Likelihood methods in randomized trials with noncompliance and subsequent nonresponse

This article considers likelihood methods for estimating the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. We first derive the observed data likelihood function as a closed form expression of the parameter given the observed data where both response and compliance state are treated as variables with missing values. Then we describe an iterative procedure which maximizes the observed data likelihood function directly to compute a maximum likelihood estimator (MLE) of the causal effect of treatment assignment. Closed form expressions at each iterative step are provided. Finally we compare the MLE with an alternative estimator where the probability distribution of the compliance state is estimated independent of the response and its missingness mechanism. Our work indicates that direct maximum likelihood inference is straightforward for this problem. Extensive simulation studies are provided to examine the finite sample performance of the proposed methods.     

Parameter estimation for the dirichlet-multinomial distribution using supplementary beta-binomial data

We develop estimates for the parameters of the Dirichlet-multinomial distribution (DMD) when there is insufficient data to obtain maximum likelihood or method of moment estimates known in the literature. We do, however, have supplemetary beta-binomial data pertaining to the marginals of the DMD, and use these data when estimating the DMD parameters. A real situation and data set are given where our estimates are applicable.     

Expected estimating equations to accommodate covariate measurement error

Estimating equations which are not necessarily likelihood-based score equations are becoming increasingly popular for estimating regression model parameters. This paper is concerned with estimation based on general estimating equations when true covariate data are missing for all the study subjects, but surrogate or mismeasured covariates are available instead. The method is motivated by the covariate measurement error problem in marginal or partly conditional regression of longitudinal data. We propose to base estimation on the expectation of the complete data estimating equation conditioned on available data. The regression parameters and other nuisance parameters are estimated simultaneously by solving the resulting estimating equations. The expected estimating equation (EEE) estimator is equal to the maximum likelihood estimator if the complete data scores are likelihood scores and conditioning is with respect to all the available data. A pseudo-EEE estimator, which requires less computation, is also investigated. Asymptotic distribution theory is derived. Small sample simulations are conducted when the error process is an order 1 autoregressive model. Regression calibration is extended to this setting and compared with the EEE approach. We demonstrate the methods on data from a longitudinal study of the relationship between childhood growth and adult obesity.     

Measurement error in the generalised linear model

This paper considers the problem of estimating the linear parameters of a Generalised Linear Model (GLM) when the explanatory variable is subject to measurement error. In this situation the induced model for dependence on the approximate explanatory variable is not usually of GLM form. However, when the distribution of measurement error is known or estimated from replicated measurements, application of the GLIM iteratively reweighted least squares algorithm with transformed data and weighting is shown to produce maximum quasi likelihood estimates in many cases. Details of this approach are given for two particular generalized linear models; simulation results illustrate the usefulness of the theory for these models.     

The robustness of the quasilikelihood estimator

The quasilikelihood estimator is widely used in data analysis where a likelihood is not available. We illustrate that with a given variance function it is not only conservative, in minimizing a maximum risk, but also robust against a possible misspecification of either the likelihood or cumulants of the model. In examples it is compared with estimators based on maximum likelihood and quadratic estimating functions.     

Analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable. We develop the essential methodology for estimating the model parameters via maximum likelihood method. The general form of the maximum likelihood estimator is obtained in color closed form. Adjusted treatment effects and adjusted covariate effects are given, too. We also provide the asymptotic distribution of the proposed estimators. A simulation study and a real world application are also performed to illustrate and evaluate the proposed methodology.     

Comparison of methods of estimation of parameters of the weibull distribution

The generalised least squares, maximum likelihood, Bain-Antle 1 and 2, and two mixed methods of estimating the parameters of the two-parameter Weibull distribution are compared. The comparison is made using (a) the observed relative efficiency of parameter estimates and (b) themean squared relative error in estimated quantiles, to summarize the results of 1000 simulated samples of sizes 10 and 25. The results are that: generalised least squares is the best method of estimating the shape parameter ß the best method of estimating the scale parameter a depends onthe size of ß for quantile estimation maximum likelihood is best Bain-Antle 2 is uniformly the worst of the methods.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Maximum likelihood estimation for mixtures of two gompertz distributions when censoring occurs

In estimating the proportion ‘cured’ after adjuvant treatment, a population of cancer patients can be assumed to be a mixture of two Gompertz subpopulations, those who will die of other causes with no evidence of disease relapse and those who will die of their primary cancer. Estimates of the parameters of the component dying of other causes can be obtained from census data, whereas maximum likelihood estimates for the proportion cured and for the parameters of the component of patients dying of cancer can be obtained from follow-up data.

This paper examines, through simulation of follow-up data, the feasibility of maximum likelihood estimation of a mixture of two Gompertz distributions when censoring occurs. Means, variances and mean square error of the maximum likelihood estimates and the estimated asymptotic variance-covariance matrix is obtained from the simulated samples. The relationship of these variances with sample size, proportion censored, mixing proportion and population parameters are considered.

