Design and Analysis of Experiments

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction: Elementary Statistical Notions

• Sampling Distributions

• Estimation Revisited

• Methods of Estimation

• Interval Estimation

• Linear Functions of Normally Distributed Variables

• Cochran’s Theorem

The following sections are included:

• Basic Assumptions and Definition of Least Squares

• Linear Functions with Zero Expectation

• Generalizations
  • Variance of the estimator
  • David-Neyman Generalization

• Estimation of σ²
  • General Case

• Observational Equations with Linear Restrictions On The Parameters
  • Tests of Linear Hypotheses
  • Applications
  • Test of hypothesis that there is no regression
  • Generalization

The following sections are included:

• Fundamental Principles

• Partitioning of Sums of Squares and Degrees of Freedom

• Experimental Error

• Assumptions Underlying the Analysis of Variance
  • Random Variables
  • Additivity
  • Non – Correlated Error
  • Homogeneity of Variance for Simplicity of Analysis of Variance
  • Normality for Valid Tests of Significance

• Analysis of Variance in the Case of Regression
  • Simple Regression
  • Multiple Regression
  • Partial Regression

• One–Way Classification

• Two-Way Classification with a Single Observation per Cell
  • Consider the hypotheses H₁
  • Consider H₂

• Testing of Individual Hypotheses

• Orthogonal Effects

• Two-Way Classification with Multiple but Equal Number of Observations in Cells

• Two-Way Classification with Multiple and Unequal Number of Observations in Cells, but There is at Least One Observation in Each Cell

• Matrix Methods (when there is no interaction)

• Case of Proportional Frequencies

• Three-Way Classification
  • Consider the hypothesis H₀: a_ij = 0
  • Consider the hypothesis

• Randomized Block Design (RBD)

• Judging the Relative Merits of Treatment

• Confidence Limits for Treatment Differences

• Uniformity Trial (cf. R.A.Fisher and Cochran)

• Latin Square Design

• Judging the Relative Merits of Treatments

• Graeco Latin Square

• Hyper – Graeco Latin Square

The following sections are included:

• One-Way Classification with a Single Concomitant Variable
  • Consider H₀: b = 0
  • Consider again H₁: α₁ = α₂ = ··· = α_k

• Two-Way Classification with One Observation per Cell and a Single Concomitant Variable
  • Consider the hypothesis H₀: b = 0
  • Consider the hypothesis H₁: α₁ = α₂ = ··· = α_p(= α, say)
  • Consider the hypothesis H₂: β₁ = β₂ = ··· = β_q = β
  • If F in (ii) and F in (iii) are significant, we proceed to test individual hypotheses such as α_i = α_j or β_l = β_k

• Some Examples of ANCOVA
  • Randomised Blocks
  • Latin Square

The following sections are included:

• Two-Way Classification with One Cell Missing
  • Analysis-Modification
  • Completed Table
  • Analysis of variance table for exact test of significance

• Two-Way Classification with Two Missing Plots

• K-Missing Plots: Yates’ Method
  • Completed Table

• Bartlett’s Method: Use of Concomitant Variables
  • Two-way classification with single missing plot

• Latin Square
  • Single missing plot
  • Several Missing Plots

• Randomized Block (r blocks and v treatments)
  • Case when the mixed plots are in the same block
  • Latin Square
  • Case when the mixed up observations correspond to the same treatment, say T_k
  • Case when the mixed up observations are in the same row
  • General Case – Iterative method of Yates
  • Bartlett’s method of Concomitant variables
  • Analysis

The following sections are included:

• Introduction

• Symmetrical Balanced Incomplete Blocks

• Resolvable Balanced Incomplete Blocks

• Affine Resolvable Balanced Incomplete Blocks
  • Efficiency of BIBD’s

• Complementary BIBD

• Analysis of BIBD Experiment
  • Intra-Block Analysis
  • Intra-Block Analysis of Variance
  • Inter-Block Information
  • Inter-Block Analysis of a BIBD
  • Inter-Block Analysis

• Missing Plot

• Principle of Connectedness

The following sections are included:

• Introduction
  • Sum Check
  • Square Sum Check

• Generalization
  • 2ⁿ Factorial Design
  • The Interaction XY
  • Generalization

• Confounding in Factorial Designs
  • An Inportant Observation

• 2³ Factorial Experiment
  • Inter-Block Information
  • Important Observations
  • Case of a 2⁴ Experiment
  • Analysis of a 2³ Factorial Experiment
  • Analysis in the Case of Partial Confounding
  • Generalization
  • Fisher’s Construction

• Fractional Replications

• Confounding in Latin Squares

• Partial Confounding

The following sections are included:

• Cyclic Groups

• Ideals

• Polynomial Rings

• Cyclotomic Polynomial

The following sections are included:

• Orthogonal Latin Squares
  • To Prove that L_i is a Latin Square
  • Key Latin Square and Key Line
  • To Construct the Orthogonal Latin Quares of Side 4
  • To Construct all 8×8 Orthogonal Latin Squares
  • Construction of A Latin Square of Side s where s is a Prime

• Construction of BIBD’s
  • Block Section and Intersection of Symmetrical BIBD’s
  • Yates Series A and Series B

• Geometric Method (Finite Projective Geometry)
  • Flats

• Finite Euclidean Geometry
  • Relation Between Euclidean and Projective Geometry
  • Construction of BIBD
  • Construction of a BIBD using Finite Euclidean Gometry

The following sections are included:

• Index