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The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.

In this study, we develop nonparametric analysis of deviance tools for generalized partially linear models based on local polynomial fitting. Assuming a canonical link, we propose expressions for both local and global analysis of deviance, which admit an additivity property that reduces to analysis of variance decompositions in the Gaussian case. Chi-square tests based on integrated likelihood functions are proposed to formally test whether the nonparametric term is significant. Simulation results are shown to illustrate the proposed chi-square tests and to compare them with an existing procedure based on penalized splines. The methodology is applied to German Bundesbank Federal Reserve data.

We present how to enhance classical generalized linear models by neural network features. On the way there, we highlight the traps and pitfalls that need to be avoided to get good statistical models. This includes the non-uniqueness of sufficiently good regression models, the balance property, and representation learning, which brings us back to the concept of the good old generalized linear models. This paper has been integrated into SSRN Manuscript 3822407.

Generalized linear models (GLMs) with error-in-covariates are useful in epidemiological research due to the ubiquity of non-normal response variables and inaccurate measurements. The link function in GLMs is chosen by the user depending on the type of response variable, frequently the canonical link function. When covariates are measured with error, incorrect inference can be made, compounded by incorrect choice of link function. In this article we propose three flexible approaches for handling error-in-covariates and estimating an unknown link simultaneously. The first approach uses a fully Bayesian (FB) hierarchical framework, treating the unobserved covariate as a latent variable to be integrated over. The second and third are approximate Bayesian approach which use a Laplace approximation to marginalize the variables measured with error out of the likelihood. Our simulation results show support that the FB approach is often a better choice than the approximate Bayesian approaches for adjusting for mea...

The generalized additive model is a well established and strong tool that allows modelling smooth effects of predictors on the response. However, if the link function, which is typically chosen as the canonical link, is misspecified, estimates can be biased. A procedure is proposed that simultaneously estimates the form of the link function and the unknown form of the predictor functions including selection of predictors. The procedure is based on boosting methodology, which obtains estimates by using a sequence of weak learners. It strongly dominates fitting procedures that are unable to modify a given link function if the true link function deviates from the fixed function. The performance of the procedure is shown in simulation studies and illustrated by real world examples.

We propose a generalized triple trends test for broad families of distributions whose canonical link follows a nonlinear dose-response model (e.g., sigmoidal \(E_{max}\) model). For the triple trends test, we also develop a procedure for calculating the sample size. The main goal of the research is to develop a likelihood inference on the effect of any given dose, which is not necessarily studied in the trial. In particular, we propose a likelihood test with superiority margins that can be calibrated for an effect size of interest. This likelihood test for statistical significance and clinical meaningfulness (TSSCM) leads to the lower effective bound (LEB), which, by construction, achieves a specified probability of coverage for the dose with an effect size of interest. The LEB is then used to define the full dose range for phase 3 trials as suggested by Temple (2004). As a result, the phase 3 trials contain critical information that allows assessment of dose(s) with optimal benefit-to-risk profiles. Thus, the clinical development strategy by Temple (2004) achieves a high probability of technical, regulatory and post-approval success. We illustrate the LEB approach with a case example and simulation studies. The LEB approach was developed in the context of the PhRMA working group on dose-response trials to implement the clinical development strategy of Temple (2004). The approach is superior to existing approaches, including those reported by the PhRMA working group.

While developing a prior distribution for any Bayesian analysis, it is important to check whether the corresponding posterior distribution becomes degenerate in the limit to the true parameter value as the sample size increases. In the same vein, it is also important to understand a more detailed asymptotic behavior of posterior distributions. This is particularly relevant in the development of many nonsubjective priors. The present paper focuses on asymptotic expansions of posteriors for generalized linear models with canonical link functions when the number of regressors grows to infinity at a certain rate relative to the growth of the sample size. These expansions are then used to derive moment matching priors in the generalized linear model setting.

