if (FALSE) {
##########################
## A Priori Power Analysis
##########################
# GOAL: Find specific sample size in each group for independent t-test
# corresponding to a power rate of .8
#
# For ease of the setup, assume the groups are the same size, and the mean
# difference corresponds to Cohen's d values of .2, .5, and .8
# This example can be solved numerically using the pwr package (see below),
# though the following simulation setup is far more general and can be
# used for any generate-analyse combination of interest
# SimFunctions(SimSolve=TRUE)
#### Step 1 --- Define your conditions under study and create design data.frame.
#### However, use NA placeholder for sample size as it must be solved,
#### and add desired power rate to object
Design <- createDesign(N = NA,
d = c(.2, .5, .8),
sig.level = .05)
Design # solve for NA's
#~~~~~~~~~~~~~~~~~~~~~~~~
#### Step 2 --- Define generate, analyse, and summarise functions
Generate <- function(condition, fixed_objects) {
Attach(condition)
group1 <- rnorm(N)
group2 <- rnorm(N, mean=d)
dat <- data.frame(group = gl(2, N, labels=c('G1', 'G2')),
DV = c(group1, group2))
dat
}
Analyse <- function(condition, dat, fixed_objects) {
p <- t.test(DV ~ group, dat, var.equal=TRUE)$p.value
p
}
Summarise <- function(condition, results, fixed_objects) {
# Must return a single number corresponding to f(x) in the
# root equation f(x) = b
ret <- c(power = EDR(results, alpha = condition$sig.level))
ret
}
#~~~~~~~~~~~~~~~~~~~~~~~~
#### Step 3 --- Optimize N over the rows in design
### (For debugging) may want to see if simulation code works as intended first
### for some given set of inputs
# runSimulation(design=createDesign(N=100, d=.8, sig.level=.05),
# replications=10, generate=Generate, analyse=Analyse,
# summarise=Summarise)
# Initial search between N = [10,500] for each row using the default
# integer solver (integer = TRUE). In this example, b = target power
solved <- SimSolve(design=Design, b=.8, interval=c(10, 500),
generate=Generate, analyse=Analyse,
summarise=Summarise)
solved
summary(solved)
plot(solved, 1)
plot(solved, 2)
plot(solved, 3)
# also can plot median history and estimate precision
plot(solved, 1, type = 'history')
plot(solved, 1, type = 'density')
plot(solved, 1, type = 'iterations')
# verify with true power from pwr package
library(pwr)
pwr.t.test(d=.2, power = .8) # sig.level/alpha = .05 by default
pwr.t.test(d=.5, power = .8)
pwr.t.test(d=.8, power = .8)
# use estimated N results to see how close power was
N <- solved$N
pwr.t.test(d=.2, n=N[1])
pwr.t.test(d=.5, n=N[2])
pwr.t.test(d=.8, n=N[3])
# with rounding
N <- ceiling(solved$N)
pwr.t.test(d=.2, n=N[1])
pwr.t.test(d=.5, n=N[2])
pwr.t.test(d=.8, n=N[3])
### failing analytic formula, confirm results with more precise
### simulation via runSimulation()
### (not required, if accuracy is important then ProBABLI should be run longer)
# csolved <- solved
# csolved$N <- ceiling(solved$N)
# confirm <- runSimulation(design=csolved, replications=10000, parallel=TRUE,
# generate=Generate, analyse=Analyse,
# summarise=Summarise)
# confirm
# Similarly, terminate if the prediction interval is consistently predicted
# to be between [.795, .805]. Note that maxiter increased as well
solved_predCI <- SimSolve(design=Design, b=.8, interval=c(10, 500),
generate=Generate, analyse=Analyse, summarise=Summarise,
maxiter=200, predCI.tol=.01)
solved_predCI
summary(solved_predCI) # note that predCI.b are all within [.795, .805]
N <- solved_predCI$N
pwr.t.test(d=.2, n=N[1])
pwr.t.test(d=.5, n=N[2])
pwr.t.test(d=.8, n=N[3])
