library(lmerTest)
library(afex)
library(ggplot2)
library(dplyr)
library(designr)
library(gridExtra)
library(MASS)We can compute power by
How to obtain the probability of a significant result?
# a) assume effect size + simulate data
x <- rep(c("x1","x2"), each=30)
y <- rep(c( 200, 220), each=30) + rnorm(60,0,50)
ggplot(data.frame(x,y), aes(x=x,y=y)) + geom_boxplot()
## [1] 0.03388535# c) do this many times
nsim <- 1000
pVal <- c()
for (i in 1:nsim) {
x <- rep(c("x1","x2"), each=30)
y <- rep(c( 200, 220), each=30) + rnorm(60,0,50)
# b) run statistical model & get p-value
pVal[i] <- t.test(y ~ x)$p.value
}
# d) check how often it is significant
(power <- mean( pVal < 0.05 ))## [1] 0.337# e)
# We could do this for different numbers of subjects and see when power is ~80%
# We could change the difference in means/residual standard deviationsWhat we usually want to achieve is a power of 80%. That’s the standard. We should plan our analysis, with the number of subjects and items to achieve this power value of 80%. That is, if the effect that we want to test really exists, we want to have an 80% chance of actually detecting the effect with our analysis. In the power analysis, we can then vary the number of subjects or items, or make different assumptions about the effect size to see what we need to achieve 80% power. This should be the basis for planing an experimental study. Note that in the simulation studies below, we will sometimes use power thresholds of lower than 80%. This should normally not be done when running empirical studies.
One more point about power: if an experimental design has low power, then this can have severe negative consequences for statistical analysis. In an extreme case where power is only 10%, then if we find a significant result, there is a very high chance that the effect is actually due to chance and that the H0 is true. Moreover, when we have low power, then the effect size estimates from significant results are guaranteed to be too large (type M = magnitude error), and there is a danger finding effects in the wrong direction (type S = sign error).
Therefore, it is important to plan studies such that they have a good power, ideally = 80%. If you want to provide evidence for the null hypothesis, then even a higher power of e.g., 95% can be beneficial.
This is the formula for a simple linear model: yi = b0 + b1 * xi + ei, where yi is the dependent variable for subject i, b0 is the intercept, b1 is the slope, xi is the value of the predictor for subject i, and ei is the random error term for subject i. We can use this to simulate data.
The predictor term: This could be a continuous variable in a linear regression analysis (e.g., xi could indicate the age of subject i). However, we can also use this formulation if the predictor is a factor, e.g., if we have two groups of subjects “x1” and “x2”. In this case, we have to use contrasts to code factors into numbers (for contrasts see Schad et al., 2020, JML). One example is the sum contrast. Here, the predictor variable is −1 for one group (e.g., for x1) and +1 for the other group (e.g., for x2).
# previous formulation
x <- rep(c("x1","x2"), each=30)
y <- rep(c( 200, 220), each=30) + rnorm(60,0,50)
# new:
x <- rep(c(-1,+1), each=30) # define predictor term: here, sum contrast coding (-1, +1)
y <- 210 + 10*x + rnorm(60,0,50) # simulate data
# run linear model
lm(y ~ x)##
## Call:
## lm(formula = y ~ x)
##
## Coefficients:
## (Intercept) x
## 207.735 -1.415##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -98.808 -30.032 3.904 37.831 70.513
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 207.735 5.649 36.77 <2e-16 ***
## x -1.415 5.649 -0.25 0.803
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 43.76 on 58 degrees of freedom
## Multiple R-squared: 0.00108, Adjusted R-squared: -0.01614
## F-statistic: 0.06273 on 1 and 58 DF, p-value: 0.8031
Other statistical software: R-packages simr and faux
The R-package designr has been written by us to create
factorial experimental designs.