Investigators commonly gather longitudinal data to assess changes in responses over time and to relate these changes to within-subject changes in predictors. With rare or expensive outcomes such as uncommon diseases and costly radiologic measurements, outcome-dependent, and more generally outcome-related, sampling plans can improve estimation efficiency and reduce cost. Longitudinal follow up of subjects gathered in an initial outcome-related sample can then be used to study the trajectories of responses over time and to assess the association of changes in predictors within subjects with change in response. In this article, we develop two likelihood-based approaches for fitting generalized linear mixed models (GLMMs) to longitudinal data from a wide variety of outcome-related sampling designs. The first is an extension of the semi-parametric maximum likelihood approach developed in Neuhaus, Scott and Wild (2002, Biometrika 89, 23-37) and Neuhaus, Scott and Wild (2006, Biometrics 62, 488-494) and applies quite generally. The second approach is an adaptation of standard conditional likelihood methods and is limited to random intercept models with a canonical link. Data from a study of attention deficit hyperactivity disorder in children motivates the work and illustrates the findings.

Generalized linear models (GLMs) are widely studied to deal with complex response variables. For the analysis of categorical dependent variables with more than two response categories, multivariate GLMs are presented to build the relationship between this polytomous response and a set of regressors. Traditional variable selection approaches have been proposed for the multivariate GLM with a canonical link function when the number of parameters is fixed in the literature. However, in many model selection problems, the number of parameters may be large and grow with the sample size. In this paper, we present a new selection criterion to the model with a diverging number of parameters. Under suitable conditions, the criterion is shown to be model selection consistent. A simulation study and a real data analysis are conducted to support theoretical findings.

We consider a generalized linear model (GLM) with canonical link function in which parameters can be subject to (i) a set of linear constraints and (ii) smoothing. We apply Lagrange methods to give a general Newton-Raphson algorithm for such a GLM in which parameters are estimated, constraints are applied and smoothing is performed simultaneously. We express the Lee-Carter model, an important model for the forecasting of human mortality, in terms of GLMs, and use our method to estimate the parameters in the model. The smoothing option allows us to improve the forecasting properties of the model. We compare the performance of (i) the Poisson model with log link for the force of mortality and (ii) the binomial model with logit link for the probability of death in a calendar year. Examples using UK Office for National Statistics data are provided.

Several two-stage multiple testing procedures have been proposed to detect gene-environment interaction in genome-wide association studies. In this article, we elucidate general conditions that are required for validity and power of these procedures, and we propose extensions of two-stage procedures using the case-only estimator of gene-treatment interaction in randomized clinical trials. We develop a unified estimating equation approach to proving asymptotic independence between a filtering statistic and an interaction test statistic in a range of situations, including marginal association and interaction in a generalized linear model with a canonical link. We assess the performance of various two-stage procedures in simulations and in genetic studies from Women's Health Initiative clinical trials. Copyright 2012, Oxford University Press.

The topic is penalized quasi-maximum likelihood estimation in generalized additive models by an approximation using a sequence of sub-models, here called blocks. The Schwarz method uses a sequence of sub-models, The technique might be useful to model comparison, where the fitted values from a sub-model are used as starting values for a larger model. We show that the algorithm method converges in canonical link of generalized additive models, and a theorem about bound condition of sub-models convergence with uncanonical link.

The Bayesian Poisson-Gamma hierarchy, leading to the negative binomial distribution, has been the standard practice in developing accident prediction models. To linearize the relationship connecting the mean of the negative binomial distribution to relevant covariates, a canonical log link has traditionally been used. Typically, little information is available regarding the choice of a particular link. To avoid link misspecification, it is proposed to nest the canonical log link model within a generalized link family and subsequently use the full Bayes method for parameter estimation, performance evaluation and inference. The proposed approach was applied to a sample of accident and traffic volume data corresponding to 99 intersections in the city of Edmonton, Alberta. The results showed that both the generalized link model and the traditional canonical link model provided adequate fit to the data. However, the Bayes factor provided a clear statistical support for the use of the generalized link approach. A procedure for link validation is also described. It allows the users (e.g., road authorities) to consider the changes in predicted accidents that will result if a generalized link is used instead of a canonical link. If a certain maximal change is tolerated, the canonical link can be used to analyze the data; otherwise the generalized link is worth the extra efforts and should be adopted. When compared with the traditional approach, the generalized link model was found to predict a lower number of accidents whenever there is a heavy traffic at the major approach, especially if combined with light flow on the minor approach. The paper concludes by identifying out areas for further research.

