# Hierarchical structure group tests in linear model Computes confidence intervals for the l1-norm of groups of linear regression coefficients in a hierarchical clustering tree. ```r clusterGroupBound(x, y, method = "average", dist = as.dist(1 - abs(cor(x))), alpha = 0.05, eps = 0.1, hcloutput, nsplit = 11, s = min(10, ncol(x) - 1), silent = FALSE, setseed = TRUE, lpSolve = TRUE) ``` ## Arguments - `x`: numeric design matrix of the regression $n * p$ with $p$ columns for $p$ predictor variables and $n$ rows corresponding to $n$ observations. - `y`: numeric response variable of length $n$. - `method`: a `character` string; the method used for constructing the hierarchical clustering tree (default: `"average"` for average linkage ) via `hclust`. Alternatively, you can provide your own hierarchical clustering through the optional argument `hcloutput`. - `dist`: a distance matrix can be specified on which the hierarchical clustering will be based (see `dist`). The default option is that the distance between variables will be calculated as 1 less the absolute correlation matrix. Alternatively, you can provide your own hierarchical clustering through the optional argument `hcloutput`. - `alpha`: numeric level in $(0, 1)$ at which the test / confidence intervals are to be constructed. - `eps`: a level of eps*alpha is used and the values of different splits are aggregated using the (1-eps) quantile. See reference below for more details. - `hcloutput`: optionally, the value of a `hclust()` call. If it is provided, the arguments `dist` and `method` are ignored. - `nsplit`: the number of data splits used. - `s`: the dimensionality of the projection that is used. Lower values lead to faster computation and if $n > 50$, then `s` is set to 50 if left unspecified, to avoid lengthy computations. - `silent`: logical enabling progress output. - `setseed`: a logical; if this is true (recommended), then the same random seeds are used for all groups, which makes the confidence intervals simultaneously valid over all groups of variables tested. - `lpSolve`: logical; only set it to false if `lpSolve()` is not working on the current machine: setting it to false will result in much slower computations; only use on small problems. ## Returns Returns a list with components - **groupNumber**: The index of the group tested in the original hierarchical clustering tree - **members**: A list containing the variables that belong into each testes group - **noMembers**: A vector containing the number of members in each group - **lowerBound**: The lower bound on the l1-norm in each group - **position**: The position on the x-axis of each group (used for plotting) - **leftChild**: Gives the index of the group that corresponds to the left child node in the tested tree (negative values correspond to leaf nodes) - **rightChild**: Same as `leftCHild` for the right child of each node - **isLeaf**: Logical vector. Is `TRUE` for a group if it is a leaf node in the tested tree or if both child nodes have a zero lower bound on their group l1-norm ## References Meinshausen, N. (2015); JRSS B, see `groupBound`. ## Author(s) Nicolai Meinshausen ## See Also Use `groupBound` to compute the lower bound for selected groups of variables whereas you use this `clusterGroupBound` to test all groups in a hierarchical clustering tree. ## Examples ```r ## Create a regression problem with correlated design (n = 10, p = 3): ## a block of size 2 and a block of size 1, within-block correlation is 0.99 set.seed(29) p <- 3 n <- 10 Sigma <- diag(p) Sigma[1,2] <- Sigma[2,1] <- 0.99 x <- matrix(rnorm(n * p), nrow = n) %*% chol(Sigma) ## Create response with active variable 1 beta <- rep(0, p) beta[1] <- 5 y <- as.numeric(x %*% beta + rnorm(n)) out <- clusterGroupBound(x, y, nsplit = 4) ## use larger value for nsplit! ## Plot and print the hierarchical group-test plot(out) print(out) out$members out$lowerBound ```

clusterGroupBound.rd function