library(designr)
# one within-subject factor
# create design
design <-
fixed.factor("X", levels=c("X1", "X2")) + # fixed effect
random.factor("Subj", instances=5) # random effect
dat <- design.codes(design) # create data frame (tibble) from experimental design
length(unique(dat$Subj)) # number of subjets## [1] 5## Subj X
## 1 Subj1 X1
## 2 Subj1 X2
## 3 Subj2 X1
## 4 Subj2 X2
## 5 Subj3 X1
## 6 Subj3 X2
## 7 Subj4 X1
## 8 Subj4 X2
## 9 Subj5 X1
## 10 Subj5 X2# one between-subject factor
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", groups="X", instances=5)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 10## Subj X
## 1 Subj01 X1
## 2 Subj02 X1
## 3 Subj03 X1
## 4 Subj04 X1
## 5 Subj05 X1
## 6 Subj06 X2
## 7 Subj07 X2
## 8 Subj08 X2
## 9 Subj09 X2
## 10 Subj10 X2# 2 (A: within-subjects) x 2 (B: between-subjects) design
design <-
fixed.factor("A", levels=c("A1", "A2")) +
fixed.factor("B", levels=c("B1", "B2")) +
random.factor("Subj", groups="B", instances=5)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 10## Subj A B
## 1 Subj01 A1 B1
## 2 Subj01 A2 B1
## 3 Subj02 A1 B1
## 4 Subj02 A2 B1
## 5 Subj03 A1 B1
## 6 Subj03 A2 B1
## 7 Subj04 A1 B1
## 8 Subj04 A2 B1
## 9 Subj05 A1 B1
## 10 Subj05 A2 B1
## 11 Subj06 A1 B2
## 12 Subj06 A2 B2
## 13 Subj07 A1 B2
## 14 Subj07 A2 B2
## 15 Subj08 A1 B2
## 16 Subj08 A2 B2
## 17 Subj09 A1 B2
## 18 Subj09 A2 B2
## 19 Subj10 A1 B2
## 20 Subj10 A2 B2# 1 factor, 2 (crossed) random effects, within-subject, between-item factor
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=5) +
random.factor("Item", groups="X", instances=2)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 5## [1] 4## Subj Item X
## 1 Subj1 Item1 X1
## 2 Subj1 Item2 X1
## 3 Subj1 Item3 X2
## 4 Subj1 Item4 X2
## 5 Subj2 Item1 X1
## 6 Subj2 Item2 X1
## 7 Subj2 Item3 X2
## 8 Subj2 Item4 X2
## 9 Subj3 Item1 X1
## 10 Subj3 Item2 X1
## 11 Subj3 Item3 X2
## 12 Subj3 Item4 X2
## 13 Subj4 Item1 X1
## 14 Subj4 Item2 X1
## 15 Subj4 Item3 X2
## 16 Subj4 Item4 X2
## 17 Subj5 Item1 X1
## 18 Subj5 Item2 X1
## 19 Subj5 Item3 X2
## 20 Subj5 Item4 X2# within-subject, within-item factor
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=5) +
random.factor("Item", instances=2)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 5## [1] 2## Subj Item X
## 1 Subj1 Item1 X1
## 2 Subj1 Item1 X2
## 3 Subj1 Item2 X1
## 4 Subj1 Item2 X2
## 5 Subj2 Item1 X1
## 6 Subj2 Item1 X2
## 7 Subj2 Item2 X1
## 8 Subj2 Item2 X2
## 9 Subj3 Item1 X1
## 10 Subj3 Item1 X2
## 11 Subj3 Item2 X1
## 12 Subj3 Item2 X2
## 13 Subj4 Item1 X1
## 14 Subj4 Item1 X2
## 15 Subj4 Item2 X1
## 16 Subj4 Item2 X2
## 17 Subj5 Item1 X1
## 18 Subj5 Item1 X2
## 19 Subj5 Item2 X1
## 20 Subj5 Item2 X2# within-subject, within-item factor, latin square design
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=5) +
random.factor("Item", instances=2) +
random.factor(c("Subj", "Item"), groups=c("X"))
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 10## [1] 4## Subj Item X
## 1 Subj01 Item1 X1
## 2 Subj01 Item2 X2
## 3 Subj01 Item3 X1
## 4 Subj01 Item4 X2
## 5 Subj02 Item1 X2
## 6 Subj02 Item2 X1
## 7 Subj02 Item3 X2
## 8 Subj02 Item4 X1
## 9 Subj03 Item1 X1
## 10 Subj03 Item2 X2
## 11 Subj03 Item3 X1
## 12 Subj03 Item4 X2
## 13 Subj04 Item1 X2
## 14 Subj04 Item2 X1
## 15 Subj04 Item3 X2
## 16 Subj04 Item4 X1
## 17 Subj05 Item1 X1
## 18 Subj05 Item2 X2
## 19 Subj05 Item3 X1
## 20 Subj05 Item4 X2
## 21 Subj06 Item1 X2
## 22 Subj06 Item2 X1
## 23 Subj06 Item3 X2
## 24 Subj06 Item4 X1
## 25 Subj07 Item1 X1
## 26 Subj07 Item2 X2
## 27 Subj07 Item3 X1
## 28 Subj07 Item4 X2
## 29 Subj08 Item1 X2
## 30 Subj08 Item2 X1
## 31 Subj08 Item3 X2
## 32 Subj08 Item4 X1
## 33 Subj09 Item1 X1
## 34 Subj09 Item2 X2
## 35 Subj09 Item3 X1
## 36 Subj09 Item4 X2
## 37 Subj10 Item1 X2
## 38 Subj10 Item2 X1
## 39 Subj10 Item3 X2
## 40 Subj10 Item4 X1Ok, let’s run a power analysis for a simple example: one within-subject factor with subject random effects.
