Donald A. Anderson

Efficient Choice Set Designs for Estimating Availability Cross-Effects Models

Batsell and Polking proposed a discrete choice model which incorporates the availability (presence or absence) of competing brands into the utility of each brand under study. The information on relative impacts of adding or deleting brands is of strategic interest, and models that do not incorporate such effects may be misleading. The designs suggested by Batsell and Polking have 2m−m−1 choice sets. Even with as few as 10=m brands, this requires over 1000 choice sets. In this paper we provide a catalog of designs for estimating cross effects models in as few as 2m−1 choice sets. This will make cross effects modelling practical in a wide range of academic and commercial settings.

Weakly resolvable IV.3 search designs for the pn factorial experiment

Abstract A series of weakly resolvable search designs for the pn factorial experiment is given for which the mean and all main effects are estimable in the presence of any number of two-factor interactions and for which any combination of three or fewer pairs of factors that interact may be detected. The designs have N = p(p–1)n+p runs except in one case where additional runs are required for detection and one case where (p−1) 2 additional runs are needed to estimate all (p–1)2 degrees of freedom for each pair of detected interactions. The detection procedure is simple enough that computations can be carried out with hand calculations.

Near Minimal Resolution IV Designs for the sn Factorial Experiment

A fractional factorial design is of resolution IV if all main effects are estimable in the presence of two-factor interactions. For the sn factorial experiment such a design requires at least N = s[(s – I)n – (s – 2)] runs. In this paper a series of resolution IV designs are given for the s” factorial, s = p α where p is prime, in N = s(s – I)n runs. A special case of the construction method produces a series of generalized foldover designs for the sn experiment, s ≥ 3 and n ≥ 3, in N = s(s – I)n + s runs. These foldover designs permit estimation of the general mean in addition to all main effects and provide s degrees of freedom for estimating error. A section on analysis is included.

A logistic model for the cumulative effects of human intervention on bald eagle habitat.

We developed a logistic model based on a conjoint analysis approach to evaluate the cumulative effects of selected forms of human disturbance on bald eagles (Haliaeetus leucocephalus) in their natural habitat. The dependent variables were amount of habitat available to bald eagles for foraging and perching as a function of 5 human disturbance factors. Application of the model to a section of the Snake River in Grand Teton National Park where detailed information on levels of human intervention were known yielded reasonable values when compared to field observations. The model is currently in use by managers of the Bridger-Teton National Forest and Grand Teton National Park

A Comparison of the Determinant, Maximum Root, and Trace Optimality Criteria

Three basic criteria, determinant, trace and maxim-urn root, are in common use for determining optimality of experimental designs. Here examples are presented where the three criteria give rise to different designs. The examples are balanced resolution IV* of the 2m series and are particularly insightful with respect to the dependence of the criteria on the correlation between estimators of the parameter.

On the construction of a class of efficient row-column designs

In this paper a method for the construction of a class of row-column designs with good statistical properties and high efficiency is presented. The class of designs produced is shown to exhibit balance, orthogonality and adjusted orthogonality. The efficiencies of these designs are investigated in detail, and they are shown to be very high, and possibly maximal in some cases.

Multidimensional balanced designs

A general construction is given for m-way completely variance balanced designs where each factor has v levels, m is any integer less than or equal to k, and N = vk, where k = 2λ+1 and v = 4λ3 is a prime power. The construction gives rise to a variety of designs, easily enumerated, with the same parameters pairwise but with differing variance properties. For m = 3 there are only two distinct designs possible, and their relative efficiency is shown to be .

The spectrum of the information matrix for sn parallel flats fractions when s is prime

Abstract A parallel flats fraction (PFF) for an sn factorial experiment, when s is a prime or prime power, is defined as the set of all solutions over GF(s) to At = ci, i = 1, 2, …, f where A is r × n of rank r. The fraction is completely determined by A and C = [c1, c2, …, cf]; hence all optimality conditions are also completely determined by A and C. It has been shown before that for s a prime the information matrix for a parallel flats fraction can be characterized in terms of the direct sum of relatively small matrices over the cyclotomic field of order s over the rationals. In this paper for s a prime we obtain the eigenvalues (the spectrum) of the information matrix and the eigenvectors directly in terms of these smaller matrices over the cyclotomic field. The usefulness of these results are in the construction of optimal fractions and the comparison of competing fractions.

THE ANALYSIS OF DISCRETE CHOICE EXPERIMENTS WITH CORRELATED ERROR STRUCTURE

In a stated preference discrete choice experiment each subject is typically presented with several choice sets, and each choice set contains a number of alternatives. The alternatives are defined in terms of their name (brand) and their attributes at specified levels. The task for the subject is to choose from each choice set the alternative with highest utility for them. The multinomial is an appropriate distribution for the responses to each choice set since each subject chooses one alternative, and the multinomial logit is a common model. If the responses to the several choice sets are independent, the likelihood function is simply the product of multinomials. The most common and generally preferred method of estimating the parameters of the model is maximum likelihood (that is, selecting as estimates those values that maximize the likelihood function). If the assumption of within-subject independence to successive choice tasks is violated (it is almost surely violated), the likelihood function is incorrect and maximum likelihood estimation is inappropriate. The most serious errors involve the estimation of the variance-covariance matrix of the model parameter estimates, and the corresponding variances of market shares and changes in market shares. In this paper we present an alternative method of estimation of the model parameter coefficients that incorporates a first-order within-subject covariance structure. The method involves the familiar log-odds transformation and application of the multivariate delta method. Estimation of the model coefficients after the transformation is a straightforward generalized least squares regression, and the corresponding improved estimate of the variance-covariance matrix is in closed form. Estimates of market share (and change in market share) follow from a second application of the multivariate delta method. The method and comparison with maximum likelihood estimation are illustrated with several simulated and actual data examples. Advantages of the proposed method are: 1) it incorporates the within-subject covariance structure; 2) it is completely data driven; 3) it requires no additional model assumptions; 4) assuming asymptotic normality, it provides a simple procedure for computing confidence regions on market shares and changes in market shares; and 5) it produces results that are asymptotically equivalent to those produced by maximum likelihood when the data are independent.

Optimal fractional factorial designs for estimating interactions of one factor with all others: 3m Series

In some experimental situations, only one factor is expected to interact with other factors. Designs which permit estimation of all main effects and the interactions of one factor ‘With All Others’, are termed WAO designs. This paper discusses the existence and construction of sm WAO designs. A series of WAO designs are presented for the 3m factorial, for m = 6, 7, ... , 14. The p non-negligible effects are estimable in 9f∗ runs, where f∗ is the smallest integer such that 9f∗ ≥p. These designs are determinant optimal within the class of parallel flats fractions and, except for the case f∗ = 9, are new. They are ideally suited for sequential experiments.