R/utilities.R

In optconerrf: Optimal Monotone Conditional Error Functions

#' Print optimal conditional error trial design
#'
#' @description
#' Print an overview of the specified design parameters.
#'
#' @param x Design object of class \code{TrialDesignOptimalConditionalError}
#' @param ... Additional arguments required for generic compatibility
#'
#' @return No return value, prints the design.
#'
#' @export
print.TrialDesignOptimalConditionalError <- function(x, ...) {
  cat("Optimal Conditional Error Function Design: \n \n")
  cat("General design parameters: \n")
  cat("  Overall significance level:", x$alpha, "\n")
  cat("  First-stage efficacy boundary (p-value scale):", x$alpha1, "\n")
  cat(
    "  Binding first-stage futility boundary (p-value scale):",
    x$alpha0,
    "\n"
  )
  if (x$minimumConditionalError > 0 || x$maximumConditionalError < 1) {
    cat(
      "  Constraints on optimal conditional error:",
      paste(
        "[",
        x$minimumConditionalError,
        ", ",
        x$maximumConditionalError,
        "]",
        sep = ""
      ),
      "\n"
    )
  }
  if (
    x$minimumSecondStageInformation > 0 || x$maximumSecondStageInformation < Inf
  ) {
    cat(
      "  Constraints on second-stage information:",
      paste(
        "[",
        x$minimumSecondStageInformation,
        ", ",
        x$maximumSecondStageInformation,
        "]",
        sep = ""
      ),
      "\n"
    )
  }

  cat("\n")

  cat("Conditional power specification: \n")
  if (is.na(x$conditionalPower)) {
    cat("  Data-dependent user-specified function \n")
  } else if (!is.na(x$conditionalPower)) {
    cat("  Target conditional power:", x$conditionalPower, "\n")
  }
  if (x$useInterimEstimate) {
    cat(
      "  Alternative: interim estimate restricted to",
      paste("[", x$delta1Min, ", ", x$delta1Max, "]", sep = ""),
      "\n"
    )
    cat(
      "  First-stage non-centrality parameter restricted to",
      paste("[", x$ncp1Min, ", ", x$ncp1Max, "]", sep = ""),
      "\n"
    )
  } else {
    cat("  Alternative:", x$delta1, "\n")
    cat("  First-stage non-centrality parameter:", x$ncp1, "\n")
  }
  cat("  First-stage information:", x$firstStageInformation, "\n")
  cat("\n")

  cat("Likelihood ratio specification: \n")
  switch(
    x$likelihoodRatioDistribution,
    fixed = {
      cat(
        "  Fixed parameter(s) in likelihood ratio: ",
        paste(format(x$deltaLR, trim = TRUE), collapse = ", "),
        "\n"
      )
      cat(
        "  Parameter weights: ",
        paste(format(x$weightsDeltaLR, trim = TRUE), collapse = ", "),
        "\n"
      )
    },
    normal = {
      cat(
        "  Normally distributed prior in likelihood ratio with mean ",
        x$deltaLR,
        " and standard deviation ",
        x$tauLR,
        "\n"
      )
    },
    unif = {
      cat(
        "  Uniformly distributed prior in likelihood ratio with maximum ",
        x$deltaMaxLR,
        "\n"
      )
    },
    exp = {
      cat(
        "  Exponentially distributed prior in likelihood ratio with mean ",
        x$kappaLR,
        "\n"
      )
    },
    maxlr = {
      cat("  Maximum likelihood ratio \n")
    }
  )

  cat("\n")
  cat("Level constant: \n")
  cat("  Constant:", x$levelConstant, "\n")
  cat(
    "  Searched on interval:",
    paste(
      "[",
      x$levelConstantMinimum,
      ", ",
      x$levelConstantMaximum,
      "]",
      sep = ""
    ),
    "\n"
  )

  if (!is.null(unlist(x$monotonisationConstants))) {
    cat("\n")
    cat("Monotonisation constants: \n")
    cat(
      "  Intervals (p-value scale):",
      paste(
        "[",
        apply(
          X = cbind(
            x$monotonisationConstants$dls,
            x$monotonisationConstants$dus
          ),
          FUN = paste,
          MARGIN = 1,
          collapse = ", "
        ),
        "]",
        sep = ""
      ),
      "\n"
    )
    cat("  Constant(s) (Q scale):", x$monotonisationConstants$qs, "\n")
  }
  if (!x$enforceMonotonicity) {
    cat("\n")
    cat("Monotonicity was not enforced \n")
  }
}


