Selection Strategy for Covariance Structure of Random Effects in Linear Mixed‐effects Models

Linear mixed‐effects models are a powerful tool for modelling longitudinal data and are widely used in practice. For a given set of covariates in a linear mixed‐effects model, selecting the covariance structure of random effects is an important problem. In this paper, we develop a joint likelihood‐based selection criterion. Our criterion is the approximately unbiased estimator of the expected Kullback–Leibler information. This criterion is also asymptotically optimal in the sense that for large samples, estimates based on the covariance matrix selected by the criterion minimize the approximate Kullback–Leibler information. Finite sample performance of the proposed method is assessed by simulation experiments. As an illustration, the criterion is applied to a data set from an AIDS clinical trial.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

Empiricial Comparison between Some Model Selection Criteria

Model selection aims to find the best model. Most of the usual criteria are based on goodness of fit and parsimony and aim to maximize a transformed version of likelihood. The situation is less clear when two models are equivalent: are they close to the unknown true model or are they far from it? Based on simulations, we study the results of Vuong's test, Cox's test, AIC and BIC and the ability of these four tests to discriminate between models.     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.     

Mean shift and influence measures in linear measurement error models with stochastic linear restrictions

We present influence diagnostics for linear measurement error models with stochastic linear restrictions using the corrected likelihood of Nakamura in 1990. The case deletion and mean shift outlier models are developed to identify outlying and influential observations. We derive a corrected score test statistic for outlier detection based on mean shift outlier models. The analogs of Cook's distance and likelihood distance are proposed to determine influential observations based on case deletion models. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics and a simulation study has been used to evaluate the performance of the proposed estimators based on the mean squares error criterion and the score test statistic. Finally, a numerical example is given to illustrate the theoretical results.     

A weighted test of internal symmetry

This study examines extensions of McNemar's Test with multinomial responses, and proposes a linear weighting scheme, based on the distance of the response change, that is applied to one of these extensions (Bowker's test). This weighted version of Bowker's test is then appropriate for ordinal response variables. A Monte Carlo simulation was conducted to examine the Type I error rate of the weighted Bowker's test for a cross-classification table based on a five-category ordinal response scale. The weighted Bowker's test was also applied to a data set involving change in student attitudes towards mathematics. The results of the weighted Bowker's test were compared with the results of Bowker's test applied to the same set of data.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

A comparison of robust versions of the AIC based on M-, S- and MM-estimators

Variable selection in the presence of outliers may be performed by using a robust version of Akaike's information criterion (AIC). In this paper, explicit expressions are obtained for such criteria when S- and MM-estimators are used. The performance of these criteria is compared with the existing AIC based on M-estimators and with the classical non-robust AIC. In a simulation study and in data examples, we observe that the proposed AIC with S and MM-estimators selects more appropriate models in case outliers are present.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

On a gmanova model likelihood ratio test criterion

For the generalized MANOVA (GMANOVA) model of Potthoff and Roy (1964), X = BξA + E, Khatri (1966) derives the likelihood ratio test criterion for test-ing the composite double linear null hypothesis CξV = 0, C,V known. This criterion plays an important role in statistics, and several authors have recently studied its further properties. However, Khatri's (1966) de-reviation of the distribution of this criterion is involved. By noting that the GMANOVA model is re-stricted MANOVA model, this paper presents an alter-native simple derivation of the distribution of this criterion. The derivation is based on the generalized Sverdrup's lemma, Kabe (1965).     

Bayesian estimation and influence diagnostics of generalized partially linear mixed-effects models for longitudinal data

This paper develops a Bayesian approach to obtain the joint estimates of unknown parameters, nonparametric functions and random effects in generalized partially linear mixed models (GPLMMs), and presents three case deletion influence measures to identify influential observations based on the φ-divergence, Cook's posterior mean distance and Cook's posterior mode distance of parameters. Fisher's iterative scoring algorithm is developed to evaluate the posterior modes of parameters in GPLMMs. The first-order approximation to Cook's posterior mode distance is presented. The computationally feasible formulae for the φ-divergence diagnostic and Cook's posterior mean distance are given. Several simulation studies and an example are presented to illustrate our proposed methodologies.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

