# Invariant Causal Prediction with hidden variables Confidence intervals for causal effects in a regression setting with possible confounders. ```r hiddenICP(X, Y, ExpInd, alpha = 0.1, mode = "asymptotic", intercept=FALSE) ``` ## Arguments - `X`: A matrix (or data frame) with the predictor variables for all experimental settings - `Y`: The response or target variable of interest. Can be numeric for regression or a factor with two levels for binary classification. - `ExpInd`: Indicator of the experiment or the intervention type an observation belongs to. Can be a numeric vector of the same length as `Y` with `K` unique entries if there are `K` different experiments (for example entry 1 for all observational data and entry 2 for intervention data). Can also be a list, where each element of the list contains the indices of all observations that belong to the corresponding grouping in the data (for example two elements: first element is a vector that contains indices of observations that are observational data and second element is a vector that contains indices of all observations that are of interventional type). - `alpha`: The level of the test procedure. Use the default `alpha`=0.1 to obtain 90% confidence intervals. - `mode`: Currently only mode "asymptotic" is implemented; the argument is thus in the current version without effect. - `intercept`: Boolean variable; if `TRUE`, an intercept is added to the design matrix (but coefficients are returned without intercept term). ## Returns A list with elements - **betahat**: The point estimator for the causal effects - **maximinCoefficients**: The value in the confidence interval for each variable effects that is closest to 0. Is hence non-zero for variables with significant effects. - **ConfInt**: The matrix with confidence intervals for the causal coefficient of all variables. First row is the upper bound and second row the lower bound. - **pvalues**: The p-values of all variables. - **colnames**: The column-names of the predictor variables. - **alpha**: The chosen level. ## References none yet. ## Author(s) Nicolai Meinshausen ## See Also `ICP` for reconstructing the parents of a variable under arbitrary interventions on all other variables (but no hidden variables). See package "backShift" for constructing point estimates of causal cyclic models in the presence of hidden variables (again under shift interventions) . ## Examples ```r ########################################## ####### 1st example: ####### Simulate data with interventions set.seed(1) ## sample size n n <- 2000 ## 4 predictor variables p <- 4 ## simulate as independent Gaussian variables X <- matrix(rnorm(n*p),nrow=n) ## divide data into observational (ExpInd=1) and interventional (ExpInd=2) ExpInd <- c(rep(1,n/2),rep(2,n/2)) ## for interventional data (ExpInd==2): change distribution nI <- sum(ExpInd==2) X[ExpInd==2,] <- X[ExpInd==2,] + matrix( 5*rt( nI*p,df=3),ncol=p) ## add hidden variables W <- rnorm(n) * 5 X <- X + outer(W, rep(1,4)) ## first two variables are the causal predictors of Y beta <- c(1,1,0,0) ## response variable Y Y <- as.numeric(X%*%beta - 2*W + rnorm(n)) ####### Compute "hidden Invariant Causal Prediction" Confidence Intervals icp <- hiddenICP(X,Y,ExpInd,alpha=0.01) print(icp) ###### Print point estimates and points in the confidence interval closest to 0 print(icp$betahat) print(icp$maximinCoefficients) cat("true coefficients are:", beta) #### compare with coefficients from a linear model cat("coefficients from linear model:") print(summary(lm(Y ~ X-1))) ########################################## ####### 2nd example: ####### Simulate model X -> Y -> Z with hidden variables, trying to ###### estimate causal effects from (X,Z) on Y set.seed(1) ## sample size n n <- 10000 ## simulate as independent Gaussian variables W <- rnorm(n) noiseX <- rnorm(n) noiseY <- rnorm(n) noiseZ <- rnorm(n) ## divide data into observational (ExpInd=1) and interventional (ExpInd=2) ExpInd <- c(rep(1,n/2),rep(2,n/2)) noiseX[ which(ExpInd==2)] <- noiseX[ which(ExpInd==2)] * 5 noiseZ[ which(ExpInd==2)] <- noiseZ[ which(ExpInd==2)] * 3 ## simulate equilibrium data beta <- -0.5 alpha <- 0.9 X <- noiseX + 3*W Y <- beta* X + noiseY + 3*W Z <- alpha*Y + noiseZ ####### Compute "Invariant Causal Prediction" Confidence Intervals icp <- hiddenICP(cbind(X,Z),Y,ExpInd,alpha=0.1) print(icp) ###### Print/plot/show summary of output (truth here is (beta,0)) print(signif(icp$betahat,3)) print(signif(icp$maximinCoefficients,3)) cat("true coefficients are:", beta,0) #### compare with coefficients from a linear model cat("coefficients from linear model:") print(summary(lm(Y ~ X + Z -1))) ```

hiddenICP function