On selecting parametric link transformation families in generalized linear models

The use of parametric link transformation families in generalized linear models (GLM) has been shown to improve substantially the fit of standard analyses using a fixed link in some data sets (see Czado, 1993, for example). When link and regression parameters are globally orthogonal (Cox and Reid, 1987), then the variance inflation of the regression parameter estimates due to the additional estimation of the link is asymptotically zero. Parameter orthogonality also induces numerical stability which is seen in the reduction of computation time required for the calculation of parameter estimates. This stability remains a desirable property even for inferences which are conditional on a fixed link value. Czado and Santner (1992b), for binomial error, and Czado (1992), for GLMs have shown that only local orthogonality can be achieved in general. This paper provides conditions on the link family to extend the notion of local orthogonality at a point to orthogonality in a neighborhood asymptotically and shows that the resulting links are location and scale invariant. General concepts for the construction of such links are given, and it is shown how they relate to link families proposed in the literature. The ideas are illustrated by two examples.     

On parameter orthogonality in symmetric and skew models

Orthogonal and partly orthogonal reparametrisations are provided for certain wide and important families of univariate continuous distributions. First, the orthogonality of parameters in location-scale symmetric families is extended to symmetric distributions involving a third parameter. This sets the scene for consideration of the four-parameter situation in which skewness is also allowed. It turns out that one specific approach to generating such four-parameter families, that of two-piece distributions with a certain parametrisation restriction, has some attractive features with regard to parameter orthogonality which, to the best of our knowledge, are not shared with other four-parameter distributions. Our work also affords partly orthogonal parametrisations of three-parameter two-piece models.     

Parameter Orthogonality in Mixed Regression Models for Survival Data

The implications of parameter orthogonality for the robustness of survival regression models are considered. The question of which of the proportional hazards or the accelerated life families of models would be more appropriate for analysis is usually ignored, and the proportional hazards family is applied, particularly in medicine, for convenience. Accelerated life models have conventionally been used in reliability applications. We propose a one-parameter family mixture survival model which includes both the accelerated life and the proportional hazards models. By orthogonalizing relative to the mixture parameter, we can show that, for small effects of the covariates, the regression parameters under the alternative families agree to within a constant. This recovers a known misspecification result. We use notions of parameter orthogonality to explore robustness to other types of misspecification including misspecified base-line hazards. The results hold in the presence of censoring. We also study the important question of when proportionality matters.     

On comparing two methods for approximate conditional inference

Approximate conditional inference for a real parameter in the presence of nuisance parameters was examined from a sample-space differential viewpoint in Fraser and Reid (1988) and a conditional inference procedure was proposed. Conditional likelihood-based inference in the same setting was discussed in Cox and Reid (1987), where emphasis was placed on orthogonalizing the nuisance parameter to the parameter of interest. In this paper the sample-space partitions of the two methods are examined for the case that the minimal sufficient statistic has the same dimension as the parameter space. The methods are identical if observed and expected information gives the same orthogonality; an example indicates how they can differ more generally. A specially chosen reparameterization provides some geometrical insight to the methods and allows a comparison in terms of score functions and locally defined orthogonal parameters.     

Asymptotic properties of a conditional maximum-likelihood estimator

Inference for a scalar interest parameter in the presence of nuisance parameters is considered in terms of the conditional maximum-likelihood estimator developed by Cox and Reid (1987). Parameter orthogonality is assumed throughout. The estimator is analyzed by means of stochastic asymptotic expansions in three cases: a scalar nuisance parameter, m nuisance parameters from m independent samples, and a vector nuisance parameter. In each case, the expansion for the conditional maximum-likelihood estimator is compared with that for the usual maximum-likelihood estimator. The means and variances are also compared. In each of the cases, the bias of the conditional maximum-likelihood estimator is unaffected by the nuisance parameter to first order. This is not so for the maximum-likelihood estimator. The assumption of parameter orthogonality is crucial in attaining this result. Regardless of parametrization, the difference in the two estimators is first-order and is deterministic to this order.     

