Causal inference (with statistical uncertainty) based on invariance: exploiting the power of heterogeneous data

Transcription

1 Causal inference (with statistical uncertainty) based on invariance: exploiting the power of heterogeneous data Peter Bühlmann joint work with Jonas Peters Nicolai Meinshausen

2 ... and designing new perturbation experiments for large-scale biological systems

3 Goal in genomics: if we would make an intervention at a single gene, what would be its effect on a phenotype of interest? want to infer/predict such effects without actually doing the intervention e.g. from observational data (from observations of a steady-state system ) but here: focus on observational and interventional data with known, specified interventions changing environments or experimental settings with vaguely or un- specified interventions it doesn t need to be genes can generalize to interventions at more than one variable/gene

4 Goal in genomics: if we would make an intervention at a single gene, what would be its effect on a phenotype of interest? want to infer/predict such effects without actually doing the intervention e.g. from observational data (from observations of a steady-state system ) but here: focus on observational and interventional data with known, specified interventions changing environments or experimental settings with vaguely or un- specified interventions it doesn t need to be genes can generalize to interventions at more than one variable/gene

5 Goal in genomics: if we would make an intervention at a single gene, what would be its effect on a phenotype of interest? want to infer/predict such effects without actually doing the intervention e.g. from observational data (from observations of a steady-state system ) but here: focus on observational and interventional data with known, specified interventions changing environments or experimental settings with vaguely or un- specified interventions it doesn t need to be genes can generalize to interventions at more than one variable/gene

6 Genomics 1. Flowering of Arabidopsis Thaliana phenotype/response variable of interest: Y = days to bolting (flowering) covariates X = gene expressions from p = genes question: infer/predict the effect of knocking-out/knocking-down (or enhancing) a single gene (expression) on the phenotype/response variable Y? randomized experiments for the top-hits from statistical analysis based on n = 47 observational data points with some success (Stekhoven, Moraes, Sveinbjörnsson, Hennig, Maathuis & PB, 2012)

7 Genomics 1. Flowering of Arabidopsis Thaliana phenotype/response variable of interest: Y = days to bolting (flowering) covariates X = gene expressions from p = genes question: infer/predict the effect of knocking-out/knocking-down (or enhancing) a single gene (expression) on the phenotype/response variable Y? randomized experiments for the top-hits from statistical analysis based on n = 47 observational data points with some success (Stekhoven, Moraes, Sveinbjörnsson, Hennig, Maathuis & PB, 2012)

8 2. Gene expressions of yeast p = 5360 genes phenotype of interest: Y = expression of first gene covariates X = gene expressions from all other genes and then phenotype of interest: Y = expression of second gene covariates X = gene expressions from all other genes and so on infer/predict the effects of a single gene knock-down on all other genes

9 Effects of single gene knock-downs on all other genes (yeast) (Maathuis, Colombo, Kalisch & PB, 2010) p = 5360 genes (expression of genes) 231 gene knock downs intervention effects the truth is known in good approximation (thanks to intervention experiments) goal: prediction of the true large intervention effects based on observational data with no knock-downs n = 63 observational data True positives 1, IDA Lasso Elastic net Random ,000 2,000 3,000 4,000 False positives

10 it was a good finding... but some criticisms apply: not very robust: depends to an unpleasant degree how we define the truth (has been found recently by J. Mooij, J. Peters and others) old, publicly available data (Hughes et al., 2000) no collaborator... just (publicily available) data }{{} this is good! we didn t use the interventional data for training

11 A new and better attempt for the same problem single gene knock-down in yeast and measure genomewide expression (observational and interventional data) collaborators: Frank Holstege, Patrick Kemmeren et al. (Utrecht) data from modern technology Kemmeren,..., and Holstege (Cell, 2014)

12 a new approach, developed with Jonas Peters and Nicolai Meinshausen (MPI Tüb. and ETH Z.) based on some invariance principle exploiting heterogeneous data (e.g. interventional or change of environment) e.g. an aspect of Big Data allowing for statistical confidence statements Peters, PB and Meinshausen (2015)

