# Defining terms in CGAIM formula Functions used to define terms within a `cgaim` formula. `g` defines an index with ridge function and `s` a smooth covariate. ```r g(..., label = NULL, acons = list(), Cmat = NULL, bvec = 0, fcons = NULL, s_opts = list()) s(x, fcons = NULL, s_opts = list()) ``` ## Arguments - `...`: Variables entering the index. May include vectors and matrices. - `label`: Character (or any object that can coerced into one) labeling the index. By default, named after the first variable in `...`. - `acons`: A list of character naming common constraints to be applied to the index weights `alpha`. See `build_constraints` for allowed constraints. - `Cmat`: A constraint matrix for alpha coefficients. Number of columns must match the number of variables in the index. - `bvec`: Numeric vector of constraint bounds. Recycled if necessary. - `fcons`: The type of shape constraint to be applied on the smooth function. See details. - `s_opts`: A named list of options to be passed to the smoothing of ridge functions. Depends on the method used to smooth additive models. See details. - `x`: Covariate on which the smooth is applied. ## Returns A matrix containing the variables passed in `...` with additional attributes: - **fcons**: The shape constraint for smoothing. - **s_opts**: Arguments passed to the smoothing function. - **label**: The label of the term. The following attributes result from a call to `g` only: - **term**: The terms in the index. - **nterms**: The number of variables in the index. - **Cmat**: The constraint matrix for alpha coefficients. - **bvec**: The associated boundary vector. ## Details These functions define nonlinear terms in the formula, with `g` defining an index created from a collection of terms passed through the `...` argument while `s` is applied to a single variable, similarly to `s` in `mgcv`. For indices, `g` allows the definition of constraints applied to the index only. This is a convenient alternative to passing the whole constraint matrix `Cmat` in `cgaim`. Constraints can be defined by a prespecified matrix through the argument `Cmat` or through the argument `acons` for common constraints (see `build_constraints`). Note that any provided `Cmat` must match the total number of variables in `...`, including potential matrix expansion and factors. Both `Cmat` and `acons` can be passed to the function, which will bind them internally. Both `g` and `s` allow the definition of shape constraints for the smooth. Eight shape-constraints are currently available: monotone increasing (`fcons = "inc"`), monotone decreasing (`fcons = "dec"`), convex (`fcons = "cvx"`), concave (`fcons = "ccv"`), increasing and convex(`fcons = "inccvx"`), decreasing and convex (`fcons = "deccvx"`), increasing and concave (`fcons = "incccv"`), decreasing and concave (`fcons = "decccv"`). Smoothing can be controlled by the `s_opts` parameter. It is a list of argument depends on the method used for smoothing. See `s` for `smooth_method = "scam"`. For `smooth_method = "cgam"`, the parameters allowed may vary according to the shape-constraint chosen. The full list can be found in `cgam`, but only the constraints beginning with `s.` are allowed for now. No parameter are necessary when `smooth_method = "scar"` (see `scar`). ## Examples ```r ## Simulate some data n <- 200 x1 <- rnorm(n) x2 <- rnorm(n) x3 <- rnorm(n) x4 <- rnorm(n) mu <- 4 * exp(8 * x1) / (1 + exp(8 * x1)) + exp(x3) y <- mu + rnorm(n) df1 <- data.frame(y, x1, x2, x3, x4) ## Fit an unconstrained the model ans <- cgaim(y ~ g(x1, x2) + g(x3, x4), data = df1) ## Fit constrained model ans2 <- cgaim(y ~ g(x1, x2, acons = list(monotone = -1)) + g(x3, x4, fcons = "cvx"), data = df1) ## Pass constraint matrices instead ans3 <- cgaim(y ~ g(x1, x2, Cmat = -diff(diag(2))) + g(x3, x4, fcons = "cvx"), data = df1) ## Label indices ans4 <- cgaim(y ~ g(x1, x2, label = "foo") + g(x3, x4, label = "bar"), data = df1) ``` ## See Also `cgaim` for fitting the CGAIM, `build_constraints` for built-in constraint matrices.