Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure

Normality and variance homogeneity assumptions are usually the main concern of parametric procedures such as in testing the equality of central tendency measures. Violation of these assumptions can seriously inflate the Type I error rates, which will cause spurious rejection of null hypotheses. Parametric procedures such as ANOVA and t-test rely heavily on the assumptions which are hardly encountered in real data. Alternatively, nonparametric procedures do not rely on the distribution of the data, but the procedures are less powerful. In order to overcome the aforementioned issues, robust procedures are recommended. S₁ statistic is one of the robust procedures which uses median as the location parameter to test the equality of central tendency measures among groups, and it deals with the original data without having to trim or transform the data to attain normality. Previous works on S₁ showed lack of robustness in some of the conditions under balanced design. Hence, the objective of this study is to improve the original S₁ statistic by substituting median with Hodges-Lehmann estimator. The substitution was also done on the scale estimator using the variance of Hodges-Lehmann as well as several robust scale estimators. To examine the strengths and weaknesses of the proposed procedures, some variables like types of distributions, number of groups, balanced and unbalanced group sizes, equal and unequal variances, and the nature of pairings were manipulated. The findings show that all proposed procedures are robust across all conditions for every group case. Besides, three proposed procedures namely S₁(MADn), S₁(Tn) and S₁(Sn) show better performance than the original S₁ procedure under extremely skewed distribution. Overall, the proposed procedures illustrate the ability in controlling the inflation of Type I error. Hence, the objective of this study has been achieved as the three proposed procedures show improvement in robustness under skewed distributions

Robust percentile bootstrap test with modified one-step M-estimator (MOM): An alternative modern statistical analysis

Normality and homoscedasticity are two main assumptions that must be fulfilled when dealing with classical parametric tests for comparing groups. Any violation of the assumptions will cause the results to be invalid. However, in reality, these assumptions are hardly achieved. To overcome such problem, this study proposed to modify a method known as Parametric Bootstrap test by substituting the usual mean, with a highly robust location measure, modified one step M-estimator (MOM). MOM is an asymmetric trimmed mean. The substitution will make the Parametric Bootstrap test more robust for comparing groups. For this study, the trimming criteria for MOM employed two highly robust scale estimators namely MADn and Tn. A simulation study was conducted to investigate on the performance of the proposed method based on Type I error rates. To highlight the strength and weakness of the method, five variables: number of groups, balanced and unbalanced sample sizes, types of distributions, variances heterogeneity and nature of pairings of sample sizes and group variances were manipulated to create various conditions which are common to real life situations.The performance of the proposed method was then compared with the most frequently used parametric and non parametric tests for two (independent sample t-test and Mann Whitney respectively) and more than two independent groups (ANOVA and Kruskal Wallis respectively). The finding of this study indicated that, for two groups, the robust Parametric Bootstrap test performed reasonably well under the conditions of heterogeneous variances with normal or skewed distributions. While for more than two groups, the test generate good Type I error control under heterogeneous variances and skewed distributions. In comparison with the parametric and non parametric methods, the proposed test outperforms its counterparts under non-normal distribution and heterogeneous variances. The performance of each procedure was also demonstrated using real data. In general, the performance of Type I error for the proposed test is very convincing even when the assumptions of normality and homoscedasticity are violated

MATS: Inference for potentially Singular and Heteroscedastic MANOVA

In many experiments in the life sciences, several endpoints are recorded per subject. The analysis of such multivariate data is usually based on MANOVA models assuming multivariate normality and covariance homogeneity. These assumptions, however, are often not met in practice. Furthermore, test statistics should be invariant under scale transformations of the data, since the endpoints may be measured on different scales. In the context of high-dimensional data, Srivastava and Kubokawa (2013) proposed such a test statistic for a specific one-way model, which, however, relies on the assumption of a common non-singular covariance matrix. We modify and extend this test statistic to factorial MANOVA designs, incorporating general heteroscedastic models. In particular, our only distributional assumption is the existence of the group-wise covariance matrices, which may even be singular. We base inference on quantiles of resampling distributions, and derive confidence regions and ellipsoids based on these quantiles. In a simulation study, we extensively analyze the behavior of these procedures. Finally, the methods are applied to a data set containing information on the 2016 presidential elections in the USA with unequal and singular empirical covariance matrices

Robust Statistical Procedures For Testing The Equality Of Central Tendency Parameters Under Skewed Distributions [QA276.A12 S531 2005 f rb].

