Table Of ContentCausal Inference: What If
Miguel A. Hernán, James M. Robins
February 21, 2020
ii Causal Inference
Contents
Introduction: Towards less casual causal inferences vii
I Causal inference without models 1
1 A definition of causal effect 3
1.1 Individual causal effects . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Average causal effects . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Measures of causal effect . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Random variability . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Causation versus association . . . . . . . . . . . . . . . . . . . . 10
2 Randomized experiments 13
2.1 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Conditional randomization . . . . . . . . . . . . . . . . . . . . . 17
2.3 Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Inverse probability weighting . . . . . . . . . . . . . . . . . . . . 20
3 Observational studies 25
3.1 Identifiability conditions . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Exchangeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Consistency: First, define the counterfactual outcome . . . . . . 31
3.5 Consistency: Second, link counterfactuals to the observed data . 35
3.6 The target trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Effect modification 41
4.1 Definition of effect modification . . . . . . . . . . . . . . . . . . . 41
4.2 Stratification to identify effect modification . . . . . . . . . . . . 43
4.3 Why care about effect modification . . . . . . . . . . . . . . . . . 45
4.4 Stratification as a form of adjustment . . . . . . . . . . . . . . . 47
4.5 Matching as another form of adjustment . . . . . . . . . . . . . . 49
4.6 Effect modification and adjustment methods . . . . . . . . . . . 50
5 Interaction 55
5.1 Interaction requires a joint intervention . . . . . . . . . . . . . . 55
5.2 Identifying interaction . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 Counterfactual response types and interaction . . . . . . . . . . . 58
5.4 Sufficient causes . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Sufficient cause interaction . . . . . . . . . . . . . . . . . . . . . 63
5.6 Counterfactuals or sufficient-component causes?. . . . . . . . . . 65
iv Causal Inference
6 Graphical representation of causal effects 69
6.1 Causal diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Causal diagrams and marginal independence . . . . . . . . . . . 71
6.3 Causal diagrams and conditional independence . . . . . . . . . . 73
6.4 Positivity and consistency in causal diagrams . . . . . . . . . . . 75
6.5 A structural classification of bias . . . . . . . . . . . . . . . . . . 78
6.6 The structure of effect modification. . . . . . . . . . . . . . . . . 80
7 Confounding 83
7.1 The structure of confounding . . . . . . . . . . . . . . . . . . . . 83
7.2 Confounding and exchangeability . . . . . . . . . . . . . . . . . . 85
7.3 Confounding and the backdoor criterion . . . . . . . . . . . . . . 87
7.4 Confounding and confounders . . . . . . . . . . . . . . . . . . . . 90
7.5 Single-world intervention graphs . . . . . . . . . . . . . . . . . . 93
7.6 Confounding adjustment . . . . . . . . . . . . . . . . . . . . . . . 94
8 Selection bias 99
8.1 The structure of selection bias . . . . . . . . . . . . . . . . . . . 99
8.2 Examples of selection bias . . . . . . . . . . . . . . . . . . . . . . 101
8.3 Selection bias and confounding . . . . . . . . . . . . . . . . . . . 103
8.4 Selection bias and censoring . . . . . . . . . . . . . . . . . . . . . 105
8.5 How to adjust for selection bias . . . . . . . . . . . . . . . . . . . 107
8.6 Selection without bias . . . . . . . . . . . . . . . . . . . . . . . . 110
9 Measurement bias 113
9.1 Measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.2 The structure of measurement error . . . . . . . . . . . . . . . . 114
9.3 Mismeasured confounders . . . . . . . . . . . . . . . . . . . . . . 116
9.4 Intention-to-treat effect: the effect of a misclassified treatment . 117
9.5 Per-protocol effect . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10 Random variability 123
10.1 Identification versus estimation . . . . . . . . . . . . . . . . . . 123
10.2 Estimation of causal effects . . . . . . . . . . . . . . . . . . . . 126
10.3 The myth of the super-population . . . . . . . . . . . . . . . . . 128
10.4 The conditionality “principle” . . . . . . . . . . . . . . . . . . . 130
10.5 The curse of dimensionality . . . . . . . . . . . . . . . . . . . . 133
II Causal inference with models 137
11 Why model? 139
11.1 Data cannot speak for themselves . . . . . . . . . . . . . . . . . 139
11.2 Parametric estimators of the conditional mean . . . . . . . . . . 141
11.3 Nonparametric estimators of the conditional mean . . . . . . . 142
11.4 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.5 The bias-variance trade-off . . . . . . . . . . . . . . . . . . . . . 145
12 IP weighting and marginal structural models 149
12.1 The causal question . . . . . . . . . . . . . . . . . . . . . . . . . 149
12.2 Estimating IP weights via modeling . . . . . . . . . . . . . . . . 150
12.3 Stabilized IP weights . . . . . . . . . . . . . . . . . . . . . . . . 153
12.4 Marginal structural models. . . . . . . . . . . . . . . . . . . . . 155
