# Estimate a PAG with the FCI Algorithm Estimate a Partial Ancestral Graph (PAG) from observational data, using the FCI (Fast Causal Inference) algorithm, or from a combination of data from different (e.g., observational and interventional) contexts, using the FCI-JCI (Joint Causal Inference) extension. ```r fci(suffStat, indepTest, alpha, labels, p, skel.method = c("stable", "original", "stable.fast"), type = c("normal", "anytime", "adaptive"), fixedGaps = NULL, fixedEdges = NULL, NAdelete = TRUE, m.max = Inf, pdsep.max = Inf, rules = rep(TRUE, 10), doPdsep = TRUE, biCC = FALSE, conservative = FALSE, maj.rule = FALSE, numCores = 1, selectionBias = TRUE, jci = c("0","1","12","123"), contextVars = NULL, verbose = FALSE) ``` ## Arguments - `suffStat`: sufficient statistics: A named `list` containing all necessary elements for the conditional independence decisions in the function `indepTest`. - `indepTest`: a `function` for testing conditional independence. The function is internally called as `indepTest(x,y, S, suffStat)`, and tests conditional independence of `x` and `y` given `S`. Here, `x` and `y` are variables, and `S` is a (possibly empty) vector of variables (all variables are denoted by their column numbers in the adjacency matrix). `suffStat` is a list with all relevant information, see above. The return value of `indepTest()` is the p-value of the test for conditional independence. - `alpha`: numeric significance level (in $(0, 1)$) for the individual conditional independence tests. - `labels`: (optional) `character` vector of variable (or node ) names. Typically preferred to specifying `p`. - `p`: (optional) number of variables (or nodes). May be specified if `labels` are not, in which case `labels` is set to `1:p`. - `skel.method`: character string specifying method; the default, `"stable"` provides an **order-independent** skeleton, see `skeleton`. - `type`: character string specifying the version of the FCI algorithm to be used. By default, it is `"normal"`, and so the normal FCI algorithm is called. If set to `"anytime"`, the Anytime FCI is called and `m.max` needs to be specified. If set to `"adaptive"`, the Adaptive Anytime FCI is called and `m.max` is not used. For more information, see Details. - `fixedGaps`: `logical` matrix of dimension p*p. If entry `[i,j]` or `[j,i]` (or both) are TRUE, the edge i-j is removed before starting the algorithm. Therefore, this edge is guaranteed to be absent in the resulting graph. - `fixedEdges`: logical matrix of dimension p*p. If entry `[i,j]` or `[j,i]` (or both) are TRUE, the edge i-j is never considered for removal. Therefore, this edge is guaranteed to be present in the resulting graph. - `NAdelete`: If indepTest returns `NA` and this option is `TRUE`, the corresponding edge is deleted. If this option is `FALSE`, the edge is not deleted. - `m.max`: Maximum size of the conditioning sets that are considered in the conditional independence tests. - `pdsep.max`: Maximum size of Possible-D-SEP for which subsets are considered as conditioning sets in the conditional independence tests. If the nodes `x` and `y` are adjacent in the graph and the size of $Possible-D-SEP(x) \ {x, y}$ is bigger than pdsep.max, the edge is simply left in the graph. Note that if pdsep.max is less than Inf, the final PAG may be a supergraph of the one computed with pdsep.max = Inf, because fewer tests may have been performed in the former. - `rules`: Logical vector of length 10 indicating which rules should be used when directing edges. The order of the rules is taken from Zhang (2008). - `doPdsep`: If `TRUE`, Possible-D-SEP is computed for all nodes, and all subsets of Possible-D-SEP are considered as conditioning sets in the conditional independence tests, if not defined otherwise in `pdsep.max`. If `FALSE`, Possible-D-SEP is not computed, so that the algorithm simplifies to the Modified PC algorithm of Spirtes, Glymour and Scheines (2000, p.84). - `biCC`: If `TRUE`, only nodes on paths between nodes `x` and `y` are considered to be in Possible-D-SEP(`x`) when testing independence between `x` and `y`. Uses biconnected components, `biConnComp` from `RBGL`. - `conservative`: Logical indicating if the unshielded triples should be checked for ambiguity the second time when v-structures are determined. For more information, see details. - `maj.rule`: Logical indicating if the unshielded triples should be checked for ambiguity the second time when v-structures are determined