Quantification of model risk that is caused by model misspecification

In this paper, we suggest a technique to quantify model risk, particularly model misspecification for binary response regression problems found in financial risk management, such as in credit risk modelling. We choose the probability of default model as one instance of many other credit risk models that may be misspecified in a financial institution. By way of illustrating the model misspecification for probability of default, we carry out quantification of two specific statistical predictive response techniques, namely the binary logistic regression and complementary log–log. The maximum likelihood estimation technique is employed for parameter estimation. The statistical inference, precisely the goodness of fit and model performance measurements, are assessed. Using the simulation dataset and Taiwan credit card default dataset, our finding reveals that with the same sample size and very small simulation iterations, the two techniques produce similar goodness-of-fit results but completely different performance measures. However, when the iterations increase, the binary logistic regression technique for balanced dataset reveals prominent goodness of fit and performance measures as opposed to the complementary log–log technique for both simulated and real datasets.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

The effects of covariate adjustment in generalized linear models

Results from classical linear regression regarding the effects of covariate adjustment, with respect to the issues of confounding, the precision with which an exposure effect can be estimated, and the efficiency of hypothesis tests for no treatment effect in randomized experiments, are often assumed to apply more generally to other types of regression models. In this paper results pertaining to several generalized linear models involving a dichotomous response variable are given, demonstrating that with respect to the issues of confounding and precision, for models having a linear or log link function the results of classical linear regression do generally apply, whereas for other models, including those having a logit, probit, log-log, complementary log-log, or generalized logistic link function, the results of classical linear regression do not always apply. It is also shown, however, that for any link function, covariate adjustment results in improved efficiency of hypothesis tests for no treatment effect in randomized experiments, and hence that the classical linear regression results regarding efficiency do apply for all models having a dichotomous response variable.     

Simple Formula for Calculating Bias‐corrected AIC in Generalized Linear Models

In real‐data analysis, deciding the best subset of variables in regression models is an important problem. Akaike's information criterion (AIC) is often used in order to select variables in many fields. When the sample size is not so large, the AIC has a non‐negligible bias that will detrimentally affect variable selection. The present paper considers a bias correction of AIC for selecting variables in the generalized linear model (GLM). The GLM can express a number of statistical models by changing the distribution and the link function, such as the normal linear regression model, the logistic regression model, and the probit model, which are currently commonly used in a number of applied fields. In the present study, we obtain a simple expression for a bias‐corrected AIC (corrected AIC, or CAIC) in GLMs. Furthermore, we provide an ‘R’ code based on our formula. A numerical study reveals that the CAIC has better performance than the AIC for variable selection.     

Establishing decision tree-based short-term default credit risk assessment models

ABSTRACT

Traditional credit risk assessment models do not consider the time factor; they only think of whether a customer will default, but not the when to default. The result cannot provide a manager to make the profit-maximum decision. Actually, even if a customer defaults, the financial institution still can gain profit in some conditions. Nowadays, most research applied the Cox proportional hazards model into their credit scoring models, predicting the time when a customer is most likely to default, to solve the credit risk assessment problem. However, in order to fully utilize the fully dynamic capability of the Cox proportional hazards model, time-varying macroeconomic variables are required which involve more advanced data collection. Since short-term default cases are the ones that bring a great loss for a financial institution, instead of predicting when a loan will default, a loan manager is more interested in identifying those applications which may default within a short period of time when approving loan applications. This paper proposes a decision tree-based short-term default credit risk assessment model to assess the credit risk. The goal is to use the decision tree to filter the short-term default to produce a highly accurate model that could distinguish default lending. This paper integrates bootstrap aggregating (Bagging) with a synthetic minority over-sampling technique (SMOTE) into the credit risk model to improve the decision tree stability and its performance on unbalanced data. Finally, a real case of small and medium enterprise loan data that has been drawn from a local financial institution located in Taiwan is presented to further illustrate the proposed approach. After comparing the result that was obtained from the proposed approach with the logistic regression and Cox proportional hazards models, it was found that the classifying recall rate and precision rate of the proposed model was obviously superior to the logistic regression and Cox proportional hazards models.     

A Semiparametric Binary Regression Model Involving Monotonicity Constraints

Abstract.  We study a binary regression model using the complementary log–log link, where the response variable Δ is the indicator of an event of interest (for example, the incidence of cancer, or the detection of a tumour) and the set of covariates can be partitioned as ( X ,  Z ) where Z (real valued) is the primary covariate and X (vector valued) denotes a set of control variables. The conditional probability of the event of interest is assumed to be monotonic in Z , for every fixed X . A finite-dimensional (regression) parameter β describes the effect of X . We show that the baseline conditional probability function (corresponding to X  =  0 ) can be estimated by isotonic regression procedures and develop an asymptotically pivotal likelihood-ratio-based method for constructing (asymptotic) confidence sets for the regression function. We also show how likelihood-ratio-based confidence intervals for the regression parameter can be constructed using the chi-square distribution. An interesting connection to the Cox proportional hazards model under current status censoring emerges. We present simulation results to illustrate the theory and apply our results to a data set involving lung tumour incidence in mice.     