Kajian ini menyelidik kesan ralat Jenis I dan kuasa keatas dua jenis kaedah teguh. Kaedah pertama dikenali sebagai statistik S1 yang julung kalinya diselidik oleh Babu et al. (1999). This study examined the effect of Type I error and power on two types of robust methods. The first method is known as the S1 statistic, which was first studied by Babu et al. (1999)

The modification and evaluation of the Alexander-Govern test in terms of power

This study centres on the comparison of independent group tests in terms of power, by using parametric method, such as the Alexander-Govern test.The Alexander-Govern (AG) test uses mean as its central tendency measure. It is a better alternative compared to the Welch test, the James test and the ANOVA, because it produces high power and gives good control of Type I error rates for a normal data under variance heterogeneity. But this test is not robust for a non-normal data. When trimmed mean was applied on the test as its central tendency measure under non-normality, the test was only robust for two group condition, but as the number of groups increased more than two groups, the test was no more robust.As a result, a highly robust estimator known as the MOM estimator was applied on the test, as its central tendency measure.This test is not affected by the number of groups, but could not control Type I error rates under skewed heavy tailed distribution.In this study, the Winsorized MOM estimator was applied in the AG test, as its central tendency measure. A simulation of 5,000 data sets were generated and analysed on the test, using the SAS package.The result of the analysis, shows that with the pairing of unbalanced sample size of (15:15:20:30) with equal variance of (1:1:1:1) and the pairing of unbalanced sample size of (15:15:20:30) with unequal variance of (1:1:1:36) with effect size index (f = 0.8), the AGWMOM test only produced a high power value of 0.9562 and 0.8336 compared to the AG test, the AGMOM test and the ANOVA respectively and the test is considered to be sufficient

H-statistic with winsorized modified one-step M-estimator as central tendency measure

Two-sample independent t-test and ANOVA are classical procedures which are widely used to test the equality of two groups and more than two groups respectively. However, these parametric procedures are easily affected by non-normality, becoming more obvious when heterogeneity of variances and unbalanced group sizes exist. It is well known that the violation in the assumption of the tests will lead to inflation in Type I error rate and decreasing in the power of test. Nonparametric procedures like Mann-Whitney and Kruskal-Wallis may be the alternative to the parametric procedures, however, loss of information occur due to the ranking data. In mitigating these problems, robust procedures can be used as the other alternative. One of the procedures is H-statistic. When used with modified one-step M-estimator (MOM), the test statistic (MOM-H) produces good control of Type I error rate even under small sample size but inconsistent under certain conditions investigated. Furthermore, power of test is low which might be due to the trimming process. In this study, MOM was winsorized (WMOM) to retain the original sample size. The Hstatistic when combines with WMOM as the central tendency measure (WMOM-H) shows better control of Type I error rate as compared to MOM-H especially under balanced design regardless of the shape of distributions. It also performs well under highly skewed and heavy tailed distribution for unbalanced design. On top of that, WMOM-H also generates better power value, as compared to MOM-H and ANOVA under most of the conditions investigated. WMOM-H also has better control of Type I error rates with no liberal value (>0.075) compared to the parametric (t-test and ANOVA) and nonparametric (Mann-Whitney and Kruskal-Wallis) procedures. In general, this study demonstrates that winsorization process (WMOM) is able to improve the performance of H-statistic in terms of controlling Type I error rate and increasing power of test

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

Impact of unbalancedness and heteroscedasticity on classic parametric significance tests of two-way fixed-effects ANOVA tests

Classic parametric statistical tests, like the analysis of variance (ANOVA), are powerful tools used for comparing population means. These tests produce accurate results provided the data satisfies underlying assumptions such as homoscedasticity and balancedness, otherwise biased results are obtained. However, these assumptions are rarely satisfied in real-life. Alternative procedures must be explored. This thesis aims at investigating the impact of heteroscedasticity and unbalancedness on effect sizes in two-way fixed-effects ANOVA models. A real-life dataset, from which three different samples were simulated was used to investigate the changes in effect sizes under the influence of unequal variances and unbalancedness. The parametric bootstrap approach was proposed in case of unequal variances and non-normality. The results obtained indicated that heteroscedasticity significantly inflates effect sizes while unbalancedness has non-significant impact on effect sizes in two-way ANOVA models. However, the impact worsens when the data is both unbalanced and heteroscedastic.StatisticsM. Sc. (Statistics

Small Area Shrinkage Estimation

The need for small area estimates is increasingly felt in both the public and private sectors in order to formulate their strategic plans. It is now widely recognized that direct small area survey estimates are highly unreliable owing to large standard errors and coefficients of variation. The reason behind this is that a survey is usually designed to achieve a specified level of accuracy at a higher level of geography than that of small areas. Lack of additional resources makes it almost imperative to use the same data to produce small area estimates. For example, if a survey is designed to estimate per capita income for a state, the same survey data need to be used to produce similar estimates for counties, subcounties and census divisions within that state. Thus, by necessity, small area estimation needs explicit, or at least implicit, use of models to link these areas. Improved small area estimates are found by "borrowing strength" from similar neighboring areas.Comment: Published in at http://dx.doi.org/10.1214/11-STS374 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org