Analytical closed-form solution for binary logit regression by categorical predictors

In contrast to the common belief that the logit model has no analytical presentation, it is possible to find such a solution in the case of categorical predictors. This paper shows that a binary logistic regression by categorical explanatory variables can be constructed in a closed-form solution. No special software and no iterative procedures of nonlinear estimation are needed to obtain a model with all its parameters and characteristics, including coefficients of regression, their standard errors and t-statistics, as well as the residual and null deviances. The derivation is performed for logistic models with one binary or categorical predictor, and several binary or categorical predictors. The analytical formulae can be used for arithmetical calculation of all the parameters of the logit regression. The explicit expressions for the characteristics of logit regression are convenient for the analysis and interpretation of the results of logistic modeling.     

Properties of maximum likelihood estimates of the ratio of parameters in ordinal response regression models

The properties of a method of estimating the ratio of parameters for ordered categorical response regression models are discussed. If the link function relating the response variable to the linear combination of covariates is unknown then it is only possible to estimate the ratio of regression parameters. This ratio of parameters has a substitutability or relative importance interpretation.

The maximum likelihood estimate of the ratio of parameters, assuming a logistic function (McCullagh, 1980), is found to have very small bias for a wide variety of true link functions. Further it is shown using Monte Carlo simulations that this maximum likelihood estimate, has good coverage properties, even if the link function is incorrectly specified. It is demonstrated that combining adjacent categories to make the response binary can result in an analysis which is appreciably less efficient. The size of the efficiency loss on, among other factors, the marginal distribution in the ordered categories     

Joint analysis of nonlinear heterogeneous longitudinal data and binary outcome: an application to AIDS clinical studies

Finite mixture models are currently used to analyze heterogeneous longitudinal data. By releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, finite mixture models not only can estimate model parameters but also cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, which might be associated with a clinically important binary outcome. This article develops a joint modeling of a finite mixture of NLME models for longitudinal data in the presence of covariate measurement errors and a logistic regression for a binary outcome, linked by individual latent class indicators, under a Bayesian framework. Simulation studies are conducted to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and logistic regression are fitted separately, followed by an application to a real data set from an AIDS clinical trial, in which the viral dynamics and dichotomized time to the first decline of CD4/CD8 ratio are analyzed jointly.     

On nonlinear models for time series

The paper is a review of nonlinear processes used in time series analysis and presents some new original results about stationary distribution of a nonlinear autoregres-sive process of the first order. The following models are considered: nonlinear autoregessive processes, threshold AR processes, threshold MA processes, bilinear models, auto-regressive models with random parameters including double stochastic models, exponential AR models, generalized threshold models and smooth transition autoregressive models, Some tests for linearity of processes are also presented.     

Optimal estimating function for estimation and prediction in semi-parametric models

In a seminal paper, Godambe [1985. The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.] introduced the ‘estimating function’ approach to estimation of parameters in semi-parametric models under a filtering associated with a martingale structure. Later, Godambe [1987. The foundations of finite sample estimation in stochastic processes II. Bernoulli, Vol. 2. V.N.V. Science Press, 49–54.] and Godambe and Thompson [1989. An extension of quasi-likelihood Estimation. J. Statist. Plann. Inference 22, 137–172.] replaced this filtering by a more flexible conditioning. Abraham et al. [1997. On the prediction for some nonlinear time-series models using estimating functions. In: Basawa, I.V., et al. (Eds.), IMS Selected Proceedings of the Symposium on Estimating Functions, Vol. 32. pp. 259–268.] and Thavaneswaran and Heyde [1999. Prediction via estimating functions. J. Statist. Plann. Inference 77, 89–101.] invoked the theory of estimating functions for one-step ahead prediction in time-series models. This paper addresses the problem of simultaneous estimation of parameters and multi-step ahead prediction of a vector of future random variables in semi-parametric models by extending the inimitable approach of 13 and 14. The proposed technique is in conformity with the paradigm of the modern theory of estimating functions leading to finite sample optimality within a chosen class of estimating functions, which in turn are used to get the predictors. Particular applications of the technique give predictors that enjoy optimality properties with respect to other well-known criteria.     

Partially Linear Additive Models with Unknown Link Functions

In this paper, we consider partially linear additive models with an unknown link function, which include single‐index models and additive models as special cases. We use polynomial spline method for estimating the unknown link function as well as the component functions in the additive part. We establish that convergence rates for all nonparametric functions are the same as in one‐dimensional nonparametric regression. For a faster rate of the parametric part, we need to define appropriate ‘projection’ that is more complicated than that defined previously for partially linear additive models. Compared to previous approaches, a distinct advantage of our estimation approach in implementation is that estimation directly reduces estimation in the single‐index model and can thus deal with much larger dimensional problems than previous approaches for additive models with unknown link functions. Simulations and a real dataset are used to illustrate the proposed model.     