using a majority rule idea, which is less strict than the standard conservative. For more information, see details. - `numCores`: Specifies the number of cores to be used for parallel estimation of `skeleton`. - `selectionBias`: If `TRUE`, allow for selection bias. If `FALSE`, selection bias is excluded by assumption and hence rules R5-R7, as in Zhang (2008), are disabled. - `jci`: String specifying the JCI assumptions that are used. It can be one of: - **"0"**: No JCI assumption is made (default), - **"1"**: JCI assumption 1 (no system variable causes any context variable), - **"12"**: JCI assumptions 1 and 2 (no system variable causes any context variable, and no system variable is confounded with any context variable), - **"123"**: JCI assumptions 1, 2 and 3 (no system variable causes any context variable, no system variable is confounded with any context variable, and all context variables are confounded but are not direct causes of each other). For more information, see Mooij et al. (2020). - `contextVars`: Subset of variable indices {1,...,p} that will be treated as context variables in the JCI extension of FCI. - `verbose`: If true, more detailed output is provided. ## Returns An object of `class` `fciAlgo` (see `fciAlgo`) containing the estimated graph (in the form of an adjacency matrix with various possible edge marks), the conditioning sets that lead to edge removals (sepset) and several other parameters. ## Details This function is a generalization of the PC algorithm (see `pc`), in the sense that it allows arbitrarily many latent and selection variables. Under the assumption that the data are faithful to a DAG that includes all latent and selection variables, the FCI algorithm (Fast Causal Inference algorithm) (Spirtes, Glymour and Scheines, 2000) estimates the Markov equivalence class of MAGs that describe the conditional independence relationships between the observed variables. Under the assumption that the data are $\sigma$-faithful to a simple (possibly cyclic) SCM that allows for latent confounding (but selection bias is absent), the FCI algorithm estimates the $\sigma$-Markov equivalence class of the DMGs (directed mixed graphs) that describe the causal relations between the observed variables and the conditional independence relationships in the observed distribution through $\sigma$-separation (Mooij and Claassen, 2020). The FCI-JCI (Joint Causal Inference) extension allows the algorithm to combine data from different contexts, for example, observational and different types of interventional data (Mooij et al., 2020). FCI estimates a partial ancestral graph (PAG ). The PAG represents a Markov equivalence class of DAGs with latent and selection variables in the acyclic case (Zhang, 2008), and a Markov equivalence class of directed graphs with latent variables (but without selection variables) in the cyclic $\sigma$-separation case. A PAG contains the following types of edges: o-o, o--, o->, -->, <->, ---. The bidirected edges come from latent confounders, and the undirected edges come from latent selection variables. The edges have the following interpretation: (i) there is an edge between `x` and `y` if and only if variables `x` and `y` are conditionally dependent given S for all sets S consisting of all selection variables and a subset of the observed variables (assuming the Markov property and faithfulness); (ii) a tail `x --* y` at x on an edge between x and y means that x is an ancestor of y or S; (iii) an arrowhead `x <-* y` at x on an edge between x and y means that x is not an ancestor of y, nor of S; (iv) a circle mark `x o-* y` at x on an edge between x and y means that there exists both a graph in the Markov equivalence class where x is ancestor of y or S, and one where x is not ancestor of y, nor of S. For further information on the interpretation of PAGs see e.g. (Zhang, 2008) and (Mooij and Claassen, 2020). The first part of the FCI algorithm is analogous to the PC algorithm. It starts with a complete undirected graph and estimates an initial skeleton using `skeleton(*, method="stable")` which produces an initial order-independent skeleton, see `skeleton` for more details. All edges of this skeleton are of the form o-o. Due to the presence of hidden variables, it is no longer sufficient to consider only subsets of the neighborhoods of nodes `x` and `y` to decide whether the edge `x o-o y` should be removed. Therefore, the initial skeleton may contain some