Nothing
R/non-overlap-measures.R
In SingleCaseES: A Calculator for Single-Case Effect Sizes
Defines functions
calc_IRD
IRD
calc_PAND
PAND
calc_PEM
PEM
calc_PND
PND
calc_Tau_BC
Tau_BC
calc_Tau_U
Tau_U
calc_Tau
Tau
calc_NAP
NAP
Documented in
IRD
NAP
PAND
PEM
PND
Tau
Tau_BC
Tau_U
#' @title Non-overlap of all pairs
#'
#' @description Calculates the non-overlap of all pairs index (Parker & Vannest,
#' 2009).
#'
#' @param SE character value indicating which formula to use for calculating the
#' standard error of NAP, with possible values \code{"unbiased"} for the
#' exactly unbiased estimator, \code{"Hanley"} for the Hanley-McNeil
#' estimator, \code{"null"} for the (known) variance under the null hypothesis
#' of no effect, or \code{"none"} to not calculate a standard error. Defaults
#' to "unbiased".
#' @param trunc_const logical value indicating whether to return the truncation
#' constant used to calculate the standard error.
#'
#' @inheritParams calc_ES
#'
#'
#' @details NAP is calculated as the proportion of all pairs of one observation
#' from each phase in which the measurement from the B phase improves upon the
#' measurement from the A phase, with pairs of data points that are exactly
#' tied being given a weight of 0.5. The range of NAP is [0,1], with a null
#' value of 0.5.
#'
#' The unbiased variance estimator was described by Sen (1967) and Mee (1990).
#' The Hanley estimator was proposed by Hanley and McNeil (1982). The null
#' variance is a known function of sample size, equal to the exact sampling
#' variance when the null hypothesis of no effect holds. When the null
#' hypothesis does not hold, the null variance will tend to over-estimate the
#' true sampling variance of NAP.
#'
#' The confidence interval for NAP is calculated based on the symmetrized
#' score-inversion method (Method 5) proposed by Newcombe (2006).
#'
#' @references
#'
#' Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under
#' a receiver operating characteristic (ROC) curve. \emph{Radiology, 143},
#' 29--36. doi:\doi{10.1148/radiology.143.1.7063747}
#'
#' Mee, W. (1990). Confidence intervals for probabilities and tolerance regions
#' based on a generalization of the Mann-Whitney statistic. \emph{Journal of the
#' American Statistical Association, 85}(411), 793-800.
#' doi:\doi{10.1080/01621459.1990.10474942}
#'
#' Newcombe, R. G. (2006). Confidence intervals for an effect size measure based
#' on the Mann-Whitney statistic. Part 2: Asymptotic methods and evaluation.
#' \emph{Statistics in Medicine, 25}(4), 559--573. doi:\doi{10.1002/sim.2324}
#'
#' Parker, R. I., & Vannest, K. J. (2009). An improved effect size for
#' single-case research: Nonoverlap of all pairs. \emph{Behavior Therapy,
#' 40}(4), 357--67. doi:\doi{10.1016/j.beth.2008.10.006}
#'
#' Sen, P. K. (1967). A note on asymptotically distribution-free confidence
#' bounds for P\{X<Y\}, based on two independent samples. \emph{The Annals of
#' Mathematical Statistics, 29}(1), 95-102.
#' \href{https://www.jstor.org/stable/25049448}{https://www.jstor.org/stable/25049448}
#'
#' @export
#'
#' @return A data.frame containing the estimate, standard error, and/or
#' confidence interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' NAP(A_data = A, B_data = B)
#'
#' # Example from Parker & Vannest (2009)
#' yA <- c(4, 3, 4, 3, 4, 7, 5, 2, 3, 2)
#' yB <- c(5, 9, 7, 9, 7, 5, 9, 11, 11, 10, 9)
#' NAP(yA, yB)
#'
NAP <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "NAP", improvement = improvement, SE = SE,
confidence = confidence, trunc_const = trunc_const)
}
calc_NAP <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE, ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
Q_mat <- matrix(sapply(B_data, function(j) (j > A_data) + 0.5 * (j == A_data)), nrow = m, ncol = n)
NAP <- mean(Q_mat)
res <- data.frame(ES = "NAP", Est = NAP, stringsAsFactors = FALSE)
if (SE != "none") {
Q1 <- sum(rowSums(Q_mat - NAP)^2) / (m * n^2)
Q2 <- sum(colSums(Q_mat - NAP)^2) / (m^2 * n)
trunc <- 0.5 / (m * n)
NAP_trunc <- min(max(NAP, trunc), 1 - trunc)
if (SE == "unbiased") {
X <- sum((Q_mat - NAP)^2) / (m * n)
V <- (NAP_trunc * (1 - NAP_trunc) + n * Q1 + m * Q2 - 2 * X) / ((m - 1) * (n - 1))
}
if (SE == "Hanley") {
V <- (NAP_trunc * (1 - NAP_trunc) + (n - 1) * Q1 + (m - 1) * Q2) / (m * n)
}
if (SE == "null") {
V <- (m + n + 1) / (12 * m * n)
}
res$SE <- sqrt(V)
if (trunc_const) res$trunc <- trunc
if (!is.null(confidence)) {
h <- (m + n) / 2 - 1
z <- qnorm(1 - (1 - confidence) / 2)
f <- function(x) m * n * (NAP - x)^2 * (2 - x) * (1 + x) -
z^2 * x * (1 - x) * (2 + h + (1 + 2 * h) * x * (1 - x))
res$CI_lower <- if (NAP > 0) uniroot(f, c(0, NAP))$root else 0
res$CI_upper <- if (NAP < 1) uniroot(f, c(NAP, 1))$root else 1
}
}
res
}
#' @title Tau (non-overlap)
#'
#' @description Calculates the Tau (non-overlap) index (Parker, Vannest, Davis,
#' & Sauber 2011).