12.5 Effect modification and marginal structural models . . . . . . . 157
CONTENTS v
12.6 Censoring and missing data . . . . . . . . . . . . . . . . . . . . 158
13 Standardization and the parametric g-formula 161
13.1 Standardization as an alternative to IP weighting . . . . . . . . 161
13.2 Estimating the mean outcome via modeling . . . . . . . . . . . 163
13.3 Standardizing the mean outcome to the confounder distribution 164
13.4 IP weighting or standardization? . . . . . . . . . . . . . . . . . 165
13.5 How seriously do we take our estimates? . . . . . . . . . . . . . 167
14 G-estimation of structural nested models 171
14.1 The causal question revisited . . . . . . . . . . . . . . . . . . . 171
14.2 Exchangeability revisited . . . . . . . . . . . . . . . . . . . . . . 172
14.3 Structural nested mean models . . . . . . . . . . . . . . . . . . 173
14.4 Rank preservation . . . . . . . . . . . . . . . . . . . . . . . . . . 175
14.5 G-estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
14.6 Structural nested models with two or more parameters . . . . . 179
15 Outcome regression and propensity scores 183
15.1 Outcome regression . . . . . . . . . . . . . . . . . . . . . . . . . 183
15.2 Propensity scores . . . . . . . . . . . . . . . . . . . . . . . . . . 185
15.3 Propensity stratification and standardization. . . . . . . . . . . 186
15.4 Propensity matching . . . . . . . . . . . . . . . . . . . . . . . . 188
15.5 Propensity models, structural models, predictive models . . . . 189
16 Instrumental variable estimation 193
16.1 The three instrumental conditions . . . . . . . . . . . . . . . . . 193
16.2 The usual IV estimand . . . . . . . . . . . . . . . . . . . . . . . 196
16.3 A fourth identifying condition: homogeneity . . . . . . . . . . . 198
16.4 An alternative fourth condition: monotonicity . . . . . . . . . . 200
16.5 The three instrumental conditions revisited . . . . . . . . . . . 204
16.6 Instrumental variable estimation versus other methods . . . . . 206
17 Causal survival analysis 209
17.1 Hazards and risks . . . . . . . . . . . . . . . . . . . . . . . . . . 209
17.2 From hazards to risks . . . . . . . . . . . . . . . . . . . . . . . . 211
17.3 Why censoring matters . . . . . . . . . . . . . . . . . . . . . . . 214
17.4 IP weighting of marginal structural models . . . . . . . . . . . . 216
17.5 The parametric g-formula . . . . . . . . . . . . . . . . . . . . . 217
17.6 G-estimation of structural nested models . . . . . . . . . . . . . 219
18 Variable selection for causal inference 223
18.1 The different goals of variable selection . . . . . . . . . . . . . . 223
18.2 Variables that induce or amplify bias . . . . . . . . . . . . . . . 225
18.3 Causal inference and machine learning . . . . . . . . . . . . . . 228
18.4 Doubly robust machine learning estimators. . . . . . . . . . . . 229
18.5 Variable selection is a difficult problem . . . . . . . . . . . . . . 230
III Causal inference from complex longitudinal data 233
19 Time-varying treatments 235
19.1 The causal effect of time-varying treatments . . . . . . . . . . . 235
19.2 Treatment strategies . . . . . . . . . . . . . . . . . . . . . . . . 236
19.3 Sequentially randomized experiments . . . . . . . . . . . . . . . 237
vi Causal Inference
19.4 Sequential exchangeability . . . . . . . . . . . . . . . . . . . . . 240
19.5 Identifiability under some but not all treatment strategies . . . 241
19.6 Time-varying confounding and time-varying confounders . . . . 245
20 Treatment-confounder feedback 247
20.1 The elements of treatment-confounder feedback . . . . . . . . . 247
20.2 The bias of traditional methods . . . . . . . . . . . . . . . . . . 249
20.3 Why traditional methods fail . . . . . . . . . . . . . . . . . . . 251
20.4 Why traditional methods cannot be fixed. . . . . . . . . . . . . 253
20.5 Adjusting for past treatment . . . . . . . . . . . . . . . . . . . . 254
21 G-methods for time-varying treatments 257
21.1 The g-formula for time-varying treatments . . . . . . . . . . . . 257
21.2 IP weighting for time-varying treatments . . . . . . . . . . . . . 261
21.3 A doubly robust estimator for time-varying treatments . . . . . 266
21.4 G-estimation for time-varying treatments . . . . . . . . . . . . . 268
21.5 Censoring is a time-varying treatment . . . . . . . . . . . . . . 275
22 Target trial emulation 277
22.1 The target trial (revisited) . . . . . . . . . . . . . . . . . . . . . 277
22.2 Causal effects in randomized trials . . . . . . . . . . . . . . . . 278
22.3 Causal effects in observational analyses that emulate a target trial281
22.4 Time zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
22.5 A unified analysis for causal inference . . . . . . . . . . . . . . . 284
References 288
INTRODUCTION: TOWARDS LESS CASUAL CAUSAL INFERENCES
CausalInference isanadmittedlypretentioustitleforabook. Causalinference
is a complex scientific task that relies on triangulating evidence from multiple
sources and on the application of a variety of methodological approaches. No
book can possibly provide a comprehensive description of methodologies for
causalinferenceacrossthesciences. TheauthorsofanyCausal Inference book
will have to choose which aspects of causal inference methodology they want
to emphasize.