#' Print simulation results
#'
#' @description
#' Print an overview of simulation results.
#'
#' @param x Simulation results object of class \code{SimulationResultsOptimalConditionalError}
#' @param ... Additional arguments required for generic compatibility
#'
#' @return No return value, prints the simulation results.
#'
#' @export
print.SimulationResultsOptimalConditionalError <- function(x, ...) {
  cat("Simulation results \n \n")
  cat(
    " Alternative: ",
    paste(format(x$alternative, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " First-stage futility: ",
    paste(format(x$firstStageFutility, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " First-stage efficacy: ",
    paste(format(x$firstStageEfficacy, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " Overall power: ",
    paste(format(x$overallPower, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(" Simulated trials per scenario: ", x$maxNumberOfIterations)
}


#' Print power results
#'
#' @description
#' Print an overview of exact power results.
#'
#' @param x Power results object of class \code{PowerResultsOptimalConditionalError}
#' @param ... Additional arguments required for generic compatibility
#'
#' @return No return value, prints the power calculation results.
#'
#' @export
print.PowerResultsOptimalConditionalError <- function(x, ...) {
  cat("Power calculation results \n \n")
  cat(
    " Alternative: ",
    paste(format(x$alternative, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " First-stage futility: ",
    paste(format(x$firstStageFutility, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " First-stage efficacy: ",
    paste(format(x$firstStageEfficacy, trim = TRUE), collapse = ", "),
    "\n"
  )
  cat(
    " Overall power: ",
    paste(format(x$overallPower, trim = TRUE), collapse = ", "),
    "\n"
  )
}


#' Plot the optimal conditional error function
#' @name plot.TrialDesignOptimalConditionalError
#'
#' @description
#' The returned plot is a \code{ggplot2} object and can be supplemented with additional layers using \code{ggplot2} commands.
#'
#' @param x Design object of class \code{TrialDesignOptimalConditionalError}.
#' @param range Numeric vector with two entries specifying the range of the x-axis of the plot.
#' @param type Type of plot to be created. Options are: \itemize{
#' \item \code{type = 1}: Plot the values of the optimal conditional error function against the first-stage p-value.
#' \item \code{type = 2}: Plot the second-stage information resulting from the optimal conditional error function against the first-stage p-value.
#' \item \code{type = 3}: Plot the likelihood ratio of the given specification of the optimal conditional error function against the first-stage p-value.
#' \item \code{type = 4}: Plot the function Q of the given specification of the optimal conditional error function against the first-stage p-value.
#' }
#' @param plotNonMonotoneFunction Logical. Should the non-monotone version of the plot be drawn? Not applicable for plot type 3. Default: \code{FALSE}.
#' @param ... Additional arguments required for generic compatibility
#'
#' @return No return value, plots the design.
#'
#' @export
plot.TrialDesignOptimalConditionalError <- function(
  x,
  range = c(0, 1),
  type = 1,
  plotNonMonotoneFunction = FALSE,
  ...
) {
  # Set a range of first-stage p-values
  firstStagePValues <- seq(from = range[1], to = range[2], length.out = 1e3)

  # Plot type 1: draw the optimal conditional error function
  if (type == 1) {
    # Calculate optimal conditional error for provided design
    optimalConditionalErrors <- getOptimalConditionalError(
      firstStagePValue = firstStagePValues,
      design = x
    )

    # Create base plot and save it for possible addition of the non-monotone function
    designPlot <- ggplot2::ggplot(data = NULL) +
      ggplot2::geom_line(
        mapping = ggplot2::aes(
          x = firstStagePValues,
          y = optimalConditionalErrors
        ),
        colour = "black",
        linetype = "solid",
        linewidth = 1.1
      ) +
      ggplot2::labs(
        x = "First-stage p-value",
        y = "Optimal Conditional Error"
      ) +
      ggplot2::theme_bw() +
      ggplot2::geom_vline(
        xintercept = x$alpha0,
        linetype = "dotted",
        col = "red",
        linewidth = 0.8
      ) +
      ggplot2::geom_vline(
        xintercept = x$alpha1,
        linetype = "dotted",
        col = "blue",
        linewidth = 0.8
      ) +
      ggplot2::xlim(c(range[1], range[2]))