#'
#' @inheritParams NAP
#'
#' @details Tau (non-overlap) a linear re-scaling of \code{\link{NAP}} to the
#' range [-1,1], with a null value of 0.
#'
#' Standard errors and confidence intervals for Tau are based on
#' transformations of the corresponding SEs and CIs for \code{\link{NAP}}.
#'
#' @references Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B.
#' (2011). Combining nonoverlap and trend for single-case research: Tau-U.
#' \emph{Behavior Therapy, 42}(2), 284--299.
#' doi:\doi{10.1016/j.beth.2010.08.006}
#'
#'
#' @export
#'
#' @return A list containing the estimate, standard error, and/or confidence
#' interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau(A_data = A, B_data = B)
#'
Tau <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau", improvement = improvement, SE = SE,
confidence = confidence, trunc_const = trunc_const)
}
calc_Tau <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", CI = TRUE,
confidence = .95, trunc_const = FALSE, ...) {
nap <- calc_NAP(A_data = A_data, B_data = B_data,
improvement = improvement,
SE = SE, CI = CI, confidence = confidence, trunc_const = trunc_const)
res <- data.frame(ES = "Tau", Est = 2 * nap$Est - 1, stringsAsFactors = FALSE)
if (SE != "none") {
res$SE <- 2 * nap$SE
if (trunc_const) res$trunc <- 2 * nap$trunc
}
if (!is.null(confidence)) {
res$CI_lower <- 2 * nap$CI_lower - 1
res$CI_upper <- 2 * nap$CI_upper - 1
}
res
}
#' @title Tau-U
#'
#' @description Calculates the Tau-U index with baseline trend correction
#' (Parker, Vannest, Davis, & Sauber 2011).
#'
#' @param A_data vector of numeric data for A phase, sorted in order of session
#' number. Missing values are dropped.
#' @inheritParams NAP
#'
#' @details Tau-U is an elaboration of the \code{\link{Tau}} that includes a
#' correction for baseline trend. It is calculated as Kendall's S statistic
#' for the comparison between the phase B data and the phase A data, plus
#' Kendall's S statistic for the A phase observations, scaled by the product
#' of the number of observations in each phase.
#'
#' Note that \code{A_data} must be ordered by session number.
#'
#' @references Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B.
#' (2011). Combining nonoverlap and trend for single-case research: Tau-U.
#' \emph{Behavior Therapy, 42}(2), 284--299.
#' doi:\doi{10.1016/j.beth.2010.08.006}
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau_U(A_data = A, B_data = B)
#'
Tau_U <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau_U", improvement = improvement)
}
calc_Tau_U <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
Q_P <- sapply(B_data, function(j) (j > A_data) - (j < A_data))
Q_B <- sapply(A_data, function(j) (j > A_data) - (j < A_data))
TauU <- (sum(Q_P) - sum(Q_B[upper.tri(Q_B)])) / (m * n)
data.frame(ES = "Tau-U", Est = TauU, stringsAsFactors = FALSE)
}
#' @title Tau-BC
#'
#' @description Calculates the baseline-corrected Tau index (Tarlow 2017).
#'
#' @param SE character value indicating which formula to use for calculating the
#' standard error of Tau-BC, with possible values \code{"unbiased"} for the
#' exactly unbiased estimator, \code{"Hanley"} for the Hanley-McNeil
#' estimator, \code{"null"} for the (known) variance under the null hypothesis
#' of no effect, or \code{"none"} to not calculate a standard error. Defaults
#' to "unbiased". Note that the "unbiased" standard error is unbiased for
#' \code{\link{Tau}}, but not necessarily unbiased for \code{\link{Tau_BC}}.
#' None of the standard error formulas account for the additional uncertainty
#' due to use of the baseline trend correction.
#' @param Kendall logical value indicating whether to use Kendall's rank
#' correlation to calculate the Tau effect size measure. If \code{TRUE}, the
#' Kendall's rank correlation (with adjustment for ties) is calculated between
#' the data and a dummy coded phase variable, which is consistent with the
#' method used in Tarlow (2017). Default is \code{FALSE}, which calculates
#' \code{\link{Tau}} (non-overlap) index (without adjustment for ties).
#' @param pretest_trend significance level for the initial baseline trend test.
#' The raw data are corrected and \code{\link{Tau_BC}} is calculated only if
#' the baseline trend is statistically significant. Otherwise,
#' \code{\link{Tau_BC}} is equal to \code{\link{Tau}}. Default is
#' \code{FALSE}, which always adjusts for the baseline trend.
#' @param report_correction logical value indicating whether to report the
#' baseline corrected slope and intercept values. Default is \code{FALSE}.
#' @param warn logical value indicating whether to print a message regarding the
#' outcome of the baseline trend test. Default is \code{TRUE}.
#' @inheritParams NAP
#'
#' @details Tau-BC is an elaboration of the \code{\link{Tau}} that includes a
#' correction for baseline trend. The calculation of Tau-BC involves two or
#' three steps, depending on the \code{pretest_trend} argument.
