Likelihood Ratio Tests For Multilevle Models Under Nonstandard Conditions

Multilevel models are mainly used to deal with hierarchically structured data, so that the outcome variable variation will be divided into within-group variation and between-group variation. This article briefly describes the development process of multilevel models and the scope of application, commonly used methods of parameter estimation and hypothesis testing; and specificly discusses the main models of the two-level models, giving the general form and the vector matrix form for the two-level models, and extending to three or more level models; and then explains relevant knowledge for model comparisons, which leads to the focus of this article, that is, likelihood ratio test.This paper studies the likelihood ratio test under nonstandard conditions, we first give a general model of the distribution function, the parameter space is Euclidean space and has a conical form, while the components for the true value of the parameters can be located on the border; and define the distribution function of the first derivative and second derivative, and under eight assumptions and derived conditions, proving three theorems so that obtain the distribution of statistics.The originality of this article is that the general model is extended to multilevel model, and the model fits hierarchically structured data, especially longitudinal data, so as to solve the hypothesis testing problem for the different distribution of random variables in the multilevel models. The specific content of this article, based on hypothesis testing under nonstandard conditions in the general model, gives similar parameters and parameter space, and hypothesis testing for variance components model with one or two random effects. The log-likelihood function will be the first order derivative and second derivative with respect to the parameters, to verify fewer restrictions. Using the conclusions of previous studies obtain for the asymptotic distribution of likelihood ratio test statistic is an equally weighted mixture of chi-squared distributions with different degrees of freedom, respectively. Case analysis shows the feasibility of the method and practical significance.