Labeling patients in electronic health records with respect to their statuses of having a disease or condition, i.e. case or control statuses, has increasingly relied on prediction models using high-dimensional variables derived from structured and unstructured electronic health record data. A major hurdle currently is a lack of valid statistical inference methods for the case probability. In this paper, considering high-dimensional sparse logistic regression models for prediction, we propose a novel bias-corrected estimator for the case probability through the development of linearization and variance enhancement techniques. We establish asymptotic normality of the proposed estimator for any loading vector in high dimensions. We construct a confidence interval for the case probability and propose a hypothesis testing procedure for patient case-control labelling. We demonstrate the proposed method via extensive simulation studies and application to real-world electronic health record data.
This short note is to point the reader to notice that the proof of high dimensional asymptotic normality of MLE estimator for logistic regression under the regime $p_n=o(n)$ given in paper: "Maximum likelihood estimation in logistic regression models with a diverging number of covariates. Electronic Journal of Statistics, 6, 1838-1846." is wrong.
This article presents methods for the construction of two-sided and one-sided simultaneous hyperbolic bands for the logistic and probit regression models when the predictor variable is restricted to a given interval. The bands are constructed based on the asymptotic properties of the maximum likelihood estimators. Past articles have considered building two-sided asymptotic confidence bands for the logistic model, such as Piegorsch and Casella (1988). However, the confidence bands given by Piegorsch and Casella are conservative under a single interval restriction, and it is shown in this articl…
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
Logistic regression with unknown sizes has many important applications in biological and medical sciences. All models about this problem in the literature are parametric ones. A semiparametric regression model is proposed. This model incorporates overdispersion due to the variation of sizes, and allows general dose-response relations. An Expectation Conditional Maximization algorithm is provided to maximize the log likelihood. The bootstrap method can be used to construct confidence intervals for regression coefficients. Simulation is performed to study the behavior of the proposed model. Two …
Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting. The equations leads to highly heteroscedastic regression. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples maximum likelihood performs best, but it is shown that in smaller samples the weighted regression outperforms maximum likelihood estimation with respect to bias and mean square error.
In the last decade, the secondary use of large data from health systems, such as electronic health records, has demonstrated great promise in advancing biomedical discoveries and improving clinical decision making. However, there is an increasing concern about biases in association studies caused by misclassification in the binary outcomes derived from electronic health records. We revisit the classical logistic regression model with misclassified outcomes. Despite that local identification conditions in some related settings have been previously established, the global identification of such …
Analyzing polytomous response from a complex survey scheme, like stratified or cluster sampling is very crucial in several socio-economics applications. We present a class of minimum quasi weighted density power divergence estimators for the polytomous logistic regression model with such a complex survey. This family of semiparametric estimators is a robust generalization of the maximum quasi weighted likelihood estimator exploiting the advantages of the popular density power divergence measure. Accordingly robust estimators for the design effects are also derived. Robust testing of general li…
Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables to easily update the estimates when the data arriv…
In cluster-specific studies, ordinary logistic regression and conditional logistic regression for binary outcomes provide maximum likelihood estimator (MLE) and conditional maximum likelihood estimator (CMLE), respectively. In this paper, we show that CMLE is approaching to MLE asymptotically when each individual data point is replicated infinitely many times. Our theoretical derivation is based on the observation that a term appearing in the conditional average log-likelihood function is the coefficient of a polynomial, and hence can be transformed to a complex integral by Cauchy's differenti…
We present new stochastic geometry theorems that give bounds on the probability that $m$ random data classes all contain a point in common in their convex hulls. We apply these stochastic separation theorems to obtain bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression. We also discuss connections to condition numbers for analysis of steepest descent algorithms in logistic regression and to the computation of centerpoints of data clouds.
