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When toxicity data are not available for a chemical mixture of concern, U.S. Environmental Protection Agency (EPA) guidelines allow risk assessment to be based on data for a surrogate mixture considered “sufficiently similar” in terms of chemical composition and component proportions. As a supplementary approach, using statistical equivalence testing logic and mixed model theory we have developed methodology to define sufficient similarity in dose—response for mixtures of many chemicals containing the same components with different ratios. Dose—response data from a mixture of 11 xenoestrogens and the endogenous hormone, 17s-estradiol are used to illustrate the method.

Fixed-ratio ray designs have been used for detecting and characterizing interactions of large numbers of chemicals in combination. Single-chemical dose-response data are used to predict an “additivity curve” along an environmentally relevant ray. A “mixture curve” is estimated from the mixture dose-response data along the ray. A test of additivity is equivalent to a test of coincidence of these two curves, which is based on the traditional hypothesis testing framework that assumes additivity in the null hypothesis and rejects with evidence of interaction. However, failure to reject may be due to lack of statistical power, making the claim of additivity problematic. As a solution we have developed rigorous methodology to test for additivity using statistical equivalence testing logic in which additivity is claimed based on pre-specified biologically important additivity margins, if the data support such a claim. Using the principle of confidence interval inclusion, a confidence region about the difference of meaningful functions of model parameters from the mixture model and that predicted under additivity is computed. When the confidence region is completely contained within the additivity margins then additivity is claimed with a Type I error rate chosen a priori to be some acceptably small value. The method is illustrated using an environmentally relevant fixed-ratio mixture of nine haloacetic acids where cytotoxic response is measured.

Exposure to mixtures of chemicals is an important and relevant environmental issue. Of particular interest is the detection and characterization of departure of biological effects from additivity. Methodology based on the assumption of additivity is used in fitting single-chemicaldata. Interactionsare determined and characterized by making comparisons between the observed and predicted responses at mixtures along a fixed ratio ray of the component substances. Two simultaneous tests are developed for testing for any departure from additivity. Multiple comparisons procedures are used to compare observed responses to that predicted under additivity. A simultaneous confidence band on the predicted responses along the mixture ray is also developed. The methods are illustrated with cytotoxicity data that arise when human epidermal keratinocytes are exposed to a mixture of arsenic, chromium, cadmium, and lead. Synergistic, antagonistic, and additive cytotoxicities were observed at different dose levels of the four-metal mixture.

Recent advances in quality technology have resulted from considering the variation of a response variable as well as its mean value. Current research dealing with the analysis of data involving noise factors has mostly been confined to models with normally distributed responses. The research described herein focuses on development of methodology for time-to-event data. Once model parameters have been estimated, optimization of the pertinent aspects of the event time distribution can be used to reduce the effect of the noise factors.

Alphabetic optimality has become an important component of experimental design in the case of the standard linear model. Many design criteria have been developed in order to produce optimal designs based on either parameter estimation or the prediction of a response. However, relatively little attention has been devoted to developing designs in the nonlinear model case (e.g. logistic regression model). D-optimality is the only criterion that has received much attention in the literature for the logistic regression model. This paper develops optimal designs for the logistic model based on the prediction of a response. Comparisons are made between different optimal designs by computing efficiencies based on certain optimality criteria. The robustness properties for some of the newly created designs are also investigated. Finally, the development of optimal designs for asymmetric regions of the logistic regression model is presented.

The delta method is utilized to construct an asymptotic 100(1 - alpha)% confidence interval for the response at the stationary point of a quadratic response surface. The end-points of the interval can be found directly, compared to other more, computationally intense procedures that result in a conservative interval. The coverage probability associated with intervals constructed in this manner from studies with relatively small numbers of observations is investigated via a simulation study. Finally, the methodology is illustrated with an example involving the evaluation of a combination of cytotoxic agents in the treatment of murine L1210 leukemia.

The dose-response surface for a combination of drugs is a multidimensional figure. Consequently, it is not possible to view such a surface using orthogonal axes when the number of dimensions exceeds 3. Parallel axes have been used to represent hyperdimensional figures. This paper reports on the use of parallel coordinate axes to plot the dose-response surface and its contours of constant response (isobols) for a combination of drugs. It is shown that patterns formed by intersecting line segments in the parallel system can aid in the interpretation of the fitted dose-response surface. More generally, analytic results are developed that permit the ready visualization and characterization of interaction effects of a polynomial model.

Sample size characteristics for the performance of time to event response surface studies are determined where both proportional hazards and quadratic surfaces apply. Single drug activity is evaluated through the use of confidence intervals and computer simulations are employed to assess design characteristics including optimum location, grouping and censoring. Confidence intervals can be reliably constructed for the smallest sample size considered (n=48). Power considerations applicable to pre clinical studies are presented with discussion of extensions including clinical applications.