13 Causal inference using invariant prediction Peters, PB and Meinshausen (2015) goal: find the causal variables (components) among a p-dimensional predictor variable X for a specific response variable Y consider data (X e, Y e ) F e with response variables Y e and predictor variables X e here: e }{{} E space of exp. sett. denotes an experimental setting heterogeneous data from different environments/experiments e E (aspects of Big Data )

14 Causal inference using invariant prediction Peters, PB and Meinshausen (2015) goal: find the causal variables (components) among a p-dimensional predictor variable X for a specific response variable Y consider data (X e, Y e ) F e with response variables Y e and predictor variables X e here: e }{{} E space of exp. sett. denotes an experimental setting heterogeneous data from different environments/experiments e E (aspects of Big Data )

15 data (X e, Y e ) F e example 1: E = {1, 2} encoding observational (1) and all potentially unspecific interventional data (2) example 2: E = {1, 2} encoding observational data (1) and (repeated) data from one specific intervention (2) example 3: E = {1, 2, 3} encoding (repeated) data from a first specific intervention (1), from a second specific intervention (2), and from all other (potentially unspecific) interventions (3)

16 important: we have access to more than say only observational data that is: we assume here some setting with e.g. observational and potentially unspecified interventional (or change of environment) data realistic in many applications but different than causal inference from observational data

17 Assumption: invariance of causal linear predictions there exists a true vector γ of linear causal coefficients such that e E : Y e = X e γ + ε e, ε e F ε S = {j; γj 0} causal predictors ε e independent of X S, and F ε is the same for all e, with mean zero and finite variance message: causal coefficients γ and causal variables X S are stable (are the same) across experimental settings! this is intuitively meaningful, and we will make it more rigorous

18 Assumption: invariance of causal linear predictions there exists a true vector γ of linear causal coefficients such that e E : Y e = X e γ + ε e, ε e F ε S = {j; γj 0} causal predictors ε e independent of X S, and F ε is the same for all e, with mean zero and finite variance message: causal coefficients γ and causal variables X S are stable (are the same) across experimental settings! this is intuitively meaningful, and we will make it more rigorous

19 invariance an important mathematical and scientific concept

20 make it more rigorous... e E : Y e = X e γ + ε e, ε e F ε the same for all e S = {j; γ j ε e independent of X S 0} causal predictors we prove this invariance assumption for structural equation models (SEMs), roughly assuming: 1. there are no interventions ( manipulations ) on the target/response variable Y 2. γ and F ε do not change under interventions ( manipulations ) less requirements than for classical, Pearl s -interventions (with truncated factorization) see next...

21 consider H 0,γ,S (E) : γ k = 0 if k / S and F ε such that e E : Y e = X e γ + ε e, ε e X e S, εe F ε the same for all e S, γ are potential causal variables and coefficients and H 0,S (E) : there exists γ such that H 0,γ,S (E) holds a set S {1,..., p} is called plausible causal predictors if H 0,S (E) holds

22 identifiable causal predictors under E: the set S(E), where S(E) = {S; H 0,S (E) holds} under the invariance Assumption we have: S(E) S and this is key to obtain confidence bounds for identifiable causal predictors

23 we also have by definition: S(E 1 ) S(E 2 ) if E 1 E 2 with more interventions more heterogeneity more diversity in Big Data we can identify more causal predictors

24 identifiable causal predictors : true causal predictors : S(E) as E S (E) as E question: when is S(E)(= S ) = S (E)?

25 consider linear Gaussian Structural Equation Model (SEM) X j β jk X k + ε j (j = 1,..., p + 1), }{{} k pa(j) Gaussian X p+1 = Y three types of interventions: classical do-interventions at single variables noise (or soft ) interventions: for e 1(= observational) ε e j A e j εe=1 j or ε e j ε e=1 j + Cj e simultaneous noise interventions where all variables are affected: all intervent. settings are pooled into a single one (e = 2), and for all j: ε e=2 j A e=2 j ε e=1 j or ε e=2 j ε e=1 j + Cj e=2

26 for all these interventions: β jk can change under interventions except for j = p + 1 corresponding to Y even for the classical do-interventions...