superfluous edges. These edges are removed in the next step of the algorithm which requires some orientations. Therefore, the v-structures are determined using the conservative method (see discussion on `conservative` below). After the v-structures have been oriented, Possible-D-SEP sets for each node in the graph are computed at once. To decide whether edge `x o-o y` should be removed, one performs conditional indepedence tests of x and y given all possible subsets of Possible-D-SEP(x) and of Possible-D-SEP(y). The edge is removed if a conditional independence is found. This produces a fully order-independent final skeleton as explained in Colombo and Maathuis (2014). Subsequently, the v-structures are newly determined on the final skeleton (using information in sepset). Finally, as many as possible undetermined edge marks (o) are determined using (a subset of) the 10 orientation rules given by Zhang (2008). The Anytime FCI algorithm was introduced by Spirtes (2001). It can be viewed as a modification of the FCI algorithm that only performs conditional independence tests up to and including order m.max when finding the initial skeleton, using the function `skeleton`, and the final skeleton, using the function `pdsep`. Thus, Anytime FCI performs fewer conditional independence tests than FCI. To use the Anytime algorithm, one sets `type = "anytime"` and needs to specify `m.max`, the maximum size of the conditioning sets. The Adaptive Anytime FCI algorithm was introduced by Colombo et. al (2012). The first part of the algorithm is identical to the normal FCI described above. But in the second part when the final skeleton is estimated using the function `pdsep`, the Adaptive Anytime FCI algorithm only performs conditional independence tests up to and including order `m.max`, where m.max is the maximum size of the conditioning sets that were considered to determine the initial skeleton using the function `skeleton`. Thus, m.max is chosen adaptively and does not have to be specified by the user. **Conservative** versions of FCI, Anytime FCI, and Adaptive Anytime FCI are computed if `conservative = TRUE` is specified. After the final skeleton is computed, all potential v-structures a-b-c are checked in the following way. We test whether a and c are independent conditioning on any subset of the neighbors of a or any subset of the neighbors of c. When a subset makes a and c conditionally independent, we call it a separating set. If b is in no such separating set or in all such separating sets, no further action is taken and the normal version of the FCI, Anytime FCI, or Adaptive Anytime FCI algorithm is continued. If, however, b is in only some separating sets, the triple a-b-c is marked ambiguous . If a is independent of c given some S in the skeleton (i.e., the edge a-c dropped out), but a and c remain dependent given all subsets of neighbors of either a or c, we will call all triples a-b-c unambiguous . This is because in the FCI algorithm, the true separating set might be outside the neighborhood of either a or c. An ambiguous triple is not oriented as a v-structure. Furthermore, no further orientation rule that needs to know whether a-b-c is a v-structure or not is applied. Instead of using the conservative version, which is quite strict towards the v-structures, Colombo and Maathuis (2014) introduced a less strict version for the v-structures called majority rule. This adaptation can be called using `maj.rule = TRUE`. In this case, the triple a-b-c is marked as ambiguous if and only if b is in exactly 50 percent of such separating sets or no separating set was found. If b is in less than 50 percent of the separating sets it is set as a v-structure, and if in more than 50 percent it is set as a non v-structure (for more details see Colombo and Maathuis, 2014). Colombo and Maathuis (2014) showed that with both these modifications, the final skeleton and the decisions about the v-structures of the FCI algorithm are fully order-independent. Note that the order-dependence issues on the 10 orientation rules are still present, see Colombo and Maathuis (2014) for more details. The FCI-JCI extension of FCI was introduced by Mooij et. al (2020). It is an implementation of the **Joint Causal Inference** (JCI) framework that reduces causal discovery from several data sets corresponding to different contexts (e.g., observational and different interventional settings, or data measured in different labs or in