An improved and efficient biased estimation technique in logistic regression model

Abstract

In this article, we propose a new improved and efficient biased estimation method which is a modified restricted Liu-type estimator satisfying some sub-space linear restrictions in the binary logistic regression model. We study the properties of the new estimator under the mean squared error matrix criterion and our results show that under certain conditions the new estimator is superior to some other estimators. Moreover, a Monte Carlo simulation study is conducted to show the performance of the new estimator in the simulated mean squared error and predictive median squared errors sense. Finally, a real application is considered.     

A new kind of stochastic restricted biased estimator for logistic regression model

In the logistic regression model, the variance of the maximum likelihood estimator is inflated and unstable when the multicollinearity exists in the data. There are several methods available in literature to overcome this problem. We propose a new stochastic restricted biased estimator. We study the statistical properties of the proposed estimator and compare its performance with some existing estimators in the sense of scalar mean squared criterion. An example and a simulation study are provided to illustrate the performance of the proposed estimator.KEYWORDS: Logistic regression, maximum likelihood estimator, mean squared error matrix, ridge regression, simulation study, stochastic restricted estimatorMathematics Subject Classifications: Primary 62J05, Secondary 62J07     

A characteristic function approach to the biased sampling model,with application to robust logistic regression

Many sampling problems from multiple populations can be considered under the semiparametric framework of the biased, or weighted, sampling model. Included under this framework is logistic regression under case–control sampling. For any model, atypical observations can greatly influence the maximum likelihood estimate of the parameters. Several robust alternatives have been proposed for the special case of logistic regression. However, some current techniques can exhibit poor behavior in many common situations. In this paper a new family of procedures are constructed to estimate the parameters in the semiparametric biased sampling model. The procedures incorporate a minimum distance approach, but are instead based on characteristic functions. The estimators can also be represented as the minimizers of quadratic forms in simple residuals, thus yielding straightforward computation. For the case of logistic regression, the resulting estimators are shown to be competitive with the existing robust approaches in terms of both robustness and efficiency, while maintaining affine equivariance. The approach is developed under the case–control sampling scheme, yet is shown to be applicable under prospective sampling logistic regression as well.     

Multiple imputation of a randomly censored covariate improves logistic regression analysis

Randomly censored covariates arise frequently in epidemiologic studies. The most commonly used methods, including complete case and single imputation or substitution, suffer from inefficiency and bias. They make strong parametric assumptions or they consider limit of detection censoring only. We employ multiple imputation, in conjunction with semi-parametric modeling of the censored covariate, to overcome these shortcomings and to facilitate robust estimation. We develop a multiple imputation approach for randomly censored covariates within the framework of a logistic regression model. We use the non-parametric estimate of the covariate distribution or the semi-parametric Cox model estimate in the presence of additional covariates in the model. We evaluate this procedure in simulations, and compare its operating characteristics to those from the complete case analysis and a survival regression approach. We apply the procedures to an Alzheimer's study of the association between amyloid positivity and maternal age of onset of dementia. Multiple imputation achieves lower standard errors and higher power than the complete case approach under heavy and moderate censoring and is comparable under light censoring. The survival regression approach achieves the highest power among all procedures, but does not produce interpretable estimates of association. Multiple imputation offers a favorable alternative to complete case analysis and ad hoc substitution methods in the presence of randomly censored covariates within the framework of logistic regression.     

The extreme residuals in logistic regression models

Goodness-of-fit tests for logistic regression models using extreme residuals are considered. Approximations to the moments of the Pearson residuals are given for model fits made by maximum likelihood, minimum chi-square and weighted least squares and used to define modified residuals. Approximations to the critical values of the extreme statistics based on the ordinary and modified Pearson residuals are developed and assessed for the case of a single explanatory variable.     

On extending bock's model of logistic regression in the analysis of categorical data

A general class of multiple logistic regression models is reviewed and an extension is proposed which leads to restricted maximum likelihood estimates of model parameters. Examples of thegeneral model are given, with an emphasis placed on the interpretation of the parameters in each case.     

The identification of logistic regression models with errors in the variables

The simple logistic regression model with normal measurement error and normal regressor is shown to be identifiable without any extra information about the measurement error. The multiple logistic regression model with more than one regressor variable measured with error is not identifiable. If the covariance matrix of the measurement error is known up to a scalar factor, the model is identified. Further we discuss why in spite of the identifiability the models cannot be estimated in a reasonable way without extra information about the measurement error.     

