Statistics 186

Spring, 2024


Causal Inference concerns the very difficult, challenging problem of addressing questions such as, "Would vaccinating children 16 and younger against COVID 19 lead to fewer deaths among public school teachers?"  and "Would providing Harvard students access to a mobile health application designed to help them manage school stress, lead to improved school performance?" This class will include 4 modules.  The first module introduces the nuanced world of causal inference along with a fundamental tool: the language of potential outcomes.   The second module covers randomized experiments and how data from randomized experiments can be used to make causal statements.  The third module introduces the rather tricky problem of using observational (non-randomized) data to attempt to make causal statements.   The final module introduces a new and challenging area in which the goal is to make causal inference about the effect of sequences of treatments. 


Professor: Susan Murphy (


Class Times: MW 4:30 pm-5:45 pm (EST) at the Science Center, room 705.  No class 2/19, 3/11, 3/13.


TF: Nathan Cheng ( )


Sections and Office Hours:

Susan Murphy’s Office Hours:  5:15pm-6:15pm EST on Thursday in 2.335 SEC except for 3/14.

Nathan Cheng’s Office Hours:  2:00pm-3pm EST on Tuesday in the Science Center 316.06 except for 3/12


Course Webpage:  


Book: Imbens, G., & Rubin, D. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139025751.   No purchase is necessary; you can download pdf copies of the book chapters from the “Library Reserves” section of the Stat 186 Canvas site.  You can also purchase the hard copy or an Adobe e-book at the Harvard Coop Bookstore.   


Other Papers:  Scientific papers may be assigned.


Recommended Texts: Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC.  See


Prerequisites: Stat110, Stat111, Stat139.  Probability and statistical inference are needed extensively, and statistical linear models are often used.


Computing and Simulation: Some homework problems will mainly involve statistical reasoning and probability whereas other homework problems will require programming in R.  Students may use other software programs such as Python and Matlab, but we will only provide support for R.  I recommend RStudio as an interface for R. Both R and RStudio are freely available.  You are welcome to use generative AI (ChatGPT, Bard, etc) for assistance in coding.


Typical Class:

4:30pm: Ensure that you have turned in the Quiz on Canvas

4:30pm: Sit with your assigned group

4:30-5:00pm: 30 Min. Lecture

5:00-5:10pm: Breakout with your group (Discuss quiz and questions posed in Lecture)

5:10-5:20pm: Class Discussion (one of the groups leads the discussion)

5:20-5:45pm: 25 Min. Lecture


Topics Covered:

Module 1: [Potential Outcomes, Assignment Mechanisms]

Dates: 1/22, 1/24

This module provides the crucial underpinning for this entire course.  This unit will assist you in thinking critically about statements made in everyday life about cause and effect. We provide a language for investigating causality—this language will help you translate statements such as “Students who walk at least 10,000 steps per day are less likely to be stressed” into mathematical statements and then if needed reframe these statements to enhance precision.   This then allows us to be precise in how we use data to investigate causality.

Assigned Reading Material: Chapters 1 & 3 of Imbens and Rubin



Module 2: [Randomized Experiments and Associated Data Analyses]

Much of our course focuses on Module 2; Module 2 will help you understand why randomized experiments facilitate causal inference and also this module will help you understand how to reason about and conduct causal inference with experimental data.  This Module provides first hints about how you might be able to conduct causal inference when you have observational data instead of experimental data.


1.      Classical Randomized Experiments

Dates: 1/29, 1/31

This section concerns experimental settings in which we determine the assignment mechanism, that is the probability distribution of the randomized treatment assignment.  We will learn about some of the pros and cons of different approaches to randomization.

Assigned Reading Material: Chapter 4 of Imbens and Rubin.  

Other interesting material: “Statistical Properties of Randomization in Clinical Trials” by Lachin, “Properties of Simple Randomization in Clinical Trials” by Lachin and “Randomization in Clinical Trials: Conclusions and Recommendations” by Lachin, Matts and Wei.  These papers are under Files on Canvas.


2.      Fisher's Approach to Causal Reasoning about Treatment Effects for the Population of N Individuals in the Sample

Dates: 1/31, 2/05, 2/07, 2/21 (Note late date!)

This section concerns finite sample inference.  In this section you will learn about how you can conduct causal inference about the N units (individuals) in the experiment (finite sample inference).  Randomization tests are crucial tools.

