# Bootstrap a Fit Smooth Bootstraps a fit nonparametric regression model to form confidence intervals (BCa or percentile) and standard error estimates. ```r ## S3 method for class 'ss' boot(object, statistic, ..., R = 9999, level = 0.95, bca = TRUE, method = c("cases", "resid", "param"), fix.lambda = TRUE, cov.mat = FALSE, boot.dist = FALSE, verbose = TRUE, parallel = FALSE, cl = NULL) ## S3 method for class 'sm' boot(object, statistic, ..., R = 9999, level = 0.95, bca = TRUE, method = c("cases", "resid", "param"), fix.lambda = TRUE, fix.thetas = TRUE, cov.mat = FALSE, boot.dist = FALSE, verbose = TRUE, parallel = FALSE, cl = NULL) ## S3 method for class 'gsm' boot(object, statistic, ..., R = 9999, level = 0.95, bca = TRUE, method = c("cases", "resid", "param"), fix.lambda = TRUE, fix.thetas = TRUE, cov.mat = FALSE, boot.dist = FALSE, verbose = TRUE, parallel = FALSE, cl = NULL) ``` ## Arguments - `object`: a fit from `ss` (smoothing spline), `sm` (smooth model), or `gsm` (generalized smooth model) - `statistic`: a function to compute the statistic (see Details) - `...`: additional arguments to `statistic` function (optional) - `R`: number of bootstrap resamples used to form bootstrap distribution - `level`: confidence level for bootstrap confidence intervals - `bca`: logical indicating whether to calculate BCa (default) or percentile intervals - `method`: resampling method used to form bootstrap distribution - `fix.lambda`: logical indicating whether the smoothing parameter should be fixed (default) or re-estimated for each bootstrap sample - `fix.thetas`: logical indicating whether the "extra" smoothing parameters should be fixed (default) or re-estimated for each bootstrap sample. Only applicable to `sm` and `gsm` objects with multiple penalized terms. - `cov.mat`: logical indicating whether the bootstrap estimate of the covariance matrix should be returned - `boot.dist`: logical indicating whether the bootstrap distribution should be returned - `verbose`: logical indicating whether the bootstrap progress bar should be printed - `parallel`: logical indicating if the `parallel` package should be used for parallel computing (of the bootstrap distribution). Defaults to FALSE, which implements sequential computing. - `cl`: cluster for parallel computing, which is used when `parallel = TRUE`. Note that if `parallel = TRUE` and `cl = NULL`, then the cluster is defined as `makeCluster(detectCores())`. ## Details The `statistic` function must satisfy the following two requirements: (1) the first input must be the `object` of class `ss`, `sm`, or `gsm` (2) the output must be a scalar or vector calculated from the `object` In most applications, the `statistic` function will be the model predictions at some user-specified `newdata`, which can be passed to `statistic` using the `...` argument. If `statistic` is not provided, then the function is internally defined to be the model predictions at an equidistance sequence (for `ss` objects) or the training data predictor scores (for `sm` and `gsm` objects). ## Returns Produces an object of class 'boot.ss', 'boot.sm', or 'boot.gsm', with the following elements: - **t0**: Observed statistic, computed using `statistic(object, ...)` - **se**: Bootstrap estimate of the standard error - **bias**: Bootstrap estimate of the bias - **cov**: Bootstrap estimate of the covariance (if `cov.mat = TRUE`) - **ci**: Bootstrap estimate of the confidence interval - **boot.dist**: Bootstrap distribution of statistic (if `boot.dist = TRUE`) - **bias.correct**: Bias correction factor for BCa confidence interval. - **acceleration**: Acceleration parameter for BCa confidence interval. The output list also contains the elements `object`, `R`, `level`, `bca`, `method`, `fix.lambda`, and `fix.thetas`, all of which are the same as the corresponding input arguments. ## References