Confidence intervals for variance component ratios in unbalanced linear mixed models

Methods for constructing confidence intervals for variance component ratios in general unbalanced mixed models are developed. The methods are based on inverting the distribution of the signed root of the log-likelihood ratio statistic constructed from either the restricted maximum likelihood or the full likelihood. As this distribution is intractable, the inversion is rather based on using a saddlepoint approximation to its distribution. Apart from Wald's exact method, the resulting intervals are unrivalled in terms of achieving accuracy in overall coverage, underage, and overage. Issues related to the proper “reference set” with which to judge the coverage as well as issues connected to variance ratios being nonnegative with lower bound 0 are addressed. Applications include an unbalanced nested design and an unbalanced crossed design.     

A note on inference for P(X < Y) for right truncated exponentially distributed data

In this paper, a likelihood based analysis is developed and applied to obtain confidence intervals and p values for the stress-strength reliability R  =  P(X  <  Y) with right truncated exponentially distributed data. The proposed method is based on theory given in Fraser et al. (Biometrika 86:249–264, 1999) which involves implicit but appropriate conditioning and marginalization. Monte Carlo simulations are used to illustrate the accuracy of the proposed method.     

Confidence Intervals of Effect Size for Randomized Comparative Parallel-Group Studies with Unequal Variances

Recent years have seen a heightened interest in estimating effect size—a common measure of effect magnitude in biomedical research—because of its direct clinical relevance. In this article, three interval estimates of effect size for randomized comparative parallel-group studies with unequal variances are discussed. Two real-life examples illustrate that confidence intervals obtained by three methods are quite different, especially when the sample sizes are small. Simulation results show that confidence intervals generated by the modified signed log-likelihood ratio method yield essentially the exact coverage probabilities, whereas the other two methods, even though they are more popular methods, yield less satisfactory results.     

Estimation for two-way analysis of variance with correlated errors

We consider some estimation and distribution problems encountered in a two way analysis of variance model with only one observation per cell, errors correlated in one level, and the variances are not necessarily equal. The independence criteria for the row and interaction mean sum of squares and distribution of the maximum likelihood estimator of the correlation coefficient are given.     

Third-order likelihood-based inference for the log-normal regression model

This paper examines the general third-order theory to the log-normal regression model. The interest parameter is its conditional mean. For inference, traditional first-order approximations need large sample sizes and normal-like distributions. Some specific third-order methods need the explicit forms of the nuisance parameter and ancillary statistic, which are quite complicated. Note that this general third-order theory can be applied to any continuous models with standard asymptotic properties. It only needs the log-likelihood function. With small sample settings, the simulation studies for confidence intervals of the conditional mean illustrate that the general third-order theory is much superior to the traditional first-order methods.     

A dynamic system for Gompertz model

Two-parameter Gompertz distribution has been introduced as a lifetime model for reliability inference recently. In this paper, the Gompertz distribution is proposed for the baseline lifetimes of components in a composite system. In this composite system, failure of a component induces increased load on the surviving components and thus increases component hazard rate via a power-trend process. Point estimates of the composite system parameters are obtained by the method of maximum likelihood. Interval estimates of the baseline survival function are obtained by using the maximum-likelihood estimator via a bootstrap percentile method. Two parametric bootstrap procedures are proposed to test whether the hazard rate function changes with the number of failed components. Intensive simulations are carried out to evaluate the performance of the proposed estimation procedure.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

Improved Likelihood Inference on Testing the Difference of Non Centrality Parameters of Two Independent Non central t Distributions with Identical Degrees of Freedom

Kim and Hayter (2008 Kim , J. , Hayter , A. J. ( 2008 ). Testing the equality of the non centrality parameters of the two non central t-distributions with identical degrees of freedom . Communications in Statistics: Simulation and Computation 37 : 1709 – 1717 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) showed that testing the difference of the non centralities of two non central t distributions, which have identical degrees of freedom, is the equivalent to testing the difference of two signal-to-noise ratios for two independent samples from different normal distributions. Moreover, from a biomedical research point of view, the above test is also equivalent to testing the existence of the effect size when the sample sizes of the control group and treatment group are the same. In this article, two third-order likelihood-based methods are proposed to approximate the p-value for this significance test problem. A real life example is used to illustrate that results from the proposed methods and the existing methods can be quite different. Simulation results show that the proposed methods give remarkable accuracy even when sample size reaches the smallest possible value.     

M-estimator-based robust estimation of the number of components of a superimposed sinusoidal signal model

In this paper, we consider the problem of estimating the number of components of a superimposed nonlinear sinusoids model of a signal in the presence of additive noise. We propose and provide a detailed empirical comparison of robust methods for estimation of the number of components. The proposed methods, which are robust modifications of the commonly used information theoretic criteria, are based on various M-estimator approaches and are robust with respect to outliers present in the data and heavy-tailed noise. The proposed methods are compared with the usual non-robust methods through extensive simulations under varied model scenarios. We also present real signal analysis of two speech signals to show the usefulness of the proposed methodology.     

Chi-Squared Goodness-of-Fit Tests: Cell Selection and Power

To use the Pearson chi-squared statistic to test the fit of a continuous distribution, it is necessary to partition the support of the distribution into k cells. A common practice is to partition the support into cells with equal probabilities. In that case, the power of the chi-squared test may vary substantially with the value of k. The effects of different values of k are investigated with a Monte Carlo power study of goodness-of-fit tests for distributions where location and scale parameters are estimated from the observed data. Allowing for the best choices of k, the Pearson and log-likelihood ratio chi-squared tests are shown to have similar maximum power for wide ranges of alternatives, but this can be substantially less than the power of other well-known goodness-of-fit tests.