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=30)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 30## [1] -1 1Next, we will use the R-function simLMM() to simulate
data for a linear mixed-effects model. simLMM() is
currently under development.
If you find any errors, please write them into a script that reproduces
the error and send it to us (e.g., to [email protected]). This would be very important
feedback! Thank you for your help.
How does the function work? The most important function arguments are the following:
simLMM(formula, data, Fixef, VC_sd, CP)
Formula is a standard formula used in the lmer function.
For the present example, the formula would be
form = ~ 1 + X + (1 + X | Subj). Since simLMM
is a simulation function, we do not have to specify a dependent
variable. This example indicates an intercept 1 and a fixed
effect for factor X. For the random effects, we see random
effects for subjects Subj; these specify a random intercept
and random slopes for factor X. Since there is only one
pipe |, we know that the random intercepts and slopes are
assumed to be correlated.
The next argument is the data data, which is a data
frame containing all the variables - this is just the same as in the
lmer function. Here, our data frame was termed
dat, i.e., data=dat.
The next argument Fixef specifies the fixed effects.
This is a vector of numbers that specify the regression coefficients /
fixed effects for each term in the formula. In our example, we have two
fixed effects, an intercept and a slope. Thus, we enter two numbers,
e.g., Fixef = c(200, 10), which specifies an intercept of
200 and a slope of 10.
The next argument VC_sd specifies the standard
deviations for the random effects. It is entered as a list. In the
present example, we have only two random effects: subjects and the
residual noise. These are assumed to be normally distributed with mean
zero and some standard deviation. For subjects we have two random
effects terms, the intercept and the random slopes. Let’s assume the
intercept varies across subjects with a standard deviation of 30 and the slope varies across subjects with
a standard deviation of 10. We combine
these two terms in one vector sd_Subj <- c(30, 10). The
second random effects term is the residual standard deviation. We here
assume the residual noise is normally distributed with a mean of 0 and some standard deviation, which we
assume here is 50. We now combine these
standard deviations into a list
VC_sd = list( c(30, 10), 50): we first enter the standard
deviations for the subject random effects as a vector, and then we enter
the standard deviation for the residual noise.
The next argument CP specifies the random effects
correlations. Here, we have different options for how to specify this.
One option is to simply specify a single number, e.g.,
CP=0.3. This will set all random effects correlations in
the model to the value 0.3. If we have
several random effects terms, e.g., for subjects and for items, we can
enter a vector of length two, where the first number would specify all
random effects correlations for subjects, and the second would specify
all random effects correlations for items. Last, we can also enter a
list of correlation matrices. E.g., when we have subjects and items, we
would have a list of length two. And the first entry would contain a
correlation matrix for the subject random effects and the second entry
would contain a correlation matrix for item random effects. Here,
because we have only one random effects correlation in the model, we
simply use the specification CP=0.3.
Taken together, this gives the following call of the
simLMM function:
simLMM(~ X + (X | Subj), data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Res), CP=0.3).
One additional argument that we can use is empirical. By
default this is set to empirical=FALSE. If we set this to
empirical=TRUE, then the fixed effects estimates in the
simulated data will exactly correspond to the numbers that we specify.
However, the random effects will not be exact. We will demonstrate this
below. If we use the default empirical=FALSE, then the
numbers (fixed-effects + random effects) that we specify are the “true”
population parameters, from which we draw a random sample in the
simulation. Note that empirical=TRUE does not work for data
from a logistic GLMM and it does not work when you have continuous
predictors, i.e., covariates, instead of/in addition to factors.
With this background, let’s simulate data for our within-subject
design. We use empirical=TRUE.