different countries) to causal discovery from the pooled data (treated as a single 'observational' data set). Two types of variables are distinguished in the JCI framework: system variables (describing aspects of the system in some environment) and context variables (describing aspects of the environment of the system). Different assumptions regarding the causal relations between context variables and system variables can be made. The most common assumption (JCI Assumption 1: 'exogeneity') is that no system variable can affect any context variable. The second, less common, assumption (JCI Assumption 2: 'complete randomized context') is that there is no latent confounding between system and context. The third assumption (JCI Assumption 3: 'generic context model') is in fact a faithfulness assumption on the context distribution. The FCI-JCI extension can be used by specifying the subset of the variables that are designated as context variables with the `contextVars` argument, and by specifying the combination of JCI assumptions that is used with the `jci` argument. For the default values of these arguments, the JCI extension is not used. The only difference between the FCI-JCI extension and the standard FCI algorithm is that the background knowledge about the PAG implied by the JCI assumptions is exploited at several points in the FCI-JCI algorithm to enforce adjacencies between context variables (in case of JCI Assumption 3) and to orient certain edges adjacent to context variables (in case of JCI Assumptions 1, 2 or 3). The current JCI framework assumes that no selection bias is present, and therefore the FCI-JCI extension should be called with `selectionBias = FALSE`. For more details on FCI-JCI, and the general JCI framework, see Mooij et al. (2020). ## References D. Colombo and M.H. Maathuis (2014). Order-independent constraint-based causal structure learning. **Journal of Machine Learning Research** 15 3741-3782. D. Colombo, M. H. Maathuis, M. Kalisch, T. S. Richardson (2012). Learning high-dimensional directed acyclic graphs with latent and selection variables. **Ann. Statist.** 40 , 294-321. M.Kalisch, M. Maechler, D. Colombo, M. H. Maathuis, P. Buehlmann (2012). Causal Inference Using Graphical Models with the R Package pcalg. **Journal of Statistical Software** 47(11) 1--26, tools:::Rd_expr_doi("10.18637/jss.v047.i11") . J. M. Mooij, S. Magliacane, T. Claassen (2020). Joint Causal Inference from Multiple Contexts. **Journal of Machine Learning Research** 21 (99), 1-108. J. M. Mooij and T. Claassen (2020). Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles. **In Proc. of the 36th Conference on Uncertainty in Artificial Intelligence (UAI-20)**, 1159-1168. P. Spirtes (2001). An anytime algorithm for causal inference. **In Proc. of the Eighth International Workshop on Artificial Intelligence and Statistics** 213-221. Morgan Kaufmann, San Francisco. P. Spirtes, C. Glymour and R. Scheines (2000). **Causation, Prediction, and Search**, 2nd edition, MIT Press, Cambridge (MA). P. Spirtes, C. Meek, T.S. Richardson (1999). In: **Computation, Causation and Discovery**. An algorithm for causal inference in the presence of latent variables and selection bias. Pages 211-252. MIT Press. T.S. Richardson and P. Spirtes (2002). Ancestral graph Markov models. **Annals of Statistics** 30 962-1030. J. Zhang (2008). On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. **Artificial Intelligence** 172 1873-1896. ## See Also `fciPlus` for a more efficient variation of FCI; `skeleton` for estimating a skeleton using the PC algorithm; `pc` for estimating a CPDAG using the PC algorithm; `pdsep` for computing Possible-D-SEP for each node and testing and adapting the graph accordingly; `qreach` for a fast way of finding Possible-D-SEP for a given node. `gaussCItest`, `disCItest`, `binCItest`, `dsepTest` and `dsepAMTest` as examples for `indepTest`. ## Author(s) Diego Colombo, Markus Kalisch (kalisch@stat.math.ethz.ch ) and Joris Mooij. ## Examples ```r ################################################## ## Example without latent variables ################################################## set.seed(42) p <- 7 ## generate and draw random DAG : myDAG <- randomDAG(p, prob = 0.4) ## find skeleton and PAG using the FCI algorithm suffStat <- list(C = cov2cor(trueCov(myDAG)), n = 10^9) res <- fci(suffStat, indepTest=gaussCItest, alpha = 0.9999, p=p, doPdsep = FALSE) ################################################## ## Example