R/bootEff-fun.R

In semEff: Automatic Calculation of Effects for Piecewise Structural Equation Models

#' @title Bootstrap Effects
#' @description Bootstrap model effects (standardised coefficients) and optional
#'   SEM correlated errors.
#' @param mod A fitted model object, or a list or nested list of such objects.
#'   Alternatively, a `"psem"` object from
#'   [`piecewiseSEM::psem()`](https://rdrr.io/cran/piecewiseSEM/man/psem.html).
#'   If model lists are unnamed, response variable names will be used.
#' @param R Number of bootstrap resamples to generate.
#' @param seed Seed for the random number generator. If not provided, a random
#'   five-digit integer is used (see Details).
#' @param type The type of bootstrapping to perform. Can be `"nonparametric"`
#'   (default), `"parametric"`, or `"semiparametric"` (the last two currently
#'   only for mixed models, via [lme4::bootMer()]).
#' @param ran.eff For nonparametric bootstrapping of mixed models, the name of
#'   the (highest-level) random effect to resample (see Details).
#' @param cor.err Optional, names of SEM correlated errors to be bootstrapped
#'   (ignored if `mod` is a `"psem"` object). Should be of the form: `c("var1 ~~
#'   var2", "var3 ~~ var4", ...)` (spaces optional), using model/response
#'   variable names.
#' @param catch.err Logical, should errors generated during model fitting or
#'   estimation be caught and `NA` returned for estimates? If `FALSE`, any such
#'   errors will cause the function to exit.
#' @param parallel The type of parallel processing to use. Can be one of
#'   `"snow"`, `"multicore"`, or `"no"` (for none).
#' @param ncpus Number of system cores to use for parallel processing. If `NULL`
#'   (default), all available cores are used.
#' @param cl Optional cluster to use if `parallel = "snow"`. If `NULL`
#'   (default), a local cluster is created using the specified number of cores.
#' @param bM.arg A named list of any additional arguments to [lme4::bootMer()].
#' @param ... Arguments to [stdEff()].
#' @details `bootEff()` uses [boot::boot()] (primarily) to bootstrap
#'   standardised effects from a fitted model or list of models (calculated
#'   using [stdEff()]). Bootstrapping is typically nonparametric, i.e. model
#'   effects are calculated from data where the rows have been randomly sampled
#'   with replacement. 10,000 such resamples should provide accurate coverage
#'   for confidence intervals in most situations, with fewer sufficing in some
#'   cases. To ensure that data is resampled in the same way across individual
#'   bootstrap operations within the same run (e.g. models in a list), the same
#'   seed is set per operation, with the value saved as an attribute to the
#'   matrix of bootstrapped values (for reproducibility). The seed can either be
#'   user-supplied or a randomly-generated five-digit number (default), and is
#'   always re-initialised on exit (i.e. `set.seed(NULL)`).
#'
#'   Where `weights` are specified, bootstrapped effects will be a weighted
#'   average across the set of candidate models for each response variable,
#'   calculated after each model is first refit to the resampled dataset
#'   (specifying `weights = "equal"` will use a simple average instead – see
#'   [avgEst()]). If no weights are specified and `mod` is a nested list of
#'   models, the function will throw an error, as it will be expecting weights
#'   for a presumed model averaging scenario. If instead the user wishes to
#'   bootstrap each individual model, they should recursively apply the function
#'   using [rMapply()] (remember to set a seed).
#'
#'   Where names of response variables with correlated errors are specified to
#'   `cor.err`, the function will also return bootstrapped Pearson correlated
#'   errors (weighted residuals) for those models. If `weights` are supplied and
#'   `mod` is a nested list, residuals will first be averaged across candidate
#'   models. If any two models (or candidate sets) with correlated errors were
#'   fit to different subsets of data observations, both models/sets are first
#'   refit to data containing only the observations in common.
#'
#'   For nonparametric bootstrapping of mixed models, resampling should occur at
#'   the group-level, as individual observations are not independent. The name
#'   of the random effect to resample must be supplied to `ran.eff`. For nested
#'   random effects, this should be the highest-level group (Davison & Hinkley,