# simulate data
fix <- c(200,10) # set fixed effects
sd_Subj <- c(30, 10) # set random effects standard deviations for subjects
sd_Res <- 50 # set residual standard deviation
dat$ysim <- simLMM(form = ~ 1 + X + (1 + X | Subj),
data = dat,
Fixef = fix,
VC_sd = list(sd_Subj, sd_Res),
CP = 0.3,
empirical = TRUE)## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + X1 + ( 1 + X1 | Subj )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## Subj (Intercept) 30.0
## X1 10.0 0.30
## Residual 50.0
## Number of obs: 60, groups: Subj, 30
## Fixed effects:
## (Intercept) X1
## 200 10We see that the simLMM function produces some output
that looks very similar to the output from a fitted lmer
object. In this output, we can check whether our specification of fixed
and random effects is what we intended, or whether we made some mistake
in our specification. We can turn this output off by specifying
verbose=FALSE. This will be useful in power simulations
where we simulate data many many times.
Next, let’s check the simulated data.
# check group means
dat %>% group_by(X) %>% summarize(M=mean(ysim)) # the group means are EXACTLY 190 and 210. This is due to empirical=TRUE.## # A tibble: 2 × 2
## X M
## <fct> <dbl>
## 1 X1 190
## 2 X2 210# check fixed effects + random effects
(m1 <- lmer(ysim ~ Xn + (Xn || Subj), data=dat, control=lmerControl(calc.derivs=FALSE))) # The fixed effects are EXACTLY as indicated. This is due to empirical=TRUE.## Warning in as_lmerModLT(model, devfun): Model may not have converged with 1
## eigenvalue close to zero: 1.7e-09## Linear mixed model fit by REML ['lmerModLmerTest']
## Formula: ysim ~ Xn + (Xn || Subj)
## Data: dat
## REML criterion at convergence: 646.0295
## Random effects:
## Groups Name Std.Dev.
## Subj (Intercept) 38.20
## Subj.1 Xn 28.89
## Residual 35.46
## Number of obs: 60, groups: Subj, 30
## Fixed Effects:
## (Intercept) Xn
## 200 10## Anova Table (Type 3 tests)
##
## Response: ysim
## Effect df MSE F ges p.value
## 1 X 1, 29 2926.46 2.05 .028 .163
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1We can see that because the setting empirical=TRUE, the
fixed effects were estimated exactly to the numbers
that we set in the specification. This suggests that the simulation
process was implemented as intended. However, the random effects terms
are not exact.
VERY IMPORTANT: In order to perform a power
analysis, we have to use empirical=FALSE. In a power
analysis, we are assuming that we randomly sample from the population
parameters. This is what would happen in a “real” experiment. And we
want to know how often an effect is significant.
Next, we perform a power analysis. We want to find out how many
subjects are needed to run this study. We assume the fixed and random
effects specified above. We simulate data from the model for different
numbers of subjects. And we then fit lmer models to the
simulated data to see how often the effect of factor X is
significant given that there is a true effect of X in the
data.
nsubj <- seq(10,100,1) # we vary the number of subjects from 10 to 100 in steps of 1
nsim <- length(nsubj) # number of simulations
COF <- COFaov <- data.frame()
warn <- c()
for (i in 1:nsim) { # i <- 1
#print(paste0(i,"/",nsim))
# create experimental design
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=nsubj[i])
dat <- design.codes(design)
nsj <- length(unique(dat$Subj))
# create contrast and store as numeric variable
contrasts(dat$X) <- c(-1, +1)
dat$Xn <- model.matrix(~X,dat)[,2]
# for each number of subjects, we run 10 simulations to obtain more stable results
for (j in 1:10) { # j <- 1
# simulate data
dat$ysim <- simLMM(~ X + (X | Subj), data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Res), CP=0.3, empirical=FALSE, verbose=FALSE)
# ANOVA
AOV <- aov_car(ysim ~ 1 + Error(Subj/X), data=dat)
AOVcof <- summary(AOV)$univariate.tests[2,]
COFaov <- rbind(COFaov,c(nsj,AOVcof))
# LMM
#LMM <- lmer(ysim ~ Xn + (Xn || Subj), data=dat,
# REML=FALSE, control=lmerControl(calc.derivs=FALSE))
# run lmer and capture any warnings
ww <- ""
suppressMessages(suppressWarnings(
LMM <- withCallingHandlers({
lmer(ysim ~ Xn + (Xn || Subj), data=dat, REML=FALSE,
control=lmerControl(calc.derivs=FALSE))
},
warning = function(w) { ww <<- w$message }
)
))
LMMcof <- coef(summary(LMM))[2,]
COF <- rbind(COF,c(nsj,LMMcof,isSingular(LMM)))
warn[i] <- ww
}
}
#load(file="data/Power0.rda")
# Results for LMMs
names(COF) <- c("nsj","Estimate","SE","df","t","p","singular")