It is well known that the exponential dispersion family (EDF) of univariate distributions is closed under Bayesian revision in the presence of natural conjugate priors. However, this is not the case for the general multivariate EDF. This paper derives a second-order approximation to the posterior likelihood of a naturally conjugated generalised linear model (GLM), i.e., multivariate EDF subject to a link function (Section 5.5). It is not the same as a normal approximation. It does, however, lead to second-order Bayes estimators of parameters of the posterior. The family of second-order approximations is found to be closed under Bayesian revision. This generates a recursion for repeated Bayesian revision of the GLM with the acquisition of additional data. The recursion simplifies greatly for a canonical link. The resulting structure is easily extended to a filter for estimation of the parameters of a dynamic generalised linear model (DGLM) (Section 6.2). The Kalman filter emerges as a special case. A secon...

The use of graphical methods for comparing the quality of prediction throughout the design space of an experiment has been explored extensively for responses modeled with standard linear models. In this paper, fraction of design space (FDS) plots are adapted to evaluate designs for generalized linear models (GLMs). Since the quality of designs for GLMs depends on the model parameters, initial parameter estimates need to be provided by the experimenter. Consequently, an important question to consider is the design's robustness to user misspecification of the initial parameter estimates. FDS plots provide a graphical way of assessing the relative merits of different designs under a variety of types of parameter misspecification. Examples using logistic and Poisson regression models with their canonical links are used to demonstrate the benefits of the FDS plots.

The Fisher information for the canonical link exponential family generalised linear mixed model is derived. The contribution from the fixed effects parameters is shown to have a particularly simple form. © 2007 Elsevier Inc. All rights reserved.

In this paper we develop an algorithm for assessing the effect of small perturbations of the data on the validity of a postulated generalized linear model. The procedure is based on the geometric notion of shape operator, a single mathematical object that gathers together all the normal curvatures of a given influence graph or hypersurface. In addition to introducing relevant theoretical notions and explaining the foundations of the local influence assessment method in a generalized linear model with a canonical link, we provide a detailed example of application together with the explicit $R$ code implementing the algorithm. En este artículo proponemos un algoritmo para valorar numéricamente la influencia local en un modelo lineal generalizado sometido a una perturbación de las observaciones. El procedimiento se basa en el concepto geométrico de operador de horma, el cual recoge en un solo objeto matemático todas las curvaturas normales de la hipersuperficie o gráfica de influencia. Además de introducir algunas nociones útiles para la teoría y de explicar los fundamentos que sustentan la valuación para el caso de un modelo lineal generalizado que posea enlace canónico, presentamos con gran detalle un ejemplo de aplicación y un breve comentario sobre la implementación computacional del algoritmo.

We consider Poisson regression with the canonical link function. This regression model is widely used in regression analysis involving count data; one important application in electrical engineering is transmission tomography. In this paper, we establish the variable selection consistency and estimation consistency of the l 1 -regularized maximum-likelihood estimator in this regression model, and characterize the asymptotic sample complexity that ensures consistency even under the compressive sensing setting (or the n ≪ p setting in high-dimensional statistics).

Generalized additive models (GAMs) based on the binomial and Poisson distributions can be used to provide flexible semi-parametric modelling of binary and count outcomes. When used with the canonical link function, these GAMs provide semi-parametrically adjusted odds ratios and rate ratios. For adjustment of other effect measures, including rate differences, risk differences and relative risks, non-canonical link functions must be used together with a constrained parameter space. However, the algorithms used to fit these models typically rely on a form of the iteratively reweighted least squares algorithm, which can be numerically unstable when a constrained non-canonical model is used. We describe an application of a combinatorial EM algorithm to fit identity link Poisson, identity link binomial and log link binomial GAMs in order to estimate semi-parametrically adjusted rate differences, risk differences and relative risks. Using smooth regression functions based on B-splines, the method provides stable convergence to the maximum likelihood estimates, and it ensures that the estimates always remain within the parameter space. It is also straightforward to apply a monotonicity constraint to the smooth regression functions. We illustrate the method using data from a clinical trial in heart attack patients.

In this article, we obtain an estimator of the regression parameters for generalized linear models, using the Jacobian technique. We restrict ourselves to the natural exponential family for the response variable and choose the conjugate prior for the natural parameter. Using the Jacobian of transformation, we obtain the posterior distribution for the canonical link function and thereby obtain the posterior mode for the link. Under the full rank assumption for the covariate matrix, we then find an estimator for the regression parameters for the natural exponential family. Then the proposed estimator is specially derived for the Poisson model with log link function, and the binomial response model with the logit link function. We also discuss extensions to the binomial response model when covariates are all positive. Finally, an illustrative real-life example is given for the Poisson model with log link. In order to estimate the standard error of our estimators, we use the Bernstein-von Mises theorem. Final...

