Design considerations for small experiments and simple logistic regression

Inference for a generalized linear model is generally performed using asymptotic approximations for the bias and the covariance matrix of the parameter estimators. For small experiments, these approximations can be poor and result in estimators with considerable bias. We investigate the properties of designs for small experiments when the response is described by a simple logistic regression model and parameter estimators are to be obtained by the maximum penalized likelihood method of Firth [Firth, D., 1993, Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38]. Although this method achieves a reduction in bias, we illustrate that the remaining bias may be substantial for small experiments, and propose minimization of the integrated mean square error, based on Firth's estimates, as a suitable criterion for design selection. This approach is used to find locally optimal designs for two support points.     

A comparison of three methods of estimating the logistic regression coefficients

A Sampling experiment performed using data collected for a large clinical trial shows that the discriminant function estimates of the logistic regression coefficients for discrete variables may be severely biased. The simulations show that the mixed variable location model coefficient estimates have bias which is of the same magnitude as the bias in the coefficient estimates obtained using conditional maximum likelihood estimates but require about one-tenth of the computer time.     

Bias Correction in the Type I Generalized Logistic Distribution

Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n ^?1; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.     

Sharpening estimators using resampling

We study ways to improve a given estimator using resampling methods like the jackknife or the bootstrap in terms of bias and the mean square error. Our key task is to devise a method to empirically check whether the bias correction employed leads to an increase or decrease in the mean square error in terms of second-order asymptotics. We derive conditions under which we can sharpen the given estimator in terms of bias and the mean square error. One may attempt to verify the condition empirically using resampling methods.     

Target estimation for the logistic regression model

The method of target estimation developed by Cabrera and Fernholz [(1999). Target estimation for bias and mean square error reduction. The Annals of Statistics, 27(3), 1080–1104.] to reduce bias and variance is applied to logistic regression models of several parameters. The expectation functions of the maximum likelihood estimators for the coefficients in the logistic regression models of one and two parameters are analyzed and simulations are given to show a reduction in both bias and variability after targeting the maximum likelihood estimators. In addition to bias and variance reduction, it is found that targeting can also correct the skewness of the original statistic. An example based on real data is given to show the advantage of using target estimators for obtaining better confidence intervals of the corresponding parameters. The notion of the target median is also presented with some applications to the logistic models.     

Bias of maximum likelihood estimators of the response standard deviation in regression

The bias of maximum likelihood estimators of the standard deviation of the response in location/scale regression models is considered. Results are obtained for a very wide family of densities for the response variable. These are used to propose point estimators with improved mean square error properties and to demonstrate the importance of bias correction in statistical inference when samples are moderately small.     

Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data

Abstract. In this article, a naive empirical likelihood ratio is constructed for a non‐parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias‐corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual‐adjusted empirical log‐likelihood ratio is shown to be asymptotically chi‐squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data‐driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation‐based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.     

Bias in Small-Sample Inference With Count-Data Models

Abstract

Both Poisson and negative binomial regression can provide quasi-likelihood estimates for coefficients in exponential-mean models that are consistent in the presence of distributional misspecification. It has generally been recommended, however, that inference be carried out using asymptotically robust estimators for the parameter covariance matrix. As with linear models, such robust inference tends to lead to over-rejection of null hypotheses in small samples. Alternative methods for estimating coefficient estimator variances are considered. No one approach seems to remove all test bias, but the results do suggest that the use of the jackknife with Poisson regression tends to be least biased for inference.     

Data-based interval estimation of classification error rates

Leave-one-out and 632 bootstrap are popular data-based methods of estimating the true error rate of a classification rule, but practical applications almost exclusively quote only point estimates. Interval estimation would provide better assessment of the future performance of the rule, but little has been published on this topic. We first review general-purpose jackknife and bootstrap methodology that can be used in conjunction with leave-one-out estimates to provide prediction intervals for true error rates of classification rules. Monte Carlo simulation is then used to investigate coverage rates of the resulting intervals for normal data, but the results are disappointing; standard intervals show considerable overinclusion, intervals based on Edgeworth approximations or random weighting do not perform well, and while a bootstrap approach provides intervals with coverage rates closer to the nominal ones there is still marked underinclusion. We then turn to intervals constructed from 632 bootstrap estimates, and show that much better results are obtained. Although there is now some overinclusion, particularly for large training samples, the actual coverage rates are sufficiently close to the nominal rates for the method to be recommended. An application to real data illustrates the considerable variability that can arise in practical estimation of error rates.     

Exact expressions for the weights used in least-squares regression estimation for the log-logistic and Weibull distribution

Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting, and this technique outperforms the maximum likelihood (ML) estimation often in small samples. This leads to a highly heteroskedastic regression problem. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples, the ML performs best, but it is shown that in smaller samples, the weighted regression outperforms the ML estimation with respect to bias and mean square error.     

Strategies for handling missing data in longitudinal studies with questionnaires

Missing data methods, maximum likelihood estimation (MLE) and multiple imputation (MI), for longitudinal questionnaire data were investigated via simulation. Predictive mean matching (PMM) was applied at both item and scale levels, logistic regression at item level and multivariate normal imputation at scale level. We investigated a hybrid approach which is combination of MLE and MI, i.e. scales from the imputed data are eliminated if all underlying items were originally missing. Bias and mean square error (MSE) for parameter estimates were examined. ML seemed to provide occasionally the best results in terms of bias, but hardly ever on MSE. All imputation methods at the scale level and logistic regression at item level hardly ever showed the best performance. The hybrid approach is similar or better than its original MI. The PMM-hybrid approach at item level demonstrated the best MSE for most settings and in some cases also the smallest bias.     

