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# Between-subjects ANOVA.
#' Necessary sample size to reach desired power for a one or two-way
#' between-subjects ANOVA using an uncertainty and publication bias correction
#' procedure
#'
#' @description \code{ss.power.ba} returns the necessary per-group sample size
#' to achieve a desired level of statistical power for a planned study testing
#' an omnibus effect using a one or two-way fully between-subjects ANOVA,
#' based on information obtained from a previous study. The effect from the
#' previous study can be corrected for publication bias and/or uncertainty to
#' provide a sample size that will achieve more accurate statistical power for
#' a planned study, when compared to approaches that use a sample effect size
#' at face value or rely on sample size only. The bias and uncertainty
#' adjusted previous study noncentrality parameter is also returned, which can
#' be transformed to various effect size metrics.
#'
#' @details Researchers often use the sample effect size from a prior study as
#' an estimate of the likely size of an expected future effect in sample size
#' planning. However, sample effect size estimates should not usually be used
#' at face value to plan sample size, due to both publication bias and
#' uncertainty.
#'
#' The approach implemented in \code{ss.power.ba} uses the observed
#' \eqn{F}-value and sample size from a previous study to correct the
#' noncentrality parameter associated with the effect of interest for
#' publication bias and/or uncertainty. This new estimated noncentrality
#' parameter is then used to calculate the necessary per-group sample size to
#' achieve the desired level of power in the planned study.
#'
#' The approach uses a likelihood function of a truncated non-central F
#' distribution, where the truncation occurs due to small effect sizes being
#' unobserved due to publication bias. The numerator of the likelihood
#' function is simply the density of a noncentral F distribution. The
#' denominator is the power of the test, which serves to truncate the
#' distribution. Thus, the ratio of the numerator and the denominator is a
#' truncated noncentral F distribution. (See Taylor & Muller, 1996, Equation
#' 2.1. and Anderson & Maxwell, 2017, for more details.)
#'
#' Assurance is the proportion of times that power will be at or above the
#' desired level, if the experiment were to be reproduced many times. For
#' example, assurance = .5 means that power will be above the desired level
#' half of the time, but below the desired level the other half of the time.
#' Selecting assurance = .5 (selecting the noncentrality parameter at the 50th
#' percentile of the likelihood distribution) results in a median-unbiased
#' estimate of the population noncentrality parameter and does not correct for
#' uncertainty. In order to correct for uncertainty, assurance > .5
#' can be selected, which corresponds to selecting the noncentrality parameter
#' associated with the (1 - assurance) quantile of the likelihood
#' distribution.
#'
#' If the previous study of interest has not been subjected to publication
#' bias (e.g., a pilot study), \code{alpha.prior} can be set to 1 to indicate
#' no publication bias. Alternative \eqn{\alpha}-levels can also be
#' accommodated to represent differing amounts of publication bias. For
#' example, setting \code{alpha.prior}=.20 would reflect less severe
#' publication bias than the default of .05. In essence, setting
#' \code{alpha.prior} at .20 assumes that studies with \eqn{p}-values less
#' than .20 are published, whereas those with larger \eqn{p}-values are not.
#'
#' In some cases, the corrected noncentrality parameter for a given level of
#' assurance will be estimated to be zero. This is an indication that, at the
#' desired level of assurance, the previous study's effect cannot be
#' accurately estimated due to high levels of uncertainty and bias. When this
#' happens, subsequent sample size planning is not possible with the chosen
#' specifications. Two alternatives are recommended. First, users can select a
#' lower value of assurance (e.g. .8 instead of .95). Second, users can reduce
#' the influence of publciation bias by setting \code{alpha.prior} at a value
#' greater than .05. It is possible to correct for uncertainty only by setting
#' \code{alpha.prior}=1 and choosing the desired level of assurance. We
#' encourage users to make the adjustments as minimal as possible.
#'
#' \code{ss.power.ba} assumes that the planned study will have equal n.
#' Unequal n in the previous study is handled in the following way for
#' between-subjects anova designs. If the user enters an N not equally
#' divisible by the number of cells, the function calculates n by dividing N
#' by the number of cells and both rounding up and rounding down the result,
#' effectively assuming equal n. The suggested sample size for the planned
#' study is calculated using both of these values of n, and the function
#' returns the larger of these two suggestions, to be conservative. The
#' adjusted noncentrality parameter that is output is the lower of the two
#' possibilities, again, to be conservative. Although equal-n previous studies
#' are preferable, this approach will work well as long as the cell sizes are
#' only slightly discrepant.