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.

This paper studies a class of AIC-like model selection criteria for a generalized linear model with the canonical link. They have the form of log L - p * C , where log L is the maximized log-likelihood, p is the number of parameters and C is a term depending on the sample size n and satisfying C / n → 0 and C / log log n → ∞ as n → ∞ . Under suitable conditions, this class of criteria is shown to be strongly consistent. A simulation study was also conducted to assess the finite-sample performance with various choices of C for variable selection in a logit model and a log-linear model.

SummaryThe paper considers canonical link generalized linear models with stratum-specific nuisance intercepts and missing covariate data. This family includes the conditional logistic regression model. Existing methods for this problem, each of which uses a conditioning argu- ment to eliminate the nuisance intercept, model either the missing covariate data or the missingness process. The paper compares these methods under a common likelihood framework. The semiparametric efficient estimator is identified, and a new estimator, which reduces dependence on the model for the missing covariate, is proposed. A simulation study compares the methods with respect to efficiency and robustness to model misspecification.

The methods of adaptive designs are some statistical methods which allow us to choose new design variables according to the information obtained from the previous design variables and their corresponding responses. They have been implemented into many practical applications such as in engineering-control problems, educational/psychological testing and biological assays. For example, in the modern educational testing theory, they apply the idea of adaptive designs to the tailored testing or the computerized adaptive testing, in which a new test item is chosen for each individual test-taker according to the estimate of his/her ability, which is based on his/her performance to the previous test items assigned to him/her. In this paper, we propose a sequential procedure for constructing fixed size confidence regions with a prescribed precision and a given coverage probability for the regression parameters of generalized linear models with adaptive designs. The procedure is based on the maximum quasi-likelihood estimate and without assuming canonical link functions for generalized linear models. Some large sample properties and simulation results of the proposed sequential procedures are obtained. The parallel results for generalized linear models with fixed or i.i.d. design variables can be obtained as corollaries.

The “leapfrog” hybrid Monte Carlo algorithm is a simple and effective MCMC method for fitting Bayesian generalized linear models with canonical link. The algorithm leads to large trajectories over the posterior and a rapidly mixing Markov chain, having superior performance over conventional methods in difficult problems like logistic regression with quasicomplete separation. This method offers a very attractive solution to this common problem, providing a method for identifying datasets that are quasicomplete separated, and for identifying the covariates that are at the root of the problem. The method is also quite successful in fitting generalized linear models in which the link function is extended to include a feedforward neural network. With a large number of hidden units, however, or when the dataset becomes large, the computations required in calculating the gradient in each trajectory can become very demanding. In this case, it is best to mix the algorithm with multivariate random walk Met...

Summary In the estimation of a population mean or total from a random sample, certain methods based on linear models are known to be automatically design consistent, regardless of how well the underlying model describes the population. A sufficient condition is identified for this type of robustness to model failure; the condition, which we call ‘internal bias calibration’, relates to the combination of a model and the method used to fit it. Included among the internally bias-calibrated models, in addition to the aforementioned linear models, are certain canonical link generalized linear models and nonparametric regressions constructed from them by a particular style of local likelihood fitting. Other models can often be made robust by using a suboptimal fitting method. Thus the class of model-based, but design consistent, analyses is enlarged to include more realistic models for certain types of survey variable such as binary indicators and counts. Particular applications discussed are the estimation of the size of a population subdomain, as arises in tax auditing for example, and the estimation of a bootstrap tail probability.

We develop analysis of deviance tools for generalized partial linear models based on local polynomial fitting. Assuming a canonical link, we propose expressions for both local and global analysis of deviance, which admit an additivity property that reduces to ANOVA decompositions in the Gaussian case. Chi-square tests based on integrated likelihood functions are proposed to formally test whether the non-parametric term is significant. Simulation results are shown to illustrate the proposed chi-square tests. The methodology is applied to German Bundesbank Federal Reserve data.