#'
#' If \code{pretest_trend = FALSE} (the default), the first step involves
#' adjusting the outcomes for baseline trend estimated using Theil-Sen
#' regression. In the second step, the residuals from Theil-Sen regression are
#' used to calculate the \code{Tau} (using either Kendall's rank correlation,
#' with adjustment for ties, or computing Tau directly, without adjustment for
#' ties).
#'
#' Alternately, \code{pretest_trend} can be set equal to a significance level
#' between 0 and 1 (e.g. \code{pretest_trend = .05}, as suggested by Tarlow
#' (2017). In this case, the first step involves a significance test for the
#' slope of the baseline trend based on Kendall's rank correlation. If the
#' slope is not significantly different from zero, then no baseline trend
#' adjustment is made and Tau-BC is set equal to \code{Tau} index. If the
#' slope is significantly different from zero, then in the second step, the
#' outcomes are adjusted for baseline trend using Theil-Sen regression. Then,
#' in the third step, the residuals from Theil-Sen regression are used to
#' calculate the \code{Tau} index. If \code{Kendall = FALSE} (the default),
#' then \code{\link{Tau}} (non-overlap) index is calculated. If \code{Kendall
#' = TRUE}, then Kendall's rank correlation is calculated, including
#' adjustment for ties, as in Tarlow (2017).
#'
#' Note that the standard error formulas are based on the standard errors for
#' \code{\link{Tau}} (non-overlap) and they do not account for the additional
#' uncertainty due to use of the baseline trend correction (nor to the
#' pre-test for statistical significance of baseline trend, if used).
#'
#' @seealso \code{\link{Tau}}, \code{\link{Tau_U}}
#'
#' @references Tarlow, K. R. (2017). An improved rank correlation effect size
#' statistic for single-case designs: Baseline corrected Tau. \emph{Behavior
#' modification, 41}(4), 427-467. doi:\doi{10.1177/0145445516676750}
#'
#' @export
#'
#' @return A list containing the estimate, standard error, and/or confidence
#' interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau_BC(A_data = A, B_data = B)
#'
#' @importFrom utils combn
#'
Tau_BC <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95,
trunc_const = FALSE,
Kendall = FALSE,
pretest_trend = FALSE,
report_correction = FALSE,
warn = TRUE
) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau_BC", improvement = improvement,
SE = SE, confidence = confidence, trunc_const = trunc_const,
Kendall = Kendall,
pretest_trend = pretest_trend,
report_correction = report_correction,
warn = warn
)
}
calc_Tau_BC <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", CI = TRUE,
confidence = .95, trunc_const = FALSE,
Kendall = FALSE,
pretest_trend = FALSE,
report_correction = FALSE,
warn = TRUE,
...) {
m <- length(A_data)
n <- length(B_data)
session_A <- 1:m
session_B <- (m + 1) : (m + n)
if (pretest_trend > 0 & pretest_trend < 1) {
if (!requireNamespace("Kendall", quietly = TRUE)) {
stop("The baseline trend test requires the Kendall package. Please install it.", call. = FALSE)
}
pval_slope_A <- Kendall::Kendall(A_data, session_A)$sl
if (pval_slope_A > pretest_trend) {
if (warn) message("The baseline trend is not statistically significant. Tau is calculated without trend correction.")
adjust <- FALSE
} else {
adjust <- TRUE
}
} else if (pretest_trend != FALSE) {
stop("The pretest_trend argument must be FALSE or a number between 0 and 1.")
} else {
adjust <- TRUE
}
# Calculate baseline trend coefficients
slopes <- apply(combn(m, 2), 2, function(x) diff(A_data[x]) / diff(session_A[x]))
slope <- median(slopes)
intercept <- median(A_data - session_A * slope)
# Adjust if necessary
if (adjust) {
A_data <- A_data - intercept - slope * session_A
B_data <- B_data - intercept - slope * session_B
}
if (Kendall) {
if (improvement == "decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
# Tarlow (2017) approach to calculating Tau
if (sd(c(A_data, B_data)) == 0) {
tau <- 0
} else {
res_Kendall <- Kendall::Kendall(c(A_data, B_data),
c(rep(0L, m), rep(1L, n)))
tau <- as.numeric(res_Kendall$tau)
}
res <- data.frame(ES = "Tau-BC", Est = tau, stringsAsFactors = FALSE)
res$SE <- sqrt((2/(m+n)) * (1-(tau^2)))
if (!is.null(confidence)) {
CI <- tau + c(-1, 1) * qnorm(1 - (1 - confidence) / 2) * res$SE
res$CI_lower <- CI[1]
res$CI_upper <- CI[2]
}
} else {
# Calculate basic Tau (no adjustment for ties)
res <- calc_Tau(A_data = A_data, B_data = B_data,
improvement = improvement, SE = SE,
CI = CI, confidence = confidence, trunc_const = trunc_const)
res$ES <- "Tau-BC"
}
if (report_correction) {
res$slope <- slope
res$intercept <- intercept
if (pretest_trend != FALSE) res$pval_slope_A <- pval_slope_A
}
return(res)
}
#' @title Percentage of non-overlapping data
#'
#' @description Calculates the percentage of non-overlapping data index
#' (Scruggs, Mastropieri, & Castro, 1987).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PND is calculated as the
#' proportion of observations in the B phase that exceed the highest
#' observation from the A phase. For an outcome where decrease is desirable,
#' PND is the proportion of observations in the B phase that are less than the
#' lowest observation from the A phase. The range of PND is [0,1].