# Alternatively, and often more realistically, wait.time can be used
# to specify how long the user is willing to wait for a final estimate.
# Solutions involving more iterations will be more accurate,
# and therefore it is recommended to run the ProBABLI root-solver as long
# the analyst can tolerate if the most accurate estimates are desired.
# Below executes the simulation for 5 minutes for each condition up
# to a maximum of 1000 iterations, terminating based on whichever occurs first
solved_5min <- SimSolve(design=Design, b=.8, interval=c(10, 500),
generate=Generate, analyse=Analyse, summarise=Summarise,
wait.time="5", maxiter=1000)
solved_5min
summary(solved_5min)
# use estimated N results to see how close power was
N <- solved_5min$N
pwr.t.test(d=.2, n=N[1])
pwr.t.test(d=.5, n=N[2])
pwr.t.test(d=.8, n=N[3])
#------------------------------------------------
#######################
## Sensitivity Analysis
#######################
# GOAL: solve effect size d given sample size and power inputs (inputs
# for root no longer required to be an integer)
# Generate-Analyse-Summarise functions identical to above, however
# Design input includes NA for d element
Design <- createDesign(N = c(100, 50, 25),
d = NA,
sig.level = .05)
Design # solve for NA's
#~~~~~~~~~~~~~~~~~~~~~~~~
#### Step 2 --- Define generate, analyse, and summarise functions (same as above)
#~~~~~~~~~~~~~~~~~~~~~~~~
#### Step 3 --- Optimize d over the rows in design
# search between d = [.1, 2] for each row
# In this example, b = target power
# note that integer = FALSE to allow smooth updates of d
solved <- SimSolve(design=Design, b = .8, interval=c(.1, 2),
generate=Generate, analyse=Analyse,
summarise=Summarise, integer=FALSE)
solved
summary(solved)
plot(solved, 1)
plot(solved, 2)
plot(solved, 3)
# plot median history and estimate precision
plot(solved, 1, type = 'history')
plot(solved, 1, type = 'density')
plot(solved, 1, type = 'iterations')
# verify with true power from pwr package
library(pwr)
pwr.t.test(n=100, power = .8)
pwr.t.test(n=50, power = .8)
pwr.t.test(n=25, power = .8)
# use estimated d results to see how close power was
pwr.t.test(n=100, d = solved$d[1])
pwr.t.test(n=50, d = solved$d[2])
pwr.t.test(n=25, d = solved$d[3])
### failing analytic formula, confirm results with more precise
### simulation via runSimulation() (not required; if accuracy is important
### PROBABLI should just be run longer)
# confirm <- runSimulation(design=solved, replications=10000, parallel=TRUE,
# generate=Generate, analyse=Analyse,
# summarise=Summarise)
# confirm
#------------------------------------------------
#####################
## Criterion Analysis
#####################
# GOAL: solve Type I error rate (alpha) given sample size, effect size, and
# power inputs (inputs for root no longer required to be an integer). Only useful
# when Type I error is less important than achieving the desired 1-beta (power)
Design <- createDesign(N = 50,
d = c(.2, .5, .8),
sig.level = NA)
Design # solve for NA's
# all other function definitions same as above
# search for alpha within [.0001, .8]
solved <- SimSolve(design=Design, b = .8, interval=c(.0001, .8),
generate=Generate, analyse=Analyse,
summarise=Summarise, integer=FALSE)
solved
summary(solved)
plot(solved, 1)
plot(solved, 2)
plot(solved, 3)
# plot median history and estimate precision
plot(solved, 1, type = 'history')
plot(solved, 1, type = 'density')
plot(solved, 1, type = 'iterations')
# verify with true power from pwr package
library(pwr)
pwr.t.test(n=50, power = .8, d = .2, sig.level=NULL)
pwr.t.test(n=50, power = .8, d = .5, sig.level=NULL)
pwr.t.test(n=50, power = .8, d = .8, sig.level=NULL)
# use estimated alpha results to see how close power was
pwr.t.test(n=50, d = .2, sig.level=solved$sig.level[1])
pwr.t.test(n=50, d = .5, sig.level=solved$sig.level[2])
pwr.t.test(n=50, d = .8, sig.level=solved$sig.level[3])
### failing analytic formula, confirm results with more precise
### simulation via runSimulation() (not required; if accuracy is important
### PROBABLI should just be run longer)
# confirm <- runSimulation(design=solved, replications=10000, parallel=TRUE,
# generate=Generate, analyse=Analyse,
# summarise=Summarise)
# confirm
}
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