An ordinal logistic regression model with misclassification of the outcome variable and categorical covariate

Ordinal outcomes are frequently employed in diagnosis and clinical trials. Clinical trials of Alzheimer's disease (AD) treatments are a case in point using the status of mild, moderate or severe disease as outcome measures. As in many other outcome oriented studies, the disease status may be misclassified. This study estimates the extent of misclassification in an ordinal outcome such as disease status. Also, this study estimates the extent of misclassification of a predictor variable such as genotype status. An ordinal logistic regression model is commonly used to model the relationship between disease status, the effect of treatment, and other predictive factors. A simulation study was done. First, data based on a set of hypothetical parameters and hypothetical rates of misclassification was created. Next, the maximum likelihood method was employed to generate likelihood equations accounting for misclassification. The Nelder-Mead Simplex method was used to solve for the misclassification and model parameters. Finally, this method was applied to an AD dataset to detect the amount of misclassification present. The estimates of the ordinal regression model parameters were close to the hypothetical parameters. β1 was hypothesized at 0.50 and the mean estimate was 0.488, β2 was hypothesized at 0.04 and the mean of the estimates was 0.04. Although the estimates for the rates of misclassification of X1 were not as close as β1 and β2, they validate this method. X 1 0-1 misclassification was hypothesized as 2.98% and the mean of the simulated estimates was 1.54% and, in the best case, the misclassification of k from high to medium was hypothesized at 4.87% and had a sample mean of 3.62%. In the AD dataset, the estimate for the odds ratio of X 1 of having both copies of the APOE 4 allele changed from an estimate of 1.377 to an estimate 1.418, demonstrating that the estimates of the odds ratio changed when the analysis includes adjustment for misclassification. ^

Logistic regression with incompletely observed binary

Logistic regression is one of the most important tools in the analysis of epidemiological and clinical data. Such data often contain missing values for one or more variables. Common practice is to eliminate all individuals for whom any information is missing. This deletion approach does not make efficient use of available information and often introduces bias.^ Two methods were developed to estimate logistic regression coefficients for mixed dichotomous and continuous covariates including partially observed binary covariates. The data were assumed missing at random (MAR). One method (PD) used predictive distribution as weight to calculate the average of the logistic regressions performing on all possible values of missing observations, and the second method (RS) used a variant of resampling technique. Additional seven methods were compared with these two approaches in a simulation study. They are: (1) Analysis based on only the complete cases, (2) Substituting the mean of the observed values for the missing value, (3) An imputation technique based on the proportions of observed data, (4) Regressing the partially observed covariates on the remaining continuous covariates, (5) Regressing the partially observed covariates on the remaining continuous covariates conditional on response variable, (6) Regressing the partially observed covariates on the remaining continuous covariates and response variable, and (7) EM algorithm. Both proposed methods showed smaller standard errors (s.e.) for the coefficient involving the partially observed covariate and for the other coefficients as well. However, both methods, especially PD, are computationally demanding; thus for analysis of large data sets with partially observed covariates, further refinement of these approaches is needed. ^

POLYCHOTOMOUS LOGISTIC REGRESSION ANALYSIS (HYPERTENSION, CORONARY HEART DISEASE)

The history of the logistic function since its introduction in 1838 is reviewed, and the logistic model for a polychotomous response variable is presented with a discussion of the assumptions involved in its derivation and use. Following this, the maximum likelihood estimators for the model parameters are derived along with a Newton-Raphson iterative procedure for evaluation. A rigorous mathematical derivation of the limiting distribution of the maximum likelihood estimators is then presented using a characteristic function approach. An appendix with theorems on the asymptotic normality of sample sums when the observations are not identically distributed, with proofs, supports the presentation on asymptotic properties of the maximum likelihood estimators. Finally, two applications of the model are presented using data from the Hypertension Detection and Follow-up Program, a prospective, population-based, randomized trial of treatment for hypertension. The first application compares the risk of five-year mortality from cardiovascular causes with that from noncardiovascular causes; the second application compares risk factors for fatal or nonfatal coronary heart disease with those for fatal or nonfatal stroke. ^

Logistic regression with Markov chains as covariates

The tobacco-specific nitrosamine 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) is an obvious carcinogen for lung cancer. Since CBMN (Cytokinesis-blocked micronucleus) has been found to be extremely sensitive to NNK-induced genetic damage, it is a potential important factor to predict the lung cancer risk. However, the association between lung cancer and NNK-induced genetic damage measured by CBMN assay has not been rigorously examined. ^ This research develops a methodology to model the chromosomal changes under NNK-induced genetic damage in a logistic regression framework in order to predict the occurrence of lung cancer. Since these chromosomal changes were usually not observed very long due to laboratory cost and time, a resampling technique was applied to generate the Markov chain of the normal and the damaged cell for each individual. A joint likelihood between the resampled Markov chains and the logistic regression model including transition probabilities of this chain as covariates was established. The Maximum likelihood estimation was applied to carry on the statistical test for comparison. The ability of this approach to increase discriminating power to predict lung cancer was compared to a baseline "non-genetic" model. ^ Our method offered an option to understand the association between the dynamic cell information and lung cancer. Our study indicated the extent of DNA damage/non-damage using the CBMN assay provides critical information that impacts public health studies of lung cancer risk. This novel statistical method could simultaneously estimate the process of DNA damage/non-damage and its relationship with lung cancer for each individual.^

Evaluation of the Hosmer -Lemeshow global goodness -of-fit statistic for mixed variable logistic regression models

