The construction of optimal designs for dose-escalation studies

Methods for the construction of A-, MV-, D- and E-optimal designs for dose-escalation studies are presented. Algebraic results proved elusive and explicit expressions for the requisite optimal designs are only given for a restricted class of traditional designs. Recourse to numerical procedures and heuristics is therefore made. Complete enumeration of all possible designs is discussed but is, as expected, highly computer intensive. Two exchange algorithms, one based on block exchanges and termed the Block Exchange Algorithm and the other a candidate-set-free algorithm based on individual exchanges and termed the Best Move Algorithm, are therefore introduced. Of these the latter is the most computationally effective. The methodology is illustrated by means of a range of carefully selected examples.

This work was supported by funds from the University of Cape Town and the National Research Foundation, South Africa. The study represents an extension of the first author’s contribution to the discussion of Bailey (2009). A part of the paper was completed while the first author was a Visiting Fellow with the Design and Analysis of Experiments Programme at the Isaac Newton Institute of Mathematical Sciences, Cambridge, UK, in 2011 and she would like to thank Rosemary Bailey for some insightful discussions during that visit. Sayi Toutou and Mark Steinhaus provided help with the computer programming and their input is appreciated.

Authors and Affiliations

1. Department of Statistical Sciences, University of Cape Town, Rondebosch, 7700, South Africa

   Linda M. Haines & Allan E. Clark

Corresponding author

Correspondence to Linda M. Haines.

Appendix

Consider an extended traditional design with information matrix L as specified in Sect. 3. Then it is straightforward to show that the n×n matrix

has eigenvalues \(\lambda _{1}= \frac{b( a + b \theta ) + m x_{2}}{m}\) with multiplicity n−1 and attendant eigenvectors (1,−1,0,…,0), (1,0,−1,…,0) through to (1,0,0,…,−1), or any basis thereof, and

with multiplicity 1 and eigenvector 1=(1,1,1,…,1)^T. It is thus clear that the matrix L itself has n−1 eigenvalues λ ₁ with eigenvectors (0,1,−1,0,…,0), (0,1,0,−1,…,0) through to (0,1,0,0,…,−1), or any basis thereof. Furthermore it is readily seen that the remaining eigenvalues of L are λ _n =(n+1)λ _n,22 with eigenvector (−n,1,…,1) and 0 with eigenvector 1. Thus the non-zero eigenvalues of the Moore-Penrose inverse of L, namely L ^g, are \(\frac{1}{\lambda _{1}}\) with multiplicity n−1 and \(\frac{1}{\lambda _{n}}\) with multiplicity 1 and the corresponding eigenvectors are the same as those of L, that is for λ ₁ and λ _n respectively (Pringle and Rayner 1971, pp. 23–24).

Consider now a pairwise comparison of treatment effects, written \(\underline{c}^{T} {\underline{\tau}}\), with variance

where Λ _g is a diagonal matrix with diagonal elements the eigenvalues of L ^g and H is an orthogonal matrix with columns the corresponding normalized eigenvectors. Specifically suppose, without loss of generality, that the matrices Λ _g and H can be expressed in partitioned form as

and

where B is an n×(n−1) matrix with columns any normalized orthogonal basis of the n−1 vectors (1,−1,0,…,0), (1,0,−1,…,0) through to (1,0,0,…,−1). Then for \(\underline{c}^{T} = \underline{c}_{01}^{T} = (1,-1, 0, \ldots, 0)\), the first column of B can be taken to be and all other columns of B are orthogonal to this vector and have first element 0. Thus it follows immediately that \(\underline{c}_{01}^{T} L^{g} \underline{c}_{01} = \frac{1}{n} ( \frac{n+1}{\lambda _{n}} + \frac{n-1}{\lambda _{1}} )\). In addition, for \(\underline{c}^{T} = \underline{c}_{12}^{T} = (0,1,-1, 0, \ldots, 0)\), the first column of B can be taken to be \(\frac{1}{\sqrt{2}} \underline{c}_{12}\) so that all other columns of B are orthogonal to \(\underline{c}_{12}\) and thus \(\underline{c}_{12}^{T} L^{g} \underline{c}_{12} = \frac{2}{\lambda _{1}}\). Similar arguments to those used for \(\underline{c}_{01}\) and \(\underline{c}_{12}\) hold for all contrast vectors of the form \(\underline{c}_{0i}\) for i=2,…,n and \(\underline{c}_{ij}\) for 1≤i<j≤n respectively.

Reprints and permissions