Repeated Measurements and Cross-Over Designs

Chapter 3k-Sample Repeated Measurements Design

3.1 INTRODUCTION

This chapter generalizes the ideas developed in Chapter 2. Let there be k populations and let k-independent random samples be taken from these k populations. Let Y_iα be the p-dimensional response vector on the αth unit from the ith population, α = 1, 2, …, N_i; i = 1, 2, …, k.

Let Y_iα ~ IMN_p(μ_i, ∑ _i). Let . It is not necessary to have the observations taken from k populations. One may take N homogeneous units, randomly subdivide them into k sets of sizes N₁, N₂, …, N_k, assign the k treatments such that the ith treatment is assigned to each of the N_i experimental units of the ith set, and take p responses at the same time or at different time intervals. In either case, three kinds of null hypotheses are tested:

1. Parallelism or interaction of the populations and responses
2. Equality of population or group effects
3. Equality of response or period effects

Let μ′_i = (μ_i1, μ_i2, …, μ_ip). The three kinds of null hypotheses can be represented as follows:

(3.1.1)

(3.1.2)

(3.1.3)

To test these three null hypotheses, we initially assume ...