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```
#' pmsampsize
#' - Calculates the minimum sample size required for developing a multivariable prediction model
#'
#' \code{pmsampsize} computes the minimum sample size required for the development of a new
#' multivariable prediction model using the criteria proposed by Riley \emph{et al}. 2018. \code{pmsampsize}
#' can be used to calculate the minimum sample size for the development of models with
#' continuous, binary or survival (time-to-event) outcomes. Riley \emph{et al}. lay out a series of
#' criteria the sample size should meet. These aim to minimise the overfitting and to ensure
#' precise estimation of key parameters in the prediction model. \cr \cr
#' For continuous outcomes, there are four criteria: \cr
#' i) small overfitting defined by an expected shrinkage of predictor effects by 10\% or less, \cr
#' ii) small absolute difference of 0.05 in the model's apparent and adjusted R-squared value, \cr
#' iii) precise estimation of the residual standard deviation, and \cr
#' iv) precise estimation of the average outcome value. \cr \cr
#' The sample size calculation requires the user to pre-specify (e.g. based on previous evidence)
#' the anticipated R-squared of the model, and the average outcome value and standard deviation
#' of outcome values in the population of interest. \cr \cr
#' For binary or survival (time-to-event) outcomes, there are three criteria: \cr
#' i) small overfitting defined by an expected shrinkage of predictor effects by 10\% or less, \cr
#' ii) small absolute difference of 0.05 in the model's apparent and adjusted Nagelkerke's R-squared
#' value, and \cr
#' iii) precise estimation (within +/- 0.05) of the average outcome risk in the
#' population for a key timepoint of interest for prediction.
#'
#' NB: When specifying a binary outcome prediction model with 12 or fewer predictor
#' parameters, an alternative approach by van Smeden et al. may be considered as
#' presented here: https://mvansmeden.shinyapps.io/BeyondEPV/
#'
#' With thanks to Gary Collins, Glen Martin & Kym Snell for helpful input & feedback
#'
#'
#' @author Joie Ensor (Keele University, j.ensor@keele.ac.uk),
#' @author Emma C. Martin (Phastar),
#' @author Richard D. Riley (Keele University)
#'
#' @param type specifies the type of analysis for which sample size is being calculated
#' \itemize{
#' \item \code{"c"} specifies sample size calculation for a prediction model with a continuous outcome
#' \item \code{"b"} specifies sample size calculation for a prediction model with a binary outcome
#' \item \code{"s"} specifies sample size calculation for a prediction model with a survival (time-to-event) outcome
#' }
#' @param rsquared specifies the expected value of the (Cox-Snell) R-squared of the new model,
#' where R-squared is the percentage of variation in outcome values explained by the model.
#' For example, the user may input the value of the (Cox-Snell) R-squared reported for a
#' previous prediction model study in the same field. If taking a value from a previous
#' prediction model development study, users should input the model's adjusted R-squared
#' value, not the apparent R-squared value, as the latter is optimistic (biased). However,
#' if taking the R-squared value from an external validation of a previous model, the
#' apparent R-squared can be used (as the validation data was not used for development, and
#' so R-squared apparent is then unbiased). Note that for binary and survival outcome
#' models, the Cox-Snell R-squared value is required; this is the generalised version of
#' the well-known R-squared for continuous outcomes, based on the likelihood. The papers
#' by Riley et al. (see references) outline how to obtain the Cox-Snell R-squared value
#' from published studies if they are not reported, using other information (such as the
#' C-statistic [see cstatistic() option below] or Nagelkerke's R-squared). Users should
#' be conservative with their chosen R-squared value; for example, by taking the R-squared
#' value from a previous model, even if they hope their new model will improve performance.
#'
#' @param parameters specifies the number of candidate predictor parameters for potential
#' inclusion in the new prediction model. Note that this may be larger than the number of
#' candidate predictors, as categorical and continuous predictors often require two or more
#' parameters to be estimated.
#'
#' @param shrinkage specifies the level of shrinkage desired at internal validation after
#' developing the new model. Shrinkage is a measure of overfitting, and can range from 0 to 1,
#' with higher values denoting less overfitting. We recommend a shrinkage = 0.9 (the
#' default in \code{pmsampsize}), which indicates that the predictor effect (beta coefficients) in
#' the model would need to be shrunk by 10\% to adjust for overfitting. See references
#' below for further information.