Davison, A. C., & Hinkley, D. V. (1997). **Bootstrap Methods and Their Application**. Cambridge University Press. tools:::Rd_expr_doi("10.1017/CBO9780511802843") Efron, B., & Tibshirani, R. J. (1994). **An Introduction to the Boostrap**. Chapman & Hall/CRC. tools:::Rd_expr_doi("10.1201/9780429246593") Moulton, L. H., & Zeger, S. L. (1991). Bootstrapping generalized linear models. **Computational Statistics & Data Analysis, 11**(1), 53-63. tools:::Rd_expr_doi("10.1016/0167-9473(91)90052-4") ## Author(s) Nathaniel E. Helwig ## Note For `gsm` objects, requesting `method = "resid"` uses a variant of the one-step technique described in Moulton and Zeger (1991), which forms the bootstrap estimates of the coefficients without refitting the model. As a result, when bootstrapping `gsm` objects with `method = "resid"`: (1) it is necessary to set `fix.lambda = TRUE` and `fix.thetas = TRUE` (2) any logical `statistic` must depend on the model `coefficients`, e.g., through the model predictions. ## See Also `ss` for fitting "ss" (smoothing spline) objects `sm` for fitting "sm" (smooth model) objects `gsm` for fitting "gsm" (generalized smooth model) objects ## Examples ```r ## Not run: ########## EXAMPLE 1 ########## ### smoothing spline # generate data set.seed(1) n <- 100 x <- seq(0, 1, length.out = n) fx <- 2 + 3 * x + sin(2 * pi * x) y <- fx + rnorm(n, sd = 0.5) # fit smoothing spline ssfit <- ss(x, y, nknots = 10) # nonparameteric bootstrap cases set.seed(0) boot.cases <- boot(ssfit) # nonparameteric bootstrap residuals set.seed(0) boot.resid <- boot(ssfit, method = "resid") # parameteric bootstrap residuals set.seed(0) boot.param <- boot(ssfit, method = "param") # plot results par(mfrow = c(1, 3)) plot(boot.cases, main = "Cases") plot(boot.resid, main = "Residuals") plot(boot.param, main = "Parametric") ########## EXAMPLE 2 ########## ### smooth model # generate data set.seed(1) n <- 100 x <- seq(0, 1, length.out = n) fx <- 2 + 3 * x + sin(2 * pi * x) y <- fx + rnorm(n, sd = 0.5) # fit smoothing spline smfit <- sm(y ~ x, knots = 10) # define statistic (to be equivalent to boot.ss default) newdata <- data.frame(x = seq(0, 1, length.out = 201)) statfun <- function(object, newdata) predict(object, newdata) # nonparameteric bootstrap cases set.seed(0) boot.cases <- boot(smfit, statfun, newdata = newdata) # nonparameteric bootstrap residuals set.seed(0) boot.resid <- boot(smfit, statfun, newdata = newdata, method = "resid") # parameteric bootstrap residuals (R = 99 for speed) set.seed(0) boot.param <- boot(smfit, statfun, newdata = newdata, method = "param") # plot results par(mfrow = c(1, 3)) plotci(newdata$x, boot.cases$t0, ci = boot.cases$ci, main = "Cases") plotci(newdata$x, boot.resid$t0, ci = boot.resid$ci, main = "Residuals") plotci(newdata$x, boot.param$t0, ci = boot.param$ci, main = "Parametric") ########## EXAMPLE 3 ########## ### generalized smooth model # generate data set.seed(1) n <- 100 x <- seq(0, 1, length.out = n) fx <- 2 + 3 * x + sin(2 * pi * x) y <- fx + rnorm(n, sd = 0.5) # fit smoothing spline gsmfit <- gsm(y ~ x, knots = 10) # define statistic (to be equivalent to boot.ss default) newdata <- data.frame(x = seq(0, 1, length.out = 201)) statfun <- function(object, newdata) predict(object, newdata) # nonparameteric bootstrap cases set.seed(0) boot.cases <- boot(gsmfit, statfun, newdata = newdata) # nonparameteric bootstrap residuals set.seed(0) boot.resid <- boot(gsmfit, statfun, newdata = newdata, method = "resid") # parameteric bootstrap residuals set.seed(0) boot.param <- boot(gsmfit, statfun, newdata = newdata, method = "param") # plot results par(mfrow = c(1, 3)) plotci(newdata$x, boot.cases$t0, ci = boot.cases$ci, main = "Cases") plotci(newdata$x, boot.resid$t0, ci = boot.resid$ci, main = "Residuals") plotci(newdata$x, boot.param$t0, ci = boot.param$ci, main = "Parametric") ## End(Not run) ```

boot function