#'   1997; Ren et al., 2010). This form of resampling will result in datasets of
#'   different sizes if observations are unbalanced across groups; however this
#'   should not generally be an issue, as the number of independent units
#'   (groups), and hence the 'degrees of freedom', remains
#'   [unchanged](https://stats.stackexchange.com/questions/46965/bootstrapping-unbalanced-clustered-data-non-parametric-bootstrap).
#'
#'   For mixed models with [non-nested random
#'   effects](https://stats.stackexchange.com/questions/228800/crossed-vs-nested-random-effects-how-do-they-differ-and-how-are-they-specified),
#'   nonparametric resampling will not be appropriate. In these cases,
#'   parametric or semiparametric bootstrapping can be performed instead via
#'   [lme4::bootMer()] (with additional arguments passed to that function as
#'   necessary). NOTE: As [lme4::bootMer()] takes only a fitted model as its
#'   first argument (i.e. no lists), any model averaging is calculated
#'   'post-hoc' using the estimates in boot objects for each candidate model,
#'   rather than during the bootstrapping process itself (i.e. the default
#'   procedure via [boot::boot()]). Results are then returned in a new boot
#'   object for each response variable or correlated error estimate.
#'
#'   If supplied a list containing both mixed and non-mixed models, [bootEff()]
#'   with nonparametric bootstrapping will still work and will treat all models
#'   as mixed models for resampling (with a warning). This is likely a
#'   relatively rare scenario, but may occur where the user decides that
#'   non-mixed models perform similarly and/or cause less fitting issues than
#'   their mixed counterparts for at least some response variables (e.g. where
#'   random effect variance estimates are at or near zero). The data will be
#'   resampled on the supplied random effect for all models. If nonparametric
#'   bootstrapping is not used in this scenario however, an error will occur, as
#'   [lme4::bootMer()] will only accept mixed models.
#'
#'   Parallel processing is used by default via the [parallel] package and
#'   option `parallel = "snow"` (and is generally recommended), but users can
#'   specify the type of parallel processing to use, or none. If `"snow"`, a
#'   cluster of workers is created using [makeCluster()], and the user can
#'   specify the number of system cores to incorporate in the cluster (defaults
#'   to all available). [bootEff()] then exports all required objects and
#'   functions to this cluster using [clusterExport()], after performing a
#'   (rough) match of all objects and functions in the current global
#'   environment to those referenced in the model call(s). Users should load any
#'   required external packages prior to calling the function.
#'
#' @note Bootstrapping mixed (or indeed any other) models may take a very long
#'   time when the number of replicates, observations, parameters, and/or models
#'   is high. To decrease processing time, it may be worth trying different
#'   optimisers and/or other options to generate faster estimates (always check
#'   results).
#' @return An object of class `"boot"` containing the bootstrapped effects, or a
#'   (named) list/nested list of such objects.
#' @references Burnham, K. P., & Anderson, D. R. (2002). *Model Selection and
#'   Multimodel Inference: A Practical Information-Theoretic Approach* (2nd
#'   ed.). Springer-Verlag. <https://link.springer.com/book/10.1007/b97636>
#'
#'   Davison, A. C., & Hinkley, D. V. (1997). *Bootstrap Methods and their
#'   Application*. Cambridge University Press.
#'
#'   Ren, S., Lai, H., Tong, W., Aminzadeh, M., Hou, X., & Lai, S. (2010).
#'   Nonparametric bootstrapping for hierarchical data. *Journal of Applied
#'   Statistics*, *37*(9), 1487–1498. \doi{10/dvfzcn}
#' @examples
#' # Bootstrap Shipley SEM (test – 1 rep)
#' # (set 'site' as group for resampling – highest-level random effect)
#' bootEff(shipley.sem, R = 1, ran.eff = "site", parallel = "no")
#'
#' # Check estimates (use saved boot object – 1000 reps)
#' lapply(shipley.sem.boot, "[[", 1)  # original
#' lapply(shipley.sem.boot, function(i) head(i$t))  # bootstrapped
#' @export
bootEff <- function(mod, R, seed = NULL,
                    type = c("nonparametric", "parametric", "semiparametric"),
                    ran.eff = NULL, cor.err = NULL, catch.err = TRUE,
                    parallel = c("snow", "multicore", "no"), ncpus = NULL,
                    cl = NULL, bM.arg = NULL, ...) {

