Relative Risk Regression for Binary Outcomes: Methods and Recommendations

Relative risks are often considered preferable to odds ratios for quantifying the association between a predictor and a binary outcome. Relative risk regression is an alternative to logistic regression where the parameters are relative risks rather than odds ratios. It uses a log link binomial generalised linear model, or log‐binomial model, which requires parameter constraints to prevent probabilities from exceeding 1. This leads to numerical problems with standard approaches for finding the maximum likelihood estimate (MLE), such as Fisher scoring, and has motivated various non‐MLE approaches. In this paper we discuss the roles of the MLE and its main competitors for relative risk regression. It is argued that reliable alternatives to Fisher scoring mean that numerical issues are no longer a motivation for non‐MLE methods. Nonetheless, non‐MLE methods may be worthwhile for other reasons and we evaluate this possibility for alternatives within a class of quasi‐likelihood methods. The MLE obtained using a reliable computational method is recommended, but this approach requires bootstrapping when estimates are on the parameter space boundary. If convenience is paramount, then quasi‐likelihood estimation can be a good alternative, although parameter constraints may be violated. Sensitivity to model misspecification and outliers is also discussed along with recommendations and priorities for future research.     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.     

Penalized pseudo-likelihood hazard estimation: A fast alternative to penalized likelihood

Penalized likelihood method has been developed previously for hazard function estimation using standard left-truncated, right-censored lifetime data with covariates, and the functional ANOVA structures built into the log hazard allows for versatile nonparametric modeling in the setting. The computation of the method can be time-consuming in the presence of continuous covariates; however, due to the repeated numerical integrations involved. Adapting a device developed by Jeon and Lin [An effective method for high dimensional log-density ANOVA estimation, with application to nonparametric graphical model building. Statist. Sinica 16, 353–374] for penalized likelihood density estimation, we explore an alternative approach to hazard estimation where the log likelihood is replaced by some computationally less demanding pseudo-likelihood. An assortment of issues are addressed concerning the practical implementations of the approach including the selection of smoothing parameters, and extensive simulations are presented to assess the inferential efficiency of the “pseudo” method as compared to the “real” one. Also noted is an asymptotic theory concerning the convergence rates of the estimates parallel to that for the original penalized likelihood estimation.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Benchmark profile and inferences for joint-exposure quantal data in quantitative risk assessment

Abstract

In risk assessment, it is often desired to make inferences on the minimum dose levels (benchmark doses or BMDs) at which a specific benchmark risk (BMR) is attained. The estimation and inferences of BMDs are well understood in the case of an adverse response to a single-exposure agent. However, the theory of finding BMDs and making inferences on the BMDs is much less developed for cases where the adverse effect of two hazardous agents is studied simultaneously. Deutsch and Piegorsch [2012. Benchmark dose profiles for joint-action quantal data in quantitative risk assessment. Biometrics 68(4):1313–22] proposed a benchmark modeling paradigm in dual exposure setting—adapted from the single-exposure setting—and developed a strategy for conducting full benchmark analysis with joint-action quantal data, and they further extended the proposed benchmark paradigm to continuous response outcomes [Deutsch, R. C., and W. W. Piegorsch. 2013. Benchmark dose profiles for joint-action continuous data in quantitative risk assessment. Biometrical Journal 55(5):741–54]. In their 2012 article, Deutsch and Piegorsch worked exclusively with the complementary log link for modeling the risk with quantal data. The focus of the current paper is on the logit link; particularly, we consider an Abbott-adjusted [A method of computing the effectiveness of an insecticide. Journal of Economic Entomology 18(2):265–7] log-logistic model for the analysis of quantal data with nonzero background response. We discuss the estimation of the benchmark profile (BMP)—a collection of benchmark points which induce the prespecified BMR—and propose different methods for building benchmark inferences in studies involving two hazardous agents. We perform Monte Carlo simulation studies to evaluate the characteristics of the confidence limits. An example is given to illustrate the use of the proposed methods.     

Threshold autoregression analysis for finite-range time series of counts with an application on measles data

This article studies the threshold autoregression analysis for the self-exciting threshold binomial autoregressive processes. Parameters' point estimation and interval estimation problems are considered via the empirical likelihood method. A new algorithm to estimate the threshold value of the threshold model is also given. Simulation study is conducted for the evaluation of the developed approach. An application on measles data is provided to show the applicability of the method.     

