class: center, top, .title-slide, title-slide # Nonparametric Bounds in Two-Sample Summary-Data Mendelian Randomization ## Some Cautionary Tales for Practice<br>.vsmall[(slides at <a href="https://rpubs.com/rmtrane/jsm_teaser" class="uri">https://rpubs.com/rmtrane/jsm_teaser</a>)] ### Ralph Møller Trane</br>.small[<a href="mailto:rtrane@wisc.edu" class="email">rtrane@wisc.edu</a>] ### University of Wisconsin–Madison<br> ### 2021-08-12 --- # Setup Does some (binary) `\(X\)` cause (binary) `\(Y\)`? (We will only consider binary `\(X\)`, `\(Y\)`.) Formally, want to learn something about `\(\text{ATE} = E[Y^1 - Y^0] = E[Y^1] - E[Y^0]\)`. (Note: `\(\text{ATE} \in [-1,1]\)`.) We can estimate the ATE if we can find an instrument `\(Z\)` such that <img src="data:image/png;base64,#short_introduction_files/figure-html/unnamed-chunk-2-1.png" height="200px" style="display: block; margin: auto;" /> Formally, `\(Z\)` should satisfy (A1) `\(Z \not\perp X\)` *(Relevance)*</br> (A2) `\(Z \perp U\)` *(Independent instrument)*</br> (A3) `\(Y^{z,x} = Y^{z',x} = Y^{x}\)` for all `\(x,z,z'\)` *(Exclusion restriction)*</br> (A4) `\(Y^{z,x} \perp Z, X | U\)` *(Conditional ignorability of `\(X,Z\)` given `\(U\)`)* --- # Nonparametric bounds Without further assumptions, we can obtain firm bounds from the distribution of `\((X,Y|Z)\)` for binary `\(Z\)`: <a name=cite-manski_nonparametric_1990></a>([Manski, 1990](#bib-manski_nonparametric_1990)): `$$\small \max \left\{\begin{array}{c} \max_z -P(Y = 0, X = 1 | Z = z) - P(Y = 1, X = 0 | Z = z) \\ \max_{z_1 \neq z_2} P(Y = 1 | Z = z_1) - P(Y = 1 | Z = z_2) - P(Y = 1, X = 0 | Z = z_1) - P(Y = 0, X = 1 | Z = z_2) \end{array}\right\} \\ \le \text{ATE} \le \\ \small \min \left\{\begin{array}{c} \min_z P(Y = 1, X = 1 | Z = z) + P(Y = 0, X = 0 | Z = z) \\ \min_{z_1 \neq z_2} P(Y = 1 | Z = z_1) - P(Y = 1 | Z = z_2) + P(Y = 1, X = 0 | Z = z_1) + P(Y = 0, X = 1 | Z = z_2) \end{array}\right\}$$` Similar bounds obtainable for general categorical `\(Z\)` <a name=cite-richardson_ace_2014></a>([Richardson and Robins, 2014](#bib-richardson_ace_2014)). These bounds are well-known. One important result: width is always less than `\(1 - ST\)`, where `\(ST = |P(X = 1|Z=1) - P(X = 1|Z=0)|\)` <a name=cite-balke_bounds_1997></a>([Balke and Pearl, 1997](#bib-balke_bounds_1997)). However, in MR studies when using genetic markers as instruments, we often rely on GWAS results, and only have `\((X|Z)\)` and `\((Y|Z)\)`. Fortunately, bounds using `\(P(X|Z)\)` and `\(P(Y|Z)\)` have been derived <a name=cite-ramsahai_causal_2012></a>([Ramsahai, 2012](#bib-ramsahai_causal_2012)), but their behavior not well-known. -- Our main question: **what can we learn from nonparametric bounds of causal effects in two-sample MR studies?** --- # Main Observation Generally, bounds from two-sample data are very wide: .pull-left[A: Two-sample IV bounds for the ATE of smoking on lung cancer.] .pull-right[B: Two-sample IV bounds for the ATE of high cholesterol on heart attack.] <img src="data:image/png;base64,#/Users/ralphtrane/Documents/ACEBounds/figures/png/example_analyses/bivariate_bounds.png" height="400" class="imgcenter"/> Note: no longer guaranteed width less than `\(1\)`!! --- # Check out the full presentation for... ... more on the width of two-sample bounds and the relationship to instrument strength ... an attempt to quantify the loss of information that is due to the two-sample design ... lessons learned that can help inform researchers on how to incorporate/utilize two-sample nonparametric bounds in practice ... more references! --- # References <a name=bib-balke_bounds_1997></a>[Balke, A. and J. Pearl](#cite-balke_bounds_1997) (1997). "Bounds on Treatment Effects from Studies with Imperfect Compliance". In: _Journal of the American Statistical Association_ 92.439, pp. 1171-1176. ISSN: 0162-1459. DOI: [10.1080/01621459.1997.10474074](https://doi.org/10.1080%2F01621459.1997.10474074). URL: [https://doi.org/10.1080/01621459.1997.10474074](https://doi.org/10.1080/01621459.1997.10474074) (visited on Feb. 05, 2020). <a name=bib-manski_nonparametric_1990></a>[Manski, C. F.](#cite-manski_nonparametric_1990) (1990). "Nonparametric Bounds on Treatment Effects". In: _The American Economic Review_ 80.2, pp. 319-323. ISSN: 0002-8282. <a name=bib-ramsahai_causal_2012></a>[Ramsahai, R. R.](#cite-ramsahai_causal_2012) (2012). "Causal Bounds and Observable Constraints for Non-Deterministic Models". In: _J. Mach. Learn. Res._ 13, pp. 829-848. ISSN: 1532-4435. <a name=bib-richardson_ace_2014></a>[Richardson, T. S. and J. M. Robins](#cite-richardson_ace_2014) (2014). "ACE Bounds; SEMs with Equilibrium Conditions". In: _Statistical Science_ 29.3, pp. 363-366. ISSN: 0883-4237. DOI: [10.1214/14-STS485](https://doi.org/10.1214%2F14-STS485). arXiv: [1410.0470](https://arxiv.org/abs/1410.0470).