#'
#' @param prevalence (binary outcome option) specifies the overall outcome proportion
#' (for a prognostic model) or
#' overall prevalence (for a diagnostic model) expected within the model development
#' dataset. This should be derived based on previous studies in the same population.
#'
#' @param cstatistic (binary outcome option) specifies the C-statistic reported in an
#' existing prediction model study to be used in conjunction with the expected
#' prevalence to approximate the Cox-Snell R-squared using the approach of Riley et al. 2020.
#' Ideally, this should be an optimism-adjusted C-statistic. The approximate Cox-Snell R-squared
#' value is used as described above for the rsquared() option, and so is treated as a baseline
#' for the expected performance of the new model.
#'
#' @param seed (binary outcome option) specifies the initial value of the random-number
#' seed used by the random-number functions when simulating data to approximate the
#' Cox-Snell R-squared based on reported C-statistic and expect prevalence as described
#' by Riley et al. 2020
#'
#' @param rate (survival outcome option) specifies the overall event rate in the population of interest,
#' for example as obtained from a previous study, for the survival outcome of interest. NB: rate must
#' be given in time units used for meanfup and timepoint options.
#'
#' @param timepoint (survival outcome option) specifies the timepoint of interest for prediction.
#' NB: time units must be the same as given for meanfup option (e.g. years, months).
#'
#' @param meanfup (survival outcome option) specifies the average (mean) follow-up time
#' anticipated for individuals in the model development dataset,
#' for example as taken from a previous study in the population of interest.
#' NB: time units must be the same as given for timepoint option.
#'
#' @param intercept (continuous outcome options) specifies the average outcome value in the population of
#' interest e.g. the average blood pressure, or average pain score.
#' This could be based on a previous study, or on clinical knowledge.
#'
#' @param sd (continuous outcome options) specifies the standard deviation (SD) of
#' outcome values in the population e.g.
#' the SD for blood pressure in patients with all other predictors set to the average.
#' This could again be based on a previous study, or on clinical knowledge.
#'
#' @param mmoe (continuous outcome options) multiplicative margin of error (MMOE)
#' acceptable for calculation of the
#' intercept. The default is a MMOE of 10\%. Confidence interval for the intercept will be
#' displayed in the output for reference. See references below for further information.
#'
#' @import stats
#'
#' @examples
#' ## Examples based on those included in two papers by Riley et al.
#' ## published in Statistics in Medicine (2018).
#' ## NB: Survival example based on Riley et al. BMJ paper (2020).
#'
#' ## Binary outcomes (Logistic prediction models)
#' # Use pmsampsize to calculate the minimum sample size required to develop a
#' # multivariable prediction model for a binary outcome using 24 candidate
#' # predictor parameters. Based on previous evidence, the outcome prevalence is
#' # anticipated to be 0.174 (17.4%) and a lower bound (taken from the adjusted
#' # Cox-Snell R-squared of an existing prediction model) for the new model's
#' # R-squared value is 0.288
#'
#' pmsampsize(type = "b", rsquared = 0.288, parameters = 24, prevalence = 0.174)
#'
#' # Now lets assume we could not obtain a Cox-Snell R-squared estimate from an existing
#' # prediction model, but instead had a C-statistic (0.89) reported for the existing prediction
#' # model. We can use this C-statistic along with the prevalence to approximate the Cox-Snell
#' # R-squared using the approach of Riley et al. (2020). Use pmsampsize with the cstatistic()
#' # option instead of rsquared() option.