COF$warning <- warn
COF$noWarning <- warn==""
COF$sign <- as.numeric(COF$p < .05 & COF$Estimate>0) # determine significant results
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
# Results for ANOVAs
names(COFaov) <- c("nsj","SS","numDF","ErrorSS","denDF","F","p")
COFaov$sign <- as.numeric(COFaov$p < .05) # determine significant results
COFaov$nsjF <- gtools::quantcut(COFaov$nsj, q=seq(0,1,length=10))
COFaov$nsjFL <- plyr::ddply(COFaov,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
#save(COF, COFaov, file="data/Power0.rda")Note that in a standard LMM analysis, we would be interested to fit a parsimonious LMM (Matuschek et al., 2017, JML). This involves careful selection of the random effects. Unless one has an automatized selection of the parsimonious LMM, this is not possible when performing a power analysis and fitting hundreds of models. An alternative possibility is to fit a model with all random slopes, but with no random effects correlations. The problem with random effects correlation is that they are often very hard to estimate from the data. When trying to estimate them, there are often failures in model fit. However, removing the random effects correlations from the fit presumably does not affect power and alpha error much (Barr et al., 2013, JML). Therefore, for power analyses, it is recommended to not estimate random effects correlations. However, normally all random slopes should be estimated, since neglecting random slopes in the estimation can sometimes heavily inflate the alpha error and lead to false positive results (i.e., if there is strong true variation in the effect; Barr et al., 2013, JML).
First, we should check whether the models were singular fits. Singular fits are cases where some of the parameter values were estimated on their boundary. For example, a random effects standard deviation can be estimated to be exactly 0, or a correlation parameter can be estimated to be exactly +1 or −1. Normally, this should be avoided in LMMs, usually by excluding the respective term from the model. However, this is not possible in power analyses due to the large number of model fits. Therefore, we here check in how many cases there are singular fits.
## [1] 0The mean of zero indicates that none of the models were singular fits. This supports the results from the analysis.
Moreover, we can check whether there were any warning messages, i.e., whether the optimizer has converged for the models.
## [1] 0.4945055This shows that 50% of fits showed no warning message, which also means that 50% of model fits did exhibit a warning message. This is very bad news. It probably means that we cannot use this power analysis. Indeed, if we would run a study, there would be a high chance that there is a problem with convergence.
## [1] ""
## [2] "Model may not have converged with 1 eigenvalue close to zero: 6.7e-10"
## [3] ""
## [4] ""
## [5] "Model failed to converge with 1 negative eigenvalue: -1.5e-06"
## [6] ""
## [7] ""
## [8] "Model may not have converged with 1 eigenvalue close to zero: -1.3e-09"
## [9] "Model failed to converge with 1 negative eigenvalue: -4.2e-07"
## [10] ""
## [11] ""
## [12] ""
## [13] "Model failed to converge with 1 negative eigenvalue: -6.0e-08"
## [14] ""
## [15] "Model may not have converged with 1 eigenvalue close to zero: -6.3e-09"
## [16] ""
## [17] "Model failed to converge with 1 negative eigenvalue: -1.8e-07"
## [18] ""
## [19] "Model failed to converge with 1 negative eigenvalue: -1.8e-07"
## [20] ""Notice that we haven’t yet implemented the detection of warning messages for the other example analyses (see below). But this should be done when conducting a power analysis. There is a paper that argues that warning messages can be largely ignored (https://debruine.github.io/lmem_sim/articles/paper.html): “as long as there are not too many of these non-convergence warnings relative to the number of runs, you can probably ignore them because the won’t affect the overall estimates”. However, with 50% warnings, we definitely have a problem. One possible option could be to see whether the average effect estimate (e.g. for the fixed effects) is the same for fits with versus without convergence warning. One reason why we have many warnings here is because we used a small number of data points to speed up the fitting algorithm.
Advice:
(This advice is off the top of the head; not based on evidence.)