#'
#' @references Scruggs, T. E., Mastropieri, M. A., & Casto, G. (1987). The
#' quantitative synthesis of single-subject research: Methodology and
#' validation. \emph{Remedial and Special Education, 8}(2), 24--43.
#' doi:\doi{10.1177/074193258700800206}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PND(A_data = A, B_data = B)
#'
PND <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PND", improvement = improvement)
}
calc_PND <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
PND <- mean(B_data > max(A_data, na.rm = TRUE), na.rm = TRUE)
data.frame(ES = "PND", Est = PND, stringsAsFactors = FALSE)
}
#' @title Percentage exceeding the median
#'
#' @description Calculates the percentage exceeding the median (PEM) index (Ma,
#' 2006).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PEM is calculated as the
#' proportion of observations in the B phase that exceed the median
#' observation from the A phase. For an outcome where decrease is desirable,
#' PEM is calculated as the proportion of observations in the B phase that are
#' less than the median observation from the A phase. Ties are counted with a
#' weight of 0.5. The range of PEM is [0,1].
#'
#' @references Ma, H.-H. (2006). An alternative method for quantitative
#' synthesis of single-subject researches: Percentage of data points exceeding
#' the median. \emph{Behavior Modification, 30}(5), 598--617.
#' doi:\doi{10.1177/0145445504272974}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PEM(A_data = A, B_data = B)
#'
PEM <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PEM", improvement = improvement)
}
calc_PEM <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
med <- median(A_data, na.rm = TRUE)
PEM <- mean((B_data > med) + 0.5 * (B_data == med), na.rm = TRUE)
data.frame(ES = "PEM", Est = PEM, stringsAsFactors = FALSE)
}
#' @title Percentage of all non-overlapping data (PAND)
#'
#' @description Calculates the percentage of all non-overlapping data index
#' (Parker, Hagan-Burke, & Vannest, 2007; Parker, Vannest, & Davis, 2011).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PAND is calculated as
#' the proportion of observations remaining after removing the fewest possible
#' number of observations from either phase so that the highest remaining
#' point from the baseline phase is less than the lowest remaining point from
#' the treatment phase. For an outcome where decrease is desirable, PAND is calculated as
#' the proportion of observations remaining after removing the fewest possible
#' number of observations from either phase so that the lowest remaining
#' point from the baseline phase is greater than the highest remaining point from
#' the treatment phase. The range of PAND depends on the number of
#' observations in each phase.
#'
#' @references Parker, R. I., Hagan-Burke, S., & Vannest, K. J. (2007).
#' Percentage of all non-overlapping data (PAND): An alternative to PND.
#' \emph{The Journal of Special Education, 40}(4), 194--204.
#' doi:\doi{10.1177/00224669070400040101}
#'
#' Parker, R. I., Vannest, K. J., & Davis, J. L. (2011). Effect size in
#' single-case research: A review of nine nonoverlap techniques.
#' \emph{Behavior Modification, 35}(4), 303--22.
#' doi:\doi{10.1177/0145445511399147}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PAND(A_data = A, B_data = B)
#'
PAND <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PAND", improvement = improvement)
}
calc_PAND <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
X <- c(-Inf, sort(A_data))
Y <- c(sort(B_data), Inf)
ij <- expand.grid(i = 1:(m + 1), j = 1:(n + 1))
ij$no_overlap <- mapply(function(i, j) X[i] < Y[j], i = ij$i, j = ij$j)
ij$overlap <- with(ij, i + n - j)
overlaps <- with(ij, max(overlap * no_overlap))
PAND <- overlaps / (m + n)
data.frame(ES = "PAND", Est = PAND, stringsAsFactors = FALSE)
}
#' @title Robust improvement rate difference
#'
#' @description Calculates the robust improvement rate difference index (Parker,
#' Vannest, & Brown, 2009). The range of IRD depends on the number of
#' observations in each phase.
#'
#' @inheritParams NAP
#'
#' @seealso \code{\link{PAND}}
#'
#' @references Parker, R. I., Vannest, K. J., & Brown, L. (2009). The
#' improvement rate difference for single-case research. \emph{Exceptional Children,
#' 75}(2), 135--150. doi:\doi{10.1177/001440290907500201}
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' IRD(A_data = A, B_data = B)
#'
IRD <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "IRD", improvement = improvement)
}
calc_IRD <- function(A_data, B_data, improvement = "increase", ...) {
pand <- calc_PAND(A_data = A_data, B_data = B_data, improvement = improvement)
m <- sum(!is.na(A_data))
n <- sum(!is.na(B_data))
IRD <- ((m+n)^2 * pand$Est - m^2 - n^2) / (2 * m * n)
data.frame(ES = "IRD", Est = IRD, stringsAsFactors = FALSE)
}
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Defines functions calc_IRD IRD calc_PAND PAND calc_PEM PEM calc_PND PND calc_Tau_BC Tau_BC calc_Tau_U Tau_U calc_Tau Tau calc_NAP NAP
Documented in IRD NAP PAND PEM PND Tau Tau_BC Tau_U
#' @title Non-overlap of all pairs
#'
#' @description Calculates the non-overlap of all pairs index (Parker & Vannest,
#' 2009).