A comparison of two varying coefficient meta-analysis methods for an average risk difference

Two interval estimation methods for a general linear function of binomial proportions have been proposed. One method [Zou GY, Huang W, Zhang X. A note on confidence interval estimation for a linear function of binomial proportions. Comput Statist Data Anal. 2009;53:1080–1085] combines Wilson interval estimates of individual proportions, and the other method [Price RM, Bonett DG. An improved confidence interval for a linear function of binomial proportions. Comput Statist Data Anal. 2004;45:449–456] uses an adjusted Wald interval. Both methods are appropriate in varying coefficient meta-analysis models where the risk differences are allowed to vary across studies. The two methods were compared in a simulation study under realistic meta-analysis conditions and the adjusted Wald method was found to have the best performance characteristics.     

A clustering cure rate model with application to a sealant study

In this paper, the destructive negative binomial (DNB) cure rate model with a latent activation scheme [V. Cancho, D. Bandyopadhyay, F. Louzada, and B. Yiqi, The DNB cure rate model with a latent activation scheme, Statistical Methodology 13 (2013b), pp. 48–68] is extended to the case where the observations are grouped into clusters. Parameter estimation is performed based on the restricted maximum likelihood approach and on a Bayesian approach based on Dirichlet process priors. An application to a real data set related to a sealant study in a dentistry experiment is considered to illustrate the performance of the proposed model.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

Modelling small and medium enterprise loan defaults as rare events: the generalized extreme value regression model

A pivotal characteristic of credit defaults that is ignored by most credit scoring models is the rarity of the event. The most widely used model to estimate the probability of default is the logistic regression model. Since the dependent variable represents a rare event, the logistic regression model shows relevant drawbacks, for example, underestimation of the default probability, which could be very risky for banks. In order to overcome these drawbacks, we propose the generalized extreme value regression model. In particular, in a generalized linear model (GLM) with the binary-dependent variable we suggest the quantile function of the GEV distribution as link function, so our attention is focused on the tail of the response curve for values close to one. The estimation procedure used is the maximum-likelihood method. This model accommodates skewness and it presents a generalisation of GLMs with complementary log–log link function. We analyse its performance by simulation studies. Finally, we apply the proposed model to empirical data on Italian small and medium enterprises.     

Confirmatory Factor Analysis of Ordinal Data Using Full‐Information Adaptive Quadrature

We conducted confirmatory factor analysis (CFA) of responses (N=803) to a self‐reported measure of optimism, using full‐information estimation via adaptive quadrature (AQ), an alternative estimation method for ordinal data. We evaluated AQ results in terms of the number of iterations required to achieve convergence, model fit, parameter estimates, standard errors (SE), and statistical significance, across four link‐functions (logit, probit, log‐log, complimentary log‐log) using 3–10 and 20 quadrature points. We compared AQ results with those obtained using maximum likelihood, robust maximum likelihood, and robust diagonally weighted least‐squares estimation. Compared to the other two link‐functions, logit and probit not only produced fit statistics, parameters estimates, SEs, and levels of significance that varied less across numbers of quadrature points, but also fitted the data better and provided larger completely standardised loadings than did maximum likelihood and diagonally weighted least‐squares. Our findings demonstrate the viability of using full‐information AQ to estimate CFA models with real‐world ordinal data.     

Goodness-of-fit statistics for log-link regression models

The use of log binomial regression, regression on binary outcomes using a log link, is becoming increasingly popular because it provides estimates of relative risk. However, little work has been done on model evaluation. We used simulations to compare the performance of five goodness-of-fit statistics applied to different models in a log binomial setting, namely the Hosmer–Lemeshow, the normalized Pearson chi-square, the normalized unweighted sum of squares, Le Cessie and van Howelingen's statistic based on smoothed residuals and the Hjort–Hosmer test. The normalized Pearson chi-square was unsuitable as the rejection rate depended also on the range of predicted probabilities. The Le Cessie and van Howelingen's test statistic had poor sampling properties when evaluating a correct model and was also considered to be unsuitable in this context. The performance of the remaining three statistics was comparable in most simulations. However, using real data the Hjort–Hosmer outperformed the other two statistics.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

