Multiple discoveries in causal inference: LATE for the party (original) (raw)

. Author manuscript; available in PMC: 2024 Jul 2.

In 1994 economists Guido Imbens and Joshua Angrist formulated a causal model to estimate local average treatment effect (LATE), which was a major factor in their winning the 2021 Nobel Prize in economics. In the same year, 1994, we published an independent formulation of the LATE applied to medical research. The parallel development of LATE in these two settings and others offers insights into the nature of statistical research and multiple discoveries.

“When the time is ripe for certain things, these things appear in different places in the manner of violets coming to light in early spring.”

Farkas Bolyai (related to the invention of non-Euclidean geometry)

What is LATE?

LATE is a method for estimating the effect of receipt of treatment, as opposed to the effect of treatment assignment, in a randomized trial with noncompliance or in a before-and-after study with different availabilities of treatment in the two time periods. To keep matters simple, we use a randomized trial as the main example, with randomization to either treatment T0 (the standard treatment, which could be no treatment) or treatment T1. An important distinction is between two-sided and one-sided noncompliance. One-sided noncompliance is the scenario in which all persons randomized to T0 receive T0 and some (but not all) persons randomized to T1 receive T0 (i.e., they do not comply with the assigned treatment). Two-sided noncompliance is the scenario in which some (but not all) persons randomized to T0 receive T1 and some (but not all) persons randomized to T1 receive T0. The LATE framework involves 3 key aspects.

(1) With two-sided noncompliance, there are 4 latent classes, later called principal strata, which usually go by the names (i) never-takers, who would receive T0 regardless of assigned treatment, (ii) always-takers, who would receive T1 regardless of assigned treatment, (iii) compliers, who would receive the assigned treatment, and (iv) defiers, who would receive the opposite of the assigned treatment (Table 1). With one-sided noncompliance there are only 2 latent classes: never-takers and compliers. Estimates of treatment effect within a latent class are causal estimates because latent class is a baseline variable. For some persons in the trial, their latent class cannot be determined. For example, a person assigned treatment T0 who receives T0 could either be a never-taker or a complier (Table 2).

Table 1.

LATE framework for binary outcome (Y=0, 1)

Table 2.

Probabilities of a binary outcome Y= (0, 1) in LATE framework

(2) The exclusion restriction assumption says that, among always-takers and never-takers, the probability of outcome depends only on the treatment received, so information on randomization group does not change this probability (Table 1).

(3) The monotonicity assumption, which is only needed for two-sided noncompliance, specifies no defiers (Table 1).

The estimate of LATE is the intent-to-treat estimate divided by the estimated difference in the fraction receiving treatment between the two randomization groups (Table 3).

Table 3.

LATE with risk difference for binary outcome (Y=0, 1)

| LATE=βC,T1–βC,T0=pr(Y=1|randomizedtoT1)–pr(Y=1|randomizedtoT0)pr(receiveT1|randomizedtoT1)–pr(receiveT1|randomizedtoT0). | | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | For two-sided noncompliance, the LATE formula is based on the following equations: pr(Y=1|randomized to T1) = π(N) β(N) + π(C) β(C, T1) + π(A) β(A), | | pr(Y=1|randomized to T0)= π(N) β(N) + π(C) β(C, T0) + π(A) β(A), | | pr(receive T1| randomized to T1) – pr(receive T1| randomized to T0) = π(A) + π(C) – π(A).For one-sided noncompliance, π(A) = 0 and pr(receive T1| randomized to T0) = 0 |

In rapidly developing fields terminology is often in flux. In biostatistics literature, an alternative name for LATE, the complier average causal effect (CACE), has gained popularity, while LATE appears in the economics literature. Because they contain the word “average”, both LATE and CACE imply a difference. To include relative risk, we take a more general view of LATE as an effect of receipt of treatment among the latent class of compliers.

Harvard Economics Department. Early 1990s

As recounted by Angrist in his Noble Prize lecture, when he and Guido Imbens overlapped as assistant professors in the Harvard Economics Department in 1990, they wanted to better understand instrumental variables, which is a statistical tool for causal inference. This led them to formulate LATE for one-sided noncompliance. Later they learned about a 1984 paper by Howard Bloom, at the Harvard School of Government, which had independently formulated LATE for one-sided noncompliance. Referring to Bloom’s work, Angrist commented that it “remarkably derived results from first principles.” In his remarks, Angrist noted that he and Imbens were “late to the partial compliance party”. Subsequently Imbens and Angrist introduced the monotonicity assumption, which led to their noteworthy 1994 paper for LATE with two-sided noncompliance.

Harvard School of Public Health.1983

In the early 1980’s while Bloom was developing LATE for one-sided noncompliance, one of us (SGB) was also independently developing LATE for one-sided noncompliance only a few miles away. SGB was a graduate student in the Department of Biostatistics at the Harvard School of Public Health when Marvin Zelen was the chair of the Department. SGB had been thinking of a new way to analyze Zelen’s randomized consent design which randomized participants to either standard of care or an offer of new treatment that led to one-sided noncompliance. Like Bloom, SGB also derived results from first principles.