#'
#' @param SE character value indicating which formula to use for calculating the
#' standard error of NAP, with possible values \code{"unbiased"} for the
#' exactly unbiased estimator, \code{"Hanley"} for the Hanley-McNeil
#' estimator, \code{"null"} for the (known) variance under the null hypothesis
#' of no effect, or \code{"none"} to not calculate a standard error. Defaults
#' to "unbiased".
#' @param trunc_const logical value indicating whether to return the truncation
#' constant used to calculate the standard error.
#'
#' @inheritParams calc_ES
#'
#'
#' @details NAP is calculated as the proportion of all pairs of one observation
#' from each phase in which the measurement from the B phase improves upon the
#' measurement from the A phase, with pairs of data points that are exactly
#' tied being given a weight of 0.5. The range of NAP is [0,1], with a null
#' value of 0.5.
#'
#' The unbiased variance estimator was described by Sen (1967) and Mee (1990).
#' The Hanley estimator was proposed by Hanley and McNeil (1982). The null
#' variance is a known function of sample size, equal to the exact sampling
#' variance when the null hypothesis of no effect holds. When the null
#' hypothesis does not hold, the null variance will tend to over-estimate the
#' true sampling variance of NAP.
#'
#' The confidence interval for NAP is calculated based on the symmetrized
#' score-inversion method (Method 5) proposed by Newcombe (2006).
#'
#' @references
#'
#' Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under
#' a receiver operating characteristic (ROC) curve. \emph{Radiology, 143},
#' 29--36. doi:\doi{10.1148/radiology.143.1.7063747}
#'
#' Mee, W. (1990). Confidence intervals for probabilities and tolerance regions
#' based on a generalization of the Mann-Whitney statistic. \emph{Journal of the
#' American Statistical Association, 85}(411), 793-800.
#' doi:\doi{10.1080/01621459.1990.10474942}
#'
#' Newcombe, R. G. (2006). Confidence intervals for an effect size measure based
#' on the Mann-Whitney statistic. Part 2: Asymptotic methods and evaluation.
#' \emph{Statistics in Medicine, 25}(4), 559--573. doi:\doi{10.1002/sim.2324}
#'
#' Parker, R. I., & Vannest, K. J. (2009). An improved effect size for
#' single-case research: Nonoverlap of all pairs. \emph{Behavior Therapy,
#' 40}(4), 357--67. doi:\doi{10.1016/j.beth.2008.10.006}
#'
#' Sen, P. K. (1967). A note on asymptotically distribution-free confidence
#' bounds for P\{X<Y\}, based on two independent samples. \emph{The Annals of
#' Mathematical Statistics, 29}(1), 95-102.
#' \href{https://www.jstor.org/stable/25049448}{https://www.jstor.org/stable/25049448}
#'
#' @export
#'
#' @return A data.frame containing the estimate, standard error, and/or
#' confidence interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' NAP(A_data = A, B_data = B)
#'
#' # Example from Parker & Vannest (2009)
#' yA <- c(4, 3, 4, 3, 4, 7, 5, 2, 3, 2)
#' yB <- c(5, 9, 7, 9, 7, 5, 9, 11, 11, 10, 9)
#' NAP(yA, yB)
#'
NAP <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "NAP", improvement = improvement, SE = SE,
confidence = confidence, trunc_const = trunc_const)
}
calc_NAP <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE, ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
Q_mat <- matrix(sapply(B_data, function(j) (j > A_data) + 0.5 * (j == A_data)), nrow = m, ncol = n)
NAP <- mean(Q_mat)
res <- data.frame(ES = "NAP", Est = NAP, stringsAsFactors = FALSE)
if (SE != "none") {
Q1 <- sum(rowSums(Q_mat - NAP)^2) / (m * n^2)
Q2 <- sum(colSums(Q_mat - NAP)^2) / (m^2 * n)
trunc <- 0.5 / (m * n)
NAP_trunc <- min(max(NAP, trunc), 1 - trunc)
if (SE == "unbiased") {
X <- sum((Q_mat - NAP)^2) / (m * n)
V <- (NAP_trunc * (1 - NAP_trunc) + n * Q1 + m * Q2 - 2 * X) / ((m - 1) * (n - 1))
}
if (SE == "Hanley") {
V <- (NAP_trunc * (1 - NAP_trunc) + (n - 1) * Q1 + (m - 1) * Q2) / (m * n)
}
if (SE == "null") {
V <- (m + n + 1) / (12 * m * n)
}
res$SE <- sqrt(V)
if (trunc_const) res$trunc <- trunc
if (!is.null(confidence)) {
h <- (m + n) / 2 - 1
z <- qnorm(1 - (1 - confidence) / 2)
f <- function(x) m * n * (NAP - x)^2 * (2 - x) * (1 + x) -
z^2 * x * (1 - x) * (2 + h + (1 + 2 * h) * x * (1 - x))
res$CI_lower <- if (NAP > 0) uniroot(f, c(0, NAP))$root else 0
res$CI_upper <- if (NAP < 1) uniroot(f, c(NAP, 1))$root else 1
}
}
res
}
#' @title Tau (non-overlap)
#'
#' @description Calculates the Tau (non-overlap) index (Parker, Vannest, Davis,
#' & Sauber 2011).
