# Estimate Feasible Intervention Strategies Estimate a family of feasible intervention strategies for a continuous treatment (and optionally time-varying covariates). The method returns, for each intervention strategy, the corresponding feasible intervention values and a summary of overlap and positivity-violation diagnostics. ```r feasible( X, Anodes = NULL, Ynodes = NULL, Lnodes = NULL, Cnodes = NULL, abar = NULL, alpha = 0.95, grid.size = 0.5, tol = 1e-2, left.boundary = NULL, right.boundary = NULL, screen = FALSE, survival = FALSE, d.method = c("hazardbinning", "binning", "parametric", "hal_density"), verbose = TRUE, ... ) ``` ## Arguments - `X`: A data frame containing all nodes in temporal order. Columns must include the treatment, outcome, covariate and censoring nodes specified in `Anodes`, `Ynodes`, `Lnodes`, and `Cnodes`. - `Anodes`: Character vector giving the column names in `X` of the (possibly time-varying) treatment nodes. These define the treatment history. - `Ynodes`: Character vector giving the column names in `X` of the outcome nodes. At least one outcome node must be specified, and all of them must occur after the first treatment node in the column ordering of `X`. - `Lnodes`: Optional character vector of confounder nodes. May be `NULL` if there are no such nodes. - `Cnodes`: Optional character vector of censoring (or competing event) nodes. May be `NULL` if there is no censoring. - `abar`: Numeric vector or matrix specifying the target interventions. * If a **vector**, each element defines a static intervention that sets the treatment to that value at **all** time points. In this case, each strategy corresponds to a single scalar target value. * If a **matrix**, rows index intervention strategies and columns index time points (one column per element in `Anodes`). Then `abar[k, t]` is the target treatment value for strategy `k` at time `t`. The argument must be numeric; `NULL` is not allowed. - `alpha`: Numeric scalar in `(0, 1)` controlling the density-truncation level. For each observation and time point, the method finds the smallest density threshold $f_\alpha$ such that at most `alpha` of the total mass lies above this threshold. Cells with density below $f_\alpha$ are treated as infeasible . Default is `0.95`. - `grid.size`: Positive numeric scalar giving the spacing of the grid used to approximate the treatment density. If `NULL`, no internal grid is constructed and `abar` itself is used as the grid of evaluation points (this is only allowed when `abar` is a vector). Default is `0.5`. - `tol`: Non-negative numeric tolerance used when combining `abar` with the grid. Points closer than `tol` to an element of `abar` are considered duplicates and are dropped from the internal grid before merging with `abar`. Default is `1e-2`. - `left.boundary`: Optional numeric scalar setting the left boundary of the grid used to approximate the treatment density. If `NULL`, the minimum of the observed treatment values and `abar` is used. - `right.boundary`: Optional numeric scalar setting the right boundary of the density grid. If `NULL`, the maximum of the observed treatment values and `abar` is used. - `screen`: Logical; if `TRUE`, use variable-screening (via internal functions from `CICI`) for the treatment models, otherwise use the full treatment model formulae. Default is `FALSE`. - `survival`: Logical; indicates whether the outcome nodes correspond to a survival-type structure. Passed to the internal model-building function. Default is `FALSE`. - `d.method`: Character string specifying the density-estimation method used to estimate the conditional treatment density. Must be one of `"hazardbinning"`, `"binning"`, `"parametric"`, or `"hal_density"`. - `verbose`: Logical; if `TRUE`, print warnings about ignored arguments passed via `...` and other diagnostic messages. Default is `TRUE`. - ``...``: Additional arguments passed to the underlying density-estimation function determined by `d.method`. For example, these may include `SL.library` for Super Learner-based methods, or tuning parameters for specific density estimators. Arguments not recognised by the chosen `d.method` are silently ignored when `verbose = FALSE` and produce a warning when `verbose = TRUE`. ## Details The main steps of the algorithm are: 1. Model specification: Treatment models for each time point are constructed via helper routines from `CICI`. If `screen = TRUE`, a screening step updates the treatment formulas before density estimation (only recommended to address computational constraints). 2. Grid construction: A grid of treatment values, `query_abar`, is formed by combining: * observed treatment values in `X[, Anodes]`, * the target values in `abar`, and * a regular grid from `left.boundary` to `right.boundary` with spacing `grid.size` (when `grid.size` is not `NULL`). If `grid.size = NULL`, the grid is restricted to the unique values in `abar` (only allowed when `abar` is a vector). 