Testing zero variance components is one of the most challenging problems in the context of linear mixed-effects (LME) models. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under this null hypothesis is incorrect because the null is on the boundary of the parameter space. During the last two decades many tests have been proposed to overcome this difficulty, but these tests cannot be easily applied for testing multiple variance components, especially for testing a subset of them. We instead introduce a simple test statistic based on the variance least square estimator of variance components. With this comes a permutation procedure to approximate its finite sample distribution. The proposed test covers testing multiple variance components and any subset of them in LME models. Interestingly, our method does not depend on the distribution of the random effects and errors except for their mean and variance. We show, via simulations, that the proposed test has good operating characteristics with respect to Type I error and power. We conclude with an application of our process using real data from a study of the association of hyperglycemia and relative hyperinsulinemia.

Introduction of complex chemical compounds in the wastewater treatment plants challenged the biological process efficiency. Photocatalyst oxidation was newly introduced to remove the trace organic contaminations. Since their surface area determined the production of hydroxyl radicals, the pure TiO 2 surface area was increased by the preparation of TiO 2 -SiO 2 nanocomposite using sol-gel technique. Later on, the X-ray diffraction, scanning electron microscopy, energy diffraction spectroscopy, Fourier transform infrared spectroscopy, UV-vis reflective dispersion spectroscopy, and adsorption of nitrogen gas were deployed for its characterization. Anatase phase was detected as the dominant form of nanocomposites made in this study. Furthermore, rise of calcination temperature, increased the nanoparticle pore size. Therefore, 600˚C was selected to gain the optimum nanoparticle pore size (around 10 nm). Finally, the photocatalytic activities of the synthesized nanoparticles were evaluated for removing Acid Blue 9 (AB9) dye. The photocatalytic performance of the most optimized ratio TiO 2 -SiO 2 nanocomposites was three times higher than that of pure TiO 2 and even commercial nanoparticle.

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.

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