Case studies dealing with heteroscedasticity in ANOVA tests

Zhang (2015a) conducted a study in trying to use a parametric bootstrap approach (PB) to find solutions on one-way ANOVA in the presence of heteroscedasticity and unequal group sizes without using transformation of data technique. Based on the parametric bootstrap test proposed by Krishnamoorthy, Lu and Mathew (2007), Zhang (2015b) further extended the PB algorithm to a multiple comparison procedure (MCP) to test the equality of factor level means and to do pairwise comparisons of two-way ANOVA in the presence of unequal group sizes and heteroscedastic variances.

Simulation studies pertaining to this effect showed that, under heteroscedasticity assumption, the parametric bootstrap test proved to be one of the best technique for testing equality of factor level means. Furthermore, Zhang (2015a) proposed a parametric bootstrap test for mul-tiple comparison in one-way ANOVA when error variances and group sizes vary. The research showed that a complete solution can be achieved when the proposed parametric bootstrap test

of multiple comparison is used together with the Krishnamoorthy, Lu and Mathew (2007) PB test. The simulation results achieved showed that the multiple comparison procedure and the Type I error of overall test were both close to nominal level.

Moreover, Zhang (2015b) had another study where a parametric bootstrap approach for simul-taneous confidence intervals was proposed for all pairwise multiple comparisons in a two-way unbalanced design with unequal variances. Similarly, simulation results depicted that the Type I error of the multiple comparison test were close to the nominal level, even for small samples.

The proposed method performed better than the Turkey-Kramer procedure under heteroscedas-tic variances and unequal group sizes.

In another study, Xu et al. (2015) proposed a parametric bootstrap (PB) test to compare it with the generalised F (GF) test for testing equal effects of factors of a two-way ANOVA model without interaction in the presence of heteroscedasticity. They used the Monte Carlo simulation to evaluate the powers of tests and the Type I error rates. In their study, it was discovered that, in the presence of heteroscedastic error variances and/ or as the number of factor levels increases, the classical F-test and the generalised F-test yield to serious Type I error properties. However, with the use of the parametric bootstrap (PB) test, the Type I error problems are kept under control. As a result of their research, Xu et al. (2015) concluded that, the parametric bootstrap (PB) test performs satisfactorily better than the generalized F (GF) test in two-way fixed effects models under heteroscedasticity, regardless of the number of factor levels involved, sample sizes or error variance values.

Xu et al. (2013), in their article, considered a two-way ANOVA model with unequal cell fre-quencies without the homoscedasticity assumption. They proposed a parametric bootstrap (PB) approach for testing main and interaction effects, and comparing it with the generalised F (GF) test. As usual, the Monte Carlo simulation was used to evaluate the The Type I error rates and powers of the tests. Their studies showed that the parametric bootstrap test per-formed satisfactorily better than the generalised F-test, even for small samples. As in the earlier studies reviewed, the results of their study indicated that the generalised F test portrayed poor

Type I error properties especially when the number of factor levels or treatment combinations increased.

Wang and Akritas (2011) developed an asymptotic theory for hypotheses testing in high-dimensional analysis of variance (HANOVA) in which the distributions are not specified at all. Most results in the literature have been restricted to observations of no more than two-way designs for continuous data. Wang and Akritas (2011) formulated a way that allowed the re-sponse variable to be either continuous, discrete or categorical. They developed an asymptotic theory to test the main and interaction effects of up to the third order in unbalanced designs with unequal error variances, arbitrary number of factors and unequal sample sizes, using two types of test statistics; one with χ² distribution to test low-dimensional parameters; and other with a limiting normal distribution for testing high-dimensional parameters.

Simulation results carried on the Arabidopsis Thaiana gene expression data show that the pro-posed test statistics performed well in both continuous and discrete HANOVA in terms of type I error accuracy, computing time and power. The ANOVA F-test was affected by unbalancedness and heteroscedasticity. The proposed test portrayed proved to be more powerful, producing reliable type I error rates as well as being computationally user-friendly when compared to the traditional HANOVA methods.

Gaugler and Akritas (2013) proposed a modification in the F-Statistic in testing the significance of the main random effects in two-factor random and mixed effects designs. Under the new test procedures that Gaugler and Akritas (2013) proposed, the symmetry assumption was not made, that is, the interaction term was not assumed independent from the main effect even though the two are uncorrelated in the random effects model. They based their asymptotic the-ory of deriving adjusted F-statistics based on the Neyman-Scott framework taking the notion that the number of factor levels in both factors can be large whereas the sizes of the groups can remain constant. As such, the test statistics can be derived by considering the difference of suitably defined mean squares (MSB-MSE^∗ for the mixed effects and MSB-MSAB for the

Using these newly proposed test statistics under fully nonparametric models, the simulations done proved beyond doubt that these proposed statistics performed sufficiently well in situations where classical F statistic seems to violate the underlying assumptions, especially balancedness, symmetry and homoscedasticity.

On a different occasion, Zhang (2012) proposed a simple and accurate approximate degrees of freedom (ADF) test to address the problem of heteroscedastic two-way ANOVA. This attempt came as a means of amending the bias of blindly employing classical F-tests especially when the ANOVA model is heteroscedastic. In the study, Zhang (2012) noted that simulations reflected that ADF test produces good results in different cell sizes whereas the classical F-tests perform badly in the presence of heteroscedasticity.

All in all, recent study shows that in the presence of heteroscedasticity, F-tests suffer from lack of power, resulting in serious biased conclusions. In an empirical study conducted by Moder (2007) on ANOVA problems, the assumption of equal variance seemed to be more problematic in ANOVA models with wider ratios of standard deviations.