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```
#' Necessary sample size to reach desired power for an independent t-test using
#' an uncertainty and publication bias correction procedure
#'
#' @description \code{ss.power.it} returns the necessary per-group sample size
#' to achieve a desired level of statistical power for a planned study using
#' an independent t-test, 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.it} uses the observed
#' \eqn{t}-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. In the two-group case, this formula reduces to the density of
#' a truncated noncentral \eqn{t}-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 corrects 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.it} assumes that the planned study will have equal n.
#' Unequal n in the previous study is handled in the following way for the
#' independent-t. If the user enters an odd value for N, no information is
#' available on the exact group sizes. The function calculates n by dividing N
#' by 2 and both rounding up and rounding down the result, thus 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. If the user enters a vector for n with two
#' different values, specific information is available on the exact group
#' sizes. n is calcualted as the harmonic mean of these two values (a measure
#' of effective sample size). Again, this is rounded both up and down. 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. When the individual group sizes of an unequal-n previous study
#' are known, we highly encourage entering these explicitly, especially if the
#' sample sizes are quite discrepant, as this allows for the greatest precision
#' in estimating an appropriate planned study n.
#'
#' @param t.observed Observed \eqn{t}-value from a previous study used to plan
#' sample size for a planned study
#' @param n Per group sample size (if equal) or the two group sample sizes of
#' the previous study (enter either a single number or a vector of length 2)
#' @param N Total sample size of the previous study, assumed equal across groups
#' if specified
#' @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.it(t.observed=3, n=20, 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.it <- function(t.observed, n, N, 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))
{
if(!is.null(N))
{
if(N <= 2) stop("Your total sample size is too small")
if(!missing(n)) stop("Because you specified 'N' you should not specify 'n'.")
if(missing(n))
{
n.ru <- ceiling(N/2)
N.ru <- 2*n.ru
DF.ru <- N.ru-2
n.rd <- floor(N/2)
N.rd <- 2*n.rd
DF.rd <- N.rd-2
}
}
}
if(missing(N))
{
if(!(length(n) %in% c(1, 2))) stop("The value of 'n' should be a vector of length two or a single value (for equal group sample sizes)")
if(length(n)==2)
{
n.1 <- n[1]
n.2 <- n[2]
n.ru <- ceiling(2/((1/n.1)+(1/n.2)))
N.ru <- 2*n.ru
DF.ru <- N.ru-2
n.rd <- floor(2/((1/n.1)+(1/n.2)))
N.rd <- 2*n.rd
DF.rd <- N.rd-2
}
if(length(n)==1)
{
n.ru <- n
N.ru <- 2*n.ru
DF.ru <- N.ru-2
n.rd <- n
N.rd <- 2*n.rd
DF.rd <- N.rd-2
}
}
NCP <- seq(from=0, to=100, by=step)
## Rounding up
d.density.ru <- dt(t.observed, df=DF.ru, ncp=NCP)
value.critical.ru <- qt(1-alpha.prior/2, df=DF.ru)
if(t.observed <= value.critical.ru) stop("Your observed t statistic is nonsignificant based on your specfied alpha of the prior study. Please increase 'alpha.prior' so 't.observed' exceeds the critical value")
area.above.critical.value.ru <- 1 - pt(value.critical.ru, df=DF.ru, ncp=NCP)
area.other.tail.ru <- pt(-1*value.critical.ru, df=DF.ru, ncp=NCP)
power.values.ru <- area.above.critical.value.ru + area.other.tail.ru
area.above.t.ru <- 1 - pt(t.observed, df=DF.ru, ncp = NCP)
area.above.t.opp.ru <- pt(-1*t.observed, df = DF.ru, ncp = NCP)
area.area.between.ru <- (area.above.critical.value.ru - area.above.t.ru) + (area.other.tail.ru - area.above.t.opp.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
denom.df <- (2*nrep)-2
diff.ru <- -1
while (diff.ru < 0)
{
criticalT <- qt(1-alpha.planned/2, df = denom.df)
powers1.ru <- 1 - pt(criticalT, df = denom.df, ncp = sqrt(nrep/n.ru)*TM.Percentile.ru)
powers2.ru <- pt(-1*criticalT, df=denom.df, ncp = sqrt(nrep/n.ru)*TM.Percentile.ru)
powers.ru <- powers1.ru + powers2.ru
diff.ru <- powers.ru - power
nrep <- nrep + 1
denom.df = (2*nrep)-2
}
}
repn.ru <- nrep-1
##
## Rounding up
d.density.rd <- dt(t.observed, df=DF.rd, ncp=NCP)
value.critical.rd <- qt(1-alpha.prior/2, df=DF.rd)
if(t.observed <= value.critical.rd) stop("Your observed t statistic is nonsignificant based on your specfied alpha of the prior study. Please increase 'alpha.prior' so 't.observed' exceeds the critical value")
area.above.critical.value.rd <- 1 - pt(value.critical.rd, df=DF.rd, ncp=NCP)
area.other.tail.rd <- pt(-1*value.critical.rd, df=DF.rd, ncp=NCP)
power.values.rd <- area.above.critical.value.rd + area.other.tail.rd
area.above.t.rd <- 1 - pt(t.observed, df=DF.rd, ncp = NCP)
area.above.t.opp.rd <- pt(-1*t.observed, df = DF.rd, ncp = NCP)
area.area.between.rd <- (area.above.critical.value.rd - area.above.t.rd) + (area.other.tail.rd - area.above.t.opp.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
denom.df <- (2*nrep)-2
diff.rd <- -1
while (diff.rd < 0)
{
criticalT <- qt(1-alpha.planned/2, df = denom.df)
powers1.rd <- 1 - pt(criticalT, df = denom.df, ncp = sqrt(nrep/n.rd)*TM.Percentile.rd)
powers2.rd <- pt(-1*criticalT, df=denom.df, ncp = sqrt(nrep/n.rd)*TM.Percentile.rd)
powers.rd <- powers1.rd + powers2.rd
diff.rd <- powers.rd - power
nrep <- nrep + 1
denom.df = (2*nrep)-2
}
}
repn.rd <- nrep-1
output.n <- max(repn.ru, repn.rd)
return(list(output.n, min(TM.Percentile.ru, TM.Percentile.rd)))
}
```

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