# Estimate an RFCI-PAG using the RFCI Algorithm Estimate an RFCI-PAG from observational data, using the RFCI-algorithm. ```r rfci(suffStat, indepTest, alpha, labels, p, skel.method = c("stable", "original", "stable.fast"), fixedGaps = NULL, fixedEdges = NULL, NAdelete = TRUE, m.max = Inf, rules = rep(TRUE, 10), conservative = FALSE, maj.rule = FALSE, numCores = 1, verbose = FALSE) ``` ## Arguments - `suffStat`: Sufficient statistics: List containing all necessary elements for the conditional independence decisions in the function `indepTest`. - `indepTest`: Predefined 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 containing all relevant elements for the conditional independence decisions. The return value of `indepTest` is the p-value of the test for conditional independence. - `alpha`: significance level (number 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`. - `fixedGaps`: A 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`: A 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. - `rules`: Logical vector of length 10 indicating which rules should be used when directing edges. The order of the rules is taken from Zhang (2009). - `conservative`: Logical indicating if the unshielded triples should be checked for ambiguity after the skeleton has been found, similar to the conservative PC algorithm. - `maj.rule`: Logical indicating if the unshielded triples should be checked for ambiguity after the skeleton has been found using a majority rule idea, which is less strict than the conservative. - `numCores`: Specifies the number of cores to be used for parallel estimation of `skeleton`. - `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 rather similar to fci . However, it does not compute any Possible-D-SEP sets and thus does not make tests conditioning on subsets of Possible-D-SEP. This makes RFCI much faster than FCI. The orientation rules for v-structures and rule 4 were modified in order to produce an RFCI-PAG which, in the oracle version, is guaranteed to have the correct ancestral relationships. The first part of the RFCI algorithm is analogous to the PC and FCI algorithm. It starts with a complete undirected graph and estimates an initial skeleton using the function `skeleton`, 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. The FCI algorithm performs independence tests conditioning on subsets of Possible-D-SEP to remove those edges. Since this procedure is computationally infeasible, the RFCI algorithm uses a different approach to remove some of those superfluous edges before orienting the v-structures and the discriminating paths in orientation rule 4. Before orienting the v-structures, we perform the following additional conditional independence tests. For each unshielded triple a-b-c in the initial skeleton, we check if both a and b and b and c are conditionally dependent given the separating of a and c (sepset(a,c)). These conditional dependencies may not have been checked while estimating the initial skeleton, since sepset(a,c) does not need to be a subset of the neighbors of a nor of the neighbors of c. If both conditional dependencies hold and b is not in the sepset(a,c), the triple is oriented as a v-structure a->b<-c. On the other hand, if an additional conditional independence relationship may be detected, say a is independent from b given the sepset(a,c), the edge between a and c is removed from the graph and the set responsible for that is saved in sepset(a,b). The removal of an edge can destroy or create new unshielded triples in the graph. To solve this problem we work with lists (for details see Colombo et al., 2012). Before orienting discriminating paths, we perform the following additional conditional independence tests. For each triple a <-* b o- *c with a -> c, the algorithm searches for a discriminating path p = for b of minimal length, and checks that the vertices in every consecutive pair (f1,f2) on p are conditionally dependent given all subsets of $sepset(d,c) \ {f1, f2}$ . If we do not find any conditional independence relationship, the path is oriented as in rule (R4). If one or more conditional independence relationships are found, the corresponding edges are removed, their minimal separating sets are stored. Conservative RFCI can be computed if the argument of `conservative` is `TRUE`. After the final skeleton is computed and the additional local tests on all unshielded triples, as described above, have been done, 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 RFCI 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 RFCI 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). The implementation uses the stabilized skeleton `skeleton`, which produces an initial order-independent skeleton. The final skeleton and edge orientations can still be order-dependent, see Colombo and Maathuis (2014). ## 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. ## See Also `fci` and `fciPlus` for estimating a PAG using the FCI algorithm; `skeleton` for estimating an initial skeleton using the RFCI algorithm; `pc` for estimating a CPDAG using the PC algorithm; `gaussCItest`, `disCItest`, `binCItest` and `dsepTest` as examples for `indepTest`. ## Author(s) Diego Colombo and Markus Kalisch (kalisch@stat.math.ethz.ch ). ## 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 RFCI algorithm suffStat <- list(C = cov2cor(trueCov(myDAG)), n = 10^9) indepTest <- gaussCItest res <- rfci(suffStat, indepTest, alpha = 0.9999, p=p, verbose=TRUE) ##################################################% -------------- ## Example with hidden variables ## Zhang (2008), Fig. 6, p.1882 ################################################## ## create the DAG : V <- LETTERS[1:5] edL <- setNames(vector("list", length = 5), 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)) ## and leave edL[[ 5 ]] empty g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed") if (require(Rgraphviz)) plot(g) ## define the latent variable L <- 1 ## 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) ## find PAG with RFCI algorithm ## as dependence "oracle", we use the true correlation matrix in ## gaussCItest() with a large "virtual sample size" and a large alpha : rfci.pag <- rfci(suffStat = list(C = true.corr, n = 10^9), indepTest = gaussCItest, alpha = 0.9999, labels = V[-L], 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 that estimated and correct PAG are in agreement: stopifnot(corr.pag == rfci.pag@amat) ```

rfci function