# Calculate P-values Based on Multi-Splitting Approach Calculate p-values and confidence intervals based on the multi-splitting approach ```r multi.split(x, y, B = 100, fraction = 0.5, ci = TRUE, ci.level = 0.95, model.selector = lasso.cv, classical.fit = lm.pval, classical.ci = lm.ci, parallel = FALSE, ncores = getOption("mc.cores", 2L), gamma = seq(ceiling(0.05 * B) / B, 1 - 1 / B, by = 1 / B), args.model.selector = NULL, args.classical.fit = NULL, args.classical.ci = NULL, return.nonaggr = FALSE, return.selmodels = FALSE, repeat.max = 20, verbose = FALSE) ``` ## Arguments - `x`: numeric design matrix (without intercept). - `y`: numeric response vector. - `B`: the number of sample-splits, a positive integer. - `fraction`: a number in $(0,1)$, the fraction of data used at each sample split for the model selection process. The remaining data is used for calculating the p-values. - `ci`: logical indicating if a confidence interval should be calculated for each parameter. - `ci.level`: (if `ci` is true:) a number in $(0,1)$, typically close to 1, the desired coverage level of the confidence intervals. - `model.selector`: a `function` to perform model selection, with default `lasso.cv`. The function must have at least two arguments, `x` (the design matrix) and `y` (the response vector). Return value is the index vector of selected columns. See `lasso.cv` and `lasso.firstq` for an example. Additional arguments can be passed via `args.model.selector`. - `classical.fit`: a `function` to calculate (classical) p-values. Default is `lm.pval`. The function must have at least two arguments, `x` (the design matrix) and `y` (the response vector), and return the vector of p-values. See `lm.pval` for an example. Additional arguments can be passed through `args.classical.fit`. - `classical.ci`: a `function` to calculate (classical) confidence intervals. Default is `lm.ci`. The function must have at least 3 arguments, `x` (the design matrix), `y` (the response vector) and `level` (the coverage level), and return the matrix of confidence intervals. See `lm.ci` for an example. Additional arguments can be passed through `args.classical.ci`. - `parallel`: logical indicating if parallelization via `mclapply` should be used. - `ncores`: number of cores used for parallelization as `mc.cores` in `mclapply()`. - `gamma`: vector of gamma-values. In case gamma is a scalar, the value $Q_j$ instead of $P_j$ is being calculated (see reference below). - `args.model.selector`: named `list` of further arguments for function `model.selector`. - `args.classical.fit`: named `list` of further arguments for function `classical.fit`. - `args.classical.ci`: named `list` of further arguments for function `classical.ci`. - `return.nonaggr`: `logical` indicating if the unadjusted p-values be returned. - `return.selmodels`: `logical` indicating if the selected models (at each split) should be returned. Necessary for the `clusterGroupTest()` part of the result. - `repeat.max`: positive integer indicating the maximal number of split trials. Should not matter in regular cases, but necessary to prevent infinite loops in borderline cases. - `verbose`: should information be printed out while computing? (logical). ## Returns - **pval.corr**: Vector of multiple testing corrected p-values. - **gamma.min**: Value of gamma where minimal p-values was attained. - **clusterGroupTest**: Function to perform groupwise tests based on hierarchical clustering. You can either provide a distance matrix and clustering method or the output of hierarchical clustering from the function `hclust` as for `clusterGroupBound`. P-values are adjusted for multiple testing. ## References Meinshausen, N., Meier, L. and , P. (2009) P-values for high-dimensional regression. **Journal of the American Statistical Association** 104 , 1671--1681. Mandozzi, J. and , P. (2015) A sequential rejection testing method for high-dimensional regression with correlated variables. To appear in the International Journal of Biostatistics. Preprint arXiv:1502.03300 ## Author(s) Lukas Meier, Ruben Dezeure, Jacopo Mandozzi ## See Also `lasso.cv`, `lasso.firstq`; `lm.pval`, `lm.ci`. ## Examples ```r n <- 40 # a bit small, to keep example "fast" p <- 256 x <- matrix(rnorm(n * p), nrow = n, ncol = p) y <- x[,1] * 2 + x[,2] * 2.5 + rnorm(n) ## Multi-splitting with lasso.firstq as model selector function ## 'q' must be specified fit.multi <- multi.split(x, y, model.selector = lasso.firstq, args.model.selector = list(q = 10)) fit.multi head(fit.multi$pval.corr, 10) ## the first 10 p-values ci. <- confint(fit.multi) head(ci.) # the first 6 stopifnot(all.equal(ci., with(fit.multi, cbind(lci, uci)), check.attributes=FALSE)) ## Use default 'lasso.cv' (slower!!) -- incl cluster group testing: system.time(fit.m2 <- multi.split(x, y, return.selmodels = TRUE))# 9 sec (on "i7") head(fit.m2$pval.corr) ## the first 6 p-values head(confint(fit.m2)) ## the first 6 95% conf.intervals ## Now do clustergroup testing clGTst <- fit.m2$clusterGroupTest names(envGT <- environment(clGTst))# about 14 if(!interactive()) # if you are curious (and advanced): print(ls.str(envGT), max = 0) stopifnot(identical(clGTst, envGT$clusterGroupTest)) ccc <- clGTst() str(ccc) ccc$hh # the clustering has.1.or.2 <- sapply(ccc$clusters, function(j.set) any(c(1,2) %in% j.set)) ccc$pval[ has.1.or.2] ## all very small ccc$pval[!has.1.or.2] ## all 1 ```

multi.split function