Maximum Likelihood Estimates Of Coefficients Research Articles

Multilevel Analysis with Few Clusters: Improving Likelihood-Based Methods to Provide Unbiased Estimates and Accurate Inference

AbstractQuantitative comparative social scientists have long worried about the performance of multilevel models when the number of upper-level units is small. Adding to these concerns, an influential Monte Carlo study by Stegmueller (2013) suggests that standard maximum-likelihood (ML) methods yield biased point estimates and severely anti-conservative inference with few upper-level units. In this article, the authors seek to rectify this negative assessment. First, they show that ML estimators of coefficients are unbiased in linear multilevel models. The apparent bias in coefficient estimates found by Stegmueller can be attributed to Monte Carlo Error and a flaw in the design of his simulation study. Secondly, they demonstrate how inferential problems can be overcome by usingrestrictedML estimators for variance parameters and at-distribution with appropriate degrees of freedom for statistical inference. Thus, accurate multilevel analysis is possible within the framework that most practitioners are familiar with, even if there are only a few upper-level units.

Firth's logistic regression with rare events: accurate effect estimates and predictions?

Firth's logistic regression has become a standard approach for the analysis of binary outcomes with small samples. Whereas it reduces the bias in maximum likelihood estimates of coefficients, bias towards one-half is introduced in the predicted probabilities. The stronger the imbalance of the outcome, the more severe is the bias in the predicted probabilities. We propose two simple modifications of Firth's logistic regression resulting in unbiased predicted probabilities. The first corrects the predicted probabilities by a post hoc adjustment of the intercept. The other is based on an alternative formulation of Firth's penalization as an iterative data augmentation procedure. Our suggested modification consists in introducing an indicator variable that distinguishes between original and pseudo-observations in the augmented data. In a comprehensive simulation study, these approaches are compared with other attempts to improve predictions based on Firth's penalization and to other published penalization strategies intended for routine use. For instance, we consider a recently suggested compromise between maximum likelihood and Firth's logistic regression. Simulation results are scrutinized with regard to prediction and effect estimation. We find that both our suggested methods do not only give unbiased predicted probabilities but also improve the accuracy conditional on explanatory variables compared with Firth's penalization. While one method results in effect estimates identical to those of Firth's penalization, the other introduces some bias, but this is compensated by a decrease in the mean squared error. Finally, all methods considered are illustrated and compared for a study on arterial closure devices in minimally invasive cardiac surgery. Copyright © 2017 John Wiley & Sons, Ltd.

On the Information Content of Partial Observations: A Bayesian Approach

On the Information Content of Partial Observations: A Bayesian Approach

A variable risk clinical prognostic compartmental model

A model for predicting the occurrence of disease endpoints from a knowledge of baseline variables and intermediate events is described and illustrated with numerical examples. Maximum likelihood equations are developed and maximum likelihood estimates of coefficients in risk functions of variables included in a stepwise upward procedure are applied to a small set of data for illustrative purposes.

A Hospital's Outputs as a Function of Supply and Demand Characteristics

The occupancy rate, the case census of necessary and discretionary patients, and the diagnosis-specific average length of stay in nonprofit hospitals are analyzed as functions of the socioeconomic and demographic characteristics of the population within the hospital's catchment area, the supply and specialty structure of the physician manpower in the area, and the hospital's capacity to provide laborand capital-intensive medical care. The data are taken from a survey of hospitals participating in the Professional Activities Study. A statistical estimating procedure is used that provides maximum-likelihood estimates of coefficients for a measurement model of demandfor inpatient care and a set of structural equations linking that demand to the hospital output measures. The results indicate that different hospital outputs respond to different aspects of the characteristics of the population-at-risk.

The maximum likelihood estimation of coefficient of diffusion in a birth and diffusion process

The individuals of a population reproduce according to the linear birth process and simultaneously diffuse on the real line according to a Brownian motion process. The maximum likelihood estimator of the coefficient of diffusion is shown to be unbiased, strongly consistent and asymptotically normal under three different sampling schemes.