Statistics 186
Spring,
2025
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 (samurphy@g.harvard.edu).
Class
Times: MW 4:30 pm-5:45 pm (EST). No class 2/17, 3/17, 3/19.
TF: ???
Sections
and Office Hours:
Susan Murphy s Office Hours: ??? 2.335 SEC
TBN s Office Hours:
Course
Webpage: Stat 186 Canvas site
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.
Other Papers: Scientific papers may be assigned.
Recommended Texts: Hern n MA, Robins JM (2020). Causal
Inference: What If. Boca Raton: Chapman & Hall/CRC. See https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/
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:
1/27, 1/29
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: 2/03, 2/05
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: 2/05, 2/10, 2/12
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 2/12 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: 02/17,
02/19, 02/24
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: 02/26,
03/03, 03/05
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: 03/10, 03/24, 03/26, 03/31
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: 04/02, 04/07
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: 04/09, 04/14, 04/16, 04/21
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 03/12 (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 4pmEST
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 1/30 and is due in Canvas at 4pm on
2/13.
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.
To help with various circumstances
(expected or unexpected), your lowest homework score will be dropped. Further you can petition me for up to 2 late
homework submissions by emailing me by the Wednesday at 4pmEST preceding the
due date for the homework. A late
homework submission is 24 hours late (so due the Friday at 4pmEST after the due
date) and is submitted directly to me.
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 12
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 TBD. 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 ????? to see if they can help and/or you can call
the HUIT help desk.