On August 12, 1983, SGB completed and printed a manuscript discussing maximum likelihood estimation of LATE for the randomized consent design with a binary outcome and one-sided noncompliance. Ultimately, the work was not published due to lack of enthusiasm from SGB’s mentors. (A copy of this manuscript can be found in the Supplementary material of Latent Class Instrumental Variables. A Clincal and Biostatistical Perspective).

Johns Hopkins Medical Institution. Circa 1991

While working at the National Institutes of Health, SGB extended the LATE formulation to two-sided noncompliance with the inclusion of the monotonicity assumption (although not by that name). Like his previous work, this extension to two-sided noncompliance involved binary endpoints and maximum likelihood estimation. In the early 1990’s, SGB gave a talk on this topic at the Johns Hopkins Medical Institutions, mentioning potential applications to randomized trials. His talk piqued the interest of another of us (KSL), an anesthesiologist at Johns Hopkins. KSL suggested applying the method to before-and-after studies on an important topic in her field. This led to the Paired Availability Design, which involved multiple before-and-after studies with different availabilities of treatments and an analysis using LATE. When data became available, we found that the Paired Availability Design yielded results that agreed with a meta-analysis of randomized trials and differed greatly from the results of a high-quality observational study that likely omitted a key confounder.

Later developments

Imbens extended LATE to regression discontinuity designs, which are designs used for causal inference based on a cutpoint of a variable, such as level of a biomarker, that triggers the offering a new treatment. SGB later extended LATE to survival analysis in cancer screening trials and adjusting for auxiliary variables and missing outcomes in cancer prevention trials. Along with Constantine Frangakis from Johns Hopkins, we extended LATE to a 3-arm randomized trial for estimating the effect of receipt of treatment under a type of partial noncompliance. One concern with applying LATE has been the generalizability from compliers to everyone in the trial. With multiple randomized trials or the paired availability design, we developed a simple way to investigate this generalizability by plotting LATE versus the difference in the fraction receiving the treatment in each trial or before-and-after study.

Three themes

Three themes emerge from this history.

First, one should not be surprised by initial skepticism of the usefulness of a new statistical method. Angrist noted some initial skepticism by a noted statistician of his LATE formulation. SGB needed a concrete example, as provided by KSL, to provide a more convincing discussion.

Second, it is easy to be unaware of independent methodological developments in other areas of application. Not surprisingly different areas of application involve different terminologies, making a literature search difficult. After publishing our paper in 1994, we learned about a 1989 paper by Thomas Permutt and J. Richard Hebel from the University of Maryland School of Medicine. They discussed the 3 key aspects of the LATE method for two-sided noncompliance as a by-product of recursive equations, a less transparent formulation that complicates extensions. We also later learned of two other 1991 independent formulations of LATE for one-sided noncompliance, one by Alfred Sommer and Scott L. Zeger from Johns Hopkins and one by Robert J. Connor, Philip C. Prorok, and Douglas L. Weed at the National Institutes of Health, both of which used relative risk.

Third, parallel developments of methodologies can involve variations (Table 4). The development of LATE in economics focused on continuous outcomes. The paired availability design focused on binary outcomes, maximum likelihood estimation, and generalizability using multiple before-and-after studies. As noted previously, some versions of LATE involved relative risks instead of risk differences or averages.

Table 4.

Overview of independent developments of LATE by different researchers

The bottom line is that multiple discoveries in statistics may be more common than many readers realize. Fortunately, in the development of statistical methods, if you miss one party, you are well on your way to attending the next one.

Footnotes

Disclaimer

The opinions expressed by the authors are their own and this material should not be interpreted as representing the official viewpoint of the U.S. Department of Health and Human Services, the National Institutes of Health, or the National Cancer Institute, or the Division of Cancer Prevention.

Contributor Information

Stuart G. Baker, National Cancer Institute, 9609 Medical Center Dr, MSC 9789, Bethesda MD 20892-9789

Karen S. Lindeman, Johns Hopkins Medical Institutions

Further Reading

1. Angrist JD (2022), “Empirical Strategies in Economics: Illuminating the Path from Cause to Effect,” Econometrica, 90, 2509–2539. [Google Scholar]
2. Baker SG and Lindeman KS (1994), “The Paired Availability Design. A Proposal for Evaluating Epidural Analgesia During Labor,” Statistics in Medicine, 13, 2269–2278. [DOI] [PubMed] [Google Scholar]
3. Baker SG, Kramer BS and Lindeman KS (2016), “Latent Class Instrumental Variables. A Clinical and Biostatistical Perspective,” Statistics in Medicine, 35,147–160. [DOI] [PMC free article] [PubMed] [Google Scholar]
4. Imbens GW, Angrist JD (1994), “Identification and Estimation of Local Average Treatment Effects,” Econometrica, 62, 467–475. [Google Scholar]