One Dimensional Root (Zero) Finding in Simulation Experiments

Function provides a stochastic root-finding approach to solve specific quantities in simulation experiments (e.g., solving for a specific sample size to meet a target power rate) using the Probablistic Bisection Algorithm with Bolstering and Interpolations (ProBABLI; Chalmers, 2024). The structure follows the three functional steps outlined in runSimulation, however portions of the design input are taken as variables to be estimated rather than fixed, where an additional constant b is required in order to solve the root equation f(x) - b = 0.

SimSolve(
  design,
  interval,
  b,
  generate,
  analyse,
  summarise,
  replications = list(burnin.iter = 15L, burnin.reps = 50L, max.reps = 500L,
    min.total.reps = 9000L, increase.by = 10L),
  integer = TRUE,
  formula = y ~ poly(x, 2),
  family = "binomial",
  parallel = FALSE,
  cl = NULL,
  save = TRUE,
  resume = TRUE,
  method = "ProBABLI",
  wait.time = NULL,
  ncores = parallelly::availableCores(omit = 1L),
  type = ifelse(.Platform$OS.type == "windows", "PSOCK", "FORK"),
  maxiter = 100L,
  check.interval = TRUE,
  verbose = TRUE,
  control = list(),
  predCI = 0.95,
  predCI.tol = NULL,
  ...
)

# S3 method for class 'SimSolve'
summary(object, tab.only = FALSE, reps.cutoff = 300, ...)

# S3 method for class 'SimSolve'
plot(x, y, ...)

Arguments

design

a tibble or data.frame object containing the Monte Carlo simulation conditions to be studied, where each row represents a unique condition and each column a factor to be varied (see createDesign). However, exactly one column of this object in each row must be specified with NA placeholders to indicate that the missing value should be estimated via the select stochastic optimizer

interval

a vector of length two, or matrix with nrow(design) rows and two columns, containing the end-points of the interval to be searched per row condition. If a vector then the interval will be used for all rows in the supplied design object

b

a single constant used to solve the root equation f(x) - b = 0

generate

generate function. See runSimulation

analyse

analysis function. See runSimulation

summarise

summary function that returns a single number corresponding to the function evaluation f(x) in the equation f(x) = b to be solved as a root f(x) - b = 0. Unlike in the standard runSimulation() definitions this input is required. For further information on this function specification, see runSimulation

replications

a named list or vector indicating the number of replication to use for each design condition per PBA iteration. By default the input is a list with the arguments burnin.iter = 15L, specifying the number of burn-in iterations to used, burnin.reps = 50L to indicate how many replications to use in each burn-in iteration, max.reps = 500L to prevent the replications from increasing higher than this number, min.total.reps = 9000L to avoid termination when very few replications have been explored (lower bound of the replication budget), and increase.by = 10L to indicate how many replications to increase per iteration after the burn-in stage. Default can overwritten by explicit definition (e.g., replications = list(increase.by = 25L)).

Vector inputs can specify the exact replications for each respective iteration. As a general rule, early iterations should be relatively low for initial searches to avoid unnecessary computations when locating the approximate location of the root, while the number of replications should gradually increase after this burn-in to reduce the sampling variability.

integer

logical; should the values of the root be considered integer or numeric? If TRUE then bolstered directional decisions will be made in the PBA function based on the collected sampling history

formula

regression formula to use when interpolate = TRUE. Default fits an orthogonal polynomial of degree 2

family

family argument passed to glm. By default the 'binomial' family is used, as this function defaults to power analysis setups where isolated results passed to summarise will return 0/1s, however other families should be used if summarise returns something else (e.g., if solving for a particular standard error then a 'gaussian' family would be more appropriate).

Note that if individual results from the analyse steps should not be used (i.e., only the aggregate from summarise is meaningful) then set control = list(summarise.reg_data = TRUE) to override the default behaviour, thereby using only the aggregate information and weights

parallel

for parallel computing for slower simulation experiments (see runSimulation for details)

cl

see runSimulation

save

logical; store temporary file in case of crashes. If detected in the working directory will automatically be loaded to resume (see runSimulation for similar behaviour)

resume

logical; if a temporary SimDesign file is detected should the simulation resume from this location? Keeping this TRUE is generally recommended, however this should be disabled if using SimSolve within runSimulation to avoid reading improper save states

method

optimizer method to use. Default is the stochastic root-finder 'ProBABLI', but can also be the deterministic options 'Brent' (which uses the function uniroot) or 'bisection' for the classical bisection method. If using deterministic root-finders then replications must either equal a single constant to reflect the number of replication to use per deterministic iteration or be a vector of length maxiter to indicate the replications to use per iteration

wait.time

(optional) argument passed to PBA to indicate the time to wait (specified in minutes if a numeric vector is passed) per row in the Design object rather than using pre-determined termination criteria based on the estimates. For example, if three three conditions were defined in Design, and wait.time="5", then the total search time till terminate after 15 minutes regardless of independently specified termination criteria in control. See timeFormater for alternative specifications

ncores

see runSimulation

type

type of cluster (see makeCluster) or plotting type to use.