Moderate sample size typical of cooperative trials yield clinically acceptable estimates. Both increasing sample size and decreasing proportion of censored data decreases variance and covariance of the unknown parameters. Useful results can be obtained with data which are as much as 50% censored. Moreover, if the sample size is sufficiently large, survival data which are as much as 70% censored can yield satisfactory results.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Coordinate-free analysis of trends in British social mobility

This paper is intended to make a contribution to the ongoing debate about declining social mobility in Great Britain by analyzing mobility tables based on data from the 1991 British Household Panel Survey and the 2005 General Household Survey. The models proposed here generalize Hauser's levels models and allow for a semi-parametric analysis of change in social mobility. The cell frequencies are assumed to be equal to the product of three effects: the effect of the father's position for the given year, the effect of the son's position for the given year, and the mobility effect related to the difference between the father's and the son's positions. A generalization of the iterative proportional fitting procedure is proposed and applied to computing the maximum likelihood estimates of the cell frequencies. The standard errors of the estimated parameters are computed under the product-multinomial sampling assumption. The results indicate opposing trends of mobility between the two timepoints. Fewer steps up or down in the society became less likely, while more steps became somewhat more likely.     

Estimation in a Multivariable Two-Phase Regression

A likelihood approach is considered for the problems of estimating the changepoint and other parameters in a multivariable two-phase regression.Methods for finding the maximum likelihood estimates are given for the cases when the covariance matrix is known, and unknown.The distribution of the usual likelihood ratio test statistic is Investigated using simulations, and a Monte-Carlo aporoach is suggested for testing for the existence of a change-point.Numerical1 Illistrute aie provided.     

Mean-squared errors of small-area estimators under a unit-level multivariate model

This work deals with estimating the vector of means of certain characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean-squared and cross-product errors of the empirical best linear unbiased predictors of the means, when model parameters are estimated either by maximum likelihood (ML) or by restricted ML. This approach has been implemented on a Monte Carlo study using social and labour data from the Spanish Labour Force Survey.     

A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

Numerical solution of likelihood equations for a first-order response surface model for mixture paired comparisons

Summary A simple procedure for numerical solution of the likelihood equations for estimating the regression parameters of a first-order response surface model for the treatment parameters of mixture paired comparison experiments is developed. It is demonstrated that, for defined rotatable designs, those regression parameters are simple functions of the main effect parameters of a corresponding factorial model with no interactions. The maximum likelihood estimators of those main effect parameters, and hence of their corresponding regression parameters, are obtained through using procedures of treatment contrasts, factorial and iterations. A numerical example is given to illustrate applications of the procedures developed in this paper.     

Asymptotic efficiency of a competing risks model: A two-sample case

A generalized Cox regression model is studied for the covariance analysis of competing risks data subject to independent random censoring. The information of the maximum partial likelihood estimates is compared with that of maximum likelihood estimates assuming a log linear hazard function.The method of generalized variance is used to define the efficiency of estimation between the two models. This is then applied to two-sample problems with two exponentially censoring rates. Numerical results are summarized ane presented graphically.The detailed results indicate that the semi-parametric model wrks well for a higher rate of censoring. A method of generalizing the result to type 1 censoring and the efficiency of estimating the coefficient of the covariate are discussecd. A brief account of using the results to help design experiments is also given.     

A Pairwise Likelihood Procedure for Analyzing Exchangeable Binary Data with Random Cluster Sizes

For the exchangeable binary data with random cluster sizes, we use a pairwise likelihood procedure to give a set of approximately optimal unbiased estimating equations for estimating the mean and variance parameters. Theoretical results are obtained establishing the large sample properties of the solutions to the estimating equations. An application to a developmental toxicity study is given. Simulation results show that the pairwise likelihood procedure is valid and performs better than the GEE procedure for the exchangeable binary data.     

Using BBPSO Algorithm to Estimate the Weibull Parameters with Censored Data

This article proposes the maximum likelihood estimates based on bare bones particle swarm optimization (BBPSO) algorithm for estimating the parameters of Weibull distribution with censored data, which is widely used in lifetime data analysis. This approach can produce more accuracy of the parameter estimation for the Weibull distribution. Additionally, the confidence intervals for the estimators are obtained. The simulation results show that the BB PSO algorithm outperforms the Newton–Raphson method in most cases in terms of bias, root mean square of errors, and coverage rate. Two examples are used to demonstrate the performance of the proposed approach. The results show that the maximum likelihood estimates via BBPSO algorithm perform well for estimating the Weibull parameters with censored data.     

ESTIMATION OF THE CONCENTRATION PARAMETERS OF THE FISHER MATRIX DISTRIBUTION ON 50(3) AND THE BINGHAM DISTRIBUTION ON Sq, q≥ 2

Two procedures are considered for estimating the concentration parameters of the Fisher matrix distribution for rotations or orientations in three dimensions. The first is maximum likelihood. The use of a convenient 1-dimensional integral representation of the normalising constant, which greatly simplifies the computation, is suggested. The second approach exploits the equivalence of the Fisher distribution for rotations in three dimensions, and the Bingham distribution for axes in four dimensions. We describe a pseudo likelihood procedure which works for the Bingham distribution in any dimension. This alternative approach does not require numerical integration. Results on the asymptotic efficiency of the pseudo likelihood estimator relative to the maximum likelihood estimator are given, and the two estimators are compared in the analysis of a well-known vectorcardiography dataset.     

Estimation of odds ratios under order restrictions

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.