#'
#' pmsampsize(type = "b", cstatistic = 0.89, parameters = 24, prevalence = 0.174)
#'
#' ## Survial outcomes (Cox prediction models)
#' # Use pmsampsize to calculate the minimum sample size required for developing
#' # a multivariable prediction model with a survival outcome using 30 candidate
#' # predictors. We know an existing prediction model in the same field has an
#' # R-squared adjusted of 0.051. Further, in the previous study the mean
#' # follow-up was 2.07 years, and overall event rate was 0.065. We select a
#' # timepoint of interest for prediction using the newly developed model of 2
#' # years
#'
#' pmsampsize(type = "s", rsquared = 0.051, parameters = 30, rate = 0.065,
#' timepoint = 2, meanfup = 2.07)
#'
#' ## Continuous outcomes (Linear prediction models)
#' # Use pmsampsize to calculate the minimum sample size required for developing
#' # a multivariable prediction model for a continuous outcome (here, FEV1 say),
#' # using 25 candidate predictors. We know an existing prediction model in the
#' # same field has an R-squared adjusted of 0.2, and that FEV1 values in the
#' # population have a mean of 1.9 and SD of 0.6
#'
#' pmsampsize(type = "c", rsquared = 0.2, parameters = 25, intercept = 1.9, sd = 0.6)
#'
#' @references Riley RD, Ensor J, Snell KIE, Harrell FE, Martin GP, Reitsma JB, et al.
#' Calculating the sample size required for developing a clinical prediction model.
#' \emph{BMJ (Clinical research ed)}. 2020
#'
#' @references Riley RD, Snell KIE, Ensor J, Burke DL, Harrell FE, Jr., Moons KG, Collins GS.
#' Minimum sample size required for developing a multivariable prediction model: Part I continuous outcomes.
#' \emph{Statistics in Medicine}. 2018 (in-press). doi: 10.1002/sim.7993
#'
#' @references Riley RD, Snell KIE, Ensor J, Burke DL, Harrell FE, Jr., Moons KG, Collins GS.
#' Minimum sample size required for developing a multivariable prediction model:
#' Part II binary and time-to-event outcomes.
#' \emph{Statistics in Medicine}. 2018 (in-press). doi: 10.1002/sim.7992
#'
#' @references van Smeden M, Moons KG, de Groot JA, et al. Sample size for binary logistic
#' prediction models: Beyond events per variable criteria.
#' \emph{Stat Methods Med Res}. 2019;28(8):2455-74
#'
#' @references Riley, RD, Van Calster, B, Collins, GS. A note on estimating the Cox-Snell R2
#' from a reported C statistic (AUROC) to inform sample size calculations for developing a
#' prediction model with a binary outcome. \emph{Statistics in Medicine}. 2020
#' @export
pmsampsize <- function(type,
rsquared = NA,
parameters,
shrinkage = 0.9,
prevalence = NA,
cstatistic = NA,
seed = 123456,
rate = NA,
timepoint = NA,
meanfup = NA,
intercept = NA,
sd = NA,
mmoe=1.1) {
# error checking
pmsampsize_errorcheck(type=type,rsquared=rsquared,parameters=parameters,shrinkage=shrinkage,cstatistic=cstatistic,
prevalence=prevalence,rate=rate,timepoint=timepoint,meanfup=meanfup,intercept=intercept,sd=sd,mmoe=mmoe)
# choose function based on analysis type
if (type == "c") out <- pmsampsize_cont(rsquared=rsquared,parameters=parameters,intercept=intercept,
sd=sd,shrinkage=shrinkage,mmoe=mmoe)
if (type == "b") {
if (is.na(cstatistic) == F) {
if (is.na(rsquared) == F) {
stop("Only one of rsquared() or cstatistic() can be specified")
}
approx_rsq <- cstat2rsq(cstatistic=cstatistic, prevalence=prevalence, seed=seed)
rsquared <- round(approx_rsq$R2.coxsnell,digits = 4)
}
else {
if (is.na(rsquared)) {
stop("One of rsquared() or cstatistic() must be specified")
}
}
out <- pmsampsize_bin(rsquared=rsquared,parameters=parameters,prevalence=prevalence,
shrinkage=shrinkage,cstatistic=cstatistic)
}
if (type == "s") out <- pmsampsize_surv(rsquared=rsquared,parameters=parameters,rate=rate,
timepoint=timepoint,meanfup=meanfup,shrinkage=shrinkage,mmoe=mmoe)
est <- out
class(est) <- "pmsampsize"
est
}
#' @export
print.pmsampsize <- function(x, ...) {
if (x$type == "continuous") {
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming MMOE <= 1.1 in estimation of intercept & residual standard deviation \n")
cat("SPP - Subjects per Predictor Parameter \n \n")
}
if (x$type == "survival") {
