Estimating high-dimensional intervention effects from observational data

arXiv:0810.4214v3 [stat.ME] 2 Sep 2009 The Annals of Statistics 2009, Vol. 37, No. 6A, 3133–3164 DOI: 10.1214/09-AOS685 c ⃝Institute of Mathematical Statistics, 2009 ESTIMATING HIGH-DIMENSIONAL INTERVENTION EFFECTS FROM OBSERVATIONAL DATA By Marloes H. Maathuis, Markus Kalisch and Peter B¨uhlmann ETH Z¨urich We assume that we have observational data generated from an unknown underlying directed acyclic graph (DAG) model. A DAG is typically not identifiable from observational data, but it is possible to consistently estimate the equivalence class of a DAG. Moreover, for any given DAG, causal effects can be estimated using intervention calculus. In this paper, we combine these two parts. For each DAG in the estimated equivalence class, we use intervention calculus to esti- mate the causal effects of the covariates on the response. This yields a collection of estimated causal effects for each covariate. We show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph. This lo- cal approach is computationally fast and feasible in high-dimensional problems. We propose to use summary measures of the set of pos- sible causal effects to determine variable importance. In particular, we use the minimum absolute value of this set, since that is a lower bound on the size of the causal effect. We demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production. 1. Introduction. Our work is motivated by the following problem in bi- ology. We want to know which genes play a role in a certain phenotype, say a disease status or, in our case, a continuous value of riboflavin (vi- tamin B2) production in the bacterium Bacillus subtilis. To be more pre- cise, our goal is to infer which genes have an effect on the phenotype in terms of an intervention. If we knocked down single genes, which of them would show a relevant or important effect on the phenotype? The difficulty is, however, that the available data are only observational. For our con- crete problem, we observe the logarithm of the riboflavin production rate as a continuous response and expression measurements from essentially the Received October 2008; revised January 2009. AMS 2000 subject classifications. 62-09, 62H99. Key words and phrases. Causal analysis, directed acyclic graph (DAG), graphical mod- eling, intervention calculus, PC-algorithm, sparsity. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2009, Vol. 37, No. 6A, 3133–3164. This reprint differs from the original in pagination and typographic detail. 1 2 M. H. MAATHUIS, M. KALISCH AND P. B¨UHLMANN whole genome of B. subtilis as high-dimensional covariates. Using such ob- servational data, we want to infer all (single gene) intervention effects. This task coincides with inferring causal effects, a well-established area in Statis- tics (e.g., [5, 8, 10, 11, 13, 18, 24, 25, 26] and [31]). We emphasize that, in our application, it is exactly the intervention or causal effect that is of in- terest, rather than a regression-type effect of association. If we can estimate the intervention effects from observational data, we can score each gene ac- cording to its potential to have an intervention (knock-down) effect on the riboflavin production rate, and the most promising candidate genes can be tested afterward in biological experiments. Pearl ([25], page 285) formulates the distinction between associational and causal concepts as follows: “an associational concept is any relationship that can be defined in terms of a joint distribution of observed variables, and a causal concept is any relationship that cannot be defined from the distribu- tion alone... . Every claim invoking causal concepts must be traced to some premises that invoke such concepts; it cannot be inferred or derived from statistical associations alone.” Thus, in order to obtain causal statements from observational data, one needs to make additional assumptions. One possibility is to assume that the data were generated by a directed acyclic graph (DAG) which is known beforehand. DAGs describe causal concepts, since they code potential causal relationships between variables: the exis- tence of a directed edge x →y means that x may have a direct causal effect on y, and the absence of a directed edge x →y means that x cannot have a direct causal effect on y (see Remark 2.3 for a definition of direct causal effect). Given a set of conditional dependencies from observational data and a corresponding DAG model, one can compute causal effects using interven- tion calculus (e.g., [24] and [25]). In this paper, we consider the problem of inferring causal information from observational data, under the assumption that the data were generated by an unknown DAG. This is a more realistic assumption, since, in many practical problems, one does not know the DAG. In this scenario, the causal

…(Full text truncated)…

This content is AI-processed based on ArXiv data.