This website provides full documentation of the code used for our paper, and step-by-step guides on how to reproduce every figure and result. For the curious, it was generated using pkgdown.

The R-package ATEBounds includes every bit of code needed to reproduce all figures and results included in our paper. Since the main purpose is to ensure the reproducibility of figures and results from this specific paper, it has not been thoroughly tested in other settings. The code has not been optimized for performance, and was mainly written and documented for internal use only. That being said, many parts of this can be directly applied to other data sets, or reused in different settings.

You can install this package using remotes::install_github("rmtrane/ATEBounds"). This site includes an introduction to finding nonparametric bounds on the ATE using this package (Get started) and documentation of the functions included. On top of that, there are a few pages with information on how to reproduce our results; see below for more. One that might be of particular interest is vignette("example_analysis"), which shows how to use this package to obtain two-sample bounds based on studies from the MR-Base database.

Nonparametric bounds of the Average Treatment Effect (ATE) are intriguing for several reasons. Getting bounds on the ATE with no further assumptions other than what is embedded in the IV model is very appealing in particular as such assumptions often limit the structure of unmeasured confounders. By definition, assumptions about unmeasured confounders cannot be checked, and it is therefore desirable to avoid such.

Many have considered nonparametric bounds in the past. Swanson et al. (2018) provide a nice overview of different bounds derived from different sets of assumptions. Arguably the weakest set of assumptions result in what is known as the Balke-Pearl bounds, first presented in Balke and Pearl (1993).

The same set of bounds can be obtained using a geometric approach as presented by Ramsahai (2012). We will rely on this work as it provides a very general approach to finding bounds. This allows us to obtain bounds in a setting where we do not have observations of (X, Y, Z), but rather observations of (X, Z) and (Y, Z) from two different data sources. (We will refer to the former as the one-sample (or trivariate) setting, and the latter as the two-sample (or bivariate) setting.) This is relevant today in many mendelian randomization (MR) analyses that have a natural desire to take advantage of the large databases readily available. These databases contain results from GWAS in the form of summary statistics. Each GWAS is essentially exploring associations between a feature of interest (smoking, obesity, incidence of heart attack, etc.) and many thousands of SNPs. When the feature of interest is binary, the summary statistics are often obtained using logistic regression.

The behavior of nonparametric bounds based on one-sample data is pretty well-studied, but little is known about the behavior of nonparametric bounds based on two-sample data. The goal of this work is to shine a light on when nonparametric bounds can be useful particularly in two-sample MR studies based on GWAS summary statistics. We do this based on two aspects of the bounds: (1) the length of the bounds and (2) whether the bounds cover the null effect of zero.

Our paper contains the following sections:

• Section 2 reviews relevant notation, definitions and assumptions, and introduces the two-sample IV bounds derived by Ramsahai (2012).
• Section 3 shows that even when adding two strong monotonicity assumptions, the width of two-sample bounds are bounded above by 2 − 2 ⋅ ST (where ST = max_{z1 ≠ z2} = |P(X = 1|Z = z₁) − P(X = 2|Z = z₂)| is the measure of strength of the instrument we consider). This is important as one-sample bounds are guaranteed to have width at most 1. We show through simulations that this upper bounds is sharp also when the two monotonicity assumptions are not imposed. Figure 1a of our paper shows this, and is created in vignette("bounds_from_bivariate"). Since our ST metric isn’t common in MR studies, we illustrate the relationship between the ST measure and coefficients from a logistic model, and show what kind of coefficient is needed for two-sample bounds to exclude the null effect 0 for various effect sizes. The latter can to some extend be thought of as a “power analysis,” except we work under the assumption that we have population based probabilities. Figures 1b and 2 are created in vignette("power_sims_figures"). Finally, we mention the use of multiple instruments. The main part of this discussion is presented in eAppendix A.5. Simulations for this eAppendix are in vignette("multiple_IVs_sims") and figures are created in vignette("multiple_IVs_figures").
• Section 4 illustrates the information loss we see when using a two-sample rather than a one-sample design. vignette("bounds_from_trivariate") shows how Figure 3 was created.
• Section 5 demonstrates our findings on two real MR studies. vignette("example_analysis") shows how we obtained the data and preprocessed it using the TwoSampleMR R-package (Hemani et al. 2018), and how Figures 4 and 5 were created. It also includes the code needed to reproduce eFigures and eTables in eAppendix A.7.

Balke, Alexander, and Judea Pearl. 1993. “Nonparametric Bounds on Causal Effects from Partial Compliance Data.” JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION.

Hemani, Gibran, Jie Zheng, Benjamin Elsworth, Kaitlin H Wade, Valeriia Haberland, Denis Baird, Charles Laurin, et al. 2018. “The MR-Base Platform Supports Systematic Causal Inference Across the Human Phenome.” Edited by Ruth Loos. eLife 7 (May): e34408. https://doi.org/10.7554/eLife.34408.

Ramsahai, Roland R. 2012. “Causal Bounds and Observable Constraints for Non-Deterministic Models.” J. Mach. Learn. Res. 13 (March): 829–48.

Swanson, Sonja A., Miguel A. Hern’an, Matthew Miller, James M. Robins, and Thomas S. Richardson. 2018. “Partial Identification of the Average Treatment Effect Using Instrumental Variables: Review of Methods for Binary Instruments, Treatments, and Outcomes.” Journal of the American Statistical Association 113 (522): 933–47. https://doi.org/10.1080/01621459.2018.1434530.