Quasi-Likelihood Regression with Multiple Indices and Smooth Link and Variance Functions

Abstract.  A flexible semi-parametric regression model is proposed for modelling the relationship between a response and multivariate predictor variables. The proposed multiple-index model includes smooth unknown link and variance functions that are estimated non-parametrically. Data-adaptive methods for automatic smoothing parameter selection and for the choice of the number of indices M are considered. This model adapts to complex data structures and provides efficient adaptive estimation through the variance function component in the sense that the asymptotic distribution is the same as if the non-parametric components are known. We develop iterative estimation schemes, which include a constrained projection method for the case where the regression parameter vectors are mutually orthogonal. The proposed methods are illustrated with the analysis of data from a growth bioassay and a reproduction experiment with medflies. Asymptotic properties of the estimated model components are also obtained.     

ON CONFIDENCE REGIONS OF EMBEDDED MODELS IN REGULAR PARAMETRIC FAMILIES (A GEOMETRIC APPROACH)

Regular parametric families are commonly encountered in statistical problems (e.g. Cox & Hinkley, 1974). In this paper, we propose a differential geometric framework for the embedded models in these families. Our framework may be regarded as an extension of that presented by Bates & Watts (1980) for nonlinear regression models. As an application, we use this geometric framework to derive three kinds of improved approximate confidence regions for the parameter and parameter subsets in terms of curvatures. The results obtained by Hamilton et al. (1982) and Hamilton (1986) are extended to embedded models in regular parametric families.     

The Cox Proportional Hazards Model with a Partly Linear Relative Risk Function

The conventional Cox proportional hazards regression model contains a loglinear relative risk function, linking the covariate information to the hazard ratio with a finite number of parameters. A generalization, termed the partly linear Cox model, allows for both finite dimensional parameters and an infinite dimensional parameter in the relative risk function, providing a more robust specification of the relative risk function. In this work, a likelihood based inference procedure is developed for the finite dimensional parameters of the partly linear Cox model. To alleviate the problems associated with a likelihood approach in the presence of an infinite dimensional parameter, the relative risk is reparameterized such that the finite dimensional parameters of interest are orthogonal to the infinite dimensional parameter. Inference on the finite dimensional parameters is accomplished through maximization of the profile partial likelihood, profiling out the infinite dimensional nuisance parameter using a kernel function. The asymptotic distribution theory for the maximum profile partial likelihood estimate is established. It is determined that this estimate is asymptotically efficient; the orthogonal reparameterization enables employment of profile likelihood inference procedures without adjustment for estimation of the nuisance parameter. An example from a retrospective analysis in cancer demonstrates the methodology.     

Classical and robust orthogonal regression between parts of compositional data

The different parts (variables) of a compositional data set cannot be considered independent from each other, since only the ratios between the parts constitute the relevant information to be analysed. Practically, this information can be included in a system of orthonormal coordinates. For the task of regression of one part on other parts, a specific choice of orthonormal coordinates is proposed which allows for an interpretation of the regression parameters in terms of the original parts. In this context, orthogonal regression is appropriate since all compositional parts – also the explanatory variables – are measured with errors. Besides classical (least-squares based) parameter estimation, also robust estimation based on robust principal component analysis is employed. Statistical inference for the regression parameters is obtained by bootstrap; in the robust version the fast and robust bootstrap procedure is used. The methodology is illustrated with a data set from macroeconomics.     

Testing inference in variable dispersion beta regressions

The class of beta regression models proposed by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions, Journal of Applied Statistics 31 (2004), pp. 799–815] is useful for modelling data that assume values in the standard unit interval (0, 1). The dependent variable relates to a linear predictor that includes regressors and unknown parameters through a link function. The model is also indexed by a precision parameter, which is typically taken to be constant for all observations. Some authors have used, however, variable dispersion beta regression models, i.e., models that include a regression submodel for the precision parameter. In this paper, we show how to perform testing inference on the parameters that index the mean submodel without having to model the data precision. This strategy is useful as it is typically harder to model dispersion effects than mean effects. The proposed inference procedure is accurate even under variable dispersion. We present the results of extensive Monte Carlo simulations where our testing strategy is contrasted to that in which the practitioner models the underlying dispersion and then performs testing inference. An empirical application that uses real (not simulated) data is also presented and discussed.     