If type used in plot then can be 'density' to plot the density of the iteration history after the burn-in stage, 'iterations' for a bubble plot with inverse replication weights. If not specified then the default PBA plots are provided (see PBA)

maxiter

the maximum number of iterations (default 100) except when wait.time is specified (automatically increased to 3000 to avoid early termination)

check.interval

logical; should an initial check be made to determine whether f(interval[1L]) and f(interval[2L]) have opposite signs? If FALSE, the specified interval is assumed to contain a root, where f(interval[1]) < 0 and f(interval[2] > 0. Default is TRUE

verbose

logical; print information to the console?

control

a list of the algorithm control parameters. If not specified, the defaults described below are used.

tol: tolerance criteria for early termination (.1 for integer = TRUE searches; .00025 for non-integer searches
rel.tol: relative tolerance criteria for early termination (default .0001)
k.success: number of consecutive tolerance successes given rel.tol and tol criteria (default is 3)
bolster: logical; should the PBA evaluations use bolstering based on previous evaluations? Default is TRUE, though only applicable when integer = TRUE
interpolate.R: number of replications to collect prior to performing the interpolation step (default is 3000 after accounting for data exclusion from burnin.iter). Setting this to 0 will disable any interpolation computations
include_reps: logical; include a column in the condition elements to indicate how many replications are currently being evaluated? Mainly useful when further tuning within each ProBABLI iteration is desirable (e.g., for increasing/decreasing bootstrap draws as the search progresses). Default is FALSE
summarise.reg_data: logical; should the aggregate results from Summarise (along with its associated weights) be used for the interpolation steps, or the raw data from the Analyse step? Set this to TRUE when the individual results from Analyse give less meaningful information

predCI

advertised confidence interval probability for final model-based prediction of target b given the root input estimate. Returned as an element in the summary() list output

predCI.tol

(optional) rather than relying on the changes between successive estimates (default), if the predicting CI is consistently within this supplied tolerance range then the search will be terminated. This provides termination behaviour based on the predicted precision of the root solutions rather than their stability history, and therefore can be used to obtain estimates with a particular level of advertised accuracy. For example, when solving for a sample size value (N) if the solution associated with b = .80 requires that the advertised 95 is consistently between [.795, .805] then predCI.tol = .01 should be used to reflect this tolerance range

...

additional arguments to be pasted to PBA

object

object of class 'SimSolve'

tab.only

logical; print only the (reduced) table of estimates?

reps.cutoff

integer indicating the rows to omit from the output if the number of replications are less than this value

x

object of class 'SimSolve'

y

design row to plot. If omitted defaults to 1

Value

the filled-in design object containing the associated lower and upper interval estimates from the stochastic optimization

Details

Root finding is performed using a progressively bolstered version of the probabilistic bisection algorithm (PBA) to find the associated root given the noisy simulation objective function evaluations. Information is collected throughout the search to make more accurate predictions about the associated root via interpolation. If interpolations fail, then the last iteration of the PBA search is returned as the best guess.

For greater advertised accuracy with ProBABLI, termination criteria can be based on the width of the advertised predicting interval (via predCI.tol) or by specifying how long the investigator is willing to wait for the final estimates (via wait.time, where longer wait times lead to progressively better accuracy in the final estimates).

References

Chalmers, R. P. (2024). Solving Variables with Monte Carlo Simulation Experiments: A Stochastic Root-Solving Approach. Psychological Methods. DOI: 10.1037/met0000689

Chalmers, R. P., & Adkins, M. C. (2020). Writing Effective and Reliable Monte Carlo Simulations with the SimDesign Package. The Quantitative Methods for Psychology, 16(4), 248-280. doi:10.20982/tqmp.16.4.p248

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples


if (FALSE) { # \dontrun{

##########################
## 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 2 minutes per condition

solved_2min <- SimSolve(design=Design[1, ], b=.8, interval=c(10, 500),
                generate=Generate, analyse=Analyse, summarise=Summarise,
                wait.time="2")
solved_2min
summary(solved_2min)

# use estimated N results to see how close power was
N <- solved_2min$N
pwr.t.test(d=.2, n=N[1])


#------------------------------------------------

#######################
## 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

} # }