#'
#' @param F.observed Observed F-value from a previous study used to plan sample
#' size for a planned study
#' @param N Total sample size of the previous study
#' @param levels.A Number of levels for factor A
#' @param levels.B Number of levels for factor B, which is NULL if a single
#' factor design
#' @param effect Effect most of interest to the planned study: main effect of A
#' (\code{factor.A}), main effect of B (\code{factor.B}), interaction
#' (\code{interaction})
#' @param alpha.prior Alpha-level \eqn{\alpha} for the previous study or the
#' assumed statistical significance necessary for publishing in the field; to
#' assume no publication bias, a value of 1 can be entered
#' @param alpha.planned Alpha-level (\eqn{\alpha}) assumed for the planned study
#' @param assurance Desired level of assurance, or the long run proportion of
#' times that the planned study power will reach or surpass desired level
#' (assurance > .5 corrects for uncertainty; assurance < .5 not recommended)
#' @param power Desired level of statistical power for the planned study
#' @param step Value used in the iterative scheme to determine the noncentrality
#' parameter necessary for sample size planning (0 < step < .01) (users should
#' not generally need to change this value; smaller values lead to more
#' accurate sample size planning results, but unnecessarily small values will
#' add unnecessary computational time)
#'
#' @return Suggested per-group sample size for planned study
#' Publication bias and uncertainty- adjusted prior study noncentrality parameter
#'
#' @export
#' @import stats
#'
#' @examples
#' ss.power.ba(F.observed=5, N=120, levels.A=2, levels.B=3, effect="factor.B",
#' alpha.prior=.05, alpha.planned=.05, assurance=.80, power=.80, step=.001)
#'
#' @author Samantha F. Anderson \email{samantha.f.anderson@asu.edu},
#' Ken Kelley \email{kkelley@@nd.edu}
#'
#' @references Anderson, S. F., & Maxwell, S. E. (2017).
#' Addressing the 'replication crisis': Using original studies to design
#' replication studies with appropriate statistical power. \emph{Multivariate
#' Behavioral Research, 52,} 305-322.
#'
#' Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample size
#' planning for more accurate statistical power: A method correcting sample
#' effect sizes for uncertainty and publication bias. \emph{Psychological
#' Science, 28,} 1547-1562.
#'
#' Taylor, D. J., & Muller, K. E. (1996). Bias in linear model power and
#' sample size calculation due to estimating noncentrality.
#' \emph{Communications in Statistics: Theory and Methods, 25,} 1595-1610.
ss.power.ba <- function(F.observed, N, levels.A, levels.B=NULL, effect=c("factor.A", "factor.B", "interaction"), alpha.prior=.05, alpha.planned=.05, assurance=.80, power=.80, step=.001)
{
if(alpha.prior > 1 | alpha.prior <= 0) stop("There is a problem with 'alpha' of the prior study (i.e., the Type I error rate), please specify as a value between 0 and 1 (the default is .05).")
if(alpha.prior == 1) {alpha.prior <- .999 }
if(alpha.planned >= 1 | alpha.planned <= 0) stop("There is a problem with 'alpha' of the planned study (i.e., the Type I error rate), please specify as a value between 0 and 1 (the default is .05).")
if(assurance >= 1)
{
assurance <- assurance/100
}
if(assurance<0 | assurance>1)
{
stop("There is a problem with 'assurance' (i.e., the proportion of times statistical power is at or above the desired value), please specify as a value between 0 and 1 (the default is .80).")
}
if(assurance <.5)
{
warning( "THe assurance you have entered is < .5, which implies you will have under a 50% chance at achieving your desired level of power" )
}
if(power >= 1) power <- power/100
if(power<0 | power>1) stop("There is a problem with 'power' (i.e., desired statistical power), please specify as a value between 0 and 1 (the default is .80).")
if(missing(N)) stop("You must specify 'N', which is the total sample size.")
if(is.null(levels.B)) if(effect=="interaction") stop("You cannot select 'effect=interaction' if you do not specify 'levels.B'.")
if(is.null(levels.B)) cells <- levels.A
if(!is.null(levels.B)) cells <- levels.A*levels.B
NCP <- seq(from=0, to=100, by=step) # sequence of possible values for the noncentral parameter.
if(is.null(levels.B)) type <- "ANOVA.one.way"
if(!is.null(levels.B)) type <- "ANOVA.two.way"
## ROUNDING UP
if(type=="ANOVA.one.way")
{
cells <- levels.A
n.ru <- ceiling(N/cells) # To ensure that the between sample size is appropriate given specifications.
N.ru <- n.ru*cells
df.numerator <- levels.A - 1
df.denominator.ru <- N.ru-levels.A
}
if(type=="ANOVA.two.way")
{
cells <- levels.A*levels.B
n.ru <- ceiling(N/cells) # To ensure that the between sample size is appropriate given specifications.