Weighted logistic regression and robust analysis of diverse toxicology data

Simpson, Carroll, Zhou and Guth (1996) developed an ordinal response regression approach to meta-analysis of data from diverse toxicology studies, applying the methodology to a database of acute inhalation studies of tetra-chloroethylene. We present an alternative analysis of the same data, with two major differences: (1) interval censored scores are assigned worst-case values, e.g., a score known to be in the interval [0,1] is set equal to 1; and (2) the response is reduced to a binary response (adverse, nonadverse). We explore the stability of the analysis by varying a robustness parameter and graphing the curves traced out by the estimates and confidence intervals.     

Two-parameter ridge estimator in the binary logistic regression

The binary logistic regression is a commonly used statistical method when the outcome variable is dichotomous or binary. The explanatory variables are correlated in some situations of the logit model. This problem is called multicollinearity. It is known that the variance of the maximum likelihood estimator (MLE) is inflated in the presence of multicollinearity. Therefore, in this study, we define a new two-parameter ridge estimator for the logistic regression model to decrease the variance and overcome multicollinearity problem. We compare the new estimator to the other well-known estimators by studying their mean squared error (MSE) properties. Moreover, a Monte Carlo simulation is designed to evaluate the performances of the estimators. Finally, a real data application is illustrated to show the applicability of the new method. According to the results of the simulation and real application, the new estimator outperforms the other estimators for all of the situations considered.     

Restricted ridge estimator in the logistic regression model

It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, Schaefer et al. presented a ridge estimator in the logistic regression model. Making use of the ridge estimator, when some linear restrictions are also present, we introduce a restricted ridge estimator in the logistic regression model. Statistical properties of this newly defined estimator will be studied and comparisons are done in the simulation study in the sense of mean squared error criterion. A real-data example and a simulation study are introduced to discuss the performance of this estimator.     

Functional logistic regression: a comparison of three methods

Functional logistic regression is becoming more popular as there are many situations where we are interested in the relation between functional covariates (as input) and a binary response (as output). Several approaches have been advocated, and this paper goes into detail about three of them: dimension reduction via functional principal component analysis, penalized functional regression, and wavelet expansions in combination with Least Absolute Shrinking and Selection Operator penalization. We discuss the performance of the three methods on simulated data and also apply the methods to data regarding lameness detection for horses. Emphasis is on classification performance, but we also discuss estimation of the unknown parameter function.     

A simpler approach for mediation analysis for dichotomous mediators in logistic regression

Mediation is a hypothesized causal chain among three variables. Mediation analysis for continuous response variables is well developed in the literature, and it can be shown that the indirect effect is equal to the total effect minus the direct effect. However, mediation analysis for categorical responses is still not fully developed. The purpose of this article is to propose a simpler method of analysing the mediation effect among three variables when the dependent and mediator variables are both dichotomous. We propose using the latent variable technique which in turn will adjust for the necessary condition that indirect effect is equal to the total effect minus the direct effect. An intensive simulation study is conducted to compare the proposed method with other methods in the literature. Our theoretical derivation and simulation study show that the proposed approach is simpler to use and at least as good as other approaches provided in the literature. We illustrate our approach to test for the potential mediators on the relationship between depression and obesity among children and adolescents compared to the method in Winship and Mare using National children health survey data 2011–2012.     

Change point problems in the model of logistic regression

The paper considers generalized maximum likelihood asymptotic power one tests which aim to detect a change point in logistic regression when the alternative specifies that a change occurred in parameters of the model. A guaranteed non-asymptotic upper bound for the significance level of each of the tests is presented. For cases in which the test supports the conclusion that there was a change point, we propose a maximum likelihood estimator of that point and present results regarding the asymptotic properties of the estimator. An important field of application of this approach is occupational medicine, where for a lot chemical compounds and other agents, so-called threshold limit values (or TLVs) are specified.We demonstrate applications of the test and the maximum likelihood estimation of the change point using an actual problem that was encountered with real data.     

Identification of multiple influential observations in logistic regression

The identification of influential observations in logistic regression has drawn a great deal of attention in recent years. Most of the available techniques like Cook's distance and difference of fits (DFFITS) are based on single-case deletion. But there is evidence that these techniques suffer from masking and swamping problems and consequently fail to detect multiple influential observations. In this paper, we have developed a new measure for the identification of multiple influential observations in logistic regression based on a generalized version of DFFITS. The advantage of the proposed method is then investigated through several well-referred data sets and a simulation study.     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

The sliced inverse regression algorithm as a maximum likelihood procedure

It is shown that the sliced inverse regression procedure proposed by Li corresponds to the maximum likelihood estimate where the observations in each slice are samples of multivariate normal distributions with means in an affine manifold.