Estimating bounds on causal effects in high-dimensional and possibly confounded systems

Abstract

We present an algorithm for estimating bounds on causal effects from observational data which combines graphical model search with simple linear regression. We assume that the underlying system can be represented by a linear structural equation model with no feedback, and we allow for the possibility of latent confounders. Under assumptions standard in the causal search literature, we use conditional independence constraints to search for an equivalence class of ancestral graphs. Then, for each model in the equivalence class, we perform the appropriate regression (using causal structure information to determine which covariates to adjust for) to estimate a set of possible causal effects. Our approach is based on the IDA procedure of Maathuis et al. (2009), which assumes that all relevant variables have been measured (i.e., no latent confounders). We generalize their work by relaxing this assumption, which is often violated in applied contexts. We validate the performance of our algorithm in simulation experiments.

Robust Causal Structure Learning with Some Hidden Variables

“…First, we see that, when there is no confounding, RFCI,LV‐IDA is capable of achieving a high precision. This is consistent with previous findings indicating that LV‐IDA is conservative but capable of recovering a small but high quality set of total causal effects (Malinsky and Spirtes, ). When h >0, the set of models that we simulate from is particularly challenging for methods relying on MAGs since nearly all pairs of observed variables are confounded.…”

“…As detailed earlier, there are other approaches which are capable of estimating DAG models and total causal effects in the presence of hidden variables, i.e. FCI‐type algorithms (Spirtes et al ., ; Colombo et al ., ; Claassen et al ., ) and LV‐IDA (Malinsky and Spirtes, ). In both our simulations and our first application, we found that such approaches are very conservative under our assumptions.…”

“…We consider the following methods, where the first‐stage tuning parameters are selected as before (when applicable): (a) GES,IDA (the CPDAG is estimated by using GES; the tuning parameter λ n is chosen with the BIC; IDA is applied with the resulting CPDAG and the sample covariance matrix ); (b) NSDIST,IDA (the DAG is estimated by using NSDIST and converted to a CPDAG; the tuning parameter for the second stage ( λ , with the notation of Han et al . ()) is chosen by using the BIC; IDA is applied with the resulting CPDAG and the sample covariance matrix ); (c) (PCA * + GES * ) , IDA (the top k principal components are first estimated from the data and regressed out; the CPDAG is estimated by using GES on the residuals; the tuning parameter λ n is chosen with perfect knowledge to maximize the average precision (in terms of causal effect recovery); IDA is applied with the resulting CPDAG and the covariance matrix of the residuals); (d) (LRpS+GES),IDA (the CPDAG is estimated by using LRpS+GES; the tuning parameter of the second stage λ n is chosen with the BIC; IDA is applied with the resulting CPDAG and the covariance matrix ); (e) RFCI,LV‐IDA (Colombo et al ., ; Malinsky and Spirtes, ) (the PAG is estimated with RFCI; the significance level α for RFCI is given by α =0.5/√ n ; LV‐IDA is applied to the resulting PAG and the sample covariance matrix ); whenever LV‐IDA outputs ‘NA’, the corresponding pair is not counted, i.e. it is neither a true positive nor false positive result) (in several cases, the LV‐IDA algorithm, when applied to a single data set, was still running after a few days of computation; given that we simulated data from hundreds of data sets, we could not experiment with many values of α ); (f) RANDOM,IDA (100 random DAGs are generated from the same model as was used in the simulation; total causal effects are then estimated based on the resulting CPDAG and the sample covariance matrix ; we report the interval that is spanned by the 2.5–97.5‐percentiles of the distribution of precisions at fixed recalls); (g) EMPTY,IDA (causal effects are computed without adjustment, which is equivalent to applying the ida function of the pcalg package to an empty graph and the sample covariance matrix). …”

Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-Based Approach

“…Finally we note that identifiability has also been studied under the assumption that the functional relationships depicted by the causal model are linear (Angrist, Imbens, and Rubin 1996;Van der Zander and Liśkiewicz 2016;Chen, Kumor, and Bareinboim 2017) or nonparametric with additive error terms (Peters, Mooij, Janzing, and Schölkopf 2014;Peña and Bendtsen 2017) and when the causal graph is not completely known (Maathuis, Kalisch, and Bühlmann 2009;Entner, Hoyer, and Spirtes 2013;Hyttinen et al 2015;Perković et al 2015;Malinsky and Spirtes 2017;Jaber, Zhang, and Bareinboim 2018). Extending the search in these directions is an interesting line of future research.…”

A Bayesian Approach for Estimating Causal Effects from Observational Data

“…From observational data alone, the causal graph can in general only be identified up to its Markov equivalence class. Several works study causal effects in the light of the limited identifiability of causal structures (Entner, Hoyer, and Spirtes 2013;Hyttinen, Eberhardt, and Järvisalo 2015;Perković et al 2018;Jaber, Zhang, and Bareinboim 2018a;2018b;Malinsky and Spirtes 2017).…”