Every Coin Has Two Sides An Introduction to Causal Inference in Pharmaceutical S.pdf
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Every Coin Has Two Sides Causal Inference in Pharmaceutical Statistics
An Introduction to Asymptotic Statistics
1
1
The Number Two
3
1.1
Two, Dos, 2, II ...
3
1.2
The Most Useful Equation in Statistics
3
1.3
Mean and Standard Deviation .
5
1.4
Why Does the Universe Choose the Number 2? .
8
1.5
Exercises
. .
12
2
Fundamentals of Asymptotic Statistics
13
2.1
I.I.D. .
13
2.1.1
Sample and Population
. .
13
2.1.2
Population and Superpopulation .
15
2.1.3
Continuous and Discrete Variables
15
2.2
Convergence .
19
2.2.1
Convergence in Probability and in Distribution .
19
2.2.2
Law of Large Numbers .
20
2.2.3
Central Limit Theorem
. .
21
2.3
Probability and Empirical Distributions .
24
2.3.1
Parametric and Nonparametric Distributions
24
2.3.2
Empirical Distribution .
24
2.3.3
Bootstrapping
.
26
2.3.4
Bijective Relationship .
27
2.4
Estimand and Estimator
28
2.4.1
Plug-in Estimator
.
28
2.4.2
Bias and Variance
.
30
2.4.3
Asymptotic and Non-Asymptotic Properties .
31
2.5
Consistency and Asymptotic Normality
.
33
2.6
Exercises
. .
35
3
Statistical Inference
37
3.1
Two Great Statisticians
. .
37
3.2
Testing
.
38
vii
viii
Contents
3.2.1
Significance Testing
. .
38
3.2.2
Permutation and P-value
.
39
3.2.3
Hypothesis Testing .
42
3.3
Confidence Interval .
44
3.3.1
Construction and Interpretation
.
44
3.3.2
Point Estimator and Standard Error .
48
3.3.3
Bootstrap Method .
49
3.4
A Unified Approach
. .
51
3.4.1
Statistical Inference Based on Normality .
52
3.4.2
Statistical Inference Based on Asymptotic Normality
53
3.4.3
The Moderate Number 2
.
55
3.5
Exercises
. .
57
4
Maximum Likelihood Estimation (MLE)
58
4.1
Parametric MLE and Nonparametric MLE
58
4.1.1
Parametric MLE
58
4.1.2
Nonparametric MLE
62
4.2
Non-asymptotic Theory of Likelihood
. .
64
4.2.1
Score and Information .
64
4.2.2
Cram´er–Rao Lower Bound
. .
65
4.3
Asymptotic Theory of Likelihood .
69
4.3.1
Consistency and Asymptotic Normality
.
69
4.3.2
Asymptotic Efficiency .
72
4.4
Right Model or Wrong Model .
73
4.4.1
Kullback–Leibler Divergence
.
75
4.4.2
Consistency and Asymptotic Normality
.
77
4.5
Exercises
. .
80
5
Minimum Loss Estimation (MLE)
82
5.1
Dependent and Independent Variables
82
5.1.1
Relationship Between Two Variables .
82
5.1.2
R2
83
5.1.3
Least Squares Estimator
84
5.2
Minimum Loss Estimator
.
85
5.2.1
Two Main Types of Dependent Variables
. .
85
5.2.2
Law of Iterative Expectations .
86
5.2.3
L2 Loss .
87
5.2.4
Binary Cross-Entropy Loss
. .
90
5.2.5
A Unified Approach
. .
93
5.3
Two Purposes
95
5.3.1
Explanation .
96
5.3.2
Prediction
96
5.3.3
Summary
97
5.4
Exercises
. .
