Correlated Bivariate Linear Growth Models: Optimal Designs for Slope Parameter Estimation

Abstract

Considered is a linear growth model involving each of two continuous measurable characteristics Y and Z. Along an equispaced time scale, experimental units (eus) are recruited/selected and measurements are recorded for both the characteristics. An eu may be enrolled at a time point i and may be kept under study continuously up to a time point j without any further “recall”. We assume the total duration of the study to be (2k+1) units of time and set the time points as $\{-k,-(k - 1),\cdots \,,-2,-1,0,1,2,\cdots \,,k - 1,k\}$ so that $-k \leq i \leq j \leq k$. Denote by t an arbitrary time point and by Y _t and Z _t random realizations of the characteristics Y and Z at time point t. Generally, four types of correlation structures are readily involved: Corr(Y _t , Z _t ); Corr(Y _t , Y _r ), r≠t, Corr(Z _t , Z _r ), r≠t, Corr(Y _t ,Z _r ), r≠t. We assume a mean model: $E(Y _{t}) =\alpha +\beta t$ and $E(Z_{t}) =\gamma +\delta t$. Our purpose is to suggest an optimal design for most efficient joint estimation of the slope parameters β and δ in the above model with suitable covariance structures. Our study is closely based on that of Abt et al. (Optimal designs in growth curve models: Part I: Correlated model for linear growth: Optimal designs for slope parameter estimation and growth prediction. J Stat Plan Infer 64:141–150, 1997). Essentially we suggest using either a single pair of time points − k and k, each with 50% recruit, or the stretch of all time points $(-k,-(k - 1),\ldots,-1,0,1,\ldots,k - 1,k)$, depending on the nature of correlations. This is based on A- and D-optimality criteria. Extensions to other mean models (such as linear–quadratic and quadratic–quadratic) are wide open.