In certain survival studies the percentage of right-hand single censoring and the frequency of survival assessment, which results in the grouping of otherwise distinct failure times, is under experimental control. It is important to determine the consequences of the use of the Peto-Breslow approximate likelihood function in the analysis of studies with grouped and censored observation. A Monte Carlo study was performed to simulate the response surface analysis of data generated under the assumption of Cox's proportional hazard model. Fortuitous reductions in the mean squared error of the model parameters with slight grouping and censoring are observed.

The approach of preliminary test estimation has been suggested by Ellerton and Myers (1977) as a means for controlling the size of the J-criterion of the estimator ŷ of the true response,ηIn this paper, the j-criterion associated with the response estimator employed by Ellerton and Myers is evaluated and comparisons made with the j-criteria of several procedures which fit one model all the time. The comparisons suggest that preliminary test estimation offers much advantage over “first-order” procedures when there are enough observations available to fit a second-order model.

Statistical methods for fitting nonlinear functions to data generated by correlated response variates are discussed. Estimation of the model parameters is performed with an iterative two-stage scheme. The estimation procedure accommodates both within-unit and between-unit variability in fitting a response surface. Under regularity conditions the procedure yields asymptotically normal, strongly consistent estimators. If desired a patterned variance-covariance matrix can be assumed and incorporated into the model. The methods are illustrated by an analysis of data from a study of the combined effects of hepatotoxins in which between- and within-subject measurements are recorded.

The purpose of this paper is to present the theory and develop an algorithm associated with the exploration of a dual response surface system. The approach is to find conditions on a set of independent or “design” variables which maximize (or minimize) a “primary response” function subject to the condition that a “constraint response” function takes on some specified or desirable value. A method is outlined whereby a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system. He thus is able to reduce to simple plotting the complex task of exploring the dual response system. The procedure that is used to generate the plots depends on the nature of the individual univariate response functions. In certain situations it becomes necessary to apply the additional constraint that the located operating conditions are a certain “distance” from the origin of the independent variables (or the cent...

In this article the problem of obtaining the maximum likelihood estimates of the parameters from a special type of linear combination of discrete probability functions is discussed. It is shown that when the sample is completely categorized the estimation problem is no more complicated that that of estimating the parameters of each of the component probability functions separately. When the sample is less than completely classified an iterative procedure must be used to obtain solutions to the likelihood equations and it is shown how the problem reduces to that of the full data case. A discussion of the asymptotic properties of the resulting estimators follows.

In this article, sufficient conditions are given for the existence of a unique solution of the likelihood equation which results from a grouped data sample. The necessary and sufficient conditions for the convergence of a sequence defined by the method of successive approximations to this unique solution are also given. Finally, it is shown that when the groups from an underlying Poisson distribution are connected the method of successive approximations will converge to the unique solution of the resulting likelihood equation regardless of the starting value chosen provided that the sample is not concentrated entirely in either or both the first and last groups.

Hazard functions in cancer chemotherapeutic situations may not be proportional, so a nonproportional hazard model has been developed. The dose-response surface is explored by regression analysis of experimental data, and after the estimation of the underlying hazard function the quality of the fit of the model is assessed. Further, treatment levels may be optimized, and estimated survival distributions can be plotted for any treatment combination. In an example of two-drug treatment of murine L1210 leukemia, statistically significant nonproportionality is determined. Analysis permits extraction of potentially important information on drug interrelationships, which has been previously unavailable.

The precision of the estimated optimum from a response-surface experiment is often indicated via a confidence region about the optimum. Sometimes, because of associated secondary responses, unconstrained optima produce unrealistic operating conditions, even when the true response surface is known. We consider confidence intervals for constrained optima for which the constraint function is known or separately estimated. An example from a cancer combination chemotherapy experiment illustrates the construction of such a region.

SUMMARY A method of calculating a confidence interval about the response at the stationary point of a response surface is presented. We show that the technique can also be used to calculate a confidence region about the location of the stationary point. The procedure is applied, via Cox's proportional-hazards model, to the analysis of survival data from a preclinical cancer chemotherapy experiment involving the combination of two drugs. It is seen that the results can be useful in determining the existence of a therapeutic synergism.

In a multivariate growth-curve model, the estimator of the parameter matrix is a function of the matrix of the sums of squares and of the cross-products due to error. However, if the assumption of a patterned covariance matrix is valid, then the parameter estimator does not depend on the error matrix. A likelihood ratio test of this patterned covariance matrix is constructed and its distribution is discussed. A numerical example is provided in which the design consists of two treatment groups, with three repeated measures being taken of the three response variables.

Response sarfxe methodology (RSM) is a collection of tools developed in the 1950s for the purpose of determining optimum operating conditions in applications in the chemical industry. This article reviews the progrrss of RSM in the general areas of experimental design and analysis and indicates how its role has been affected by advanccs in other fields of applied statistics. Current areas of research in RSM are highlighted. and areas for future research are discussed.