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming 0.05 margin of error in estimation of overall risk at time point =", x$timepoint, " \n")
cat("NB: Events per Predictor Parameter (EPP) assumes overall event rate =", x$rate, " \n \n")
}
if (x$type == "binary") {
if (is.na(x$cstatistic) == F) {
cat("Given input C-statistic =", x$cstatistic, " & prevalence =", x$prevalence, " \n")
cat("Cox-Snell R-sq =", x$rsquared, " \n \n")
}
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming 0.05 margin of error in estimation of intercept \n")
cat("NB: Events per Predictor Parameter (EPP) assumes prevalence =", x$prevalence, " \n \n")
}
print(x$results_table)
if (x$type == "continuous") {
cat(" \n Minimum sample size required for new model development based on user inputs =", x$sample_size, " \n \n ")
cat("* 95% CI for intercept = (",x$int_lci,", ", x$int_uci, "), for sample size n = ",x$sample_size,sep = "")
}
if (x$type == "survival") {
cat(" \n Minimum sample size required for new model development based on user inputs = ", x$sample_size, ", \n ",sep = "")
cat("corresponding to", x$tot_per_yrs_final, "person-time** of follow-up, with", ceiling(x$events), "outcome events \n ")
cat("assuming an overall event rate =", x$rate, "and therefore an EPP =", x$EPP, " \n \n ")
cat("* 95% CI for overall risk = (",x$int_lci,", ", x$int_uci, "), for true value of ", x$int_cuminc, " and sample size n = ",x$sample_size,sep = "")
cat(" \n **where time is in the units mean follow-up time was specified in")
}
if (x$type == "binary") {
cat(" \n Minimum sample size required for new model development based on user inputs = ", x$sample_size, ", \n ",sep = "")
cat("with ", ceiling(x$events), " events (assuming an outcome prevalence = ", x$prevalence, ") and an EPP = ", x$EPP, " \n \n ",sep = "")
}
}
#' @export
summary.pmsampsize <- function(object, ...) {
if (object$type == "continuous") {
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming MMOE <= 1.1 in estimation of intercept & residual standard deviation \n")
cat("SPP - Subjects per Predictor Parameter \n \n")
}
if (object$type == "survival") {
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming 0.05 margin of error in estimation of overall risk at time point =", object$timepoint, " \n")
cat("NB: Events per Predictor Parameter (EPP) assumes overall event rate =", object$rate, " \n \n")
}
if (object$type == "binary") {
if (is.na(object$cstatistic) == F) {
cat("Given input C-statistic =", object$cstatistic, " & prevalence =", object$prevalence, " \n")
cat("Cox-Snell R-sq =", object$rsquared, " \n \n")
}
cat("NB: Assuming 0.05 acceptable difference in apparent & adjusted R-squared \n")
cat("NB: Assuming 0.05 margin of error in estimation of intercept \n")
cat("NB: Events per Predictor Parameter (EPP) assumes prevalence =", object$prevalence, " \n \n")
}
print(object$results_table)
if (object$type == "continuous") {
cat(" \n Minimum sample size required for new model development based on user inputs =", object$sample_size, " \n \n ")
cat("* 95% CI for intercept = (",object$int_lci,", ", object$int_uci, "), for sample size n = ",object$sample_size,sep = "")
}
if (object$type == "survival") {
cat(" \n Minimum sample size required for new model development based on user inputs = ", object$sample_size, ", \n ",sep = "")
cat("corresponding to", object$tot_per_yrs_final, "person-time** of follow-up, with", ceiling(object$events), "outcome events \n ")
cat("assuming an overall event rate =", object$rate, "and therefore an EPP =", object$EPP, " \n \n ")
cat("* 95% CI for overall risk = (",object$int_lci,", ", object$int_uci, "), for true value of ", object$int_cuminc, " and sample size n = ",object$sample_size,sep = "")
cat(" \n **where time is in the units mean follow-up time was specified in")
}
if (object$type == "binary") {
cat(" \n Minimum sample size required for new model development based on user inputs = ", object$sample_size, ", \n ",sep = "")
cat("with", ceiling(object$events), "events (assuming an outcome prevalence =", object$prevalence, ") and an EPP =", object$EPP, " \n \n ")
}
}
```

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