#'
#' @inheritParams NAP
#'
#' @details Tau (non-overlap) a linear re-scaling of \code{\link{NAP}} to the
#' range [-1,1], with a null value of 0.
#'
#' Standard errors and confidence intervals for Tau are based on
#' transformations of the corresponding SEs and CIs for \code{\link{NAP}}.
#'
#' @references Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B.
#' (2011). Combining nonoverlap and trend for single-case research: Tau-U.
#' \emph{Behavior Therapy, 42}(2), 284--299.
#' doi:\doi{10.1016/j.beth.2010.08.006}
#'
#'
#' @export
#'
#' @return A list containing the estimate, standard error, and/or confidence
#' interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau(A_data = A, B_data = B)
#'
Tau <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95, trunc_const = FALSE) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau", improvement = improvement, SE = SE,
confidence = confidence, trunc_const = trunc_const)
}
calc_Tau <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", CI = TRUE,
confidence = .95, trunc_const = FALSE, ...) {
nap <- calc_NAP(A_data = A_data, B_data = B_data,
improvement = improvement,
SE = SE, CI = CI, confidence = confidence, trunc_const = trunc_const)
res <- data.frame(ES = "Tau", Est = 2 * nap$Est - 1, stringsAsFactors = FALSE)
if (SE != "none") {
res$SE <- 2 * nap$SE
if (trunc_const) res$trunc <- 2 * nap$trunc
}
if (!is.null(confidence)) {
res$CI_lower <- 2 * nap$CI_lower - 1
res$CI_upper <- 2 * nap$CI_upper - 1
}
res
}
#' @title Tau-U
#'
#' @description Calculates the Tau-U index with baseline trend correction
#' (Parker, Vannest, Davis, & Sauber 2011).
#'
#' @param A_data vector of numeric data for A phase, sorted in order of session
#' number. Missing values are dropped.
#' @inheritParams NAP
#'
#' @details Tau-U is an elaboration of the \code{\link{Tau}} that includes a
#' correction for baseline trend. It is calculated as Kendall's S statistic
#' for the comparison between the phase B data and the phase A data, plus
#' Kendall's S statistic for the A phase observations, scaled by the product
#' of the number of observations in each phase.
#'
#' Note that \code{A_data} must be ordered by session number.
#'
#' @references Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B.
#' (2011). Combining nonoverlap and trend for single-case research: Tau-U.
#' \emph{Behavior Therapy, 42}(2), 284--299.
#' doi:\doi{10.1016/j.beth.2010.08.006}
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau_U(A_data = A, B_data = B)
#'
Tau_U <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau_U", improvement = improvement)
}
calc_Tau_U <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
Q_P <- sapply(B_data, function(j) (j > A_data) - (j < A_data))
Q_B <- sapply(A_data, function(j) (j > A_data) - (j < A_data))
TauU <- (sum(Q_P) - sum(Q_B[upper.tri(Q_B)])) / (m * n)
data.frame(ES = "Tau-U", Est = TauU, stringsAsFactors = FALSE)
}
#' @title Tau-BC
#'
#' @description Calculates the baseline-corrected Tau index (Tarlow 2017).
#'
#' @param SE character value indicating which formula to use for calculating the
#' standard error of Tau-BC, with possible values \code{"unbiased"} for the
#' exactly unbiased estimator, \code{"Hanley"} for the Hanley-McNeil
#' estimator, \code{"null"} for the (known) variance under the null hypothesis
#' of no effect, or \code{"none"} to not calculate a standard error. Defaults
#' to "unbiased". Note that the "unbiased" standard error is unbiased for
#' \code{\link{Tau}}, but not necessarily unbiased for \code{\link{Tau_BC}}.
#' None of the standard error formulas account for the additional uncertainty
#' due to use of the baseline trend correction.
#' @param Kendall logical value indicating whether to use Kendall's rank
#' correlation to calculate the Tau effect size measure. If \code{TRUE}, the
#' Kendall's rank correlation (with adjustment for ties) is calculated between
#' the data and a dummy coded phase variable, which is consistent with the
#' method used in Tarlow (2017). Default is \code{FALSE}, which calculates
#' \code{\link{Tau}} (non-overlap) index (without adjustment for ties).
#' @param pretest_trend significance level for the initial baseline trend test.
#' The raw data are corrected and \code{\link{Tau_BC}} is calculated only if
#' the baseline trend is statistically significant. Otherwise,
#' \code{\link{Tau_BC}} is equal to \code{\link{Tau}}. Default is
#' \code{FALSE}, which always adjusts for the baseline trend.
#' @param report_correction logical value indicating whether to report the
#' baseline corrected slope and intercept values. Default is \code{FALSE}.
#' @param warn logical value indicating whether to print a message regarding the
#' outcome of the baseline trend test. Default is \code{TRUE}.
#' @inheritParams NAP
#'
#' @details Tau-BC is an elaboration of the \code{\link{Tau}} that includes a
#' correction for baseline trend. The calculation of Tau-BC involves two or
#' three steps, depending on the \code{pretest_trend} argument.
#'
#' If \code{pretest_trend = FALSE} (the default), the first step involves
#' adjusting the outcomes for baseline trend estimated using Theil-Sen
#' regression. In the second step, the residuals from Theil-Sen regression are
#' used to calculate the \code{Tau} (using either Kendall's rank correlation,
#' with adjustment for ties, or computing Tau directly, without adjustment for
#' ties).
