This article proposes a robust model for parameter estimation when modeling in the presence of latent heterogeneity of population. Mixture of regression models are used for modeling when the data are a mixture of subgroups to allow for heterogeneity of the population. Parameter estimates from standard mixture of linear regression models are sensitive to atypical observations. To study mixtures of linear regression models, we introduce a class of robust estimators, called S-estimators. We investigate their breakdown point in mixture of linear regression models. It is expected that the robust S-estimators can achieve the high breakdown point in the contaminated data from the heterogenous populations. This model presents a unified, robust framework and parameter estimation is achieved via an expectation-conditional maximization (ECM) algorithm. This new family of robust mixture models is validated through Monte Carlo simulations. The application to real life data sets has shown that use of robust S-estimators in mixture of linear regression models accommodates the outliers producing stable estimates as compared to maximum likelihood estimators from mixture of linear regression models.

Mixture of regression models are used for modeling when the data are a mixture of subgroups to allow for heterogeneity of the population. In the presence of outliers, standard mixture of linear regression models fail. To study mixture of linear regression models in this setting, we introduce a class of robust estimators, called S-estimators. We investigate their breakdown point in mixture of linear regression models. It is expected that the S-estimators can achieve the highest possible breakdown point in the contaminated data in this setting.

This article addresses the use of different biweight functions in high breakdown mixture discriminant analysis approach of Bashir and Carter ( 2005). The translated biweight S function of Rocke ( 1996) is used in the robust estimation of the mixture distributions parameters. As in discriminant analysis, the main purpose is to classify the test observations with greater accuracy, so the estimators producing smaller errors of misclassification are preferred. In the simulation studies, the use of Tukey's biweight function was proved to be more effective in producing better classification results as compared to the translated biweight function.

Robust S-estimation is proposed for multivariate Gaussian mixture models generalizing the work of Hastie and Tibshirani (J. Roy. Statist. Soc. Ser. B 58 (1996) 155). In the case of Gaussian Mixture models, the unknown location and scale parameters are estimated by the EM algorithm. In the presence of outliers, the maximum likelihood estimators of the unknown parameters are affected, resulting in the misclassification of the observations. The robust S-estimators of the unknown parameters replace the non-robust estimators from M-step of the EM algorithm. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using robust S-estimators as compared to the standard maximum likelihood estimators.

ABSTRACT In the case of a large number of feature vector variables, using multivariate Gaussian mixture models, discrimination in a reduced subspace is studied, generalizing Hastie and Tibshirani's (1996) work, to a situation in which the outliers are present in the data. In the case of the Gaussian Mixture models, the reduced rank discriminant analysis is equivalent to the weighted rank k linear discriminant analysis (LDA). The reduced rank solution in the mixtures of multivariate Gaussian models was obtained from the full rank robust mixture solution. The classification in the new dimensions was compared with the discriminant analysis approach based on the original coordinates, using robust S-estimators. In most of the cases, the robust reduced rank mixture discriminant analysis (mda) performed better for the test data. However, for the case of common component covariance being diagonal, the robust reduced rank mixture discriminant analysis performed better than the robust full rank mixture discriminant analysis producing smaller errors in classification.

This article presents a new empirical Bayes estimator (EBE) and a shrinkage estimator for determining the relative potency from several multivariate bioassays by incorporating prior information on the model parameters based on Jeffreys' rules. The EBE can account for any extra variability among the bioassays, and if this extra variability is 0, then the EBE reduces to the maximum likelihood estimator for combinations of multivariate bioassays. The shrinkage estimator turns out to be a compromise of the prior information and the estimator from each multivariate bioassay, with the weights depending on the prior variance.

This paper develops a general multivariate iteratively re-weighted least squares (MIRLS) algorithm to fit multinomial data for a large class of link functions. The algorithm is shown to be very useful for the generalized linear model when applied to ordered data using either the logit or the normit link. Since binomial dose-response experiments are often monitored over time, then these dependent binomial responses can be modelled by using either link function. Examples from acute toxicity and teratology illustrate the algorithm. (C) 1998 John Wiley & Sons, Ltd.

This article examines the finite and infinite sample properties of the shrinkage estimator, motivated by a Bayesian argument, for the log relative potency, proposed in an earlier paper by Kim, Carter, and Hubert. This estimator can be written in closed form and is shown to have finite mean and finite variance in finite samples. As a consequence, this shrinkage estimator has finite frequentist risk, which is an improvement over the usual maximum likelihood estimator, for all finite sample sizes. Furthermore, it is shown that this estimator asymptotically behaves the same as the usual maximum likelihood estimator.

The added variable plots proposed by Pregibon and Wang and the partial residual plots proposed by Pregibon and Landwehr and co-workers for generalized linear models have been found useful for examining the relationship between the dependent variable and an independent variable; however, they can give misleading impressions about influential observations. Alternative procedures in which the plots are based on the full rather than the reduced model, or on weighted rather than unweighted variables, prove to be more informative about such observations, giving information which is consistent with Pregibon's influence diagnostics in generalized linear models.

