# simulation_functions

We present examples of setting up a simulation function for various models and R packages. These are written in a basic way to serve as blueprints for further customization.

The general layout of a simulation function is:

simfun <- function(N) {
# Generate a data set Test the hypothesis
}

For all examples in this vignette, we use an alpha level of .01 and a desired power of .95.

For further guidance on how to use the package and the find.design function specifically, see the Readme.md file

# T-Test

library(mlpwr)

We test for one sample whether the mean differs from 0. The true effect size is Cohen’s d = .3. In this scenario, this means that the true mean difference is .3.

simfun_ttest <- function(N) {
# Generate a data set
dat <- rnorm(n = N, mean = 0.3)
# Test the hypothesis
res <- t.test(dat)
res$p.value < 0.01 } Example Use. The boundaries for the sample size are set to 100 and 300 for the lower and upper boundaries, respectively. The desired power is set to .95. res <- find.design(simfun = simfun_ttest, boundaries = c(100, 300), power = 0.95) # ANOVA library(mlpwr) We test for a mean difference among n.groups groups. Each group consists of n participants. simfun_anova <- function(n, n.groups) { # Generate a data set groupmeans <- rnorm(n.groups, sd = 0.2) # generate groupmeans using cohen's f=.2 dat <- sapply(groupmeans, function(x) rnorm(n, mean = x, sd = 1)) # generate data dat <- dat |> as.data.frame() |> gather() # format # Test the hypothesis res <- aov(value ~ key, data = dat) # perform ANOVA summary(res)[[1]][1, 5] < 0.01 # extract significance } Example Use. The boundaries are defined as a list here to account for the multidimensional set of design parameters. Also, we define a cost function to enable differentiating designs with equal power. res <- find.design(simfun = simfun_anova, costfun = function(n, n.groups) 5 * n + 20 * n.groups, boundaries = list(n = c(10, 150), n.groups = c(5, 30)), power = 0.95) # Generalized Linear Models library(mlpwr) We consider generalized linear models using the stats package and the glm function. ## Using a fitted model We use an “original” data set and fit a generalized linear model assuming a Poisson distributed criterion counts. Here, counts is the criterion, treatment and outcome are the predictors. dat.original <- data.frame(counts = c(18, 17, 15, 20, 10, 20, 25, 13, 12), treatment = gl(3, 1, 9), outcome = gl(3, 3)) mod.original <- glm(counts ~ outcome + treatment, data = dat.original, family = poisson) # setting up the generalized linear model summary(mod.original) We use the parameters from the original model in the simfun. simfun_glm1 <- function(N) { # generate data dat <- data.frame(outcome = gl(3, 1, ceiling(N/3)), treatment = gl(3, ceiling(N/3)))[1:N, ] # predictors a <- predict(mod.original, newdata = dat, type = "response") # criterion 'raw' dat$counts <- rpois(N, a)  # criterion applying poisson distribution

# test hypothesis
mod <- glm(counts ~ outcome + treatment, data = dat,
family = poisson)  # fit a glm
summary(mod)$coefficients["treatment2", "Pr(>|z|)"] < 0.01 # test the coefficient of the treatment } Example Use res <- find.design(simfun = simfun_glm1, boundaries = c(20, 100), power = 0.95) ## Specifying parameters manually We use a logistic regression and generate the data using hand-specified parameters and the logistic function. logistic <- function(x) 1/(1 + exp(-x)) # logistic function We test if the second predictor is significant. simfun_glm2 <- function(N) { # generate data dat <- data.frame(pred1 = rnorm(N), pred2 = rnorm(N)) beta <- c(1.2, 0.8) # parameter weights prob <- logistic(as.matrix(dat) %*% beta) # get probability dat$criterion <- runif(N) < prob  # draw according to probability

# test hypothesis
mod <- glm(criterion ~ pred1 + pred2, data = dat,
family = binomial)  # fit a glm
summary(mod)$coefficients["pred2", "Pr(>|z|)"] < 0.01 # test the coefficient of the predictor } Example Use res <- find.design(simfun = simfun_glm2, boundaries = c(90, 200), power = 0.95) # Item Response Theory Models We first load the necessary packages to fit the utilized models and generate specific artificial data sets. library(mlpwr) library(mirt) We use the mirt package to show an example that applies an item response theory model. See ?simdata for additional options and examples to generate data with the mirt package. ## Testing a Rasch against a 2PL model We first generate data from a 2PL model. Then we want to check whether the 2PL model actually shows a better fit to the data better than the simpler Rasch model. We use a likelihood ratio test for this purpose. simfun_irt1 <- function(N) { # generate data dat <- simdata(a = c(1.04, 1.2, 1.19, 0.61, 1.31, 0.83, 1.46, 1.27, 0.51, 0.81), d = c(0.06, -1.79, -1.15, 0.88, -0.2, -1.87, 1.23, -0.08, -0.71, 0.6), N = N, itemtype = "2PL") # uses a 2PL model with a and d parameters # test hypothesis mod <- mirt(dat) # Fit 2PL Model constrained <- "F = 1-4 CONSTRAIN = (1-4, a1)" # specifying that the slopes should be kept # equal for items 1 to 4 mod_constrained <- mirt(dat, constrained) # Fit 2PL with equal slopes res <- anova(mod_constrained, mod) # perform model comparison res$p[2] < 0.01  # extract significance
}

Example Use

res <- find.design(simfun = simfun_irt1, boundaries = c(100,
500), power = 0.95, evaluations = 500)

## Testing for DIF

We check for differential item functioning in one item. Again, we generate data using the 2PL model, but using different parameters for two different groups in this case. The parameters for the first item are different in one group compared to other. We again apply an likelihood ratio test to test the null hypothesis that all item parameters are actually equal in both groups.

