Fundamentals of Causal Inference

With R

Babette A. Brumback

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN: 9780367705053
• eBook ISBN: 9781000470338
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

"Overall, this textbook is a perfect guide for interested researchers and students who wish to understand the rationale and methods of causal inference. Each chapter provides an R implementation of the introduced causal concepts and models and concludes with appropriate exercises."-An-Shun Tai & Sheng-Hsuan Lin, in Biometrics

One of the primary motivations for clinical trials and observational studies of humans is to infer cause and effect. Disentangling causation from confounding is of utmost importance. Fundamentals of Causal Inference explains and relates different methods of confounding adjustment in terms of potential outcomes and graphical models, including standardization, difference-in-differences estimation, the front-door method, instrumental variables estimation, and propensity score methods. It also covers effect-measure modification, precision variables, mediation analyses, and time-dependent confounding. Several real data examples, simulation studies, and analyses using R motivate the methods throughout. The book assumes familiarity with basic statistics and probability, regression, and R and is suitable for seniors or graduate students in statistics, biostatistics, and data science as well as PhD students in a wide variety of other disciplines, including epidemiology, pharmacy, the health sciences, education, and the social, economic, and behavioral sciences.

Beginning with a brief history and a review of essential elements of probability and statistics, a unique feature of the book is its focus on real and simulated datasets with all binary variables to reduce complex methods down to their fundamentals. Calculus is not required, but a willingness to tackle mathematical notation, difficult concepts, and intricate logical arguments is essential. While many real data examples are included, the book also features the Double What-If Study, based on simulated data with known causal mechanisms, in the belief that the methods are best understood in circumstances where they are known to either succeed or fail. Datasets, R code, and solutions to odd-numbered exercises are available on the book's website at www.routledge.com/9780367705053. Instructors can also find slides based on the book, and a full solutions manual under 'Instructor Resources'.

1Introduction

DOI: 10.1201/9781003146674-1

1.1 A Brief History

The history of causal inference is long and complex, with many thinkers from a variety of disciplines having written on the topic. A good overview from a statistician’s perspective is provided by Holland (1986). Here, we briefly touch upon some prominent milestones in the field. Our earliest stop is from Aristotle (350BC), who enumerated four types of causes.

1. What a thing is made of, e.g. the porcelain of a cup
2. The form of a thing, e.g. the octave the relation of 2:1
3. “The primary source of the change or coming to rest; e.g. the man who gave advice is a cause, the father is cause of the child, and generally what makes of what is made and what causes change of what is changed.”
4. “That for the sake of which a thing is done, e.g. health is the cause of walking about. (‘Why is he walking about?’ we say. ‘To be healthy,’ and, having said that, we think we have assigned the cause.)”

In statistics, causes are of Aristotle’s type 3, “what causes change of what is changed.” We are commonly interested in the effect of an intervention, or the cause of a health or societal problem. Type 4 causes are interesting to ponder, as these might be taken to imply that the future can cause the past, in a certain sense. However, if we translate the definition so that our intention to achieve the end is the cause, then our intention (to be healthy) can be posited to exist prior to, and thus be a type 3 cause of, the action (walking) which in turn causes (again in the sense of type 3) the end (health). If divine intervention were possible, it could be that the intention belonged to a supreme being, who then manipulated the sequence of events at least as often as necessary to reach the desired end, or fate. In that semi-deterministic or deterministic world, whether the future caused the past or the past caused the future would be irrelevant. We would be watching a movie directed by the supreme being, needlessly concerning ourselves with optimal decision-making.

Hume (1738) famously wrote in 1738 that cause and effect is not much more than the constant conjunction of events.

“We have no other notion of cause and effect, but that of certain objects, which have been always conjoined together, and which in all past instances have been found inseparable. We cannot penetrate into the reason of the conjunction. We only observe the thing itself, and always find that, from the constant conjunction, the objects require a union in the imagination… Thus, though causation be a philosophical relation, as implying contiguity, succession, and constant conjunction, yet it is only so far as it is a natural relation, and produces a union among our ideas, that we are able to reason upon it, or draw any inference from it.”

It is noteworthy that Hume did not propose any methods to verify a causal relationship other than to observe contiguity, succession, and constant conjunction. Nevertheless, shortly afterwards in 1747, James Lind undertook what is widely recognized today as one of the very first medical trials, proving a causal relationship between eating oranges and lemon and recovering from scurvy. From Brown (2005), we read that Lind took aside twelve men with advanced symptoms of scurvy “as similar as I could have them.” Six pairs of men aboard the HMS Salisbury were thus experimented upon. The first pair were given slightly alcoholic cider. The second were given an elixir of vitriol. The third pair took vinegar. The fourth drank sea water. The fifth were fed two oranges and one lemon daily for six days, when the meager supply ran out. The sixth were given a medicinal paste and cream of tartar, which is a mild laxative. The pair who were fed the oranges and the lemons were nearly recovered after only a week. Those who had drunk the cider responded favorably, but at the end of two weeks they were still too weak to return to duty. The other four pairs did not experience good effects.

By 1846, Mill (1846), in his five canons, proposed five methods for proving cause and effect:

“First Canon (the Method of Agreement): If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.

Second Canon (the Method of Difference): If an instance in which the phenomenon-under investigation occurs, and an instance in which it does not occur, have every circumstance save one in common, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or cause, or a necessary part off the cause, off the phenomenon.

Third Canon (the Joint Method ...

Table of contents

• Cover
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• 1 Introduction
• 2 Conditional Probability and Expectation
• 3 Potential Outcomes and the Fundamental Problem of Causal Inference
• 4 Effect-Measure Modification and Causal Interaction
• 5 Causal Directed Acyclic Graphs
• 6 Adjusting for Confounding: Backdoor Method via Standardization
• 7 Adjusting for Confounding: Difference-in-Differences Estimators
• 8 Adjusting for Confounding: Front-Door Method
• 9 Adjusting for Confounding: Instrumental Variables
• 10 Adjusting for Confounding: Propensity-Score Methods
• 11 Gaining Efficiency with Precision Variables
• 12 Mediation
• 13 Adjusting for Time-Dependent Confounding
• Appendix
• Bibliography
• Index