Testing exponentiality using transformed data

In this paper, we introduce a test for uniformity and use it as the second stage of an exact goodness-of-fit test of exponentiality. By simulation, the powers of the proposed test under various alternatives are compared with exponentiality test based on Kullback–Leibler information proposed by Ebrahimi et al. [N. Ebrahimi, M. Habibullah, and E.S. Soofi, Testing exponentiality based on Kullback–Leiber information, J. R. Statist. Soc. Ser. B 54 (1992), pp. 739–748]. The results are impressive, i.e. the proposed test has higher power than the test based on entropy.     

Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

Testing a linear ARMA model against threshold-ARMA models: A Bayesian approach

We introduce a Bayesian approach to test linear autoregressive moving-average (ARMA) models against threshold autoregressive moving-average (TARMA) models. First, the marginal posterior densities of all parameters, including the threshold and delay, of a TARMA model are obtained by using Gibbs sampler with Metropolis–Hastings algorithm. Second, reversible-jump Markov chain Monte Carlo (RJMCMC) method is adopted to calculate the posterior probabilities for ARMA and TARMA models: Posterior evidence in favor of TARMA models indicates threshold nonlinearity. Finally, based on RJMCMC scheme and Akaike information criterion (AIC) or Bayesian information criterion (BIC), the procedure for modeling TARMA models is exploited. Simulation experiments and a real data example show that our method works well for distinguishing an ARMA from a TARMA model and for building TARMA models.     

A test for equality of variances with censored samples

A variance homogeneity test for type II right-censored samples is proposed. The test is based on Bartlett's statistic. The asymptotic distribution of the statistic is investigated. The limiting distribution is that of a linear combination of i.i.d. chi-square variables with 1 degree of freedom. By using simulation, the critical values of the null distribution of the modified Bartlett's statistic for testing the homogeneity of variances of two normal populations are obtained when the sample sizes and censoring levels are not equal. Also, we investigate the properties of the proposed test (size, power and robustness). Results show that the distribution of the test statistic depends on the censoring level. An example of the use of the new methodology in animal science involving reproduction in ewes is provided.     

Some robust anova procedures under heteroscedasticity and nonnormality

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber‘s M-estimators with the Brown-Forsythe test, several robust procedures were developed; these procedures were compared through computer simulation studies with the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test. The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions.     

The Finite-Sample Performance of White's Test for Heteroskedasticity Under Stochastic Regressors

We study the finite-sample properties of White's test for heteroskedasticity in stochastic regression models where explanatory variables are random and not given. We investigate by simulation the effect of non independence of explanatory variables and error term and heteroskedasticity on White's test. A standard bootstrap method in the computationally convenient form is found to work well with respect to the size and power.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

Criteria for Linear Model Selection Based on Kullback's Symmetric Divergence

Model selection criteria are frequently developed by constructing estimators of discrepancy measures that assess the disparity between the 'true' model and a fitted approximating model. The Akaike information criterion (AIC) and its variants result from utilizing Kullback's directed divergence as the targeted discrepancy. The directed divergence is an asymmetric measure of separation between two statistical models, meaning that an alternative directed divergence can be obtained by reversing the roles of the two models in the definition of the measure. The sum of the two directed divergences is Kullback's symmetric divergence. In the framework of linear models, a comparison of the two directed divergences reveals an important distinction between the measures. When used to evaluate fitted approximating models that are improperly specified, the directed divergence which serves as the basis for AIC is more sensitive towards detecting overfitted models, whereas its counterpart is more sensitive towards detecting underfitted models. Since the symmetric divergence combines the information in both measures, it functions as a gauge of model disparity which is arguably more balanced than either of its individual components. With this motivation, the paper proposes a new class of criteria for linear model selection based on targeting the symmetric divergence. The criteria can be regarded as analogues of AIC and two of its variants: 'corrected' AIC or AICc and 'modified' AIC or MAIC. The paper examines the selection tendencies of the new criteria in a simulation study and the results indicate that they perform favourably when compared to their AIC analogues.