Assigned Reading Material: Chapter 5 of Imbens and Rubin; Section 5.1 is critical but very dense.   I suggest reading Section 5.1 again and again as you go through the other sections so that you gradually began to understand Section 5.1.  For 2/21 assigned reading is Meaning of Power.pdf on Canvas

Under Files>Readings

Other interesting material: “Statistical Properties of Randomization in Clinical Trials” by Lachin, “Properties of Simple Randomization in Clinical Trials” by Lachin and “Randomization in Clinical Trials: Conclusions and Recommendations” by Lachin, Matts and Wei.  These papers are under Files on Canvas.


3.      Neyman's Approach to Causal Reasoning about Treatment Effects for the Population of N Units in the Sample and it's Extension to using the Sample to Conduct Causal Inference about Treatment Effects in a Large Population.  

Dates:  02/12, 02/14, 02/26

Often we aim to use the sample of N units (individuals) to inform decisions about a larger population (should we provide the new take-home chemotherapy to all adolescents recovering from leukemia as opposed to the current take-home chemotherapy?).  You will learn why this causal inference both requires more assumptions and at the same is less restrictive than finite sample causal inference.   You will learn first statistical approaches to conducting this type of causal inference.

Assigned Reading Material: Chapter 6 of Imbens and Rubin


4.      Using Regression to Conduct Causal Inference. 

Dates:   02/28, 03/04, 03/18  

Regression is one of the earliest and continues to be one of the most common tools used to conduct causal inference. In regression we often add outside knowledge about the form of the mean of the outcome conditional on covariates.  In this section you will learn how you can use covariates to improve your ability to detect and conduct inference about causal effects.  You will learn about the consequences of miss-specifying the regression.

Assigned Reading Material: Chapter 7 of Imbens and Rubin; additional reading material may be assigned.



Module 3: [Observational Studies]

In many areas of science, experiments are unethical, for example, we might be interested in the causal effect of parental divorce on children’s elementary school performance.  Or for monetary or societal reasons, data from experiments is not available. These are all settings in which the “assignment mechanism” is unknown.  In this module you will learn about first approaches to conducting causal inference in these thorny problems. 


1.      Unconfounded Treatment Assignment. 

Dates:   03/20, 03/25, 03/27, 04/01

In this section you will learn about the critical role the propensity score plays in conducting causal inference, in particular for use in settings in which science along with high quality observational data can be harnessed to explain the assignment mechanism.

Assigned Reading Material: Chapter 12 of Imbens and Rubin; additional reading material may be assigned.


2.      Estimating the Propensity Score.

Dates: 04/03, 04/08

In the analysis of observational data with propensity scores you will need to estimate the propensity score.   You will learn about methods for doing this and, how to think about estimation when the goal is to reduce confounding as opposed to fitting a good model.

Assigned Reading Material. Chapter 13 Imbens and Rubin; additional reading material may be assigned.


3.      Using the Propensity Score to Conduct Causal Inference in Observational Studies.

Tentative Dates: 04/10, 04/15, 04/17, 04/22

Here we discuss how to use the propensity score via stratification/blocking in causal inference.  If time permits, we will discuss a second approach, namely propensity score weighting.

Assigned Reading Material: Chapter 17 Imbens and Rubin; additional reading material may be assigned.



Module 4: [Dynamic Treatment Regimes & Sequential Experimentation]

Tentative Date: 3 classes TBD

This is causal inference on steroids!! In this module you will learn about how to reason about potential outcomes when treatments are sequential. When treatments are sequential, it is easy for the analysis method to accidentally introduce confounding even though the treatments are randomized.

Assigned Reading Material: MRTs for Developing Digital Interventions. This paper is written for behavioral scientists.


Grading: Course grades will be based on a weighted average of homework scores (30%), quizzes (10%), participation (10%), midterm exam on 03/06 (20%) and a final exam on a date to be determined (30%). Additional information about each of these components is below.  The course is letter-graded by default, but you have my permission to switch to SAT/UNSAT grading if you prefer. If you are considering SAT/UNSAT you should discuss it with your advisor, and check whether it would count for what you want it to count for. A grade of SAT corresponds to a letter grade of C- or above.