98
Contents
ix
II
An Introduction to Causal Inference
101
6
Potential Outcomes
103
6.1
Two Sets of Guidance Documents . 103
6.2
Central Questions
. 103
6.2.1
Cause and Effect 103
6.2.2
Baseline and Endpoint . 105
6.3
Potential Outcomes . 105
6.3.1
Two Spans of 50 Years . 105
6.3.2
Two Arms 106
6.3.3
Two Worlds . 106
6.3.4
Tea Tasting Example Revisited 108
6.4
Assumptions
. . 109
6.4.1
SUTVA . 109
6.4.2
Connecting the Two Worlds 110
6.5
Exercises
. . 111
7
Study Designs
112
7.1
Interventional or Non-Interventional
. 112
7.1.1
Interventional Study 113
7.1.2
Non-Interventional Study . 114
7.2
Randomized Controlled Trials . 115
7.2.1
Randomization and Blinding
. 115
7.2.2
Efficacy and Safety . 116
7.2.3
Internal Validity and External Validity 116
7.3
Observational Studies
. 118
7.3.1
Simpson’s Paradox and Berkson’s Paradox
. 118
7.3.2
Confounding Bias and Selection Bias . 119
7.3.3
DAG and SCM . 120
7.3.4
Time-Independent and Time-Varying Confounding . 122
7.4
Concurrent Control and External Control 125
7.5
Exercises
. . 126
8
Estimand
127
8.1
Causal Estimand and Statistical Estimand 127
8.2
Identification for an RCT
. 128
8.2.1
Causal Estimand 128
8.2.2
Statistical Estimand 129
8.2.3
Estimator 131
8.3
Two Identifiability Assumptions for an NIS
. 134
8.3.1
An Assumption Implied by the DAG . 135
8.3.2
An Assumption Not Implied by the DAG 136
8.4
Two Identification Strategies
. 137
8.4.1
Standardization Strategy
. 137
8.4.2
Weighting Strategy . 137
x
Contents
8.5
Two Subpopulations 138
8.5.1
ATT and ATC . 139
8.5.2
Identification 140
8.6
Exercises
. . 142
9
Estimator
143
9.1
Two Models . 143
9.1.1
Outcome Model
. . 143
9.1.2
Treatment Model 144
9.2
Two Estimators . 144
9.2.1
G-Computation Estimator . 145
9.2.2
IPW Estimator . 148
9.3
Doubly-Robust Estimator . 151
9.3.1
A Class of Point Estimators 151
9.3.2
Minimum-Variance “Estimator” 151
9.3.3
AIPW Estimator 152
9.3.4
“One Plus One Equals Three”
. . 155
9.3.5
Tea Tasting Example Continued
. 155
9.4
Exercises
. . 159
10 Sensitivity Analysis
160
10.1 Causal and Statistical Assumptions 160
10.2 Two Causal Assumptions
. 161
10.2.1 Variable Selection
. 162
10.2.2 Sensitivity Analysis for Unmeasured Confounding
. 163
10.2.3 Sensitivity Analysis for Positivity Violation . 167
10.3 On Statistical Assumptions 169
10.4 Exercises
. . 169
III
An Introduction to Semiparametric Statistics
171
11 Regular and Asymptotically Linear Estimator
173
11.1 Regularity 174
11.1.1 Revisiting MLE . 174
11.1.2 Super-Efficiency
. . 175
11.1.3 Regularity 176
11.2 Parametric Modeling 178
11.2.1 Estimand and Estimator 178
11.2.2 Regular and Asymptotically Linear Estimator 180
11.2.3 Efficient Estimator . 181
11.3 Nonparametric Modeling 182
11.3.1 Estimand and Estimator 182
11.3.2 Parametric Submodels . 183
11.3.3 Fundamental Theorem of Regularity . 184
11.3.4 First Application of the FTR . 185
Contents
xi
11.4 Fundamentals of Semiparametric Theory . 186
11.4.1 Generalized Cram´er–Rao Lower Bound
. 186
11.4.2 Least Favorable Submodel . 187
11.4.3 Efficient Influence Function 187
11.4.4 Efficient Estimator . 188
11.5 Exercises
. . 188
12 Efficient Influence Function
190
12.1 Average Treatment Effect Revisited 190
12.1.1 Two Versions of a Statistical Estimand 190
12.1.2 Two Approaches to Deriving the EIF . 190
12.2 Approach One to Deriving the EIF 191
12.2.1 Saturated Model 191
12.2.2 MLE’s Influence Function . 192
12.3 Approach Two to Deriving the EIF 195