    # In addition to the monotone function, the non-monotone one should be drawn
    if (plotNonMonotoneFunction) {
      # There are no monotonisation constants -> monotonisation is not required
      if (is.null(unlist(x$monotonisationConstants))) {
        warning(
          "No monotonisation required. Displaying monotone function only."
        )
        designPlot # Display base plot
      } else {
        # Plotting the non-monotone function only makes sense if the provided design is enforced to be monotone
        if (x$enforceMonotonicity) {
          # Here, a new design must be created, modifying the old one will not work as intended (Operation is not "copy-on-modify")
          # All fields apart from enforceMonotonicity are copied from the original design object.
          # (There may be a better way to implement this)
          # suppressWarnings() is used because this code is expected to always produce a warning
          secondDesign <- suppressWarnings(new(
            "TrialDesignOptimalConditionalError",
            alpha = x$alpha,
            alpha1 = x$alpha1,
            alpha0 = x$alpha0,
            conditionalPower = x$conditionalPower,
            conditionalPowerFunction = x$conditionalPowerFunction,
            delta1 = x$delta1,
            firstStageInformation = x$firstStageInformation,
            useInterimEstimate = x$useInterimEstimate,
            likelihoodRatioDistribution = x$likelihoodRatioDistribution,
            deltaLR = x$deltaLR,
            weightsDeltaLR = x$weightsDeltaLR,
            tauLR = x$tauLR,
            kappaLR = x$kappaLR,
            deltaMaxLR = x$deltaMaxLR,
            minimumConditionalError = x$minimumConditionalError,
            maximumConditionalError = x$maximumConditionalError,
            minimumSecondStageInformation = x$minimumSecondStageInformation,
            maximumSecondStageInformation = x$maximumSecondStageInformation,
            levelConstantMinimum = x$levelConstantMinimum,
            levelConstantMaximum = x$levelConstantMaximum,
            ncp1Min = x$ncp1Min,
            ncp1Max = x$ncp1Max,
            enforceMonotonicity = FALSE
          ))

          # Calculate optimal conditional error for the new design
          nonMonoOptimalConditionalErrors <- getOptimalConditionalError(
            firstStagePValue = firstStagePValues,
            design = secondDesign
          )

          # Add non-monotone optimal conditional error to base plot and display it
          designPlot +
            ggplot2::geom_line(
              mapping = ggplot2::aes(
                x = firstStagePValues,
                y = nonMonoOptimalConditionalErrors
              ),
              colour = "gray",
              linetype = "dashed",
              linewidth = 1.1
            )
        } else {
          warning(
            "When using plotNonMonotoneFunction=TRUE, x should provide a monotone function. Consider setting enforceMonotonicity=TRUE in design object."
          )
          designPlot # Display base plot
        }
      }
    } else {
      designPlot # Display base plot
    }
  } else if (type == 2) {
    # Plot type 2: draw second-stage information
    # Calculate second-stage information
    secondStageInformation <- getSecondStageInformation(
      firstStagePValue = firstStagePValues,
      design = x
    )

    # Create base plot and save it for possible addition of the non-monotone function
    informationPlot <- ggplot2::ggplot() +
      ggplot2::geom_line(
        mapping = ggplot2::aes(
          x = firstStagePValues,
          y = secondStageInformation
        ),
        colour = "black",
        linetype = "solid",
        linewidth = 1.1
      ) +
      ggplot2::labs(x = "First-stage p-value", y = "Second-stage information") +
      ggplot2::theme_bw() +
      ggplot2::geom_vline(
        xintercept = x$alpha0,
        linetype = "dotted",
        col = "red",
        linewidth = 0.8
      ) +
      ggplot2::geom_vline(
        xintercept = x$alpha1,
        linetype = "dotted",
        col = "blue",
        linewidth = 0.8
      ) +
      ggplot2::xlim(c(range[1], range[2]))

