Introduction to Modern Randomization-Based Design and Analysis for Causal Inference.pdf
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Introduction to Modern Randomization-Based Design and Analysis for Causal Inference
Understanding Experimental Design: Fundamental
Concepts
1
1.1
Experiments: Basic terms and definitions
     .
2
1.1.1
Factors, levels, treatments        
2
1.1.2
Experimental units         
3
1.1.3
Response variables – outcomes
      .
4
1.2
Potential outcomes and causal estimands
     .
4
1.3
The Assignment mechanism and the three fundamental
principles of experimentation

7
1.4
Focus of this book: ONE principle, MANY designs   
10
1.4.1
Designing a good experiment      
10
1.4.2
A unified principle for analysis
      .
11
1.4.3
Three versions of randomization inference    .
12
1.4.4
Illustration: a New York school experiment

13
1.4.5
Leveraging today’s computational power   
16
1.5
Glossary
              .
17
1.6
Problems
              .
17
2
CRD with one factor, two levels
20
2.1
Introduction to completely randomized experiments (CRD) .
20
2.2
Examples of two-armed CRD         
21
2.3
Potential outcomes and estimands
       .
21
2.4
The completely randomized assignment mechanism
  .
23
2.4.1
Observed and missing potential outcomes    .
24
2.4.2
Inferring causal effects         
25
2.5
Randomization test for the sharp null hypothesis of no
treatment effect
            .
26
2.5.1
Choice of test statistic and definition of extremeness .
27
2.5.2
Computing the randomization distribution of the test
statistic under the sharp null and calculating the
resulting p-value
          .
29
vii
viii
Contents
2.5.3
Randomization test in the tomato experiment   .
30
2.5.4
Caveat about use of multiple test statistics

31
2.5.5
Implementing the randomization test in R    .
34
2.5.6
Fiducial intervals           .
40
2.6
Neymanian inference: Testing the null hypothesis on average
treatment effects             .
43
2.6.1
A bigger picture of Neymanian inference: choice of test
statistic            
46
2.6.2
Estimation of sampling variance

47
2.7
Optimal allocation of units to treatments
     .
48
2.8
Experiments with binary outcomes        .
49
2.9
Problems
              .
51
3
Better comparisons using blocking
55
3.1
Am I making a fair comparison? A hypothetical example

55
3.2
Completely Randomized Designs (CRD) to Randomized Block
Designs (RBD): the motivation         .
56
3.3
Blocks, potential outcomes and causal estimands–general
notation
              .
59
3.4
Assignment mechanism and observed outcomes in RBD
.
61
3.4.1
Estimators of treatment effects       .
64
3.5
Testing Fisher’s sharp null hypothesis in an RBD

66
3.5.1
Neymanian inference from two-treatment RBD
experiments            
72
3.6
Matched-pair designs

75
3.6.1
Some examples of matched-pair experiments
  .
76
3.6.2
Potential outcomes and causal effects
    .
76
3.6.3
Assignment mechanism in matched-pair experiments .
77
3.6.4
Fisherian inference from matched-pair experiments
79
3.6.5
Randomization test of the asthma treatment
experiment

81
3.6.6
Neymanian inference from matched-pair experiments .
82
3.7
Problems
              .
84
4
Beyond blocking: acceptable versus unacceptable allocations
89
4.1
Acceptable randomizations: a stem-cell experiment   
89
4.2
Framework for design and analysis        .
95
4.3
Mahalanobis distance as the acceptance criterion    .
100
4.4
Example: Shadish experiment
        .
106
4.5
Summary
              .
108
4.6
Problems
              .
109
Contents
ix
5
Randomized experiments with J(> 2) treatment arms
113
5.1
Potential outcomes and estimands
       .
114
5.2
Assignment mechanism: determining the appropriate
acceptability criterion

117
5.3
Observed outcomes and estimators in a CRD     .
119
5.4
Inference from CRD experiments        
121
5.4.1
Fisherian inference with J treatments from a CRD
121
5.4.2
Neymanian inference for treatment contrasts from CRD 136
5.5
Randomization-based inference from RBD: two-way
decomposition

