Prediction and Estimation of Growth Curves with Special Covariance Structures

Abstract

The model considered here is a generalized multivariate analysis-of-variance model useful especially for many types of growth-curve problems, including biological growth and technology substitutions. It is defined as Y_{p × N} = X_p×mτ_m×rA_r×N + ∈_p×N , where τ is unknown and X and A are known design matrices of ranks m < p and r < N, respectively. Further, the columns of ∈ are independent p-variate normal, with mean vector 0 and common covariance matrix Σ. In general, p is the number of time (or spatial) points observed on each of the N cases, (m — 1) is the degree of polynomial, and r is the number of groups. This article focuses mainly on predicting future observations, including partially observed vectors and future values on a given number of cases, using their past observations. These prediction problems, as well as the estimation of parameters, are considered for two parsimonious covariance structures that are very important in practice. The results are illustrated with three real data sets. Some of these results compare favorably with those in the literature. Let V = (V ^(1)′, V ^(2)′)′ be a set of p × K future observations drawn from the generalized growth-curve model, where V ⁽¹⁾ is p_i × K (i = 1, 2) and p_i + p ₂ = p. The first problem is to predict V ⁽²⁾, given V ⁽¹⁾ and Y. The second problem is to predict y, given Y, where y is a set of n(≤N) future q-dimensional observations whose previous p-dimensional observations are a subset of Y.

Key Words:

• Uniform covariance structure
• Serial covariance structure
• Conditional prediction
• Extended prediction
• Maximum likelihood estimates
• Minimum predictive variance estimates
• Biological data sets