Kristi Kuljus

Comparing Experimental Designs for Benchmark Dose Calculations for Continuous Endpoints

The BMD (benchmark dose) method that is used in risk assessment of chemical compounds was introduced by Crump (1984) and is based on dose-response modeling. To take uncertainty in the data and model fitting into account, the lower confidence bound of the BMD estimate (BMDL) is suggested to be used as a point of departure in health risk assessments. In this article, we study how to design optimum experiments for applying the BMD method for continuous data. We exemplify our approach by considering the class of Hill models. The main aim is to study whether an increased number of dose groups and at the same time a decreased number of animals in each dose group improves conditions for estimating the benchmark dose. Since Hill models are nonlinear, the optimum design depends on the values of the unknown parameters. That is why we consider Bayesian designs and assume that the parameter vector has a prior distribution. A natural design criterion is to minimize the expected variance of the BMD estimator. We present an example where we calculate the value of the design criterion for several designs and try to find out how the number of dose groups, the number of animals in the dose groups, and the choice of doses affects this value for different Hill curves. It follows from our calculations that to avoid the risk of unfavorable dose placements, it is good to use designs with more than four dose groups. We can also conclude that any additional information about the expected dose-response curve, e.g., information obtained from studies made in the past, should be taken into account when planning a study because it can improve the design.

Theory of Segmentation

1.1 Preliminaries In this chapter we focus on what Rabiner in his popular tutorial (Rabiner, 1989) calls “uncovering the hidden part of the model” or “Problem 2”, that is, hidden path inference. We consider a hidden Markov model (X,Y) = {(Xt,Yt)}t∈Z, where Y = {Yt}t∈Z is an unobservable, or hidden, homogeneous Markov chain with a finite state space S = {1, . . . ,K}, transition matrix P = (pi,j)i,j∈S and, whenever relevant, the initial probabilities πs = P(Y1 = s), s ∈ S. A reader interested in extensions to the continuous case is referred to (Cappe et al., 2005; Chigansky & Ritov, 2010). The Markov chain will be further assumed to be of the first order, bearing in mind that a higher order chain can always be converted to a first order one by expanding the state space. To simplify the mathematics, we assume that the Markov chain Y is stationary and ergodic. This assumption is needed for the asymptotic results in Section 3, but not for the finite time-horizon in Section 2. In fact, Section 2 does not even require the assumption of homogeneity. The second component X = {Xt}t∈Z is an observable process with Xt taking values in X that is typically a subspace of the Euclidean space, i.e. X ⊂ Rd. The process X can be thought of as a noisy version of Y. In order for (X,Y) to be a hidden Markov model, the following properties need to be satisfied:

Asymptotic linearity of a linear rank statistic in the case of symmetric nonidentically distributed variables

Let Y 1, …, Y n be independent but not identically distributed random variables with densities f 1, …, f n symmetric around zero. Suppose c 1, n , …, c n, n are given constants such that ∑ i c i, n =0 and . Denote the rank of Y i −Δ c i, n for any Δ∈ℝ by R(Y i −Δ c i, n ) and let a n (i) be a score defined via a score function ϕ. We study the linear rank statistic and prove that S n (Δ) is asymptotically uniformly linear in the parameter Δ in any interval [−C, C], C>0.

Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.