27 Theorem (Peter, PB and Meinshausen, 2015) if there is: S(E) = S a single do-intervention for each variable other than Y and E = p a single noise intervention for each variable other than Y and E = p a simultaneous noise intervention for each variable other than Y and E = 2 the conditions can be relaxed such that it is not necessary to intervene at all the variables

28 Statistical confidence sets for causal predictors recap: H 0,γ,S (E) : γ k = 0 if k / S and F ε such that e E : Y e = X e γ + ε e, ε e X e S, εe F ε the same for all e γ is a candidate for being a causal coefficient vector for a set S (not necessarily the true set S of causal predictors) consider plausible causal coefficients for S: Γ S (E) = {γ; H 0,γ,S (E) holds} global plausible causal coefficients: Γ(E) = S Γ S (E)

29 we have, similarly as before: Γ(E 2 ) Γ(E 1 ) if E 1 E 2 that is: Γ(E) is shrinking as we enlarge E with more interventions more heterogeneity more diversity in Big Data we have less plausible coefficients... and in some cases eventually only the causal ones (see identifiability result before)

30 confidence sets with invariant prediction 1. for each S {1,..., p} construct a set ˆΓ S (E) e.g., as follows: test H0,S (E): βpred(s) e }{{} regr. parameter β and σ e (S) }{{} resid. se. σ e E constancy of regression parameter and its residual error across e E 2. set ˆΓ(E) = 3. define Ŝ(E) = if otherwise S {1,...,p} γ ˆΓ(E) H0,S (E) is rejected at level α/2 set ˆΓ S (E) = ˆΓS (E) = classical CI for regr. param. based on X S ˆΓ S (E) {k; γ k 0} (est. plausible causal coeff.) (common part of plaus. causal coeff.)

31 Theorem (Peters, PB and Meinshausen, 2015) assume : P[γ ˆΓ S (E)] 1 α correct confidence interval in a regression problem then: P[γ ˆΓ(E)] 1 α P[Ŝ(E) S ] 1 α on the safe side (conservative) we do not need to care about identifiability: if the effect is not identifiable, the method will not wrongly claim an effect the first result on statistical confidence for causal predictors (route via graphical modeling for confidence sets seems awkward)

32 Theorem (Peters, PB and Meinshausen, 2015) assume : P[γ ˆΓ S (E)] 1 α correct confidence interval in a regression problem then: P[γ ˆΓ(E)] 1 α P[Ŝ(E) S ] 1 α on the safe side (conservative) we do not need to care about identifiability: if the effect is not identifiable, the method will not wrongly claim an effect the first result on statistical confidence for causal predictors (route via graphical modeling for confidence sets seems awkward)

33 Empirical results: simulations 100 different scenarios, 1000 data sets per scenario: E = 2, n obs = n interv {100,..., 500}, p {5,..., 40} SUCCESS PROBABILITY Marginal Marginal Regression Regression Ges Ges Gies (unknown) Gies (unknown) Gies (known) Gies (known) Lingam Lingam Invariant prediction Invariant prediction power to detect causal predictors FWER Marginal Marginal Regression Regression Ges Ges nown) Gies (unknown) nown) Gies (known) gam Lingam ction Invariant prediction familywise error rate: P[Ŝ(E) S ], aimed at 0.05

34 Gene perturbation experiments Kemmeren et al. (2014): genome-wide mrna expressions in yeast: p = 6170 genes n obs = 160 observational samples of wild-types n int = 1479 interventional samples each of them corresponds to a single gene deletion strain for our method: we use E = 2 (observational and interventional data) training-test data: training: all observational and 2/3 of interventional data test: other 1/3 of gene deletion interventions repeat this for the three blocks of interventional data since every interventional data point is used once as a response variable: we use coverage 1 α/n int with α 0.01 and n int = 1479