We consider the problem of link prediction, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analysed in a high-dimensional regime under a structural assumption. The minimax rate for the Frobenius-norm risk is established and a combinatorial estimator based on the penalised maximum likelihood approach is shown to achieve it. Furthermore, it is shown that this rate cannot be attained by any (randomised) algorithm computable in polynomial time un…
In this paper we discuss how to evaluate the differences between fitted logistic regression models across sub-populations. Our motivating example is in studying computerized diagnosis for learning disabilities, where sub-populations based on gender may or may not require separate models. In this context, significance tests for hypotheses of no difference between populations may provide perverse incentives, as larger variances and smaller samples increase the probability of not-rejecting the null. We argue that equivalence testing for a prespecified tolerance level on population differences inc…
We obtain new mathematical properties of the exponentiated odd log-logistic family of distributions, and of its special case named the exponentiated odd log-logistic Weibull, and its log transformed. A new location and scale regression model is constructed, and some simulations are carried out to verify the behavior of the maximum likelihood estimators, and of the modified deviance-based residuals. The methodology is applied to the Japanese-Brazilian emigration data.
This paper looks at effects, due to the boundary, on inference in logistic regression. It shows that first -- and, indeed, higher -- order asymptotic results are not uniform across the model. Near the boundary, effects such as high skewness, discreteness and collinearity dominate, any of which could render inference based on asymptotic normality suspect. A highly interpretable diagnostic tool is proposed allowing the analyst to check if the boundary is going to have an appreciable effect on standard inferential techniques.
We want to analyse EEG recordings in order to investigate the phonemic categorization at a very early stage of auditory processing. This problem can be modelled by a supervised classification of functional data. Discrimination is explored via a logistic functional linear model, using a wavelet representation of the data. Different procedures are investigated, based on penalized likelihood and principal component reduction or partial least squares reduction.
In order to overcome multicollinearity, we propose a stochastic restricted Liu-type max- imum likelihood estimator by incorporating Liu-type maximum likelihood estimator (Inan and Erdo- gan, 2013) to the logistic regression model when the linear restrictions are stochastic. We also discuss the properties of the new estimator. Moreover, we give a method to choose the biasing parameter in the new estimator. Finally, a simulation study is given to show the performance of the new estimator.
Consider a set of categorical variables $\mathcal{P}$ where at least one, denoted by $Y$, is binary. The log-linear model that describes the counts in the resulting contingency table implies a specific logistic regression model, with the binary variable as the outcome. Extending results in Christensen (1997), by also considering the case where factors present in the contingency table disappear from the logistic regression model, we prove that the Maximum Likelihood Estimate (MLE) for the parameters of the logistic regression equals the MLE for the corresponding parameters of the log-linear mod…
Finite-sample performance of the maximum likelihood estimator in logistic regression
Department of Statistics, University of California, Berkeley, USA（加州大学伯克利分校统计学系）；Department of Statistics, CREST/ENSAE Paris, Palaiseau, France（巴黎中央银行与应用经济学院统计学系）
This paper derives a new family of estimators, namely the minimum density power divergence estimators, as a robust generalization of the maximum likelihood estimator for the polytomous logistic regression model. Based on these estimators, a family of Wald-type test statistics for linear hypotheses is introduced. Robustness properties of both the proposed estimators and the test statistics are theoretically studied through the classical influence function analysis. Appropriate real life examples are presented to justify the requirement of suitable robust statistical procedures in place of the l…
Compared to the conditional mean as a simple point estimator, the conditional density function is more informative to describe the distributions with multi-modality, asymmetry or heteroskedasticity. In this paper, we propose a novel parametric conditional density estimation method by showing the connection between the general density and the likelihood function of inhomogeneous Poisson process models. The maximum likelihood estimates can be obtained via weighted logistic regressions, and the computation can be significantly relaxed by combining a block-wise alternating maximization scheme and …