We consider the problem of estimating and detecting outliers in count time series data following a log-linear observation driven model. Log-linear models for count time series arise naturally because they correspond to the canonical link function of the Poisson distribution. They yield both positive and negative dependence, and covariate information can be conveniently incorporated. Within this framework, we establish test procedures for detection of unusual events ('interventions') leading to different kinds of outliers, we implement joint maximum likelihood estimation of model parameters and outlier sizes and we derive formulae for correcting the data for detected interventions. The effectiveness of the proposed methodology is illustrated with two real data examples. The first example offers a fresh data analytic point of view towards the polio data. Our methodology identifies different forms of outliers in these data by an observation-driven model. The second example deals with some campylobacterosis data which we analyzed in a previous communication, by a different model. The results are reconfirmed by the new model that we put forward in this communication. The reliability of the procedure is verified using artificial data examples. © 2012 SAGE Publications. 12 4 299 322 Cited By :9

We present tight surrogate regret bounds for the class of proper (i.e., Fisher consistent) losses. The bounds generalise the margin-based bounds due to Bartlett et al. (2006). The proof uses Taylor's theorem and leads to new representations for loss and regret and a simple proof of the integral representation of proper losses. We also present a different formulation of a duality result of Bregman divergences which leads to a simple demonstration of the convexity of composite losses using canonical link functions.

Generalized linear models (GLMs) outline a wide class of regression models where the effect of the explanatory variables on the mean of the response variable is modelled throughout the link function. The choice of the link function is typically overlooked in applications and the canonical link is commonly used. The estimation of GLMs with unspecified link function is discussed, where the linearity assumption between the link and the linear predictor is relaxed and the unspecified relationship is modelled flexibly by means of P-splines. An estimating algorithm is presented, alternating estimation of two working GLMs up to convergence. The method is applied to the analysis of quit behavior of production workers where the logit, probit and clog-log links do not appear to be appropriate.

Generalized linear models (GLMs) allow for a wide range of statistical models for regression data. In particular, the logistic model is usually applied for binomial observations. Canonical links for GLMs such as the logit link in the binomial case, are often used because in this case minimal sufficient statistics for the regression parameter exist which allow for simple interpretation of the results. However, in some applications, the overall fit as measured by the p-values of goodness-of-fit statistics (as the residual deviance) can be improved significantly by the use of a noncanonical link. In this case, the interpretation of the influence of the covariables is more complicated compared to GLMs with canonical link functions. It will be illustrated through simulation that the p-value associated with the common goodness-of-link tests is not appropriate to quantify the changes to mean response estimates and other quantities of interest when switching to a noncanonical link. In particular, the rate of misspecifications becomes considerably large, when the inverse information value associated with the underlying parametric link model increases. This shows that the classical tests are often too sensitive, in particular, when the number of observations is large. The consideration of a generalized p-value function is proposed instead, which allows the exact quantification of a suitable distance to the canonical model at a controlled error rate. Corresponding tests for validating or discriminating the canonical model can easily performed by means of this function. Finally, it is indicated how this method can be applied to the problem of overdispersion.

In full exponential-family models, such as generalized linear models with canonical link, use of the Jeffreys invariant prior as a penalty function removes the leading term from the asymptotic bias of maximum likelihood estimates. In Poisson log linear and binomial logistic linear models, the posterior mode can be calculated using the standard iterative weighted least-squares algorithm with an appropriate adjustment to the working vector at each cycle. The required adjustment is a simple function of case leverages. Implementation is particularly straightforward in GLIM4, and illustrates the power of the new ‘OWN algorithm’ facility.

Use of the Jeffreys invariant prior in generalized linear models has been studied recently by a number of authors. In models with canonical link, the posterior mode has asymptotic bias of smaller order than that of the maximum likelihood estimate. In this paper a new algorithm for calculation of the posterior mode is developed. The algorithm makes use of the iterative weighted least-squares method commonly used for maximum likelihood calculations, so that implementation is possible in standard regression software, such as GLIM. Comparison is made with a more ‘direct’ approach using Newton’s method, found to be easily implemented in the object-oriented modelling environment of LISP-STAT.

This paper considers testing the goodness-of-fit of regression models. Emphasis is on a goodness-of-fit test for generalized linear models with canonical link function and known dispersion parameter. The test is based on the score test for extra variation in a random effects model. By choosing a suitable form for the dispersion matrix, a goodness-of-fit test statistic is obtained which is quite similar to test statistics based on non-parametric kernel methods. We consider the distribution of the test statistic and discuss the choice of the dispersion matrix. The testing method can handle models with continuous and discrete covariates. Corrections for bias when parameters are estimated are available and extensions to models with unknown dispersion parameters, and more general nonlinear models are discussed. The proposed goodness-of-fit method is demonstrated in a simulation study and on real data of bone marrow transplant patients. The individual contributions of observations to the test statistic are used to perform residual analyses.