#'
#' Alternately, \code{pretest_trend} can be set equal to a significance level
#' between 0 and 1 (e.g. \code{pretest_trend = .05}, as suggested by Tarlow
#' (2017). In this case, the first step involves a significance test for the
#' slope of the baseline trend based on Kendall's rank correlation. If the
#' slope is not significantly different from zero, then no baseline trend
#' adjustment is made and Tau-BC is set equal to \code{Tau} index. If the
#' slope is significantly different from zero, then in the second step, the
#' outcomes are adjusted for baseline trend using Theil-Sen regression. Then,
#' in the third step, the residuals from Theil-Sen regression are used to
#' calculate the \code{Tau} index. If \code{Kendall = FALSE} (the default),
#' then \code{\link{Tau}} (non-overlap) index is calculated. If \code{Kendall
#' = TRUE}, then Kendall's rank correlation is calculated, including
#' adjustment for ties, as in Tarlow (2017).
#'
#' Note that the standard error formulas are based on the standard errors for
#' \code{\link{Tau}} (non-overlap) and they do not account for the additional
#' uncertainty due to use of the baseline trend correction (nor to the
#' pre-test for statistical significance of baseline trend, if used).
#'
#' @seealso \code{\link{Tau}}, \code{\link{Tau_U}}
#'
#' @references Tarlow, K. R. (2017). An improved rank correlation effect size
#' statistic for single-case designs: Baseline corrected Tau. \emph{Behavior
#' modification, 41}(4), 427-467. doi:\doi{10.1177/0145445516676750}
#'
#' @export
#'
#' @return A list containing the estimate, standard error, and/or confidence
#' interval.
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' Tau_BC(A_data = A, B_data = B)
#'
#' @importFrom utils combn
#'
Tau_BC <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased", confidence = .95,
trunc_const = FALSE,
Kendall = FALSE,
pretest_trend = FALSE,
report_correction = FALSE,
warn = TRUE
) {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "Tau_BC", improvement = improvement,
SE = SE, confidence = confidence, trunc_const = trunc_const,
Kendall = Kendall,
pretest_trend = pretest_trend,
report_correction = report_correction,
warn = warn
)
}
calc_Tau_BC <- function(A_data, B_data,
improvement = "increase",
SE = "unbiased", CI = TRUE,
confidence = .95, trunc_const = FALSE,
Kendall = FALSE,
pretest_trend = FALSE,
report_correction = FALSE,
warn = TRUE,
...) {
m <- length(A_data)
n <- length(B_data)
session_A <- 1:m
session_B <- (m + 1) : (m + n)
if (pretest_trend > 0 & pretest_trend < 1) {
if (!requireNamespace("Kendall", quietly = TRUE)) {
stop("The baseline trend test requires the Kendall package. Please install it.", call. = FALSE)
}
pval_slope_A <- Kendall::Kendall(A_data, session_A)$sl
if (pval_slope_A > pretest_trend) {
if (warn) message("The baseline trend is not statistically significant. Tau is calculated without trend correction.")
adjust <- FALSE
} else {
adjust <- TRUE
}
} else if (pretest_trend != FALSE) {
stop("The pretest_trend argument must be FALSE or a number between 0 and 1.")
} else {
adjust <- TRUE
}
# Calculate baseline trend coefficients
slopes <- apply(combn(m, 2), 2, function(x) diff(A_data[x]) / diff(session_A[x]))
slope <- median(slopes)
intercept <- median(A_data - session_A * slope)
# Adjust if necessary
if (adjust) {
A_data <- A_data - intercept - slope * session_A
B_data <- B_data - intercept - slope * session_B
}
if (Kendall) {
if (improvement == "decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
# Tarlow (2017) approach to calculating Tau
if (sd(c(A_data, B_data)) == 0) {
tau <- 0
} else {
res_Kendall <- Kendall::Kendall(c(A_data, B_data),
c(rep(0L, m), rep(1L, n)))
tau <- as.numeric(res_Kendall$tau)
}
res <- data.frame(ES = "Tau-BC", Est = tau, stringsAsFactors = FALSE)
res$SE <- sqrt((2/(m+n)) * (1-(tau^2)))
if (!is.null(confidence)) {
CI <- tau + c(-1, 1) * qnorm(1 - (1 - confidence) / 2) * res$SE
res$CI_lower <- CI[1]
res$CI_upper <- CI[2]
}
} else {
# Calculate basic Tau (no adjustment for ties)
res <- calc_Tau(A_data = A_data, B_data = B_data,
improvement = improvement, SE = SE,
CI = CI, confidence = confidence, trunc_const = trunc_const)
res$ES <- "Tau-BC"
}
if (report_correction) {
res$slope <- slope
res$intercept <- intercept
if (pretest_trend != FALSE) res$pval_slope_A <- pval_slope_A
}
return(res)
}
#' @title Percentage of non-overlapping data
#'
#' @description Calculates the percentage of non-overlapping data index
#' (Scruggs, Mastropieri, & Castro, 1987).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PND is calculated as the
#' proportion of observations in the B phase that exceed the highest
#' observation from the A phase. For an outcome where decrease is desirable,
#' PND is the proportion of observations in the B phase that are less than the
#' lowest observation from the A phase. The range of PND is [0,1].
#'
#' @references Scruggs, T. E., Mastropieri, M. A., & Casto, G. (1987). The
#' quantitative synthesis of single-subject research: Methodology and
#' validation. \emph{Remedial and Special Education, 8}(2), 24--43.