This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp `phase transition'. We introduce an explicit boundary curve $h_{\text{MLE}}$, parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes $n$ and number of features $p$ proportioned in such a way that $p/n \rightarrow κ$, we show that if the problem is sufficiently high dimensional in the sense tha…
In the last decades, it has been discussed the use of epidemiological prevalence ratio (PR) rather than odds ratio as a measure of association to be estimated in cross-sectional studies. The main difficulties in use of statistical models for the calculation of PR are convergence problems, availability of adequate tools and strong assumptions. The goal of this study is to illustrate how to estimate PR and its confidence interval directly from logistic regression estimates. We present three examples and compare the adjusted estimates of PR with the estimates obtained by use of log-binomial, robu…
This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that t…
Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-s…
This article presents methods for constructing an asymptotic hyperbolic band under the multiple logistic regression model when the predictor variables are restricted to a specific region $\mathscr{X}$. Scheffé's method yields unnecessarily wide, and hence conservative, bands if the predictor variables can be restricted to a certain region. Piegorsch and Casella (1988) developed a procedure to build an asymptotic confidence band for the multiple logistic regression model over particular regions. Those regions are shown to be special cases of the region $\mathscr{X}$, which was first investigate…
Feature selection is a standard approach to understanding and modeling high-dimensional classification data, but the corresponding statistical methods hinge on tuning parameters that are difficult to calibrate. In particular, existing calibration schemes in the logistic regression framework lack any finite sample guarantees. In this paper, we introduce a novel calibration scheme for $\ell_1$-penalized logistic regression. It is based on simple tests along the tuning parameter path and is equipped with optimal guarantees for feature selection. It is also amenable to easy and efficient implement…
This is a note on logistic regression models and logistic kernel machine models. It contains derivations to some of the expressions in a paper -- SNP Set Analysis for Detecting Disease Association Using Exon Sequence Data -- submitted to BMC proceedings by these authors.
School of Engineering, Computing and Mathematics（工程、计算与数学学院）；Oxford Brookes University（牛津布鲁克斯大学）
The notion of belief likelihood function of repeated trials is introduced, whenever the uncertainty for individual trials is encoded by a belief measure (a finite random set). This generalises the traditional likelihood function, and provides a natural setting for belief inference from statistical data. Factorisation results are proven for the case in which conjunctive or disjunctive combination are employed, leading to analytical expressions for the lower and upper likelihoods of `sharp' samples in the case of Bernoulli trials, and to the formulation of a generalised logistic regression frame…
We propose an optimal estimating equation for logistic regression with linked data while accounting for false positives. It builds on a previous solution but estimates the regression coefficients with a smaller variance, in large samples.
This study aims to show the fundamental difference between logistic regression and Bayesian classifiers in the case of exponential and unexponential families of distributions, yielding the following findings. First, the logistic regression is a less general representation of a Bayesian classifier. Second, one should suppose distributions of classes for the correct specification of logistic regression equations. Third, in specific cases, there is no difference between predicted probabilities from correctly specified generative Bayesian classifier and discriminative logistic regression.
This paper considers the Liu estimator in the multinomial logistic regression model. We propose some different estimators of the biasing parameter. The mean square error (MSE) is considered as the performance criterion. In order to compare the performance of the estimators, we performed a Monte Carlo simulation study. According to the results of the simulation study, we found that increasing the correlation between the independent variables and the number of regressors has a negative effect on the MSE. However, when the sample size increases the MSE decreases even when the correlation between …
Explicitly accounting for all applicable independent variables, even when the model being tested does not, is critical in testing goodness-of-fit for logistic regression. This can increase statistical power by orders of magnitude.
This paper infers a single parameter in non-sparse logistic regression models. By transforming the null hypothesis into a moment condition, we construct the test statistic and obtain the asymptotic null distribution. Numerical experiments show that our method performs well.