3. Density estimation: For each time point, the conditional treatment density is evaluated on the grid for each observation using the specified `d.method`. The resulting matrices are normalised so that each row integrates to one over the grid (accounting for bin width). 4. Feasibility threshold: For each observation and time point, a density threshold $f_\alpha$ is computed such that the cumulative mass below the sorted densities first exceeds `1 - alpha`. Cells with density below $f_\alpha$ are flagged as infeasible . 5. Feasible mapping: For each grid cell with density below $f_\alpha$, the algorithm finds the closest grid cell with density at or above $f_\alpha$ (in terms of grid index) and maps its value to that cell. This defines a feasible intervention that avoids low-density regions. 6. Summary: For each time point $t$ and each intervention strategy (row of `abar`), the method collects: * the mean feasible value across individuals (column `Feasible`), * the proportion of cells below the density threshold (column `Low`, interpreted as `%infeasible` in the associated S3 methods), * the corresponding target value `Abar` at time $t$, and * the strategy index `Strategy`. These are combined into a data frame stored as the `"summary"` attribute of the returned object. Plotting and printing methods are available for visual and tabular diagnostics; see `plot.feasible`, `print.feasible`, and `summary.feasible`. ## Returns An object of class `"feasible"` with the following components: * `feasible`: a list of length equal to the number of strategies (rows of `abar`). Each element is a matrix with one column per time point, containing the feasible intervention values for that strategy and time point across observations. * `low_matrix`: a list of length equal to the number of time points. Each element is a logical matrix indicating, on the internal grid, which cells were marked as below the density threshold. The object additionally has a `"summary"` attribute, a data frame with at least the columns `time`, `Strategy`, `Abar`, `Feasible`, and `Low`, which is accessed and formatted by `summary.feasible` and `print.feasible`. ## See Also `plot.feasible`, `print.feasible`, `summary.feasible` ## Examples ```r data(EFV) Lnodes <- c("adherence.1","weight.1", "adherence.2","weight.2", "adherence.3","weight.3", "adherence.4","weight.4") Ynodes <- c("VL.0","VL.1","VL.2","VL.3","VL.4") Anodes <- c("efv.0","efv.1","efv.2","efv.3","efv.4") ## ------------------------------------------------------------------ ## Example 1: Hazard binning with default grid ## Static grid of targets (vector abar) over the full support of efv.* ## ------------------------------------------------------------------ abar_static <- seq(0, 10, by = 1) m_hazard <- feasible( X = EFV, Anodes = Anodes, Lnodes = Lnodes, Ynodes = Ynodes, d.method = "hazardbinning", # long computation, but appropriate abar = abar_static, grid.size = 0.5, left.boundary = 0, right.boundary = 10 ) ## Individual-level feasible values (one matrix per strategy): ## rows = individuals, columns = time points feasible_matrix <- m_hazard$feasible # pass on to gofrmula/sgf lapply(feasible_matrix, head) ## Inspect feasability of strategies m_hazard # see also ?print.feasible summary(m_hazard) # see also ?summary.feasible ## ------------------------------------------------------------------ ## Example 2: Parametric density, using abar as the grid ## Here grid.size = NULL, so only the target values are used as grid ## ------------------------------------------------------------------ abar_param <- seq(0, 10, by = 2) m_param <- feasible( X = EFV, Anodes = Anodes, Lnodes = Lnodes, Ynodes = Ynodes, # fast, but useful for reasonably symmetric distributions d.method = "parametric", abar = abar_param, grid.size = NULL, left.boundary = 0, right.boundary = 10 ) ## Inspect feasability of strategies m_param # see also ?print.feasible summary(m_param) # see also ?summary.feasible ## ------------------------------------------------------------------ ## Example 3: Matrix abar with non-constant strategies over time ## Each row is a strategy, each column corresponds to efv.0, ..., efv.4 ## ------------------------------------------------------------------ abar_matrix <- rbind( c(0, 2, 4, 6, 8), # strategy 1 c(9, 6, 2, 1, 0), # strategy 2 c(1, 3, 5, 7, 9) # strategy 3 ) m_matrix <- feasible( X = EFV, Anodes = Anodes, Lnodes = Lnodes, Ynodes = Ynodes, d.method = "parametric", abar = abar_matrix, grid.size = 1, left.boundary = 0, right.boundary = 10 ) ## Inspect feasability of strategies m_matrix # see also ?print.feasible summary(m_matrix) # see also ?summary.feasible ```

feasible function