35 Results 8 genes are found as significant causal predictors method invar.pred. GIES IDA marg.corr. rand.guess. no. true pos * (out of 8) *: quantiles for selecting true positives among 8 random draws 2 (95%), 3 (99%) our invariant prediction method has most power! and it should exhibit control against false positive selections

36 a more detailed view ACTIVITY GENE observational training data interventional training data (interv. on genes other than 5954 and 4710) ACTIVITY GENE interventional test data point (intervention on gene 5954) 1st sign. gene successful ACTIVITY GENE ACTIVITY GENE ACTIVITY GENE 5954 ACTIVITY GENE observational training data interventional training data (interv. on genes other than 3729 and 3730) ACTIVITY GENE interventional test data point (intervention on gene 3729) 2nd sign. gene successful ACTIVITY GENE ACTIVITY GENE ACTIVITY GENE 3729 ACTIVITY GENE observational training data ACTIVITY GENE 3672 interventional training data (interv. on genes other than 3672 and 1475) ACTIVITY GENE 3672 ACTIVITY GENE interventional test data point (intervention on gene 3672) ACTIVITY GENE rd sign. gene not successful (poorer mod. fit?)

37 Concluding thoughts robustness : the main requirement is the invariance assumption e E : Y e = X e γ }{{} =X e S γ +ε e, ε e F ε, ε e X S when making it a definition of causality: essentially no other major conditions required (except hidden confounders) when proving it: have shown that it holds for SEM s under various (and weaker) notions of (potentially unspecified) interventions can also prove it for some hidden variable IV-type models (Peters, PB and Meinshausen, 2015) the invariance assumption is (probably) a rather robust assumption

38 statistical confidence statements are essential for improving replicability we haven t included non-linear or non-gaussian structures cf. Imoto (2002), Mooij et al. (2009), Peters et al. (2014), Shimizu (2005) improved identifiability invariance assumption for non-linear additive noise models: e E : Y e = f (X e S ) + εe, ε e F ε, ε e X S

39 due to robustness and generality of heterogeneous data ( change of environment ) progress towards more confirmatory causal inference? if it isn t confirmatory: potentially useful for prioritization of future experiments experimental validations are important (simple organisms in biology are ideal for pursuing this!)

40 Thank you! R-package: pcalg (Kalisch, Mächler, Colombo, Maathuis & PB, ) R-package: InvariantCausalPrediction (Meinshausen, 2014) References to some of our own work: Peters, J., Bühlmann, P. and Meinshausen, N. (2015). Causal inference using invariant prediction: identification and confidence intervals. Preprint arxiv: Hauser, A. and Bühlmann, P. (2015). Jointly interventional and observational data: estimation of interventional Markov equivalence classes of directed acyclic graphs. Journal of the Royal Statistical Society, Series B, 77, Bühlmann, P., Peters, J. and Ernest, J. (2014). CAM: Causal Additive Models, high-dimensional order search and penalized regression. Annals of Statistics 42, Peters, J. and Bühlmann, P. (2014). Identifiability of Gaussian structural equation models with equal error variances. Biometrika 101, van de Geer, S. and Bühlmann, P. (2013). l0-penalized maximum likelihood for sparse directed acyclic graphs. Annals of Statistics 41, Kalisch, M., Mächler, M., Colombo, D., Maathuis, M.H. and Bühlmann, P. (2012). Causal inference using graphical models with the R package pcalg. Journal of Statistical Software 47 (11), Stekhoven, D.J., Moraes, I., Sveinbjörnsson, G., Hennig, L., Maathuis, M.H. and Bühlmann, P. (2012). Causal stability ranking. Bioinformatics 28, Maathuis, M.H., Colombo, D., Kalisch, M. and Bühlmann, P. (2010). Predicting causal effects in large-scale systems from observational data. Nature Methods 7, Maathuis, M.H., Kalisch, M. and Bühlmann, P. (2009). Estimating high-dimensional intervention effects from observational data. Annals of Statistics 37,