A new approach for feature extraction from time series is proposed in this paper. This approach consists of a specific regression model incorporating a discrete hidden logistic process. The model parameters are estimated by the maximum likelihood method performed by a dedicated Expectation Maximization (EM) algorithm. The parameters of the hidden logistic process, in the inner loop of the EM algorithm, are estimated using a multi-class Iterative Reweighted Least-Squares (IRLS) algorithm. A piecewise regression algorithm and its iterative variant have also been considered for comparisons. An ex…
In several FICO studies logistic regression has been shown to be a very competitive technology for developing unrestricted scoring models, especially for performance metrics like ROC area. Application of logistic regression has been hampered by the lack of software to handle complex score engineering such as shape and pattern constraints. The purpose of this paper is to develop a sequential quadratic programming algorithm for score engineered logistic regression. This approach is based on a simple Taylor series expansion of minus log likelihood, which is locally quadratic. and fits in with the…
Logistic Regression Model for Differentially-Private Matrix Masked Data
School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学院）；Department of Mathematics, Northeastern University（东北大学数学系）；Health Informatics Institute University of South Florida（南佛罗里达大学健康信息学研究所）
In this paper we propose a principal component Liu-type logistic estimator by combining the principal component logistic regression estimator and Liu-type logistic estimator to overcome the multicollinearity problem. The superiority of the new estimator over some related estimators are studied under the asymptotic mean squared error matrix. A Monte Carlo simulation experiment is designed to compare the performances of the estimators using mean squared error criterion. Finally, a conclusion section is presented.
Imperial College London（帝国理工学院）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
Variational Bayes (VB) is a popular scalable alternative to Markov chain Monte Carlo for Bayesian inference. We study a mean-field spike and slab VB approximation of widely used Bayesian model selection priors in sparse high-dimensional logistic regression. We provide non-asymptotic theoretical guarantees for the VB posterior in both $\ell_2$ and prediction loss for a sparse truth, giving optimal (minimax) convergence rates. Since the VB algorithm does not depend on the unknown truth to achieve optimality, our results shed light on effective prior choices. We confirm the improved performance o…
Recently, a very attractive logistic regression inference method for exponential family Gibbs spatial point processes was introduced. We combined it with the technique of quadratic tangential variational approximation and derived a new Bayesian technique for analysing spatial point patterns. The technique is described in detail, and demonstrated on numerical examples.
Tel Aviv University（特拉维夫大学）；The Open University of Israel（以色列开放大学）
In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there exist two regimes corresponding to small and large number of classes. The bounds can be reduced under the additional low noise condition. To…
Logistic regression has been widely applied in the field of biomedical research for a long time. In some applications, covariates of interest have a natural structure, such as being a matrix, at the time of collection. The rows and columns of the covariate matrix then have certain physical meanings, and they must contain useful information regarding the response. If we simply stack the covariate matrix as a vector and fit the conventional logistic regression model, relevant information can be lost, and the problem of inefficiency will arise. Motivated from these reasons, we propose in this pap…
This article develops the theoretical framework needed to study the multinomial logistic regression model for complex sample design with pseudo minimum phi-divergence estimators. Through a numerical example and simulation study new estimators are proposed for the parameter of the logistic regression model with overdispersed multinomial distributions for the response variables, the pseudo minimum Cressie-Read divergence estimators, as well as new estimators for the intra-cluster correlation coefficient. The results show that the Binder's method for the intra-cluster correlation coefficient exhi…
Logistic regression is an important statistical tool for assessing the probability of an outcome based upon some predictive variables. Standard methods can only deal with precisely known data, however many datasets have uncertainties which traditional methods either reduce to a single point or completely disregarded. In this paper we show that it is possible to include these uncertainties by considering an imprecise logistic regression model using the set of possible models that can be obtained from values from within the intervals. This has the advantage of clearly expressing the epistemic un…
We consider Bayesian logistic regression models with group-structured covariates. In high-dimensional settings, it is often assumed that only small portion of groups are significant, thus consistent group selection is of significant importance. While consistent frequentist group selection methods have been proposed, theoretical properties of Bayesian group selection methods for logistic regression models have not been investigated yet. In this paper, we consider a hierarchical group spike and slab prior for logistic regression models in high-dimensional settings. Under mild conditions, we esta…
Department of Indutrial Engineering and Operations Research, University of California, Berkeley（加州大学伯克利分校工业工程与运筹学系）
In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be e…