Mixed Discrete and Continuous Cox Regression Model

The Cox (1972) regression model is extended to include discrete and mixed continuous/discrete failure time data by retaining the multiplicative hazard rate form of the absolutely continuous model. Application of martingale arguments to the regression parameter estimating function show the Breslow (1974) estimator to be consistent and asymptotically Gaussian under this model. A computationally convenient estimator of the variance of the score function can be developed, again using martingale arguments. This estimator reduces to the usual hypergeometric form in the special case of testing equality of several survival curves, and it leads more generally to a convenient consistent variance estimator for the regression parameter. A small simulation study is carried out to study the regression parameter estimator and its variance estimator under the discrete Cox model special case and an application to a bladder cancer recurrence dataset is provided.     

Varying Dispersion Diagnostics for Inverse Gaussian Regression Models

Homogeneity of dispersion parameters is a standard assumption in inverse Gaussian regression analysis. However, this assumption is not necessarily appropriate. This paper is devoted to the test for varying dispersion in general inverse Gaussian linear regression models. Based on the modified profile likelihood (Cox & Reid, 1987), the adjusted score test for varying dispersion is developed and illustrated with Consumer- Product Sales data (Whitmore, 1986) and Gas vapour data (Weisberg, 1985). The effectiveness of orthogonality transformation and the properties of a score statistic and its adjustment are investigated through Monte Carlo simulations.     

Parametric link modification of both tails in binary regression

Common binary regression models such as logistic or probit regression have been extended to include parametric link transformation families. These binary regression models with parametric link are designed to avoid possible link misspecification and improve fit in some data sets. One and two parameter link families have been proposed in the literature (for a review see Stukel (1988)). However in real data examples published so far only one parameter link families have found to improve the fit significantly. This paper introduces a two parameter link family involving the modification of both tails of the link. An analysis based on computationally tractable Bayesian inference involving Monte Carlo sampling algorithms is presented extending earlier work of Czado (1992, 1993b). Finally, the usefulness of the two tailed link modification will be demonstrated in an example where single tail modification can be significantly improved upon by using a two tailed modification.     

The Cox–Aalen model for left-truncated and right-censored data

We analyze left-truncated and right-censored (LTRC) data using an additive-multiplicative Cox–Aalen model proposed by Scheike and Zhang (2002), which extends the Cox regression model as well as the additive Aalen model. Based on the conditional likelihood function, we derive the weighted least-squared (WLS) estimators for the regression parameters and cumulative intensity functions of the model. The estimators are shown to be consistent and asymptotically normal. A simulation study is conducted to investigate the performance of the proposed estimators.     

Series estimation for single‐index models under constraints

In this paper, a semi‐parametric single‐index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pending issue in the literature. Meanwhile, the link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimisation with the constraint of identification condition for the index parameter, which addresses an important problem in the literature of single‐index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimates are demonstrated through an extensive Monte Carlo study and an empirical example.     

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.     

Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

A NOTE ON MULTIVARIATE LOGISTIC MODELS FOR CONTINGENCY TABLES

The log-linear model is a tool widely accepted for modelling discrete data given in a contingency table. Although its parameters reflect the interaction structure in the joint distribution of all variables, it does not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters, recently introduced by Glonek & McCullagh (1995), which have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. Glonek & McCullagh give the link between the cell probabilities and the multivariate logistic parameters, in an algebraic fashion. The present paper focuses on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff-Nielsen, 1978) is used; this also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. The paper emphasises the analysis of independence hypotheses in margins of a contingency table.     

Extending the Scope of Inverse Regression Methods in Sufficient Dimension Reduction

In the area of sufficient dimension reduction, two structural conditions are often assumed: the linearity condition that is close to assuming ellipticity of underlying distribution of predictors, and the constant variance condition that nears multivariate normality assumption of predictors. Imposing these conditions are considered as necessary trade-off for overcoming the “curse of dimensionality”. However, it is very hard to check whether these conditions hold or not. When these conditions are violated, some methods such as marginal transformation and re-weighting are suggested so that data fulfill them approximately. In this article, we assume an independence condition between the projected predictors and their orthogonal complements which can ensure the commonly used inverse regression methods to identify the central subspace of interest. The independence condition can be checked by the gridded chi-square test. Thus, we extend the scope of many inverse regression methods and broaden their applicability in the literature. Simulation studies and an application to the car price data are presented for illustration.