Repeated Measurements and Cross-Over Designs
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Preface xi
1. Introduction 1
1.1 Introduction 1
1.2 One-Sample RMD 2
1.3 k-Sample RMD 4
1.4 Split-Plot Designs 7
1.5 Growth Curves 13
1.6 Cross-Over Designs 14
1.7 Two-Period Cross-Over Designs 18
1.8 Modifications in Cross-Over Designs 19
1.9 Nonparametric Methods 22
References 23
2. One-Sample Repeated Measurement Designs 25
2.1 Introduction 25
2.2 Testing for Sphericity Condition 26
2.3 Univariate ANOVA for One-Sample RMD 29
2.4 Multivariate Methods for One-Sample RMD 32
2.5 Univariate ANOVA Under Nonsphericity Condition 34
2.6 Numerical Example 35
2.7 Concordance Correlation Coefficient 41
2.8 Multiresponse Concordance Correlation Coefficient 44
2.9 Repeated Measurements with Binary Response 47
References 51
3. k-Sample Repeated Measurements Design 53
3.1 Introduction 53
3.2 Test for the Equality of Dispersion Matrices and Sphericity Condition of k-Dispersion Matrices 54
3.3 Univariate ANOVA for k-Sample RMD 57
3.4 Multivariate Methods for k-Sample RMD 60
3.5 Numerical Example 63
3.6 Multivariate Methods with Unequal Dispersion Matrices 67
3.7 Analysis with Ordered Categorical Response 72
References 75
4. Growth Curve Models 77
4.1 Introduction 77
4.2 Sigmoidal Curves 78
4.3 Analysis of Mixed Models 84
4.4 Simple Linear Growth Curve Model 90
4.5 Nonlinear Growth Curve Model 92
4.6 Numerical Example 93
4.7 Joint Action Models 100
References 103
5. Cross-Over Designs without Residual Effects 105
5.1 Introduction 105
5.2 Fixed Effects Analysis of CODWOR 107
5.3 Connectedness in CODWOR 113
5.4 Orthogonality in CODWOR 115
5.5 Latin Square Designs 116
5.6 Youden Square Design and Generalization 118
5.7 F-Squares 123
5.8 Lattice Square Designs 123
5.9 Analysis of CODWOR when the Units Effects Are Random 125
5.10 Numerical Example 127
5.11 Orthogonal Latin Squares 131
References 133
6. Cross-Over Designs with Residual Effects 135
6.1 Introduction 135
6.2 Analysis of CODWR 136
6.3 BRED 143
6.4 PBCOD(m) 148
6.5 Numerical Example 152
6.6 Analysis with Unit (or Subject) Effects Random 156
6.7 Concluding Remarks 159
References 160
7. Two-Period Cross-Over Designs with Residual Effects 163
7.1 Introduction 163
7.2 Two-Period Two-Treatment CODWR Analysis: Parametric Methods 164
7.2.1 Analysis of the design based on the model (7.2.9) 167
7.2.2 Decomposition of the model (7.2.9) into intra- and interunit components 169
7.2.3 Estimating direct effects contrast using cross-over nature of the treatments 170
7.2.4 Modified two-period two-treatment design 171
7.2.5 Cost analysis 171
7.3 Two-Period Two-Treatment CODWR Analysis: Nonparametric Methods 173
7.4 Two-Period t Treatment Cross-Over Design 174
7.5 Numerical Examples 177
References 186
8. Other Cross-Over Designs with Residual Effects 189
8.1 Introduction 189
8.2 Extra-Period Designs 191
8.2.1 Residual effect of a treatment effect on itself is the same as residual effect on other treatments 192
8.2.2 Residual effect of a treatment on itself is different from the residual effect on other treatments 193
8.3 Residual Effects Proportional to Direct Effects 194
8.4 Undiminished Residual Effects Designs 195
8.5 Treatment Balanced Residual Effects Designs 197
8.6 A General Linear Model for CODWR 199
8.7 Nested Design 201
8.8 Split-Plot Type CODWR 203
8.9 CODWR in Circular Arrangement 205
8.10 Numerical Examples 207
References 213
9. Some Constructions of Cross-Over Designs 215
9.1 Introduction 215
9.2 Galois Fields 215
9.3 Generalized Youden Designs 217
9.4 Williams' Balanced Residual Effects Designs 221
9.5 Other Balanced Residual Effects Designs 226
9.6 Combinatorially Overall Balanced Residual Effects Designs 229
9.7 Construction of Treatment Balanced Residual Effects Designs 231
9.8 Some Construction of PBCOD (m) 232
9.9 Construction of Complete Set of MOLS and Patterson's BRED 234
9.10 Balanced Circular Arrangements 235
9.11 Concluding Remarks 236
References 237
Index 245
Chapter 1
Introduction
1.1 Introduction
In experimental work, treatment or treatments are given to units and one or several observations are recorded from each unit. The experimental unit differs from problem to problem. In agricultural experiments, the unit is a plot of land; in preclinical trials, the unit is an animal; in clinical trials, the unit is a subject; in industrial experiments, the unit is a piece of equipment. Treatments are those introduced by the investigator into the experiment to study their effects. In certain experiments, only one observation will be taken on each unit, while in other experiments, several readings will be taken from each unit. In cases where several measurements are made, either they will all be taken at the same time as in a standard SAT consisting of essay/writing, critical reading, and math comprehension or they will be taken over a period of time as in several tests given in a course. In this monograph, we confine ourselves to the designs and analysis of experiments where several observations are taken from each unit.