Quizzes: Assigned Readings are listed above in each section.  To help with various circumstances (expected or unexpected), your lowest three (3) quizzes will be dropped.  Monday’s Quiz is available on Canvas starting at 4:30pm EST on Sunday and closing at 4:30pm EST on Monday at the beginning of class: similarly, Wednesday’s Quiz is available on Canvas starting at 4:30pm EST on Tuesday and closing at 4:30pm EST on Wednesday at the beginning of class. Once you start the quiz, you have 10 minutes to complete the quiz. Collaboration on Quizzes is not permitted. Use of generative AI is not permitted.


Homework: Problem sets will be assigned on every other Thursday 4pm via Canvas and will be due two weeks later on the following Thursday at 4pmEST in Canvas. The first assignment will appear in Canvas at 4pm on 1/25 and is due in Canvas at 4pm on 2/08.


Homework must be submitted via the Canvas course website; no submissions on paper or by email will be accepted. Your submission must be a single PDF file, no more than 20 MB in size, except that computer code can be uploaded in a separate supplementary file if that is more convenient for you (i.e., a .R or .Rmd  file with your R code). The outputs from your code, e.g., plots and summary statistics, must still be in your main PDF file. Your homework can be typeset, written using a tablet, or scanned from handwritten work, but must be clear and easily legible (not blurry or faint), and correctly rotated (e.g., not upside down). Always check your submission: download it after uploading it in Canvas, and make sure that it is the correct file and that it got uploaded successfully. 


Late homework submissions are not accepted. To help with various circumstances (expected or unexpected), your 2 lowest homework scores will be dropped.  


Unless otherwise specified, please show your work, simplify fully, and give clear, careful justifications for your answers (using words and sentences to explain your logic, not just formulas).


Homework Collaboration Policy: Beginning the first week, every other week students are randomly divided into collaborative groups of people on Thursdays at 4pm.  This is your discussion + homework group for next two weeks.  Each student individually submits their homework solution with a list of who was in their assigned collaboration group. You must write up your solutions yourself and in your own words. Copying someone else's solution or a solution from generative AI, or just making trivial changes to other’s solutions, including a generative AI solution for the sake of not copying verbatim, is not acceptable. For example, in problems where you must make up a story or example, two students should not have the exact same answer, or almost the same answer except one has an example with dogs chasing cats and the other has an example with cats chasing mice, with the same structure and the same numbers. I highly recommend starting problem sets early enough so that you have time to work hard on the problems on your own first, before discussing them with your group.


Using generative AI tools such as ChatGPT to help with homework is allowed, but not recommended.  In my own testing of ChatGPT (with GPT-4), it could do some Stat 186 problems correctly but also made many mistakes. It could be a useful tool for suggesting ideas (and to chat about the material with or compare answers with) but it is error-prone. Furthermore, working hard on the homework problems is crucial for learning the material and preparing for the midterm and final, so even if ChatGPT did excellent work on the Stat 186 homeworks, relying on it too much would likely be harmful for overall understanding and for performance in the class.  In any case, your solutions must reflect your own understanding of the material, explained in your own way, rather than being copied from any other source.


Participation: Active participation is expected, through attending class (Mondays and Wednesdays), completing quizzes and engaging in discussions. Stat 186 is a challenging course covering subtle concepts, so let's all try to help create a supportive, collaborative community.


Midterm: The Midterm is on Wednesday, March 6 at 4:30pmET in our classroom. You can bring one 8.5 by 11 sheet of notes (using both back and front) with you to the midterm. Otherwise, the midterm is closed book, no internet access and no computer access.


There is no makeup midterm.  To help with various circumstances (expected or unexpected), if you miss the midterm, your score on your final exam will count for 50% of your grade.  


Final:  The Final is May 3, 2024, 9am-12noon in Northwest B_108. You can bring two 8.5 by 11 sheets of notes (using both back and front) with you to the final. Otherwise, the final is closed book, no internet access, and no computer access.


Accommodations: Harvard University’s goal is to remove barriers for disabled students related to inaccessible elements of instruction or design in this course. If reasonable accommodations are necessary to provide access, please contact the Disability Access Office (DAO). Accommodations do not alter fundamental requirements of the course and are not retroactive. Students should request accommodations as early as possible, since they may take time to implement. Students should notify DAO at any time during the semester if adjustments to their communicated accommodation plan are needed


Where to get tech help: The Academic Resources Center has resources. For tech help you can chat with Nathan to see if he can help and/or you can call the HUIT help desk.