If a simulation passes (1),
Ignore the <=1% problem cases
Determine power for critical effects, VCs, and CPs — assuming that you want to determine the number of Subj and number of Item for 80% power for the effect sizes you expect.
Because this is an artificial example, we now proceed to plot the percentage of significant results as a function of the number of subjects.
p1 <- ggplot(data=COF)+
geom_smooth(aes(x=nsj, y=sign))+
geom_point( stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line( stat="summary", aes(x=nsjFL, y=sign))
p2 <- ggplot(data=COFaov)+
geom_smooth(aes(x=nsj, y=sign))+
geom_point( stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line( stat="summary", aes(x=nsjFL, y=sign))
grid.arrange(p1,p2, nrow=1)
We can also run a local regression analysis (i.e., the smoothed regression line) manually and determine the number of subjects that we need to obtain a power of 60%.
# determine number of subjects needed for a power of 60%
m0 <- loess(sign ~ nsj, data=COF)
COF$pred <- predict(m0)
idx <- COF$pred>0.6
min(COF$nsj[idx])## [1] 71We now create a 2 x 3 within-subject design with random effects for subjects.
design <-
fixed.factor("A", levels=c("A1", "A2")) +
fixed.factor("B", levels=c("B1", "B2", "B3")) +
random.factor("Subj", instances=60) #+
#random.factor("Items", instances=5)
dat <- design.codes(design)
nrow(dat)## [1] 360## [1] 60## # A tibble: 12 × 3
## Subj A B
## <fct> <fct> <fct>
## 1 Subj01 A1 B1
## 2 Subj01 A2 B1
## 3 Subj01 A1 B2
## 4 Subj01 A2 B2
## 5 Subj01 A1 B3
## 6 Subj01 A2 B3
## 7 Subj02 A1 B1
## 8 Subj02 A2 B1
## 9 Subj02 A1 B2
## 10 Subj02 A2 B2
## 11 Subj02 A1 B3
## 12 Subj02 A2 B3## [1] -1 1## 2-1 3-2
## 1 -0.6666667 -0.3333333
## 2 0.3333333 -0.3333333
## 3 0.3333333 0.6666667# Recommendation: hypr-package for contrasts + Schad et al. (2020) JML
dat$An <- model.matrix(~A,dat)[,2]
dat[,c("Bn1","Bn2")] <- model.matrix(~B,dat)[,2:3]Note that we use a sliding difference contrast
(contr.sdif) for the 3-level factor B. The first of these
contrasts tests the difference between B2 and B1, the second contrast
tests the difference between B3 and B2.
One option would now be to specify the relevant fixed and random
effects that relate to the set contrasts. This would imply defining a
fixed effect for the intercept, one for the effect of factor A, two for
the effect of factor B (which is coded via two contrasts), and two
effects for the interaction of A x B. The fixed-effects part of the
formula would look like this:
~ An + Bn1 + Bn2 + An:Bn1 + An:Bn2, and we could specify
fixed effects as Fixef=c(200,0,0,0,-1,-1). A similar
specification would be needed for the random effects part. This would be
a good approach.
One issue here might be that we might find it difficult to specify
our expectations in terms of these fixed effects. Maybe we find it
easier to specify the expected group means. We can use a trick to
specify the group means instead of the fixed effects written down above.
We create one large factor that has each design cell as a separate
factor level:
dat$F <- factor(paste0(dat$A, ".", dat$B)).
Now, in the specification of the model formula, we can remove the
intercept term (i.e., via -1 or 0 +). If we
would do so in a call to lmer (or equivalently to
lm), then the fixed effects would simply provide estimates
of the condition means, i.e., the mean responses for each factor level.
We can take a look at how this factor is coded in the design matrix by
using the function model.matrix().
## [1] "A1.B1" "A1.B2" "A1.B3" "A2.B1" "A2.B2" "A2.B3"## FA1.B1 FA1.B2 FA1.B3 FA2.B1 FA2.B2 FA2.B3
## 1 1 0 0 0 0 0
## 2 0 0 0 1 0 0
## 3 0 1 0 0 0 0
## 4 0 0 0 0 1 0
## 5 0 0 1 0 0 0
## 6 0 0 0 0 0 1We can see that in each row all columns are coded as 0, except for one column, which is coded as
1 - this column indicates the factor
level that this row corresponds to. For example, the first data point is
from the design cell A1 and B1. The data point
in the second row is from design cell A2 and
B1. This way, when we specify the fixed-effects, we can
specify the condition means, i.e., the means of the dependent variable
for each factor level.