  if (missing(R))
    stop("Number of bootstrap resamples (R) must be specified.")

  m <- mod; type <- match.arg(type); re <- ran.eff; ce <- cor.err;
  parallel <- match.arg(parallel); nc <- ncpus;

  # Convert psem object to list of models
  if (class(m)[1] == "psem") {
    m <- m[sapply(m, isMod)]
    ce <- do.call(c, m[sapply(m, class) == "formula.cerror"])
  }

  # Set model names if absent (use response names)
  mn <- names(m)
  mnz <- !nzchar(mn) | is.na(mn)
  if (isList(m) && (is.null(mn) || any(mnz))) {
    
    # Response variable names
    rn <- sapply(m, function(i) {
      if (isList(i)) i <- i[[1]]
      names(model.frame(i, data = getData(i)))[1]
    })
    
    # Replace NULL/zero-length/NA names
    names(m) <- if (!is.null(mn)) ifelse(mnz, rn, mn) else rn

  }

  # Arguments to stdEff()
  a <- list(...)

  # Weights (for model averaging)
  w <- a$weights; a$weights <- NULL
  if (isList(m) && (is.null(w) || all(w == "equal"))) {
    if (is.null(w) && any(sapply(m, isList)))
      stop("'weights' must be supplied for model averaging (or specify 'equal').")
    w <- lapply(m, function(i) w)
  }

  # Refit model(s) with any supplied data
  upd <- function(m, d) {
    upd <- function(m) {
      eval(update(m, data = d, evaluate = FALSE))
    }
    rMapply(upd, m, SIMPLIFY = FALSE)
  }
  d <- a$data; a$data <- NULL
  if (!is.null(d)) m <- upd(m, d)

  # Environment to look for model data
  main.env <- environment()
  env <- a$env; a$env <- NULL
  if (is.null(env)) env <- parent.frame()
  if (!is.null(d)) env <- main.env

  # Mixed models?
  mer <- unlist(rMapply(isMer, m))
  if (any(mer) && !all(mer))
    warning("Mixed and non-mixed models together in list. Resampling will treat all models as mixed.")
  mer <- any(mer)
  mer2 <- isTRUE(if (mer) {
    pb <- type %in% c("parametric", "semiparametric")
    if (!pb && is.null(re))
      stop("Name of random effect to resample must be specified to 'ran.eff' (or use parametric bootstrapping).")
    mer && pb
  })
  if (mer2) {

    # Modified bootMer() function
    bootMer2 <- function(...) {

      # Set up function call with specified arguments
      C <- match.call()
      n <- length(C)
      a <- c(list(FUN = s, nsim = R, seed = NULL, type = type,
                  parallel = parallel, ncpus = nc, cl = cl), bM.arg)
      for (i in 1:length(a)) {
        C[n + i] <- a[i]
        names(C)[n + i] <- names(a)[i]
      }
      C[[1]] <- as.name("bootMer")

      # Run
      set.seed(seed)
      eval.parent(C)

    }

    # Create boot statistic object for later assignment
    s <- NULL

  }

  # Names of models with correlated errors (list)
  any.ce <- isList(m) && !is.null(ce)
  if (any.ce) {
    cv <- sapply(ce, function(i) {
      gsub(" ", "", unlist(strsplit(i, "~~")))
    }, simplify = FALSE)
    cv <- cv[sapply(cv, function(i) {
      all(i %in% names(m)) && all(i %in% names(w))
    })]
    if (length(cv) < 1)
      stop("Names of variable(s) with correlated errors missing from list of models and/or weights.")
  }

  # Set up parallel processing
  if (parallel != "no") {

    # No. cores to use
    if (is.null(nc)) nc <- parallel::detectCores()

    if (parallel == "snow") {

      # Create local cluster using system cores
      if (is.null(cl)) {
        cl <- parallel::makeCluster(getOption("cl.cores", nc))
      }

      # Export required objects/functions to cluster
      # (search global env. for objects in model call(s))
      P <- function(...) {
        paste(..., collapse = " ")
      }
      mc <- P(unlist(rMapply(function(i) P(getCall(i)), m)))
      o <- unlist(lapply(search(), ls))
      o <- o[sapply(o, function(i) grepl(i, mc, fixed = TRUE))]
      o <- c(o, ls("package:semEff"))
      parallel::clusterExport(cl, o)

    }

  }

  # Generate a seed for bootstrapping
  if (is.null(seed)) {
    set.seed(NULL)
    seed <- sample(10000:99999, 1)
  }

  # Function to bootstrap effects
  bootEff <- function(m, w) {

    # Data to resample (x)
    # (mixed model: x = highest-level random effect)
    d <- getData(m, subset = TRUE, merge = TRUE, env = env)
    if (!mer2) x <- if (mer) unique(d[re]) else d