A Numerical Method for Estimating the Variance of Age at Maximum Growth Rate in Growth Models

Studies on maturation and body composition mention age at peak height velocity (PHV) as an important measure that could predict adulthood outcome. The age at PHV is often derived from growth models such as the triple logistic fitted to the stature (height) data. Theoretically, for a well-behaved growth function, age at PHV could be obtained by setting the second derivative of the growth function to zero and solving for age. Such a solution obviously depends on the parameters of the growth function. Therefore, the uncertainty in the estimation of age at PHV resulting from the uncertainty in the estimation of the growth model, need to be accounted for in the models in which it is used as a predictor. Explicit expressions for the age at PHV and, consequently the variance of the estimate of the age at PHV, do not exist for some of the commonly used nonlinear growth functions, such as the triple logistic function. Once an estimate of this variance is obtained, it could be incorporated in subsequent modeling either through measurement error models or by using the inverse variances as weights. A numerical method for estimating the variance is implemented. The accuracy of this method is demonstrated through comparisons in models where explicit solution for the variance exists. The method of estimating the variance is illustrated by applying to growth data from the Fels study and subsequently used as weights in modeling two adulthood outcomes from the same study.     

Nonlinear GCV and quasi-GCV for shrinkage models

The generalized cross-validation (GCV) method has been a popular technique for the selection of tuning parameters for smoothing and penalty, and has been a standard tool to select tuning parameters for shrinkage models in recent works. Its computational ease and robustness compared to the cross-validation method makes it competitive for model selection as well. It is well known that the GCV method performs well for linear estimators, which are linear functions of the response variable, such as ridge estimator. However, it may not perform well for nonlinear estimators since the GCV emphasizes linear characteristics by taking the trace of the projection matrix. This paper aims to explore the GCV for nonlinear estimators and to further extend the results to correlated data in longitudinal studies. We expect that the nonlinear GCV and quasi-GCV developed in this paper will provide similar tools for the selection of tuning parameters in linear penalty models and penalized GEE models.     

A Discussion on Nonlinear Models for Price Decisions in Rating-Based Product Preference Models

The purpose of this article is to discuss the application of nonlinear models to price decisions in the framework of rating-based product preference models. As revealed by a comparative simulation study, when a nonlinear model is the true model, the traditional linear model fails to properly describe the true pattern. It appears to be unsatisfactory in comparison with nonlinear models, such as logistic and natural spline, which offer some advantages, the most important being the ability to take into account more than just linear and/or monotonic effects. Consequently, when we model the product preference with a nonlinear model, we are potentially able to detect its ‘best’ price level, i.e., the price at which consumer preference towards a given attribute is at its maximum. From an application point of view, this approach is very flexible in price decisions and may produce original managerial suggestions which might not be revealed by traditional methods.     

Influence Diagnostics of Semiparametric Nonlinear Reproductive Dispersion Models

This article proposes a semiparametric nonlinear reproductive dispersion model (SNRDM) which is an extension of nonlinear reproductive dispersion model and semiparametric regression model. Maximum penalized likelihood estimators (MPLEs) of unknown parameters and nonparametric functions in SNRDMs are presented. Some novel diagnostic statistics such as Cook distance and difference deviance for parametric and nonparametric parts are developed to identify influence observations in SNRDMs on the basis of case-deletion method, and some formulae readily computed with the MPLEs algorithm for diagnostic measures are given. The equivalency of case-deletion models and mean-shift outlier models in SNRDM is investigated. A simulation study and a real example are used to illustrate the proposed diagnostic measures.     

Inferences from logistic regression models in the presence of small samples,rare events,nonlinearity, and multicollinearity with observational data

The logistic regression model has been widely used in the social and natural sciences and results from studies using this model can have significant policy impacts. Thus, confidence in the reliability of inferences drawn from these models is essential. The robustness of such inferences is dependent on sample size. The purpose of this article is to examine the impact of alternative data sets on the mean estimated bias and efficiency of parameter estimation and inference for the logistic regression model with observational data. A number of simulations are conducted examining the impact of sample size, nonlinear predictors, and multicollinearity on substantive inferences (e.g. odds ratios, marginal effects) when using logistic regression models. Findings suggest that small sample size can negatively affect the quality of parameter estimates and inferences in the presence of rare events, multicollinearity, and nonlinear predictor functions, but marginal effects estimates are relatively more robust to sample size.     