The performance of the Hosmer-Lemeshow global goodness-of-fit statistic for logistic regression models was explored in a wide variety of conditions not previously fully investigated. Computer simulations, each consisting of 500 regression models, were run to assess the statistic in 23 different situations. The items which varied among the situations included the number of observations used in each regression, the number of covariates, the degree of dependence among the covariates, the combinations of continuous and discrete variables, and the generation of the values of the dependent variable for model fit or lack of fit.^ The study found that the $\rm\ C$g* statistic was adequate in tests of significance for most situations. However, when testing data which deviate from a logistic model, the statistic has low power to detect such deviation. Although grouping of the estimated probabilities into quantiles from 8 to 30 was studied, the deciles of risk approach was generally sufficient. Subdividing the estimated probabilities into more than 10 quantiles when there are many covariates in the model is not necessary, despite theoretical reasons which suggest otherwise. Because it does not follow a X$\sp2$ distribution, the statistic is not recommended for use in models containing only categorical variables with a limited number of covariate patterns.^ The statistic performed adequately when there were at least 10 observations per quantile. Large numbers of observations per quantile did not lead to incorrect conclusions that the model did not fit the data when it actually did. However, the statistic failed to detect lack of fit when it existed and should be supplemented with further tests for the influence of individual observations. Careful examination of the parameter estimates is also essential since the statistic did not perform as desired when there was moderate to severe collinearity among covariates.^ Two methods studied for handling tied values of the estimated probabilities made only a slight difference in conclusions about model fit. Neither method split observations with identical probabilities into different quantiles. Approaches which create equal size groups by separating ties should be avoided. ^

Regularized logistic regression and multi-objective variable selection for classifying MEG data

This paper addresses the question of maximizing classifier accuracy for classifying task-related mental activity from Magnetoencelophalography (MEG) data. We propose the use of different sources of information and introduce an automatic channel selection procedure. To determine an informative set of channels, our approach combines a variety of machine learning algorithms: feature subset selection methods, classifiers based on regularized logistic regression, information fusion, and multiobjective optimization based on probabilistic modeling of the search space. The experimental results show that our proposal is able to improve classification accuracy compared to approaches whose classifiers use only one type of MEG information or for which the set of channels is fixed a priori.

System failure prediction through rare-events elastic-net logistic regression

Predicting failures in a distributed system based on previous events through logistic regression is a standard approach in literature. This technique is not reliable, though, in two situations: in the prediction of rare events, which do not appear in enough proportion for the algorithm to capture, and in environments where there are too many variables, as logistic regression tends to overfit on this situations; while manually selecting a subset of variables to create the model is error- prone. On this paper, we solve an industrial research case that presented this situation with a combination of elastic net logistic regression, a method that allows us to automatically select useful variables, a process of cross-validation on top of it and the application of a rare events prediction technique to reduce computation time. This process provides two layers of cross- validation that automatically obtain the optimal model complexity and the optimal mode l parameters values, while ensuring even rare events will be correctly predicted with a low amount of training instances. We tested this method against real industrial data, obtaining a total of 60 out of 80 possible models with a 90% average model accuracy.

LOGITCPRPLOT: Stata module to graph component-plus-residual plot for logistic regression

logitcprplot can be used after logistic regression for graphing a component-plus-residual plot (a.k.a. partial residual plot) for a given predictor, including a lowess, local polynomial, restricted cubic spline, fractional polynomial, penalized spline, regression spline, running line, or adaptive variable span running line smooth

RRLOGIT: Stata module to estimate logistic regression for randomized response data

rrlogit fits a maximum-likelihood logistic regression for randomized response data.

A review of two journals found that articles using multivariable logistic regression frequently did not report commonly recommended assumptions

Background and Objective: To examine if commonly recommended assumptions for multivariable logistic regression are addressed in two major epidemiological journals. Methods: Ninety-nine articles from the Journal of Clinical Epidemiology and the American Journal of Epidemiology were surveyed for 10 criteria: six dealing with computation and four with reporting multivariable logistic regression results. Results: Three of the 10 criteria were addressed in 50% or more of the articles. Statistical significance testing or confidence intervals were reported in all articles. Methods for selecting independent variables were described in 82%, and specific procedures used to generate the models were discussed in 65%. Fewer than 50% of the articles indicated if interactions were tested or met the recommended events per independent variable ratio of 10: 1. Fewer than 20% of the articles described conformity to a linear gradient, examined collinearity, reported information on validation procedures, goodness-of-fit, discrimination statistics, or provided complete information on variable coding. There was no significant difference (P >.05) in the proportion of articles meeting the criteria across the two journals. Conclusion: Articles reviewed frequently did not report commonly recommended assumptions for using multivariable logistic regression. (C) 2004 Elsevier Inc. All rights reserved.

Optimal crossover designs for logistic regression models in pharmacodynamics

Pharmacodynamics (PD) is the study of the biochemical and physiological effects of drugs. The construction of optimal designs for dose-ranging trials with multiple periods is considered in this paper, where the outcome of the trial (the effect of the drug) is considered to be a binary response: the success or failure of a drug to bring about a particular change in the subject after a given amount of time. The carryover effect of each dose from one period to the next is assumed to be proportional to the direct effect. It is shown for a logistic regression model that the efficiency of optimal parallel (single-period) or crossover (two-period) design is substantially greater than a balanced design. The optimal designs are also shown to be robust to misspecification of the value of the parameters. Finally, the parallel and crossover designs are combined to provide the experimenter with greater flexibility.

Modeling Data for Complications in Diabetics Using Logistic Regression

2010 Mathematics Subject Classification: 62P10.

Sequential Logistic Regression: A Method to Reveal Subtlety in Self-Efficacy

This paper uses self-efficacy to predict the success of women in introductory physics. We show how sequential logistic regression demonstrates the predictive ability of self-efficacy, and reveals variations with type of physics course. Also discussed are the sources of self-efficacy that have the largest impact on predictive ability.