12.3.1 Propensity Score Function is Known . 195
12.3.2 Propensity Score Function is Unknown 196
12.4 Exercises
. . 200
13 A Convenient Approach
201
13.1 Geometric Viewpoint 201
13.1.1 Hilbert Space and Tangent Space . 202
13.1.2 Existence and Uniqueness . 203
13.1.3 Projection 204
13.2 A Convenient Approach to Deriving EIFs
. . 204
13.2.1 Approach 204
13.2.2 Proof
. . 205
13.2.3 Notation . 208
13.3 EIFs for ATT and ATC
. . 208
13.3.1 ATT . 208
13.3.2 ATC . 210
13.4 Exercises
. . 211
14 Missing Data
213
14.1 Two Missing Mechanisms
. 213
14.2 Two Versions of a Statistical Estimand 216
14.2.1 Standardization Strategy
. 216
14.2.2 Weighting Strategy . 217
14.3 EIF for Missing Data Handling 218
14.4 What If the Missingness Were Viewed Differently? . 221
14.5 Exercises
. . 224
15 Longitudinal Data
226
15.1 Longitudinal Study . 226
15.1.1 Estimand 227
15.1.2 Identifiability Assumptions
. . 228
xii
Contents
15.2 Two Identification Strategies
. 233
15.2.1 Standardization Strategy
. 233
15.2.2 Weighting Strategy . 235
15.3 Two Series of Models 236
15.3.1 Propensity Score Models 236
15.3.2 Outcome Regression Models 237
15.4 EIFs for the Values of Treatment Regimes 239
15.4.1 Static Treatment Regimes . 239
15.4.2 Dynamic Treatment Regimes . 241
15.5 Exercises
. . 243
IV
An Introduction to Targeted Learning
245
16 Super Learning
247
16.1 The Mysterious Number 2 . 247
16.1.1 Model Selection and Variable Selection 247
16.1.2 AIC . 248
16.2 Cross-Validation
. . 251
16.2.1 Training Error and Test Error . 251
16.2.2 Leave-One-Out Cross-Validation
. 253
16.2.3 K-Fold Cross-Validation 256
16.2.4 Tuning 256
16.3 Super Learner 257
16.3.1 Discrete Super Learner . 258
16.3.2 Super Learner 259
17 Targeted Learning
262
17.1 From Estimand to Estimator
. 262
17.1.1 Estimand 262
17.1.2 Efficient Influence Function 263
17.1.3 Two Paths to Double Robustness . 263
17.2 Understanding TMLE . 265
17.2.1 Why an Update Is Needed . 265
17.2.2 How the Update Is Performed . 266
17.2.3 Brief Roadmap for TMLE . 271
17.3 Using TMLE
. . 272
17.3.1 ATE for a Binary Outcome 272
17.3.2 ATE for a Bounded Continuous Outcome 274
17.3.3 ATT and ATC . 277
17.4 Two Extensions . 284
17.4.1 Missing Data 284
17.4.2 Longitudinal Data . 287
17.5 Exercises
. . 289
Contents
xiii
18 Implementation
291
18.1 Two Software Environments 291
18.2 SAS Procedures for TMLE
. . 292
18.2.1 Procedure SUPERLEARNER . 292
18.2.2 Procedure CAEFFECT 293
18.3 R Packages for TMLE . 294
18.3.1 Package “tmle” . 296
18.3.2 Package “ltmle”
. . 298
18.4 Exercises
. . 302
19 Intercurrent Events
304
19.1 Missing Data and Intercurrent Events
. . 304
19.2 Binary and Continuous Outcomes . 305
19.2.1 Treatment Policy Strategy . 306
19.2.2 Hypothetical Strategy . 306
19.2.3 Composite Variable Strategy
. 307
19.2.4 While-on-Treatment Strategy . 308
19.2.5 Principal Stratum Strategy 308
19.3 Time-to-Event Outcome 309
19.3.1 Two Population-level Summaries . 310
19.3.2 Handling Intercurrent Events . 312
19.4 Exercises
. . 315
20 Fusion of Two Cultures
317
20.1 Two Cultures of Statistical Modeling . 317
20.2 Fusion of Two Cultures
. . 318
20.2.1 Target Estimand 318
20.2.2 Machine Learning
. 319
20.2.3 Targeted Learning . 319
20.3 Data Fusion . 319
20.3.1 Single-Arm Trial with an External RWD Control 320
20.3.2 RCT Augmented with RWD
. 323
20.4 Final Remarks
. 324
20.5 Exercises
. . 325
Appendix
327
Bibliography
363
Index
371