# Analysis and Design of Certain Quantitative Multiresponse Experiments

### International Series of Monographs in Applied Statistics and Biometry

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Analysis and Design of Certain Quantitative Multiresponse Experiments highlights (i) the need for multivariate analysis of variance (MANOVA); (ii) the need for multivariate design for multiresponse experiments; and (iii) the actual procedures and interpretation that have been used for this purpose by the authors. The development in this monograph is such that the theory and methods of uniresponse analysis and design stay very close to classical ANOVA. The book first discusses the multivariate aspect of linear models for location type of parameters, but under a univariate design, i.e. one in which each experimental unit is measured or studied with respect to all the responses. Separate chapters cover point estimation of location parameters; testing of linear hypotheses; properties of test procedures; and confidence bounds on a set of parametric functions. Subsequent chapters discuss a graphical internal comparison method for analyzing certain kinds of multiresponse experimental data; two classes of multiresponse designs, i.e. designated hierarchical and p-block designs; and the construction of various kinds of multiresponse designs.

## Table of Contents

Preface

Notation

Chapter I. Introduction

1. Some General Remarks on Multi-factor Multi-response Experiments

2. The Restricted Scope of this Monograph

3. One Continuous Response and One Relevant Unstructured Factor: Fixed-effects Model and Random-effects Model of ANOVA

4. One Continuous Response and Two or More Unstructured Factors in cross Classification: The Two Models of ANOVA

5. One Continuous Response and One or More Structured Factors

6. Multi-response Experiments and the Need for a Multivariate Development

7. Formal versus Informal Approach

8. A Brief Chapter-wise Description of the Scope of the Present Monograph

Literature

Chapter II. Linear Models

1. Some Further Remarks on Structured-unstructured Categorization for Factors and Responses

2. Linear Models

2.a. Some General Remarks

2.b. Some Special Models

3. Remarks on the Multivariate Extension

Literature

Chapter III. Point Estimation of Location Parameters

1. Introduction

2.a. Single Linear Estimation for the Uni-response Case

2.b. Formula for Estimation of Treatment Effects under Block Designs — Some Examples

2.c. Comparison between Two Different Designs in Terms of Single Linear Estimation

3.a. Simultaneous Linear Estimation for the Uni-response Case

3.b. Interpretations of the Criteria and their Use in Comparison of Designs

4.a. Single Linear Estimation for the Multi-response Case

4.b. The Same Problem under a Somewhat More General Model

5. Some Examples of Single Linear Estimation for the Multi-response Case

6. Simultaneous Linear Estimation for the Multi-response Case

7. Summary

Literature

Chapter IV. Testing of Linear Hypotheses

1. The Fixed-effects Model of MANOVA

2. Linear Hypotheses under the Fixed-effects Model of MANOVA

3 . Three Current Test Procedures

4. The Union-intersection Principle and its Impact on MANOVA and Design of Experiments

4.a. Bilinear Decomposition of the H0 of MANOVA and Some General Considerations

4.b. Some Examples of Response-wise Infinite and Contrast-wise Finite Decomposition

4.c. An Example of Response-wise Finite and Contrast-wise Infinite Decomposition: The Step-down Procedure

4.d. Some Examples of Response-wise Finite and Contrast-wise Finite Decomposition

4.e. The Main Motivation behind the Decomposition and the Union-intersection Procedures

5. Some Supplementary Mathematical and Statistical Remarks

5.a. On Availability of Tables for the Use of the Various Test Procedures

5.b. A Remark on the Distribution of the Fi's of the Step-down Procedure

6. Applications of the Model (4.2) to Some Growth Curve Problems

7. Some Numerical Examples

7.a. The Doubly Infinite Decomposition

7.b. Response-wise Infinite and Contrast-wise Finite Decomposition

7.c. Response-wise Finite and Contrast-wise Infinite Decomposition

Literature

Chapter V. Properties of the Test Procedures

1. Introductory Remarks

2. Intrinsic Properties of Group I Procedures

2.a. Monotonicity Property for the A-criterion

2.b. Monotonicity Property for the Trace Criterion

2.C. Monotonicity Property for the Largest-root Test

3. Some Remarks on the Admissibility of the Different Procedures

4. Some Results from a Monte Carlo Study

4.a. Specifics of the Monte Carlo Study

4.b. Summary of Findings

Literature

Chapter VI. Confidence Bounds

1. General Principles

l.a. Distinction between General Confidence Regions and Simultaneous Confidence Intervals

l.b. Intervals Based on (H0,H)