    # Bootstrap statistic (s)
    stat <- function(x, i) {
      xi <- if (mer) {
        do.call(rbind, lapply(x[i, ], function(j) {
          d[d[, re] == j, ]
        }))
      } else x[i, ]
      do.call(stdEff, c(list(m, w, xi), a))
    }
    stat2 <- function(x) {
      do.call(stdEff, c(list(x), a))
    }
    s <- if (mer2) stat <- stat2 else stat
    if (catch.err) {
      s <- function(...) {
        tryCatch(stat(...), error = function(e) NA)
      }
    }
    if (mer2) assign("s", s, main.env)

    # Perform bootstrap
    B <- if (!mer2) {
      set.seed(seed)
      boot::boot(x, s, R, parallel = parallel, ncpus = nc, cl = cl)
    } else {
      if (isList(m)) {

        # Bootstrap
        B <- lapply(m, bootMer2)

        # Weighted average of effects
        bn <- a$term.names
        b <- lapply(B, "[[", 1)
        b <- avgEst(b, w, bn)

        # Weighted average of bootstrapped effects
        bb <- t(sapply(1:R, function(i) {
          bb <- lapply(B, function(j) j$t[i, ])
          avgEst(bb, w, bn)
        }))
        if (ncol(bb) == R) bb <- t(bb)

        # Output new boot object
        B <- B[[1]]
        B$t0 <- b; B$t <- bb
        B$call <- NULL; B$statistic <- NULL; B$mle <- NULL
        B

      } else bootMer2(m)
    }

    # Throw warning if any model fits produced errors
    n.err <- sum(apply(rbind(B$t0, B$t), 1, function(i) any(is.na(i))))
    if (n.err > 0)
      warning(paste(n.err, "model fit(s) or parameter estimation(s) failed. NAs reported/generated."))

    # Set attributes and output
    colnames(B$t) <- names(B$t0)
    attributes(B$t)[c("sim", "seed", "n")] <- c(B$sim, seed, nrow(B$data))
    B

  }

  # Bootstrapped effects
  BE <- rMapply(bootEff, m, w, SIMPLIFY = FALSE)

  # Add bootstrapped correlated errors
  if (any.ce) {

    # Data used to fit models
    if (is.null(d)) d <- getData(m, merge = TRUE, env = env)
    obs <- rownames(d)

    # Function to get (weighted) (avg.) resids from model/boot obj./list
    res <- function(x, w = NULL) {
      f <- function(m) {
        if (!isGls(m) && !isGlm(m)) {
          resid(m, type = "deviance")
        } else resid(m)
      }
      if (isList(x)) {
        if (all(sapply(x, isBoot))) {
          r <- lapply(x, "[[", 1)
          r <- avgEst(r, w)
          rb <- t(sapply(1:R, function(i) {
            rb <- lapply(x, function(j) j$t[i, ])
            avgEst(rb, w)
          }))
          list(r, rb)
        } else {
          r <- lapply(x, f)
          avgEst(r, w)
        }
      } else {
        if (isBoot(x)) list(x$t0, x$t) else f(x)
      }
    }

    # Bootstrapped correlated errors
    BCE <- Map(function(i, j) {

      # Model(s), weights, and data 1
      m1 <- m[[i[1]]]; w1 <- w[[i[1]]]
      d1 <- getData(m1, subset = TRUE, merge = TRUE, env = env)

      # Model(s), weights, and data 2
      m2 <- m[[i[2]]]; w2 <- w[[i[2]]]
      d2 <- getData(m2, subset = TRUE, merge = TRUE, env = env)

      # Data to resample (x)
      # (mixed model: x = highest-level random effect)
      o <- obs %in% rownames(d1) & obs %in% rownames(d2)
      if (!all(o)) d <- d[o, ]
      if (!mer2) {
        x <- if (mer) unique(d[re]) else d
      } else {
        if (!all(o)) {m1 <- upd(m1, d); m2 <- upd(m2, d)}
      }

      # Bootstrap statistic (s)
      stat <- function(x, i) {
        xi <- if (mer) {
          do.call(rbind, lapply(x[i, ], function(j) {
            d[d[, re] == j, ]
          }))
        } else x[i, ]
        r1 <- res(upd(m1, xi), w1)
        r2 <- res(upd(m2, xi), w2)
        cor(r1, r2)
      }
      s <- if (mer2) stat <- res else stat
      if (catch.err) {
        na <- if (mer2) rep(NA, nrow(d)) else NA
        s <- function(...) {
          tryCatch(stat(...), error = function(e) na)
        }
      }
      if (mer2) assign("s", s, main.env)