While it is absolutely necessary to take several readings on a unit in some experiments, it is desirable to do so in other investigational settings. Consider an animal feeding experiment where four feeds, A, B, C, and D, are tested. One may plan an experiment using 16 cows in the total experiment in which each cow receives one of the four feeds, with four cows for each feed. Or the experiment may be planned with only four cows in the experiment with each cow receiving each of the four feeds at different time intervals. In the latter scenario, using only 4 cows rather than 16 cows is not only economical but also eliminates the cow-to-cow variability in testing the feeds. However, the experiment with four cows will take a longer time to complete.
The class of designs where several observations are taken on each unit can be broadly referred to as repeated measurement designs (RMD). These can be subclassified as
- One-sample RMD
- k-Sample RMD (or profile analysis)
- Cross-over designs (or change-over designs) without residual effects (CODWOR) of the treatments like Latin square designs, Youden square designs, and Lattice square designs
- Cross-over designs with residual effects (CODWR) of the treatments like two-period cross-over designs of Grizzle (1965) and balanced residual effects designs (BRED) of Williams (1949)
The standard split-plot design in certain situations can also be considered as an RMD. We will elaborate on these designs in the remaining chapters.
1.2 One-Sample RMD
In this setting, a random sample of N experimental units will be taken from a population and p responses will be taken at the same time or at different times on each experimental unit. Another scenario for this design is that N homogeneous units will be treated alike at the beginning of the experiment and p responses will be recorded on each unit at the same time or at different times.
Let Y′α = (Yα1, Yα2, …, Yαp) be the vector of the p responses on the αth experimental unit for α = 1, 2, …, N. Let us assume that Yα are independently and identically distributed as multivariate normal with mean vector μ′ = (μ1, μ2, …, μp) and positive definite dispersion matrix ∑. Both μ and ∑ are unknown.
The null hypothesis of interest in this case is
(1.2.1)The matrix ∑ is said to satisfy the circularity condition or sphericity condition if
(1.2.2)where d is a scalar, Ip–1 is an identity matrix of order p − 1, and P1 is a (p − 1) × p matrix such that
(1.2.3)is an orthogonal matrix, Jm,n being an m × n matrix with 1’s everywhere. If α′ = (α1, α2, …, αp), ∑ of the form
(1.2.4)clearly satisfies the sphericity condition. In particular, a complete symmetric matrix ∑ of the form aIp + bJp,p satisfies the sphericity condition. The matrix ∑ of Equation (1.2.4) is said to satisfy the Huynh–Feldt condition, which will be discussed in Section 2.5.
In Chapter 2, we will show that the null hypothesis (1.2.1) can be tested by the standard univariate procedures if ∑ satisfies the sphericity condition. If ∑ does not satisfy the sphericity condition, multivariate methods using Hotelling’s T2 will be used to test the null hypothesis (1.2.1), and these methods will also be described in Chapter 2.
We will now provide three practical problems:
Example 1.2.1
Three test scores were obtained for 10 randomly selected students in a large elementary statistics course. The methods to test the equality of performance in the three tests for a similar group of students are discussed in Chapter 2.
Example 1.2.2
Rao (1973) discussed an example in which observations were taken on 28 trees for thickness of cork borings in four directions: North (N), East (E), South (S), and West (W). To test the null hypothesis that the mean thickness of cork borings is the same in the four directions, the methods discussed in Chapter 2 are used.
Example 1.2.3
In a noisy industrial surrounding, one can test the possible loss of hearing due to the outside noise level. For this purpose, audiogram results can be taken of a homogeneous group of employees over specified time intervals and the data can be analyzed by one-sample RMD methods discussed in Chapter 2.
1.3 k-Sample RMD
In this setting, we have k distinct populations and we draw k-independent random samples from these populations. Let Ni be the sample size of the sample taken from the ith population (i = 1, 2, …, k) and let . Let Y′ij = (Yij1, Yij2, …, Yijp) be the vector of p responses taken on the jth selected unit from the ith population (j = 1, 2, …, Ni; i = 1, 2, …, k).
Alternatively, this design arises by taking N homogeneous experimental units and applying the ith treatment to Ni randomly selected units at the beginning of the experiment (i = 1, 2, …, k). The p-dimensional response vector Y′ij = (Yij1, Yij2, …, Yijp) can then be recorded on the jth unit receiving the ith treatment (j = 1, 2, …, Ni; i = 1, 2, …, k).
In each of these cases, we assume that Yij are independently and identically distributed multivariate normal with mean vector μ′i = (μi1, μi2, …, μip) and positive definite dispersion matrix ∑, for j = 1, 2, …, Ni, i = 1, 2, …, k. Both μi and ∑ are unknown.
In this problem, there are three different null hypotheses of interest to the experimenter and they are
(1.3.1) (1.3.2) (1.3.3)Here, μi can be interpreted as the profile of the ith population (i = 1, 2, …, k). The null hypothesis H0c then implies that we are testing the parallelism of the k profiles. If H0c is retained, the parallelism hypothesis is not rejected and the profiles will appear as in Figure 1.3.1.
Figure 1.3.1 Parallel profiles.
When H0c is rejected, the profiles may be either intersecting one another (Figure 1.3.2) or the slopes may be different between the responses (Figure 1.3.3).
Figure 1.3.2 Intersecting nonparallel profiles.
Figure 1.3.3 Nonparallel profiles with different slopes.
In experimental work, H0c is the null hypothesis of testing the interaction effects between the treatments and the responses.
If H0c is not rejected, then one will be interested to test H0a and/or H0b. In H0a, we are testing the average of p responses to be constant from population to population...