This paper extends the theory of graduation by parametric formulae to include dynamic estimation methods. This is an application of the Kaiman filter and allows the parameters of the curve fitted to vary with age. The amount of variation is determined by the amount of smoothing required, and the method can be regarded as a combination of curve fitting and sequential smoothing, each of which has been used separately for performing graduations. In practice, a dynamic straight line can always be used for the graduation and the method has a sensible logical interpretation. these papers concentrate on the use of parametric modelling methods to obtain a reasonably close fit to actual experience, giving due weight to model parsimony and smoothness. In doing this, it is usual to fit a curve (a straight line, cubic, quartic, etc.) which is assumed to apply to the data over the whole range of ages considered. It is sometimes the case that a straight line produces an adequate fit (to the transformations of the death rate considered in this paper), but greater flexibility can be obtained by using higher order polynomials, non-linear regression functions or, within the context of generalised linear models, non- canonical link functions. This paper proposes a different method of obtaining greater flexibility than can be obtained by simply fitting a straight line. The technique proposed has greater heuristic appeal than fitting higher order polynomials, and the form of the flexibility used appears to be very natural in the context of graduation. It involves a sophisticated smoothing procedure (retaining the parametric form of the regression), which can also be related to more basic methods of smoothing which seem natural for graduation, and have been used in the past. When, for example

Let W be a normal random variable with mean μand known variance σ2. Conditions on the function f(·) are given under which there exists an unbiased estimator, f(W), of f(μ) for all real μ. In particular it is shown that f(·) must be an entire function over the complex plane. Infinite series solutions for F(·) are obtained which are shown to be valid under growth conditions of the derivatives, fk( ·), of f(·). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurement-error models as a means of finding unbiased score functions when measurement error is normally distributed. Relative efficiencies comparing the proposed method to the use of conditional scores (Stefanski and Carroll, 1987) are given for the Poisson regression model with canonical link.

We consider the distribution of the (standardized) ML-estimator of the unknown parameter vector in a Generalized Linear Model with canonical link function. It will be shown that its (parametric) Bootstrap estimator is consistent under the same assumptions needed by Fahrmeir & Kaufmann (1985, 1986) to show its asymptotic normality.

The simplest and most common way of producing confidence intervals for the parameters of a generalized linear model is by means of the standard errors of the estimates. However, for a number of reasons it is desirable to base the confidence intervals on the change in log-likelihood. This paper presents a simple way of computing profile-likelihood-based confidence intervals for canonical link models in GLIM. Recent work has made it possible to compute approximate conditional intervals for these models. The implementation of these calculations as GLIM macros is discussed. The relevant macros are provided, and their use is illustrated with data from three biomedical studies.

The main purpose of this paper is to unify and extend the existing theory of ‘estimated zeroes’ in log-linear and logit models. To this end it is shown, that every GLM can be embedded in a larger model with a compact parameter space and a continuous likelihood (a ‘CGLM’). Clearly in a CGLM the MLE always exists, solving a major data-analysis problem. In the mean value parametrization the construction of the CGLM is remarkably simple; in thes-parametrization it is more complex. Estimated expected values are always finite, but the MLE need not correspond with a finite β, as is well known for estimated zeroes in log-linear models. The boundary distributions of CGLMs are classified in four categories: ‘inadmissable’, ‘degenerate’, ‘Chentsov’, and ‘constrained’. For a large class of GLMs, including all GLMs with canonical link functions and probit models, the MLE in the corresponding CGLM exists and is unique. This seems to be new for log-linear models. We give equivalent algebraic and geometric conditions (in the vein of Haberman (1974, 1977) and Albert & Anderson (1984) respectively), necessary for the existence of the MLE in the GLM (corresponding to a finite β). For a large class of GLMs these conditions are also sufficient. This too seems new for log-linear models.

We consider the fitting of generalized linear models in which the link function is assumed to be unknown, and propose the following computational method: First, estimate regression coefficients using the canonical link. Then, estimate the link via a kernel smoother, treating the direction in the predictor space determined by the regression coefficients as known. Then reestimate the direction using the estimated link and alternate between these two steps. We show that under fairly general conditions, $n^{1/2}$-consistent estimates of the direction are obtained. A small Monte Carlo study is presented.