    if (plotNonMonotoneFunction) {
      # There are no monotonisation constants -> monotonisation is not required
      if (is.null(unlist(x$monotonisationConstants))) {
        warning(
          "No monotonisation required. Displaying monotone function only."
        )
        informationPlot # Display base plot
      } else {
        # Plotting the non-monotone function only makes sense if the provided design is enforced to be monotone
        if (x$enforceMonotonicity) {
          # Here, a new design must be created, modifying the old one will not work as intended (Operation is not "copy-on-modify")
          # All fields apart from enforceMonotonicity are copied from the original design object.
          # (There may be a better way to implement this)
          # suppressWarnings() is used because this code is expected to always produce a warning
          secondDesign <- suppressWarnings(new(
            "TrialDesignOptimalConditionalError",
            alpha = x$alpha,
            alpha1 = x$alpha1,
            alpha0 = x$alpha0,
            conditionalPower = x$conditionalPower,
            conditionalPowerFunction = x$conditionalPowerFunction,
            delta1 = x$delta1,
            firstStageInformation = x$firstStageInformation,
            useInterimEstimate = x$useInterimEstimate,
            likelihoodRatioDistribution = x$likelihoodRatioDistribution,
            deltaLR = x$deltaLR,
            weightsDeltaLR = x$weightsDeltaLR,
            tauLR = x$tauLR,
            kappaLR = x$kappaLR,
            deltaMaxLR = x$deltaMaxLR,
            minimumConditionalError = x$minimumConditionalError,
            maximumConditionalError = x$maximumConditionalError,
            minimumSecondStageInformation = x$minimumSecondStageInformation,
            maximumSecondStageInformation = x$maximumSecondStageInformation,
            levelConstantMinimum = x$levelConstantMinimum,
            levelConstantMaximum = x$levelConstantMaximum,
            ncp1Min = x$ncp1Min,
            ncp1Max = x$ncp1Max,
            enforceMonotonicity = FALSE
          ))

          # Calculate optimal conditional error for the new design
          nonMonoSecondStageInformation <- getSecondStageInformation(
            firstStagePValue = firstStagePValues,
            design = secondDesign
          )

          # Add non-monotone optimal conditional error to base plot and display it
          informationPlot +
            ggplot2::geom_line(
              mapping = ggplot2::aes(
                x = firstStagePValues,
                y = nonMonoSecondStageInformation
              ),
              colour = "gray",
              linetype = "dashed",
              linewidth = 1.1
            )
        } else {
          warning(
            "When using plotNonMonotoneFunction=TRUE, x should provide a monotone function. Consider setting enforceMonotonicity=TRUE in design object."
          )
          informationPlot # Display base plot
        }
      }
    } else {
      informationPlot # Display base plot
    }
  } else if (type == 3) {
    # Plot type 3: likelihood ratio
    # Calculate likelihood ratio values
    likelihoodRatios <- getLikelihoodRatio(
      firstStagePValue = firstStagePValues,
      design = x
    )

    likelihoodRatioPlot <- ggplot2::ggplot() +
      ggplot2::geom_line(
        mapping = ggplot2::aes(x = firstStagePValues, y = likelihoodRatios),
        colour = "black",
        linetype = "solid",
        linewidth = 1.1
      ) +
      ggplot2::labs(x = "First-stage p-value", y = "Likelihood ratio") +
      ggplot2::theme_bw() +
      ggplot2::geom_vline(
        xintercept = x$alpha0,
        linetype = "dotted",
        col = "red",
        linewidth = 0.8
      ) +
      ggplot2::geom_vline(
        xintercept = x$alpha1,
        linetype = "dotted",
        col = "blue",
        linewidth = 0.8
      ) +
      ggplot2::xlim(c(range[1], range[2]))

    likelihoodRatioPlot
  } else if (type == 4) {
    # Plot type 4: Q
    firstStagePValues <- seq(
      from = max(range[1], x$alpha1),
      to = min(range[2], x$alpha0),
      length.out = 1e3
    )
    # Calculate Q values
    Q <- getMonotoneFunction(
      x = firstStagePValues,
      fun = getQ,
      design = x
    )
    QPlot <- ggplot2::ggplot() +
      ggplot2::geom_line(
        mapping = ggplot2::aes(x = firstStagePValues, y = Q),
        colour = "black",
        linetype = "solid",
        linewidth = 1.1
      ) +
      ggplot2::labs(x = "First-stage p-value", y = "Q") +
      ggplot2::theme_bw() +
      ggplot2::geom_vline(
        xintercept = x$alpha0,
        linetype = "dotted",
        col = "red",
        linewidth = 0.8
      ) +
      ggplot2::geom_vline(
        xintercept = x$alpha1,
        linetype = "dotted",
        col = "blue",
        linewidth = 0.8
      ) +
      ggplot2::xlim(c(max(range[1], x$alpha1), min(x$alpha0, range[2])))