The title of this introduction reflects our own choices: a book that helps
scientists–especially health and social scientists–generate and analyze data
to make causal inferences that are explicit about both thecausal question and
the assumptions underlying the data analysis. Unfortunately, the scientific
literature is plagued by studies in which the causal question is not explicitly
stated and the investigators’ unverifiable assumptions are not declared. This
casual attitude towards causal inference has led to a great deal of confusion.
For example, it is not uncommon to find studies in which the effect estimates
are hard to interpret because the data analysis methods cannot appropriately
answer the causal question (were it explicitly stated) under the investigators’
assumptions (were they declared).
Inthisbook,westresstheneedtotakethecausalquestionseriouslyenough
toarticulateit,andtodelineatetheseparaterolesofdataandassumptionsfor
causalinference. Oncethesefoundationsareinplace,causalinferencesbecome
necessarily less casual, which helps prevent confusion. The book describes
variousdataanalysisapproachesthatcanbeusedtoestimatethecausaleffect
of interest under a particular set of assumptions when data are collected on
each individual in a population. A key message of the book is that causal
inference cannot be reduced to a collection of recipes for data analysis.
The book is divided in three parts of increasing difficulty: Part I is about
causalinferencewithoutmodels(i.e.,nonparametricidentificationofcausalef-
fects), Part II is about causal inference with models (i.e., estimation of causal
effects with parametric models), and Part III is about causal inference from
complex longitudinal data (i.e., estimation of causal effects of time-varying
treatments). Throughoutthetext,wehaveinterspersedFinePointsandTech-
nical points that elaborate on certain topics mentioned in the main text. Fine
Points are designed to be accessible to all readers while Technical Points are
designed for readers with intermediate training in statistics. The book pro-
vides a cohesive presentation of concepts of, and methods for, causal inference
that are currently scattered across journals in several disciplines. We expect
that the book will be of interest to anyone interested in causal inference, e.g.,
epidemiologists, statisticians, psychologists, economists, sociologists, political
scientists, computer scientists...
Importantly, this is not a philosophy book. We remain agnostic about
metaphysical concepts like causality and cause. Rather, we focus on the iden-
tification and estimation of causal effects in populations, that is, numerical
quantities that measure changes in the distribution of an outcome under dif-
ferentinterventions. Forexample,wediscusshowtoestimatetheriskofdeath
viii Causal Inference
in patients with serious heart failure if they received a heart transplant versus
if they did not receive a heart transplant. Our main goal is to help decision
makers make better decisions–actionable causal inference.
Wearegratefultomanypeoplewhohavemadethisbookpossible. Stephen
Cole,SanderGreenland,JayKaufman,EleanorMurray,SonjaSwanson,Tyler
VanderWeele, and Jan Vandenbroucke provided detailed comments. Goodarz
Danaei, Kosuke Kawai, Martin Lajous, and Kathleen Wirth helped create the
NHEFSdataset. ThesamplecodeinPartIIwasdevelopedbyRogerLoganin
SAS,EleanorMurrayandRogerMurrayinStata,JoyShiandSeanMcGrathin
R,andJamesFiedlerinPython. RogerLoganhasalsobeenourLaTeXwizard.
Randall Chaput helped create the figures in Chapters 1 and 2. Josh McKible
designed the book cover. Rob Calver, our patient publisher, encouraged us to
write the book and supported our decision to make it freely available online.
Inaddition,multiplecolleagueshavehelpedusimprovethebookbydetect-
ingtyposandidentifyingunclearpassages. WeespeciallythankKafuiAdjaye-
Gbewonyo, Álvaro Alonso, Katherine Almendinger, Ingelise Andersen, Juan
José Beunza, Karen Biala, Joanne Brady, Alex Breskin, Shan Cai, Yu-Han
Chiu, Alexis Dinno, James Fiedler, Birgitte Frederiksen, Tadayoshi Fushiki,
Leticia Grize, Dominik Hangartner, John Jackson, Luke Keele, Laura Khan,
Dae Hyun Kim, Lauren Kunz, Martín Lajous, Angeliki Lambrou, Wen Wei
Loh, Haidong Lu, Mohammad Ali Mansournia, Giovanni Marchetti, Lauren
McCarl, Shira Mitchell, Louis Mittel, Hannah Oh, Ibironke Olofin, Robert
Paige, Jeremy Pertman, Melinda Power, Bruce Psaty, Brian Sauer, Tomohiro
Shinozaki, Ian Shrier, Yan Song, Øystein Sørensen, Etsuji Suzuki, Denis Tal-
bot, Mohammad Tavakkoli, Sarah Taubman, Evan Thacker, Kun-Hsing Yu,
Vera Zietemann, Jessica Young, and Dorith Zimmermann.
Part I
Causal inference without models