A Characterization For Gpsd Signals In Additive Noise

In this paper we consider a convoluted generalized power series distibution and characterize the distributions by soiutions to system of differential equations. Characterization results are derived for Poisson, binomial, geometric and Pascal (negative binomial) as special cases and later unified with Samaniego [1976, 1980] and Samaniego and Gong [1979]

receiveddate="Oct1985" reviseddate="Jun1986"     

New large-sample confidence intervals for a linear combination of binomial proportions

In this paper, we consider the problem wherein one desires to estimate a linear combination of binomial probabilities from k>2$k>2$ independent populations. In particular, we create a new family of asymptotic confidence intervals, extending the approach taken by Beal [1987. Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples. Biometrics 73, 941–950] in the two-sample case. One of our new intervals is shown to perform very well when compared to the best available intervals documented in Price and Bonett [2004. An improved confidence interval for a linear function of binomial proportions. Comput. Statist. Data Anal. 45, 449–456]. Furthermore, our interval estimation approach is quite general and could be extended to handle more complicated parametric functions and even to other discrete probability models in stratified settings. We illustrate our new intervals using two real data examples, one from an ecology study and one from a multicenter clinical trial.     

Modelling and forecasting mortality distributions in England and Wales using the Lee-Carter model

Lee and Carter proposed in 1992 a non-linear model m_xt = exp (a_x + b_x k_t + ε_xt) for fitting and forecasting age-specific mortality rates at age x and time t. For the model parameter estimation, they employed the singular value decomposition method to find a least squares solution. However, the singular value decomposition algorithm does not provide the standard errors of estimated parameters, making it impossible to assess the accuracy of model parameters. This article describes the Lee-Carter model and the technical procedures to fit and extrapolate this model. To estimate the precision of the parameter estimates of the Lee-Carter model, we propose a binomial framework, whose parameter point estimates can be obtained by the maximum likelihood approach and interval estimates by a bootstrap approach. This model is used to fit mortality data in England and Wales from 1951 to 1990 and to forecast mortality change from 1991 to 2020. The Lee-Carter model fits these mortality data very well with R² being 0.9980. The estimated overall age pattern of mortality a_x is very robust whereas there is considerable uncertainty in b_x (changes in the age pattern over time) and k_t (overall change in mortality). The fitted log age-specific mortality rates have been declining linearly from 1951 to 1990 at different paces and the projected rates will continue to decline in such a way in the 30 years prediction period.     

A Bayesian Approach for Zero-Inflated Count Regression Models by Using the Reversible Jump Markov Chain Monte Carlo Method and an Application

In this study, estimation of the parameters of the zero-inflated count regression models and computations of posterior model probabilities of the log-linear models defined for each zero-inflated count regression models are investigated from the Bayesian point of view. In addition, determinations of the most suitable log-linear and regression models are investigated. It is known that zero-inflated count regression models cover zero-inflated Poisson, zero-inflated negative binomial, and zero-inflated generalized Poisson regression models. The classical approach has some problematic points but the Bayesian approach does not have similar flaws. This work points out the reasons for using the Bayesian approach. It also lists advantages and disadvantages of the classical and Bayesian approaches. As an application, a zoological data set, including structural and sampling zeros, is used in the presence of extra zeros. In this work, it is observed that fitting a zero-inflated negative binomial regression model creates no problems at all, even though it is known that fitting a zero-inflated negative binomial regression model is the most problematic procedure in the classical approach. Additionally, it is found that the best fitting model is the log-linear model under the negative binomial regression model, which does not include three-way interactions of factors.     

On a modified version of intervened negative binomial distribution

In this paper, a generalized version of the intervened negative binomial distribution of Kumar and Sreeja [On intervened negative binomial distribution and some of its properties. Statistica. 2012;72:395–404] is considered and studied some of its properties. Certain methods of estimation of the parameters of the distribution are discussed and illustrated with the help of real life data sets. A test procedure is suggested for testing the intervention parameter and a simulation study is conducted for examining the performance of the estimators.     

A Generalized Chain Binomial Model with Aggregated Data

In large cohort studies it can be impractical to report individual data that only summary or aggregated data are available. Using aggregated data from Bernoulli trials is expected to result in overdispersion so that a quasi-binomial approach would seem feasible. We show that when applied to aggregated data arising from cohorts of individuals according to a chain binomial model, the quasi-binomial model results in biased estimates. We propose an alternate calibration estimator and demonstrate its improved performance by simulations. The calibration method is then applied to model the probability of leaving a personal emergency link service in Hong Kong.