Lasso penalized semiparametric regression on high-dimensional recurrent event data via coordinate descent

This paper studies a fast computational algorithm for variable selection on high-dimensional recurrent event data. Based on the lasso penalized partial likelihood function for the response process of recurrent event data, a coordinate descent algorithm is used to accelerate the estimation of regression coefficients. This algorithm is capable of selecting important predictors for underdetermined problems where the number of predictors far exceeds the number of cases. The selection strength is controlled by a tuning constant that is determined by a generalized cross-validation method. Our numerical experiments on simulated and real data demonstrate the good performance of penalized regression in model building for recurrent event data in high-dimensional settings.     

Diagnostics of multiple group influential observations for logistic regression models

In this paper, two new multiple influential observation detection methods, GCD.GSPR and mCD*, are introduced for logistic regression. The proposed diagnostic measures are compared with the generalized difference in fits (GDFFITS) and the generalized squared difference in beta (GSDFBETA), which are multiple influential diagnostics. The simulation study is conducted with one, two and five independent variable logistic regression models. The performance of the diagnostic measures is examined for a single contaminated independent variable for each model and in the case where all the independent variables are contaminated with certain contamination rates and intensity. In addition, the performance of the diagnostic measures is compared in terms of the correct identification rate and swamping rate via a frequently referred to data set in the literature.     

Generalized symmetrical partial linear model

In this work, we propose a new model called generalized symmetrical partial linear model, based on the theory of generalized linear models and symmetrical distributions. In our model the response variable follows a symmetrical distribution such a normal, Student-t, power exponential, among others. Following the context of generalized linear models we consider replacing the traditional linear predictors by the more general predictors in whose case one covariate is related with the response variable in a non-parametric fashion, that we do not specified the parametric function. As an example, we could imagine a regression model in which the intercept term is believed to vary in time or geographical location. The backfitting algorithm is used for estimating the parameters of the proposed model. We perform a simulation study for assessing the behavior of the penalized maximum likelihood estimators. We use the quantile residuals for checking the assumption of the model. Finally, we analyzed real data set related with pH rivers in Ireland.     

Bayeslan Credit Ratings

In this contribution we aim at improving ordinal variable selection in the context of causal models for credit risk estimation. In this regard, we propose an approach that provides a formal inferential tool to compare the explanatory power of each covariate and, therefore, to select an effective model for classification purposes. Our proposed model is Bayesian nonparametric thus keeps the amount of model specification to a minimum. We consider the case in which information from the covariates is at the ordinal level. A noticeable instance of this regards the situation in which ordinal variables result from rankings of companies that are to be evaluated according to different macro and micro economic aspects, leading to ordinal covariates that correspond to various ratings, that entail different magnitudes of the probability of default. For each given covariate, we suggest to partition the statistical units in as many groups as the number of observed levels of the covariate. We then assume individual defaults to be homogeneous within each group and heterogeneous across groups. Our aim is to compare and, therefore select, the partition structures resulting from the consideration of different explanatory covariates. The metric we choose for variable comparison is the calculation of the posterior probability of each partition. The application of our proposal to a European credit risk database shows that it performs well, leading to a coherent and clear method for variable averaging of the estimated default probabilities.     

The generalized log-gamma mixture model with covariates: local influence and residual analysis

In a sample of censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse, may be indicated by a relatively high number of individuals with large censored survival times. In this paper the generalized log-gamma model is modified for the possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model. The authors would like to thank the editor and referees for their helpful comments. This work was supported by CNPq, Brazil.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

A Bayesian approach for generalized linear models with explanatory biomarker measurement variables subject to detection limit: an application to acute lung injury

Biomarkers have the potential to improve our understanding of disease diagnosis and prognosis. Biomarker levels that fall below the assay detection limits (DLs), however, compromise the application of biomarkers in research and practice. Most existing methods to handle non-detects focus on a scenario in which the response variable is subject to the DL; only a few methods consider explanatory variables when dealing with DLs. We propose a Bayesian approach for generalized linear models with explanatory variables subject to lower, upper, or interval DLs. In simulation studies, we compared the proposed Bayesian approach to four commonly used methods in a logistic regression model with explanatory variable measurements subject to the DL. We also applied the Bayesian approach and other four methods in a real study, in which a panel of cytokine biomarkers was studied for their association with acute lung injury (ALI). We found that IL8 was associated with a moderate increase in risk for ALI in the model based on the proposed Bayesian approach.     

