# 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 fciPlus(suffStat, indepTest, alpha, labels, p, verbose=TRUE, selectionBias = TRUE, jci = c("0","1","12","123"), contextVars = NULL) ``` ## 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`. - `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`: logical indicating if progress of the algorithm should be printed. The default is true, which used to be hard coded previously. ## Details A (possibly much faster) variation of FCI (Fast Causal Inference). For details, please see the references, and also `fci`. ## 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. ## References T. Claassen, J. Mooij, and T. Heskes (2013). Learning Sparse Causal Models is not NP-hard. In **UAI 2013, Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence** ## Author(s) Emilija Perkovic, Markus Kalisch (kalisch@stat.math.ethz.ch ) and Joris Mooij. ## See Also `fci` for estimating a PAG using the FCI algorithm. ## Examples ```r ################################################## ## Example without latent variables ################################################## ## generate a random DAG ( p = 7 ) set.seed(42) p <- 7 myDAG <- randomDAG(p, prob = 0.4) ## find PAG using the FCI+ algorithm on "Oracle" suffStat <- list(C = cov2cor(trueCov(myDAG)), n = 10^9) m.fci <- fciPlus(suffStat, indepTest=gaussCItest, alpha = 0.9999, p=p) summary(m.fci) if (require(Rgraphviz)) { par(mfrow=c(1,2)) plot(myDAG) plot(m.fci) } ################################################## ## 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 <- fciPlus(suffStat, indepTest, alpha = 0.5, labels = V, # selectionBias=FALSE,verbose=TRUE) # Derive PAG with FCI+-JCI, the Joint Causal Inference extension of FCI # (assuming no selection bias, and all three JCI assumptions) # fcijci.pag <- fciPlus(suffStat, indepTest, alpha = 0.5, labels = V, # selectionBias=FALSE, contextVars=c(1,2,3), jci="123", verbose=TRUE) # 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('Ident. absence (-1) and presence (+1) of pairwise latent confounding from FCI+-JCI PAG:\n') #print(pag2conf(fcijci.pag@amat)) ```

fciPlus function