#' doi:\doi{10.1177/074193258700800206}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PND(A_data = A, B_data = B)
#'
PND <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PND", improvement = improvement)
}
calc_PND <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
PND <- mean(B_data > max(A_data, na.rm = TRUE), na.rm = TRUE)
data.frame(ES = "PND", Est = PND, stringsAsFactors = FALSE)
}
#' @title Percentage exceeding the median
#'
#' @description Calculates the percentage exceeding the median (PEM) index (Ma,
#' 2006).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PEM is calculated as the
#' proportion of observations in the B phase that exceed the median
#' observation from the A phase. For an outcome where decrease is desirable,
#' PEM is calculated as the proportion of observations in the B phase that are
#' less than the median observation from the A phase. Ties are counted with a
#' weight of 0.5. The range of PEM is [0,1].
#'
#' @references Ma, H.-H. (2006). An alternative method for quantitative
#' synthesis of single-subject researches: Percentage of data points exceeding
#' the median. \emph{Behavior Modification, 30}(5), 598--617.
#' doi:\doi{10.1177/0145445504272974}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PEM(A_data = A, B_data = B)
#'
PEM <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PEM", improvement = improvement)
}
calc_PEM <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
med <- median(A_data, na.rm = TRUE)
PEM <- mean((B_data > med) + 0.5 * (B_data == med), na.rm = TRUE)
data.frame(ES = "PEM", Est = PEM, stringsAsFactors = FALSE)
}
#' @title Percentage of all non-overlapping data (PAND)
#'
#' @description Calculates the percentage of all non-overlapping data index
#' (Parker, Hagan-Burke, & Vannest, 2007; Parker, Vannest, & Davis, 2011).
#'
#' @inheritParams NAP
#'
#' @details For an outcome where increase is desirable, PAND is calculated as
#' the proportion of observations remaining after removing the fewest possible
#' number of observations from either phase so that the highest remaining
#' point from the baseline phase is less than the lowest remaining point from
#' the treatment phase. For an outcome where decrease is desirable, PAND is calculated as
#' the proportion of observations remaining after removing the fewest possible
#' number of observations from either phase so that the lowest remaining
#' point from the baseline phase is greater than the highest remaining point from
#' the treatment phase. The range of PAND depends on the number of
#' observations in each phase.
#'
#' @references Parker, R. I., Hagan-Burke, S., & Vannest, K. J. (2007).
#' Percentage of all non-overlapping data (PAND): An alternative to PND.
#' \emph{The Journal of Special Education, 40}(4), 194--204.
#' doi:\doi{10.1177/00224669070400040101}
#'
#' Parker, R. I., Vannest, K. J., & Davis, J. L. (2011). Effect size in
#' single-case research: A review of nine nonoverlap techniques.
#' \emph{Behavior Modification, 35}(4), 303--22.
#' doi:\doi{10.1177/0145445511399147}
#'
#'
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' PAND(A_data = A, B_data = B)
#'
PAND <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "PAND", improvement = improvement)
}
calc_PAND <- function(A_data, B_data, improvement = "increase", ...) {
if (improvement=="decrease") {
A_data <- -1 * A_data
B_data <- -1 * B_data
}
m <- length(A_data)
n <- length(B_data)
X <- c(-Inf, sort(A_data))
Y <- c(sort(B_data), Inf)
ij <- expand.grid(i = 1:(m + 1), j = 1:(n + 1))
ij$no_overlap <- mapply(function(i, j) X[i] < Y[j], i = ij$i, j = ij$j)
ij$overlap <- with(ij, i + n - j)
overlaps <- with(ij, max(overlap * no_overlap))
PAND <- overlaps / (m + n)
data.frame(ES = "PAND", Est = PAND, stringsAsFactors = FALSE)
}
#' @title Robust improvement rate difference
#'
#' @description Calculates the robust improvement rate difference index (Parker,
#' Vannest, & Brown, 2009). The range of IRD depends on the number of
#' observations in each phase.
#'
#' @inheritParams NAP
#'
#' @seealso \code{\link{PAND}}
#'
#' @references Parker, R. I., Vannest, K. J., & Brown, L. (2009). The
#' improvement rate difference for single-case research. \emph{Exceptional Children,
#' 75}(2), 135--150. doi:\doi{10.1177/001440290907500201}
#'
#' @export
#'
#' @return Numeric value
#'
#' @examples
#' A <- c(20, 20, 26, 25, 22, 23)
#' B <- c(28, 25, 24, 27, 30, 30, 29)
#' IRD(A_data = A, B_data = B)
#'
IRD <- function(A_data, B_data, condition, outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase") {
calc_ES(A_data = A_data, B_data = B_data,
condition = condition, outcome = outcome,
baseline_phase = baseline_phase,
intervention_phase = intervention_phase,
ES = "IRD", improvement = improvement)
}
calc_IRD <- function(A_data, B_data, improvement = "increase", ...) {
pand <- calc_PAND(A_data = A_data, B_data = B_data, improvement = improvement)
m <- sum(!is.na(A_data))
n <- sum(!is.na(B_data))
IRD <- ((m+n)^2 * pand$Est - m^2 - n^2) / (2 * m * n)
data.frame(ES = "IRD", Est = IRD, stringsAsFactors = FALSE)
}
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