Maximum Likelihood Estimation of Logistic Regression Parameters under Two-phase, Outcome-dependent Sampling

Outcome-dependent sampling increases the efficiency of studies of rare outcomes, examples being case—control studies in epidemiology and choice–based sampling in econometrics. Two-phase or double sampling is a standard technique for drawing efficient stratified samples. We develop maximum likelihood estimation of logistic regression coefficients for a hybrid two-phase, outcome–dependent sampling design. An algorithm is given for determining the estimates by repeated fitting of ordinary logistic regression models. Simulation results demonstrate the efficiency loss associated with alternative pseudolikelihood and weighted likelihood methods for certain data configurations. These results provide an efficient solution to the measurement error problem with validation sampling based on a discrete surrogate.     

Reducing Bias and Mean Squared Error Associated With Regression-Based Odds Ratio Estimators

Ratio estimators of effect are ordinarily obtained by exponentiating maximum-likelihood estimators (MLEs) of log-linear or logistic regression coefficients. These estimators can display marked positive finite-sample bias, however. We propose a simple correction that removes a substantial portion of the bias due to exponentiation. By combining this correction with bias correction on the log scale, we demonstrate that one achieves complete removal of second-order bias in odds ratio estimators in important special cases. We show how this approach extends to address bias in odds or risk ratio estimators in many common regression settings. We also propose a class of estimators that provide reduced mean bias and squared error, while allowing the investigator to control the risk of underestimating the true ratio parameter. We present simulation studies in which the proposed estimators are shown to exhibit considerable reduction in bias, variance, and mean squared error compared to MLEs. Bootstrapping provides further improvement, including narrower confidence intervals without sacrificing coverage.     

Editorial collaborators

A simulation study of the binomial-logit model with correlated random effects is carried out based on the generalized linear mixed model (GLMM) methodology. Simulated data with various numbers of regression parameters and different values of the variance component are considered. The performance of approximate maximum likelihood (ML) and residual maximum likelihood (REML) estimators is evaluated. For a range of true parameter values, we report the average biases of estimators, the standard error of the average bias and the standard error of estimates over the simulations. In general, in terms of bias, the two methods do not show significant differences in estimating regression parameters. The REML estimation method is slightly better in reducing the bias of variance component estimates.     

Bootstrap methods for bias correction and confidence interval estimation for nonlinear quantile regression of longitudinal data

This paper examines the use of bootstrapping for bias correction and calculation of confidence intervals (CIs) for a weighted nonlinear quantile regression estimator adjusted to the case of longitudinal data. Different weights and types of CIs are used and compared by computer simulation using a logistic growth function and error terms following an AR(1) model. The results indicate that bias correction reduces the bias of a point estimator but fails for CI calculations. A bootstrap percentile method and a normal approximation method perform well for two weights when used without bias correction. Taking both coverage and lengths of CIs into consideration, a non-bias-corrected percentile method with an unweighted estimator performs best.     

Testing Inference from Logistic Regression Models in Data with Unobserved Heterogeneity at Cluster Levels

Clustering due to unobserved heterogeneity may seriously impact on inference from binary regression models. We examined the performance of the logistic, and the logistic-normal models for data with such clustering. The total variance of unobserved heterogeneity rather than the level of clustering determines the size of bias of the maximum likelihood (ML) estimator, for the logistic model. Incorrect specification of clustering as level 2, using the logistic-normal model, provides biased estimates of the structural and random parameters, while specifying level 1, provides unbiased estimates for the former, and adequately estimates the latter. The proposed procedure appeals to many research areas.     

Delete-m Jackknife for Unequal m

In this paper, the delete-mj jackknife estimator is proposed. This estimator is based on samples obtained from the original sample by successively removing mutually exclusive groups of unequal size. In a Monte Carlo simulation study, a hierarchical linear model was used to evaluate the role of nonnormal residuals and sample size on bias and efficiency of this estimator. It is shown that bias is reduced in exchange for a minor reduction in efficiency. The accompanying jackknife variance estimator even improves on both bias and efficiency, and, moreover, this estimator is mean-squared-error consistent, whereas the maximum likelihood equivalents are not.     

Analysis of ordered probit model with surrogate response data and measurement error in covariates

ABSTRACT

Often in data arising out of epidemiologic studies, covariates are subject to measurement error. In addition ordinal responses may be misclassified into a category that does not reflect the true state of the respondents. The goal of the present work is to develop an ordered probit model that corrects for the classification errors in ordinal responses and/or measurement error in covariates. Maximum likelihood method of estimation is used. Simulation study reveals the effect of ignoring measurement error and/or classification errors on the estimates of the regression coefficients. The methodology developed is illustrated through a numerical example.     

Logistic discrimination using robust estimators: An influence function approach

Logistic regression is frequently used for classifying observations into two groups. Unfortunately there are often outlying observations in a data set and these might affect the estimated model and the associated classification error rate. In this paper, the authors study the effect of observations in the training sample on the error rate by deriving influence functions. They obtain a general expression for the influence function of the error rate, and they compute it for the maximum likelihood estimator as well as for several robust logistic discrimination procedures. Besides being of interest in their own right, the influence functions are also used to derive asymptotic classification efficiencies of different logistic discrimination rules. The authors also show how influential points can be detected by means of a diagnostic plot based on the values of the influence function