      # Perform bootstrap
      B <- if (!mer2) {
        set.seed(seed)
        boot::boot(x, s, R, parallel = parallel, ncpus = nc, cl = cl)
      } else {

        # Bootstrapped resids for model 1
        B1 <- rMapply(bootMer2, m1, SIMPLIFY = FALSE)
        R1 <- res(B1)
        r1 <- R1[[1]]; rb1 <- R1[[2]]

        # Bootstrapped resids for model 2
        B2 <- rMapply(bootMer2, m2, SIMPLIFY = FALSE)
        R2 <- res(B2)
        r2 <- R2[[1]]; rb2 <- R2[[2]]

        # Correlate (and add to new boot object)
        B <- if (isList(B1)) B1[[1]] else B1
        B$t0 <- cor(r1, r2)
        B$t <- matrix(sapply(1:R, function(i) cor(rb1[i, ], rb2[i, ])))
        B$call <- NULL; B$statistic <- NULL; B$mle <- NULL
        B

      }

      # Throw a warning if any model fits produced errors
      n.err <- sum(apply(rbind(B$t0, B$t), 1, function(i) any(is.na(i))))
      if (n.err > 0)
        warning(paste(n.err, "model fit(s) or parameter estimation(s) failed. NAs reported/generated."))

      # Set attributes and output
      names(B$t0) <- j; colnames(B$t) <- j
      attributes(B$t)[c("sim", "seed", "n")] <- c(B$sim, seed, nrow(B$data))
      B

    }, cv, names(cv))

    # Append to bootstrapped effects
    if (!isList(BE)) BE <- list(BE)
    BE <- c(BE, BCE)

  }

  # Output results
  if (parallel == "snow") parallel::stopCluster(cl)
  set.seed(NULL)
  BE

}


#' @title Bootstrap Confidence Intervals
#' @description Calculate confidence intervals from bootstrapped model effects.
#' @param mod A fitted model object. Alternatively, a boot object (class
#'   `"boot"`), containing bootstrapped model effects. Can also be a list or
#'   nested list of such objects.
#' @param conf A numeric value specifying the confidence level for the
#'   intervals.
#' @param type The type of confidence interval to return (defaults to `"bca"` –
#'   see Details). See [boot::boot.ci()] for further options.
#' @param digits The number of decimal places to return for numeric values.
#' @param bci.arg A named list of any additional arguments to [boot::boot.ci()],
#'   excepting argument `index`.
#' @param ... Arguments to [bootEff()].
#' @details `bootCI()` uses [boot::boot.ci()] to calculate confidence intervals
#'   of the specified type and level calculated from bootstrapped model effects.
#'   If a model or models is supplied, bootstrapping will first be performed via
#'   [bootEff()].
#'
#'   Nonparametric bias-corrected and accelerated confidence intervals (BC*a*;
#'   Efron, 1987) are calculated by default, which should provide the most
#'   accurate coverage across a range of bootstrap sampling distributions (Puth
#'   et al., 2015). They will, however, be
#'   [inappropriate](https://stackoverflow.com/questions/7588388/adjusted-bootstrap-confidence-intervals-bca-with-parametric-bootstrap-in-boot)
#'   for parametric resampling – in which case the default will be set to the
#'   bootstrap percentile method instead (`"perc"`).
#'
#'   Effects and confidence intervals are returned in a summary table, along
#'   with the bootstrap standard errors (standard deviations of the samples) and
#'   the bootstrap biases (sample means minus original estimates). Effects for
#'   which the confidence intervals do not contain zero are highlighted with a
#'   star (i.e. 'significant' at the `conf` level).
#'
#' @note All bootstrapped confidence intervals will tend to underestimate the
#'   true nominal coverage to some extent when sample size is small (Chernick &
#'   Labudde, 2009), so the appropriate caution should be exercised in
#'   interpretation in such cases. Comparison of different interval types may be
#'   informative. For example, normal-theory based intervals may outperform
#'   bootstrap percentile methods when n < 34 (Hesterberg, 2015). Ultimately
#'   however, the bootstrap is [not a solution to small sample
#'   size](https://stats.stackexchange.com/questions/112147/can-bootstrap-be-seen-as-a-cure-for-the-small-sample-size).
#' @return A summary table of the effects and bootstrapped confidence intervals
#'   (data frame), or a list or nested list of same.
#' @references Chernick, M. R., & Labudde, R. A. (2009). Revisiting Qualms about
#'   Bootstrap Confidence Intervals. *American Journal of Mathematical and
#'   Management Sciences*, *29*(3–4), 437–456. \doi{10/c8zv}
#'
#'   Efron, B. (1987). Better Bootstrap Confidence Intervals. *Journal of the
#'   American Statistical Association*, *82*(397), 171–185. \doi{10/gfww2z}
#'
#'   Hesterberg, T. C. (2015). What Teachers Should Know About the Bootstrap:
#'   Resampling in the Undergraduate Statistics Curriculum. *The American
#'   Statistician*, *69*(4), 371–386. \doi{10/gd85v5}
#'
#'   Puth, M.-T., Neuhäuser, M., & Ruxton, G. D. (2015). On the variety of
#'   methods for calculating confidence intervals by bootstrapping. *Journal of
#'   Animal Ecology*, *84*(4), 892–897. \doi{10/f8n9rq}
#' @examples
#' # CIs calculated from bootstrapped SEM
#' (shipley.sem.ci <- bootCI(shipley.sem.boot))
#'
#' # From original SEM (models)
#' # (not typically recommended – better to use saved boot objects)
#' # system.time(
#' #   shipley.sem.ci <- bootCI(shipley.sem, R = 1000, seed = 13,
#' #                            ran.eff = "site")
#' # )
#' @export
bootCI <- function(mod, conf = 0.95, type = "bca", digits = 3, bci.arg = NULL,
                   ...) {

