###tr

###X new###5.88###6.10###5.71###6.08###5.73###6.76

Conclusions: When the variances are homogeneous any mean can be used. When the variances are not homogeneous the use of trimmed mean and modified mean could be more reliable than the others. The usage of modified mean may be more appropriate especially when there is an outlier problem with small sample size (nless than 10) or small deviation in homogeneity of variances under the distribution which has not very large skewness and kurtosis values.

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