Estimating consumer acceptance limits

A methodology is developed for estimating consumer acceptance limits on a sensory attribute of a manufactured product. In concept these limits are analogous to engineering tolerances. The method is based on a generalization of Stevens' Power Law. This generalized law is expressed as a nonlinear statistical model. Instead of restricting the analysis to this particular case, a strategy is discussed for evaluating nonlinear models in general since scientific models are frequently of nonlinear form. The strategy focuses on understanding the geometrical contrasts between linear and nonlinear model estimation and assessing the bias in estimation and the departures from a Gaussian sampling distribution. Computer simulation is employed to examine the behavior of nonlinear least squares estimation. In addition to the usual Gaussian assumption, a bootstrap sample reuse procedure and a general triangular distribution are introduced for evaluating the effects of a non-Gaussian or asymmetrical error structure. Recommendations are given for further model analysis based on the simulation results. In the case of a model for which estimation bias is not a serious issue, estimating functions of the model are considered. Application of these functions to the generalization of Stevens’ Power Law leads to a means for defining and estimating consumer acceptance limits, The statistical form of the law and the model evaluation strategy are applied to consumer research data. Estimation of consumer acceptance limits is illustrated and discussed.     

Exploratory model analysis using cdf knotting with applications to distinguishability,limiting forms,and moment ratios to the three parameter weibull family

An exploratory model analysis device we call CDF knotting is introduced. It is a technique we have found useful for exploring relationships between points in the parameter space of a model and global properties of associated distribution functions. It can be used to alert the model builder to a condition we call lack of distinguishability which is to nonlinear models what multicollinearity is to linear models. While there are simple remedial actions to deal with multicollinearity in linear models, techniques such as deleting redundant variables in those models do not have obvious parallels for nonlinear models. In some of these nonlinear situations, however, CDF knotting may lead to alternative models with fewer parameters whose distribution functions are very similar to those of the original overparameterized model. We also show how CDF knotting can be exploited as a mathematical tool for deriving limiting distributions and illustrate the technique for the 3-parameterWeibull family obtaining limiting forms and moment ratios which correct and extend previously published results. Finally, geometric insights obtained by CDF knotting are verified relative to data fitting and estimation.     

Target estimation for the logistic regression model

The method of target estimation developed by Cabrera and Fernholz [(1999). Target estimation for bias and mean square error reduction. The Annals of Statistics, 27(3), 1080–1104.] to reduce bias and variance is applied to logistic regression models of several parameters. The expectation functions of the maximum likelihood estimators for the coefficients in the logistic regression models of one and two parameters are analyzed and simulations are given to show a reduction in both bias and variability after targeting the maximum likelihood estimators. In addition to bias and variance reduction, it is found that targeting can also correct the skewness of the original statistic. An example based on real data is given to show the advantage of using target estimators for obtaining better confidence intervals of the corresponding parameters. The notion of the target median is also presented with some applications to the logistic models.     

A Robust Procedure in Nonlinear Models for Repeated Measurements

Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such information might involve direct knowledge of the actual form of the true model or might be represented by a set of differential equations that the model must satisfy. We develop M-procedures for estimating parameters and testing hypotheses of interest about these parameters in nonlinear regression models for repeated measurement data. Under regularity conditions, the asymptotic properties of the M-procedures are presented, including the uniform linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed with iterative procedures, similar to Newton–Raphson and Fisher's scoring methods. The methodology is illustrated by using a multivariate logistic regression model with real data, along with a simulation study.     

Nonlinear regression modeling via the lasso-type regularization

We consider the problem of constructing nonlinear regression models with Gaussian basis functions, using lasso regularization. Regularization with a lasso penalty is an advantageous in that it estimates some coefficients in linear regression models to be exactly zero. We propose imposing a weighted lasso penalty on a nonlinear regression model and thereby selecting the number of basis functions effectively. In order to select tuning parameters in the regularization method, we use a deviance information criterion proposed by Spiegelhalter et al. (2002), calculating the effective number of parameters by Gibbs sampling. Simulation results demonstrate that our methodology performs well in various situations.     

Instrumental variable estimation in nonlinear measurement error models

This paper presents an extension of instrumental variable estimation to nonlinear regression models. For the linear model, the extended estimator is equivalent to the two-stage least squares estimator. The extended estimator is consistent for an important class of nonlinear models, including the logistic model, under relatively weak assumptions on the distribution of the measurement error. An example and simulation study are presented for the logistic regression model. The simulations suggest the estimator is reasonably efficient.     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.     

Invariant tests based on M-estimators,estimating functions,and the generalized method of moments

We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)?type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)?type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.     

Average Most Powerful Tests for a Segmented Regression

To improve the goodness of fit between a regression model and observations, the model can be complicated; however, that can reduce the statistical power when the complication does not lead significantly to an improved model. In the context of two-phase (segmented) logistic regressions, the model evaluation needs to include testing for simple (one-phase) versus two-phase logistic regression models. In this article, we propose and examine a class of likelihood ratio type tests for detecting a change in logistic regression parameters that splits the model into two-phases. We show that the proposed tests, based on Shiryayev–Roberts type statistics, are on average the most powerful. The article argues in favor of a new approach for fixing Type I errors of tests when the parameters of null hypotheses are unknown. Although the suggested approach is partly based on Bayes–Factor-type testing procedures, the classical significance levels of the proposed tests are under control. We demonstrate applications of the average most powerful tests to an epidemiologic study entitled “Time to pregnancy and multiple births.”