140
5.5.1
Example: simplified stem cell experiment
   .
143
5.5.2
Unreplicated RBD         
145
5.6
Classical and modern strategies to avoid undesirable allocations 147
5.6.1
Latin square designs and Fisherian inference
  .
150
5.7
Problems
              .
155
6
The 22 factorial experiment
163
6.1
Factorial effects: Causal estimands in a 22 experiment

163
6.2
Assignment mechanisms, observed outcomes and unbiased
estimators

169
6.3
Computing estimated factorial effects using R

172
6.4
Orthogonal decomposition of data in 22 experiments   .
174
6.4.1
Special case of balanced 22 (or 2K experiments in
general

177
6.4.2
Decomposition for an unreplicated 22 experiment  .
178
6.5
Fisherian inference from 22 experiments      
179
6.5.1
Randomization test for sharp null hypothesis of
non-zero treatment effects        
183
6.5.2
Fiducial intervals for 22 factorial experiments   .
188
6.6
Neymanian inference from completely randomized 22
experiments
             .
191
6.7
Analysis of the education experiment
      .
193
6.8
Problems
              .
199
7
2K factorial designs
202
7.1
Factorial effects in a 2K experiment

202
7.2
Design of 2K experiments: the assignment mechanism

207
7.3
Observed outcomes in 2K experiments       .
213
7.3.1
Estimation of factorial effects      
214
7.3.2
ANOVA decomposition in 2K factorial experiments
216
7.4
Fisherian inference for 2K experiments

217
7.5
Neymanian inference from completely randomized 2K
experiments
             .
222
7.6
Factorial experiments with binary outcomes
    .
225
x
Contents
7.6.1
Example: audit experiment to identify effect of racial
discrimination

226
7.7
Optimal allocation of units to treatments
     .
230
7.8
Problems
              .
232
8
Design and analysis of factorial experiments with constraints 237
8.1
Introduction
             .
237
8.2
Two-level fractional factorial Designs
      .
238
8.2.1
Understanding fractional factorial with a 23−1
experiment

239
8.2.2
Criteria for choosing “good” fractions: The maximum
resolution criterion         
244
8.2.3
Clear effects, word-length pattern and the minimum
aberration criterion         
247
8.2.4
Factorial effects, assignment mechanism and inference
248
8.3
Split-plot Designs

252
8.3.1
Estimands, treatment assignment and estimators  .
253
8.3.2
Fisherian inference for split-plot designs   
256
8.4
Problems
              .
259
9
Model-based analysis of designed experiments and
superpopulation inference
262
9.1
Introduction
             .
262
9.2
Potential outcomes: transitioning from fixed quantities to ran-
dom variables

262
9.3
Potential outcomes, finite population and super population
estimands               .
264
9.4
A naive approach and the desirability for models    .
266
9.5
Model for potential outcomes, exchangeability of units and
prior distribution

269
9.6
Posterior distribution of parameters and
superpopulation inference         
271
9.7
Model for imputing missing outcomes and finite population
inference
              .
275
9.8
Independent normal potential outcomes model

278
9.9
Dependent potential outcomes
        .
284
9.10 2K factorial experiment
          .
288
9.11 Problems
              .
291
10 Including covariates in the design, model-based analysis and
subsequent inference
295
10.1 A naive strategy for including covariates in the model and
inference
              .
295
10.2 Inference for superpopulation and finite population estimands
using models that involve covariates

296
Contents
xi
10.3 Covariate-based model for normally distributed outcomes: an
illustration

298
10.3.1 Inference for the finite population estimand   
306
10.4 Role of design in Bayesian inference

309
10.5 Problems
              .
310
11 Appendix
312
11.1 Basic combinatorics
           .
312
11.2 Probability distributions and random variables

313
11.2.1 Classical definition of probability      
313
11.2.2 Random variables and probability distributions  
313
11.2.3 Expectation, variance and covariance     
314
11.2.4 Some probability distributions used in this book
.
316
11.3 A Short Elementary Tutorial on Frequentist
Statistical Inference
           .
317
11.4 Bayesian inference

319
11.4.1 Conjugate prior distributions      
319
11.4.2 Bayesian Point and Interval Estimation

320
11.5 Simplified proof of Neyman’s variance result (2.24)   
321
Bibliography
325
Index
333