    if (
      plotNonMonotoneFunction &&
        x$enforceMonotonicity &&
        !is.null(unlist(x$monotonisationConstants))
    ) {
      nonMonoQ <- getQ(
        firstStagePValue = firstStagePValues,
        design = x
      )

      QPlot <- QPlot +
        ggplot2::geom_line(
          mapping = ggplot2::aes(
            x = firstStagePValues,
            y = nonMonoQ
          ),
          colour = "gray",
          linetype = "dashed",
          linewidth = 1.1
        )
    }
    QPlot
  }
}

#' Summary of the optimal conditional error trial design
#'
#' @description
#' Provide an overview of the operating characteristics of the optimal conditional error trial design.
#'
#' @param object Design object of class \code{TrialDesignOptimalConditionalError}
#' @param ... Additional arguments required for generic compatibility
#'
#' @return No return value, prints a summary of design performance.
#'
#' @export
summary.TrialDesignOptimalConditionalError <- function(object, ...) {
  cat("Summary of the Optimal Conditional Error Function Design: \n \n")
  cat("General design parameters: \n")
  cat("  Overall significance level:", object$alpha, "\n")
  cat("  First-stage efficacy boundary (p-value scale):", object$alpha1, "\n")
  cat(
    "  Binding first-stage futility boundary (p-value scale):",
    object$alpha0,
    "\n"
  )
  cat("\n")

  cat("Second-stage information: \n")
  cat(
    "  Expected second-stage information (delta=0):",
    getExpectedSecondStageInformation(
      design = object,
      likelihoodRatioDistribution = "fixed",
      deltaLR = 0
    ),
    "\n"
  )
  if (object$useInterimEstimate == FALSE & length(object$weightsDeltaLR) <= 1) {
    cat(
      "  Expected second-stage information (delta=delta1=",
      object$delta1,
      "): ",
      getExpectedSecondStageInformation(
        design = object,
        likelihoodRatioDistribution = "fixed",
        deltaLR = object$delta1
      ),
      "\n",
      sep = ""
    )
  }
  if (object$useInterimEstimate == TRUE) {
    cat(
      "  Expected second-stage information (delta=delta1Min=",
      object$delta1Min,
      "): ",
      getExpectedSecondStageInformation(
        design = object,
        likelihoodRatioDistribution = "fixed",
        deltaLR = object$delta1Min
      ),
      "\n",
      sep = ""
    )
  }
  cat(
    "  Expected second-stage information (Given likelihood ratio distr.):",
    getExpectedSecondStageInformation(design = object),
    "\n"
  )
  if (object$likelihoodRatioDistribution != "maxlr") {
    if (object$likelihoodRatioDistribution == "fixed") {
      if (length(object$weightsDeltaLR) <= 1) {
        delta <- object$deltaLR
      } else {
        delta <- object$deltaLR %*% object$weightsDeltaLR
      }
    }
    if (object$likelihoodRatioDistribution == "normal") {
      delta <- object$deltaLR
    }
    if (object$likelihoodRatioDistribution == "exp") {
      delta <- object$kappaLR
    }
    if (object$likelihoodRatioDistribution == "unif") {
      delta <- object$deltaMaxLR / 2
    }
    cat(
      "  Expected second-stage information (delta=Mean of given likelihood ratio distr.=",
      delta,
      "): ",
      getExpectedSecondStageInformation(
        design = object,
        likelihoodRatioDistribution = "fixed",
        deltaLR = delta
      ),
      "\n",
      sep = ""
    )
  }
  if (!is.na(object$conditionalPower)) {
    cat(
      "  Maximum second-stage information:",
      getSecondStageInformation(
        design = object,
        firstStagePValue = object$alpha0
      )
    )
  }
  cat("\n \n")