1.c. Bounds on the "Partials"

l.d. The Nature of the Interval Estimation in Terms of the Percentage Points of the Distribution Function Used

1.e. An Additional Requirement beyond what is Suggested by the Pair (H0,H)

1.f. Intervals not Prima Facie Based on any (H0,H)

2. Bounds Connected with a Linear Hypothesis against a Linear Alternative under the Normal Multivariate Linear Model I

2.a. General Features

2.b. Nature of Bounds Associated with a Bilinear Decomposition

3.a. Exact Expressions for Bounds Associated with the Doubly Infinite Bilinear Decomposition

3.a.l. The Case u = 1

3.a.2. The Problem of Two Mean Vectors: Two-sample Case

3.a.3. The Problem of Two Mean Vectors: Single-sample Case

3.b. Three Examples of Bounds Associated with the Response-wise Infinite and Contrast-wise Finite Decomposition

3.c. Bounds Associated with the Response-wise Finite and Contrast-wise Infinite Decomposition

3.d. Bounds Associated with the Doubly Finite Bilinear Decomposition

4. Supplementary Mathematical and Statistical Remarks

4.a. On the Derivation of the Bounds

4.b. On the Lengths of the Confidence Intervals

5. Some Examples of the Total and Partial Distance Functions: Block-treatment Designs

6. Numerical Examples

Literature

Chapter VII. Graphical Methods and Internal Comparisons

1. Internal Comparisons: General Philosophy

2. A Multi-response Graphical Internal Comparisons Procedure

3. Outline of the Procedure

4. Discussion of Certain Features of the Method

5. Extensions Based on an Analog of the ANOVA Identity

6. Numerical Examples

Literature

Chapter VIII. Hierarchical and p-Block Multi-response Designs and their Analysis

1. Introductory Remarks on Multi-response Designs in General

2. General Description of Hierarchical Designs and Some Formal Problems under that Class

3. A Sketch of the Mathematical Justification of the Procedure Proposed in Section 2

4. Multi-response Designs with p-Block Systems

5. Testability Conditions of Hierarchical Designs with p-Block Systems

6. Nature of the Multivariate Designs for an Important Special Case

Literature

Chapter IX. Incomplete Multi-response Designs

1. Introduction

2. A Simple Example of an IM Design

3. Procedure for Analysis of General Multi-response Design

4. Concluding Remarks

Literature

Chapter X. Construction of Multi-response Designs

1. Introduction

2. Multi-response Designs Arising in Factorial Experiments

2.a. Not Each Response is Sensitive to Every Factor

2.b. Different Sets of Effects are Important for Different Factors

3. Multi-response Designs of the Lock-treatment Type

3.a. Hierarchical Designs

3.b. Regular Incomplete Multi-response Designs

Literature

Bibliography

Appendix A. Some Specific Matrix Results

1. Conditional or Generalized Inverses of Certain Patterned Matrices

2. Some Transformations and Representations of Matrices

3. Some Properties of Direct Products of Matrices

Literature

Appendix B. Tables and/or Charts Associated with the Procedures

Appendix C

I. Computer Programs for Multivariate Analysis of Variance

A. The Eight Operators

B. The Main Program

C. The Second Program

D. FORTRAN IV Routines for the MANOVA Operators

E. Example

F. FORTRAN Listings for the MANOVA Operators

G. Programs to be Supplied by the User

H. Composing FORTRAN IV Programs to Use the MANOVA Operators

II. Computer Program for Internal Comparisons

Literature

Author Index

Subject Index

## Product details

- No. of pages: 314
- Language: English
- Copyright: © Pergamon 1971
- Published: January 1, 1971
- Imprint: Pergamon
- eBook ISBN: 9781483157894

## About the Authors

### S. N. Roy

### R. Gnanadesikan

### J. N. Srivastava

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