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Bootstrap is a widely applicable computational statistical method. The focus of this paper lies in developing a bootstrap method for linear mixed models under homoscedasticity violation (heteroscedasticity) in variance of errors. We assume that the form of heteroscedasticity is unknown. To generate bootstrap response data as close as possible to the actual response data, we transform the marginal residuals to ensure that the variance of the modified marginal residuals in bootstrap samples is an unbiased estimator for the variance of the response variable in the linear mixed models. The consistency is proved, implying that the parameter estimators by the proposed method are asymptotically unbiased. The simulations are conducted with varying error terms and sample sizes to show improvement of the proposed method in parameter estimation. The simulation results are compared to demonstrate that the proposed method outperforms the other considered approaches including the wild method in the heteroscedastic linear mixed models under small samples and performs competitively with the parametric method and both methods outperform the other considered methods in homoscedastic linear mixed models. The proposed bootstrap method is applied to squid data for illustration of its effectiveness. [ABSTRACT FROM AUTHOR]

The second version of Kaiser's Measure of Sampling Adequacy (MSA2) has been widely applied to assess the factorability of data in psychological research. The MSA2 is developed in the population and little is known about its behavior in finite samples. If estimated MSA2s are biased due to sampling errors, misleading inferences on the factorability of data are likely to occur. This study investigates the effect of sampling error on MSA2 estimation by systematically examining the accuracy and fluctuations of MSA2 estimates with simulated continuous and ordered categorical data. Features manipulated included the number of factors, number of variables per factor, factor loading, inter-factor correlations, sample size, number of response categories, skewness of variables, and types of correlation analyzed, Pearson and polychoric correlations. Results revealed that the MSA2s were underestimated due to the effect of sampling error. Severely biased MSA2 occurred when analyzing a large number of weakly correlated variables with insufficient participants. The underestimation of MSA2 became worse with categorized data. Polychoric correlations yielded slightly more accurate but relatively unstable MSA2 estimates compared to Pearson correlations. In practice, researchers need to bear in mind the downward bias of MSA2 estimates and interpret the value of sample MSA2s with caution. [ABSTRACT FROM AUTHOR]