Events per variable for risk differences and relative risks using pseudo-observations

A method based on pseudo-observations has been proposed for direct regression modeling of functionals of interest with right-censored data, including the survival function, the restricted mean and the cumulative incidence function in competing risks. The models, once the pseudo-observations have been computed, can be fitted using standard generalized estimating equation software. Regression models can however yield problematic results if the number of covariates is large in relation to the number of events observed. Guidelines of events per variable are often used in practice. These rules of thumb for the number of events per variable have primarily been established based on simulation studies for the logistic regression model and Cox regression model. In this paper we conduct a simulation study to examine the small sample behavior of the pseudo-observation method to estimate risk differences and relative risks for right-censored data. We investigate how coverage probabilities and relative bias of the pseudo-observation estimator interact with sample size, number of variables and average number of events per variable.     

Goodness-of-fit Tests for GEE with Correlated Binary Data

The marginal logistic regression, in combination with GEE, is an increasingly important method in dealing with correlated binary data. As for independent binary data, when the number of possible combinations of the covariate values in a logistic regression model is much larger than the sample size, such as when the logistic model contains at least one continuous covariate, many existing chi-square goodness-of-fit tests either are not applicable or have some serious drawbacks. In this paper two residual based normal goodness-of-fit test statistics are proposed: the Pearson chi-square and an unweighted sum of residual squares. Easy-to-calculate approximations to the mean and variance of either statistic are also given. Their performance, in terms of both size and power, was satisfactory in our simulation studies. For illustration we apply them to a real data set.     

A comparison of ordinal logistic regression models using Classical and Bayesian approaches in an analysis of factors associated with diabetic retinopathy

Estimating the risk factors of a disease such as diabetic retinopathy (DR) is one of the important research problems among bio-medical and statistical practitioners as well as epidemiologists. Incidentally many studies have focused in building models with binary outcomes, that may not exploit the available information. This article has investigated the importance of retaining the ordinal nature of the response variable (e.g. severity level of a disease) while determining the risk factors associated with DR. A generalized linear model approach with appropriate link functions has been studied using both Classical and Bayesian frameworks. From the result of this study, it can be observed that the ordinal logistic regression with probit link function could be more appropriate approach in determining the risk factors of DR. The study has emphasized the ways to handle the ordinal nature of the response variable with better model fit compared to other link functions.     

Goodness-of-link testing in ordinal regression models

McCullagh (1980) presented a comprehensive review of regression models for ordinal response variables. In these models, the functional relationship between the covariates and the response categories is dependent on the link function. This paper shows that discrimination between links is feasible when the response variable is ordinal. Using the log-gamma distribution of Prentice (1974), a generalized link function is constructed which allows discrimination between the probit, log-log, and complementary log-log links. Sample-size considerations are noted, and examples are presented.     

A new algorithm to estimate monotone nonparametric link functions and a comparison with parametric approach

The generalized linear model (GLM) is a class of regression models where the means of the response variables and the linear predictors are joined through a link function. Standard GLM assumes the link function is fixed, and one can form more flexible GLM by either estimating the flexible link function from a parametric family of link functions or estimating it nonparametically. In this paper, we propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be monotone. It is equivalent to fit the generalized single index model with monotonicity constraint. We also conduct extensive simulation studies to compare our nonparametric approach for estimating link function with various parametric approaches, including traditional logit, probit and robit link functions, and two recently developed link functions, the generalized extreme value link and the symmetric power logit link. The simulation study shows that the link function estimated nonparametrically by our proposed algorithm performs well under a wide range of different true link functions and outperforms parametric approaches when they are misspecified. A real data example is used to illustrate the results.     

A Model Validation Procedure when Covariate Data are Missing at Random

Abstract. In the presence of missing covariates, standard model validation procedures may result in misleading conclusions. By building generalized score statistics on augmented inverse probability weighted complete‐case estimating equations, we develop a new model validation procedure to assess the adequacy of a prescribed analysis model when covariate data are missing at random. The asymptotic distribution and local alternative efficiency for the test are investigated. Under certain conditions, our approach provides not only valid but also asymptotically optimal results. A simulation study for both linear and logistic regression illustrates the applicability and finite sample performance of the methodology. Our method is also employed to analyse a coronary artery disease diagnostic dataset.     

Logistic回归的双层变量选择研究

变量选择是统计建模的重要环节,选择合适的变量可以建立结构简单、预测精准的稳健模型。本文在logistic回归下提出了新的双层变量选择惩罚方法——adaptive Sparse Group Lasso(adSGL),其独特之处在于基于变量的分组结构作筛选,实现了组内和组间双层选择。该方法的优点是对各单个系数和组系数采取不同程度的惩罚,避免了过度惩罚大系数,从而提高了模型的估计和预测精度。求解的难点是惩罚似然函数不是严格凸的,因此本文基于组坐标下降法求解模型,并建立了调整参数的选取准则。模拟分析表明,对比现有代表性方法Sparse Group Lasso、Group Lasso及Lasso,adSGL法不仅提高了双层选择精度,而且降低了模型误差。最后本文将adSGL法应用到信用卡信用评分研究,对比logistic回归,它具有更高的分类精度和稳健性。