  cat("Power and stopping probabilities: \n")
  if (is.na(object$conditionalPower)) {
    cat("  Conditional power: Data-dependent user-specified function \n")
  } else if (!is.na(object$conditionalPower)) {
    cat("  Conditional power (fixed):", object$conditionalPower, "\n")
  }
  if (object$useInterimEstimate == FALSE & length(object$weightsDeltaLR) <= 1) {
    powerResults <- getOverallPower(
      design = object,
      alternative = object$delta1
    )
    cat(
      "  Overall power (delta1=",
      object$delta1,
      "): ",
      powerResults$overallPower,
      "\n",
      sep = ""
    )
    cat(
      "  Efficacy stopping probability (delta1=",
      object$delta1,
      "): ",
      powerResults$firstStageEfficacy,
      "\n",
      sep = ""
    )
    cat(
      "  Futility stopping probability (delta1=",
      object$delta1,
      "): ",
      powerResults$firstStageFutility,
      "\n",
      sep = ""
    )
  }
  if (object$useInterimEstimate == TRUE) {
    powerResults <- getOverallPower(
      design = object,
      alternative = object$delta1Min
    )
    cat(
      "  Overall power (delta1=delta1Min= ",
      object$delta1Min,
      "): ",
      powerResults$overallPower,
      "\n",
      sep = ""
    )
    cat(
      "  Efficacy stopping probability (delta1=delta1Min= ",
      object$delta1Min,
      "): ",
      powerResults$firstStageEfficacy,
      "\n",
      sep = ""
    )
    cat(
      "  Futility stopping probability (delta1=delta1Min= ",
      object$delta1Min,
      "): ",
      powerResults$firstStageFutility,
      "\n",
      sep = ""
    )
  }
  if (object$likelihoodRatioDistribution != "maxlr") {
    if (object$likelihoodRatioDistribution == "fixed") {
      if (length(object$weightsDeltaLR) <= 1) {
        delta1 <- object$deltaLR
      } else {
        delta1 <- as.numeric(object$deltaLR %*% object$weightsDeltaLR)
      }
    }
    if (object$likelihoodRatioDistribution == "normal") {
      delta1 <- object$deltaLR
    }
    if (object$likelihoodRatioDistribution == "exp") {
      delta1 <- object$kappaLR
    }
    if (object$likelihoodRatioDistribution == "unif") {
      delta1 <- object$deltaMaxLR / 2
    }
    powerResults <- getOverallPower(design = object, alternative = delta1)
    cat(
      "  Overall power (delta1=Mean of given likelihood ratio distr.= ",
      delta1,
      "): ",
      powerResults$overallPower,
      "\n",
      sep = ""
    )
    cat(
      "  Efficacy stopping probability (delta1=Mean of given likelihood ratio distr.= ",
      delta1,
      "): ",
      powerResults$firstStageEfficacy,
      "\n",
      sep = ""
    )
    cat(
      "  Futility stopping probability (delta1=Mean of given likelihood ratio distr.= ",
      delta1,
      "): ",
      powerResults$firstStageFutility,
      "\n",
      sep = ""
    )
  }
  cat("\n")
}


#' Simple range check for numeric variables
#' @description
#' This function performs very basic range checks for numeric variables and throws
#' an error if the range is violated. A custom hint may be added to the message.
#'
#' @param variable The (named) variable to be checked.
#' @param range A vector of length 2 giving the minimum and maximum of the allowed range.
#' @param allowedEqual Logical. Are the borders of range valid values?
#' @param hint Additional message that may be printed after the error.
#'
.rangeCheck <- function(variable, range, allowedEqual, hint = "") {
  if (allowedEqual) {
    if (any(variable < range[1]) || any(variable > range[2])) {
      variableName <- deparse(substitute(variable))
      stop(paste0(
        variableName,
        " must lie in [",
        range[1],
        ", ",
        range[2],
        "]. ",
        hint
      ))
    }
  } else {
    if (any(variable <= range[1]) || any(variable >= range[2])) {
      variableName <- deparse(substitute(variable))
      stop(paste0(
        variableName,
        " must lie in ]",
        range[1],
        ", ",
        range[2],
        "[. ",
        hint
      ))
    }
  }
}