with hidden variables ## Zhang (2008), Fig. 6, p.1882 ################################################## ## create the graph g p <- 4 L <- 1 # '1' is latent V <- c("Ghost", "Max","Urs","Anna","Eva") edL <- setNames(vector("list", length=length(V)), V) edL[[1]] <- list(edges=c(2,4),weights=c(1,1)) edL[[2]] <- list(edges=3,weights=c(1)) edL[[3]] <- list(edges=5,weights=c(1)) edL[[4]] <- list(edges=5,weights=c(1)) g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed") ## compute the true covariance matrix of g cov.mat <- trueCov(g) ## delete rows and columns belonging to latent variable L true.cov <- cov.mat[-L,-L] ## transform covariance matrix into a correlation matrix true.corr <- cov2cor(true.cov) ## The same, for the following three examples indepTest <- gaussCItest suffStat <- list(C = true.corr, n = 10^9) ## find PAG with FCI algorithm. ## As dependence "oracle", we use the true correlation matrix in ## gaussCItest() with a large "virtual sample size" and a large alpha: normal.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L], verbose=TRUE) ## find PAG with Anytime FCI algorithm with m.max = 1 ## This means that only conditioning sets of size 0 and 1 are considered. ## As dependence "oracle", we use the true correlation matrix in the ## function gaussCItest with a large "virtual sample size" and a large ## alpha anytime.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L], type = "anytime", m.max = 1, verbose=TRUE) ## find PAG with Adaptive Anytime FCI algorithm. ## This means that only conditining sets up to size K are considered ## in estimating the final skeleton, where K is the maximal size of a ## conditioning set found while estimating the initial skeleton. ## As dependence "oracle", we use the true correlation matrix in the ## function gaussCItest with a large "virtual sample size" and a large ## alpha adaptive.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L], type = "adaptive", verbose=TRUE) ## define PAG given in Zhang (2008), Fig. 6, p.1882 corr.pag <- rbind(c(0,1,1,0), c(1,0,0,2), c(1,0,0,2), c(0,3,3,0)) ## check if estimated and correct PAG are in agreement all(corr.pag == normal.pag @ amat) # TRUE all(corr.pag == anytime.pag @ amat) # FALSE all(corr.pag == adaptive.pag@ amat) # TRUE ij <- rbind(cbind(1:4,1:4), c(2,3), c(3,2)) all(corr.pag[ij] == anytime.pag @ amat[ij]) # TRUE ################################################## ## Joint Causal Inference Example ## Mooij et al. (2020), Fig. 43(a), p. 97 ################################################## # Encode MAG as adjacency matrix p <- 8 # total number of variables V <- c("Ca","Cb","Cc","X0","X1","X2","X3","X4") # 3 context variables, 5 system variables # amat[i,j] = 0 iff no edge btw i,j # amat[i,j] = 1 iff i *-o j # amat[i,j] = 2 iff i *-> j # amat[i,j] = 3 iff i *-- j amat <- rbind(c(0,2,2,2,0,0,0,0), c(2,0,2,0,2,0,0,0), c(2,2,0,0,2,2,0,0), c(3,0,0,0,0,0,2,0), c(0,3,3,0,0,3,0,2), c(0,0,3,0,2,0,0,0), c(0,0,0,3,0,0,0,2), c(0,0,0,0,2,0,3,0)) rownames(amat)<-V colnames(amat)<-V # Make use of d-separation oracle as "independence test" indepTest <- dsepAMTest suffStat<-list(g=amat,verbose=FALSE) # Derive PAG that represents the Markov equivalence class of the MAG with the FCI algorithm # (assuming no selection bias) fci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V, verbose=TRUE, selectionBias=FALSE) # Derive PAG with FCI-JCI, the Joint Causal Inference extension of FCI # (assuming no selection bias, and all three JCI assumptions) fcijci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V, verbose=TRUE, contextVars=c(1,2,3), jci="123", selectionBias=FALSE) # Report results cat('True MAG:\n') print(amat) cat('PAG output by FCI:\n') print(fci.pag@amat) cat('PAG output by FCI-JCI:\n') print(fcijci.pag@amat) # Read off causal features from the FCI PAG cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI PAG:\n') print(pag2anc(fci.pag@amat)) cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI PAG:\n') print(pag2edge(fci.pag@amat)) cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI PAG:\n') print(pag2conf(fci.pag@amat)) # Read off causal features from the FCI-JCI PAG cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI-JCI PAG:\n') print(pag2anc(fcijci.pag@amat)) cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI-JCI PAG:\n') print(pag2edge(fcijci.pag@amat)) cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI-JCI PAG:\n') print(pag2conf(fcijci.pag@amat)) ```

fci function