Toxicologists frequently conduct toxicity experiments in which different treatment conditions are applied to groups of animals and the resulting mortality in each group is measured at a number of discrete time points over the course of the experiment. In this article, we develop and extend a number of diagnostic tools for the detection of mean misspecification, or systematic departures of the mean-link specification, in cumulative multinomial generalized linear models fit to such data. Several real data sets are used to illustrate these diagnostics. These tools help the analyst to differentiate between two sources of lack of fit in such models: mean misspecification and extra-multinomial variation or overdispersion.

Classical multivariate regression techniques can lead to an asymptotically efficient restricted estimator of the regression matrix if there is evidence that the regression matrix is of reduced rank. A sublethal joint toxicity experiment involving two agents is used to illustrate this approach.

A general point and interval estimator of the log relative potency is exhibited in closed form, which incorporates prior information on the log relative potency for symmetrical as well as asymmetrical parallel line bioassay. The point estimator turns out to be a weighted average of the usual estimator and the prior mean where the weights are determined by the prior variance.

Carter and Hubert (1984, Biometrics 40, 699-706) propose and apply an approach based on multivariate growth-curve theory to quantal bioassay over time. Their experimental data are reanalyzed using a fairly standard approach based on models for grouped response time data. This alternate approach is considerably more direct and theoretically more satisfactory. Practical advantages include direct assessment of model adequacy and parsimonious modelling of extra-multinomial variation.

SYNOPTIC ABSTRACTAs uniformly most powerful tests rarely exist in multivariate analysis, competing statistics have been proposed for the same testing problem. In this paper first a summary of the numerical results on the power functions of some tests of hypotheses is given, then theoretical properties of competing statistics are compared for a variety of problems.

A general linear model approach to quantitative parabolic bioassays with multivariate responses is proposed. The point and interval estimator of the relative potency and an associated test for validity is presented. The relationships with known univariate methods are illustrated. Keywords: relative potencygeneral linear modelsmaximum likelihood

A purpose of comparative parallel-line quantitative assays is to estimate the relative potency of a test drug relative to a standard drug. In a multivariate parallel-line assay several response variables will be measured. One such example occurs when a quantitative response variable is measured over time. In this paper we derive a closed form for the likelihood ratio tests for two hypotheses of interest. We first test for equal slopes; then we test to see if the potency is constant for all response variables. Finally, we present a procedure for obtaining point and interval estimators for relative potency. The procedure allows for covariables and block effects to be included in the analysis. The multivariate approach of this paper has been applied to a single-agent multivariate quantal assay, where the estimation of the median effective dose is the primary purpose. An analysis for this situation appears in Carter and Hubert (1984) and Hubert (1984). Suppose that a standard drug is administered to m experimental units in dosages di1, dim, and a test drug is administered to n experimental units in dosages d21, ..., d2n. The dose metameter is defined to be xij = logio(dij). A response variable Yij is then measured over p characteristics producing the response y[! = (yl, . . . , yp). A general linear model for the experiment may be written as

The typical quantal bioassay is used to estimate that concentration which will affect a given response in lOOp% of the subjects of interest. When p = .50 we speak of EC50 for median effective concentration, or LC50 for a lethal effect. More explicitly, such a controlled laboratory experiment usually involves subjecting nj experimental units to a concentration cj and recording the number, Zj, of experimental units affected by this concentration level. For an increasing sequence of concentrations Cj, j = 1, 2, . . . , d, with different corresponding sets of nj subjects, a sequence, zj, of the number of subjects responding, is obtained. When a lethal effect occurs, the ratios zj/nj = pj, say, become mortality rates and the problem is to estimate the concentration, LC50, which corresponds to p = .50. Various univariate methods can be used to solve this problem, the most popular being probit analysis; see, for example, Finney (1971) or Hubert (1984). If the responses are recorded after a time of t hours, the parameter to be estimated is referred to as 'the t-hour LC50'. When the experiment is monitored over a sequence of time points, the corresponding response variables become functions of concentration and time. However, the application of univariate probit analysis at each time point neglects the dependency in time and also the possible interaction of time and concentration on the response. The analysis of indirect quantitative assays with correlated data has been described by Box and Hay (1953). Their analysis assumes a repeated-measures model; that is, the correlations between observations on the same experimental unit are assumed to be equal. Elashoff (1981) studied the effect of heterogeneity of variances and correlations on this analysis. V0lund (1980) treated the case in which these correlations are arbitrary, thereby extending the work of Rao (1954). Multivariate indirect quantal assays have been studied by Kolakowski and Bock (1981) and by Kooijman (1981). Kolakowski and Bock (1981) assumed a latent structural model with independence among the responses over time. Kooijrman (1981) assumed that the mortality counts follow a multinomial distribution with the probability of death being a function of time and concentration. Such a model does not differentiate between sampling units and experimental units. For example, if concentrations of copper are administered to tanks of 20 fish, the model requires the assumption that there is no error variation among tanks of fish. Some work on the inclusion of this extra variation in the model for experiments involving litters of subjects has been done by Segreti and Munson (1981).