  m <- mod

  # Create boot object(s) from model(s) if necessary
  is.B <- all(unlist(rMapply(isBoot, m)))
  B <- if (!is.B) bootEff(m, ...) else m

  # Function
  bootCI <- function(B) {

    # Change default CI type for parametric bootstrapping
    if (B$sim == "parametric" && type[1] == "bca") {
      message("Percentile confidence intervals used for parametric bootstrap samples.")
      type <- "perc"
    }

    # Effects/bias/standard errors
    e <- B$t0
    bi <- colMeans(B$t, na.rm = TRUE) - e
    se <- apply(B$t, 2, sd, na.rm = TRUE)
    se[is.na(e)] <- NA

    # Confidence intervals
    ci <- sapply(1:length(e), function(i) {
      if (!is.na(e[i])) {
        if (e[i] != 0) {
          ci <- suppressWarnings(
            do.call(
              boot::boot.ci,
              c(list(B, conf, type, i), bci.arg)
            )
          )
          tail(as.vector(ci[[4]]), 2)
        } else rep(0, 2)
      } else rep(NA, 2)
    })

    # Combine into table
    e <- data.frame("Effect" = e, "Bias" = bi, "Std. Err." = se,
                    "Lower CI" = ci[1, ], "Upper CI" = ci[2, ],
                    check.names = FALSE)
    e <- round(e, digits)

    # Add significance stars
    stars <- apply(e, 1, function(i) {
      if (!any(is.na(i))) {
        i <- c(i[1], tail(i, 2))
        if (all(i > 0) || all(i < 0)) "*" else ""
      } else ""
    })
    e <- cbind(e, " " = stars)

    # Format table (columns, borders, spaces, etc.)
    e <- format(e, nsmall = digits)
    e <- cbind(" " = rownames(e), "|",
               e[1], "|", e[2], "|", e[3], "|", e[4:5], "|", e[6],
               fix.empty.names = FALSE)
    b <- mapply(function(i, j) {
      n1 <- nchar(j)
      n2 <- max(sapply(i, nchar), n1, 3)
      b <- if (n1 > 1) rep("-", n2) else ""
      paste(b, collapse = "")
    }, e, names(e))
    e <- rbind(b, e)
    e[1] <- format(e[1], justify = "left")
    rownames(e) <- 1:nrow(e)

    # Set attributes and output
    class(e) <- c("bootCI", class(e))
    attr(e, "ci.conf") <- conf
    attr(e, "ci.type") <- type
    e

  }

  # Apply recursively
  rMapply(bootCI, B, SIMPLIFY = FALSE)

}


#' @title Print `"bootCI"` Objects
#' @description A [print()] method for an object of class `"bootCI"`.
#' @param x An object of class `"bootCI"`.
#' @param ... Further arguments passed to or from other methods. Not currently
#'   used.
#' @return A summary table of the effects and bootstrapped confidence intervals
#'   (data frame).
# S3 method for class 'bootCI'
#' @export
print.bootCI <- function(x, ...) {
  print.data.frame(x, row.names = FALSE)
}