N.ru <- n.ru*cells
if(effect=="factor.A") df.numerator <- levels.A - 1
if(effect=="factor.B") df.numerator <- levels.B - 1
if(effect=="interaction") df.numerator <- (levels.A - 1)*(levels.B - 1)
df.denominator.ru <- N.ru - cells
}
f.density.ru <- df(F.observed, df1=df.numerator, df2=df.denominator.ru, ncp=NCP) # density of F using F observed
critF.ru <- qf(1-alpha.prior, df1=df.numerator, df2=df.denominator.ru)
if(F.observed <= critF.ru) stop("Your observed F statistic is nonsignificant based on your specfied alpha of the prior study. Please increase 'alpha.prior' so 'F.observed' exceeds the critical value")
power.values.ru <- 1 - pf(critF.ru, df1=df.numerator, df2=df.denominator.ru, ncp = NCP) # area above critical F
area.above.F.ru <- 1 - pf(F.observed, df1=df.numerator, df2=df.denominator.ru, ncp = NCP) # area above observed F
area.area.between.ru <- power.values.ru - area.above.F.ru
TM.ru <- area.area.between.ru/power.values.ru
TM.Percentile.ru <- min(NCP[which(abs(TM.ru-assurance)==min(abs(TM.ru-assurance)))])
if(TM.Percentile.ru==0) stop("The corrected noncentrality parameter is zero. Please either choose a lower value of assurance and/or a higher value of alpha for the prior study (e.g. accounting for less publication bias)")
if(TM.Percentile.ru > 0)
{
nrep <- 2
if(type=="ANOVA.two.way") denom.df <- nrep*cells-(levels.A*levels.B)
if(type=="ANOVA.one.way") denom.df <- nrep*cells-levels.A
diff.ru <- -1
while(diff.ru < 0 )
{
criticalF <- qf(1-alpha.planned, df1 = df.numerator, df2 = denom.df)
powers.ru <- 1 - pf(criticalF, df1 = df.numerator, df2 = denom.df, ncp = (nrep/n.ru)*TM.Percentile.ru)
diff.ru <- powers.ru - power
nrep <- nrep + 1
if(type=="ANOVA.two.way") denom.df <- nrep*cells-(levels.A*levels.B)
if(type=="ANOVA.one.way") denom.df <- nrep*cells-levels.A
}
}
repn.ru <- nrep-1
##
## ROUNDING DOWN
##
if(type=="ANOVA.one.way")
{
cells <- levels.A
n.rd <- floor(N/cells) # To ensure that the between sample size is appropriate given specifications.
N.rd <- n.rd*cells
df.numerator <- levels.A - 1
df.denominator.rd <- N.rd-levels.A
}
if(type=="ANOVA.two.way")
{
cells <- levels.A*levels.B
n.rd <- floor(N/cells) # To ensure that the between sample size is appropriate given specifications.
N.rd <- n.rd*cells
if(effect=="factor.A") df.numerator <- levels.A - 1
if(effect=="factor.B") df.numerator <- levels.B - 1
if(effect=="interaction") df.numerator <- (levels.A - 1)*(levels.B - 1)
df.denominator.rd <- N.rd - cells
}
f.density.rd <- df(F.observed, df1=df.numerator, df2=df.denominator.rd, ncp=NCP) # density of F using F observed
critF.rd <- qf(1-alpha.prior, df1=df.numerator, df2=df.denominator.rd)
if(F.observed <= critF.rd) stop("Your observed F statistic is nonsignificant based on your specfied alpha of the prior study. Please increase 'alpha.prior' so 'F.observed' exceeds the critical value")
power.values.rd <- 1 - pf(critF.rd, df1=df.numerator, df2=df.denominator.rd, ncp = NCP) # area above critical F
area.above.F.rd <- 1 - pf(F.observed, df1=df.numerator, df2=df.denominator.rd, ncp = NCP) # area above observed F
area.area.between.rd <- power.values.rd - area.above.F.rd
TM.rd <- area.area.between.rd/power.values.rd
TM.Percentile.rd <- min(NCP[which(abs(TM.rd-assurance)==min(abs(TM.rd-assurance)))])
if(TM.Percentile.rd==0) stop("The corrected noncentrality parameter is zero. Please either choose a lower value of assurance and/or a higher value of alpha for the prior study (e.g. accounting for less publication bias)")
if(TM.Percentile.rd > 0)
{
nrep <- 2
if(type=="ANOVA.two.way") denom.df <- nrep*cells-(levels.A*levels.B)
if(type=="ANOVA.one.way") denom.df <- nrep*cells-levels.A
diff.rd <- -1
while(diff.rd < 0 )
{
criticalF <- qf(1-alpha.planned, df1 = df.numerator, df2 = denom.df)
powers.rd <- 1 - pf(criticalF, df1 = df.numerator, df2 = denom.df, ncp = (nrep/n.rd)*TM.Percentile.rd)
diff.rd <- powers.rd - power
nrep <- nrep + 1
if(type=="ANOVA.two.way") denom.df <- nrep*cells-(levels.A*levels.B)
if(type=="ANOVA.one.way") denom.df <- nrep*cells-levels.A
}
}
repn.rd <- nrep-1
output.n <- max(repn.ru, repn.rd)
return(list(output.n, min(TM.Percentile.rd, TM.Percentile.ru)))
}
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