We optimize for the sizes of the two groups as the two design parameters. We assume that the second group involves higher costs than the first.

costfun_irt2 <- function(N1, N2) 5 * N1 + 7 * N2  # specifying a cost function 
simfun_irt2 <- function(N1, N2) {

# generate data
a1 <- a2 <- c(1.04, 1.2, 1.19, 0.61, 1.31, 0.83,
1.46, 1.27, 0.51, 0.81)  # specifying the slope
d1 <- d2 <- c(0.06, -1.79, -1.15, 0.88, -0.2, -1.87,
1.23, -0.08, -0.71, 0.6)  # specifying the difficulty
a2[1] <- a2[1] + 0.3  # the slope is different for the first item in group2
d2[1] <- d2[1] + 0.5  # the difficulty is different for the first item in group2
dat1 <- simdata(a = a1, d = d1, N = N1, itemtype = "2PL")  # creating artificial data for both groups
dat2 <- simdata(a = a2, d = d2, N = N2, itemtype = "2PL")
dat <- as.data.frame(rbind(dat1, dat2))  # combining the data sets into one object
group <- c(rep("1", N1), rep("2", N2))  # create a variable that indicates the group membership

# fit models
mod1 <- multipleGroup(dat, 1, group = group)  # fit model with different parameters for each group
mod2 <- multipleGroup(dat, 1, group = group, invariance = c("slopes",
"intercepts"))  # fit model with the same parameters for each group

# test hypothesis
res <- anova(mod2, mod1)

# extract significance
res$p[2] < 0.01 } Example Use res <- find.design(simfun = simfun_irt2, boundaries = list(N1 = c(100, 700), N2 = c(100, 700)), costfun = costfun_irt2, power = 0.95) # Multilevel Models We first load the necessary packages to fit the utilized models and generate specific artificial data sets. library(mlpwr) library(lme4) library(lmerTest) We consider multilevel models using the lme4 and lmerTest packages. We use the glmer function for model fit and the simulate function to generate data. For further options for data generation in multilevel models see ?simulate.merMod. ## Specifying parameters manually We generate data using manually specified standard deviation of the random effects and parameter weights. We generate and fit the data according to a generalized linear mixed effects model with a poisson distributed criterion variable. simfun_multi1 <- function(N) { # generate data params <- list(theta = 0.5, beta = c(2, -0.2, -0.4, -0.6)) # specifying the standard deviation of the random effects and parameter weights dat <- expand.grid(herd = 1:ceiling(N/4), period = factor(1:4))[1:N, ] # creating predictors dat$x <- simulate(~period + (1 | herd), newdata = dat,
family = poisson, newparams = params)[[1]]  # creating criterion

# test hypothesis
mod <- glmer(x ~ period + (1 | herd), data = dat,
family = poisson)  # fit model
pvalues <- summary(mod)[["coefficients"]][2:4,
"Pr(>|z|)"]  # extract p-values
any(pvalues < 0.01)  # test hypothes that any is significant
}

Example Use

res <- find.design(simfun = simfun_multi1, boundaries = c(100,
500), power = 0.95)

## Using a fitted model

We generate data from a fitted generalized linear mixed-effects model. We apply a mixed effects logistic regression in this case. It has two predictors and a random intercepts for each country.

logistic <- function(x) 1/(1 + exp(-x))

N.original <- 300
n.countries.original <- 20

# generate original data
dat.original <- data.frame(country = rep(1:n.countries.original,
length.out = N.original), pred1 = rnorm(N.original),
pred2 = rnorm(N.original))  # creating predictors
country.intercepts <- rnorm(n.countries.original, sd = 0.5)  # creating random intercepts
dat.original$intercepts <- country.intercepts[dat.original$country]  # add interecepts to data
beta <- c(1, 0.4, -0.3)  # parameter weights
prob <- logistic(as.matrix(dat.original[c("intercepts",
"pred1", "pred2")]) %*% as.matrix(beta))  # get probability
dat.original$criterion <- runif(N.original) < prob # draw according to probability # fit original model to obtain parameters mod.original <- glmer(criterion ~ pred1 + pred2 + 0 + (1 | country), data = dat.original, family = binomial) In the simulation function, we generate criterion data using the original model. Design parameters are the number of participant per country n and the number of countries n.countries. We test the hypothesis that the second predictor is significant. simfun_multi2 <- function(n, n.countries) { # generate data dat <- data.frame(country = rep(1:n.countries, length.out = n * n.countries), pred1 = rnorm(n * n.countries), pred2 = rnorm(n * n.countries)) dat$criterion <- simulate(mod.original, nsim = 1,
newdata = dat, allow.new.levels = TRUE, use.u = FALSE) |>
unlist()  # criterion data from the fitted model

# test hypothesis
mod <- glmer(criterion ~ pred1 + pred2 + 0 + (1 |
country), data = dat, family = binomial)
summary(mod)[["coefficients"]]["pred2", "Pr(>|z|)"] <
0.01  # check if significant
}

As a cost function, we can use

costfun_multi2 <- function(n, n.countries) 5 * m +
100 * n.countries

Example Use

res <- find.design(simfun = simfun_multi2, boundaries = list(n = c(10,
40), n.countries = c(5, 20)), costfun = costfun_multi2,
power = 0.95)