We look at the levels of factor F to make sure we
consider the correct order of levels. Then, we specify that the first
design cell (A1 and B1) should have a mean of
190 (e.g., ms). The second design cell
(A1 and B2) should have a mean of 200. And so forth. Based on this reasoning,
we can define the 6 means for each cell in the 2 x 3 design:
fix <- c(190,200,210,210,200,190).
Next, we also have to specify the random effects standard deviations.
Let’s assume we don’t have a lot of specific information. We can use the
same coding as in the fixed effects term, by again removing the
intercept: (-1 + F | Subj). This way, we can specify the
standard deviation of each cell mean across subjects. Then we could
simply assume that the standard deviation in each design cell is 30:
sd_Subj <- c( 30, 30, 30, 30, 30, 30).
We again set the standard deviation of the random noise to 50.
An important aspect in this coding is the random effects correlation.
This now specifies how strongly responses in the different design cells
are correlated with each other across subjects. Note that this is very
different from the usual specification, where the random effects
correlations assess the correlation between effects, e.g., intercepts
and slopes. I guess that the random effects correlations in the present
setting should be relatively high, and set it to CP=0.85.
Of course, it would make sense to investigate this in real data to see
what a realistic value would be.
## [1] "A1.B1" "A1.B2" "A1.B3" "A2.B1" "A2.B2" "A2.B3"fix <- c(190,200,210,210,200,190) # set fixed effects
sd_Subj <- c( 30, 30, 30, 30, 30, 30) # set random effects standard deviations for subjects
sd_Res <- 50 # set residual standard deviation
form <- ~ -1 + F + (-1 + F | Subj)
dat$ysim <- simLMM(form, data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Res), CP=0.85, empirical=TRUE)## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 0 + FA1.B1 + FA1.B2 + FA1.B3 + FA2.B1 + FA2.B2 + FA2.B3 + ( 0 + FA1.B1 + FA1.B2 + FA1.B3 + FA2.B1 + FA2.B2 + FA2.B3 | Subj )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## Subj FA1.B1 30.0
## FA1.B2 30.0 0.85
## FA1.B3 30.0 0.85 0.85
## FA2.B1 30.0 0.85 0.85 0.85
## FA2.B2 30.0 0.85 0.85 0.85 0.85
## FA2.B3 30.0 0.85 0.85 0.85 0.85 0.85
## Residual 50.0
## Number of obs: 360, groups: Subj, 60
## Fixed effects:
## FA1.B1 FA1.B2 FA1.B3 FA2.B1 FA2.B2 FA2.B3
## 190 200 210 210 200 190# check group means
dat %>% group_by(A) %>% summarize(M=mean(ysim)) # the data show no main effect of A## # A tibble: 2 × 2
## A M
## <fct> <dbl>
## 1 A1 200
## 2 A2 200## # A tibble: 3 × 2
## B M
## <fct> <dbl>
## 1 B1 200
## 2 B2 200
## 3 B3 200## `summarise()` has grouped output by 'A'. You can override using the `.groups`
## argument.## # A tibble: 6 × 3
## # Groups: A [2]
## A B M
## <fct> <fct> <dbl>
## 1 A1 B1 190
## 2 A1 B2 200
## 3 A1 B3 210
## 4 A2 B1 210
## 5 A2 B2 200
## 6 A2 B3 190
# check fixed effects + random effects
#summary(lmer(ysim ~ -1 + F + (-1 + F | Subj), data=dat))
summary(lmer(ysim ~ An*(Bn1+Bn2) + (An*(Bn1+Bn2) || Subj), data=dat, control=lmerControl(calc.derivs=FALSE)))## boundary (singular) fit: see help('isSingular')## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: ysim ~ An * (Bn1 + Bn2) + (An * (Bn1 + Bn2) || Subj)
## Data: dat
## Control: lmerControl(calc.derivs = FALSE)
##
## REML criterion at convergence: 3879.6
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.48609 -0.50094 -0.06283 0.56271 2.51543
##
## Random effects:
## Groups Name Variance Std.Dev.
## Subj (Intercept) 1.054e+03 32.463326
## Subj.1 An 5.888e+01 7.673168
## Subj.2 Bn1 9.929e+02 31.509743
## Subj.3 Bn2 8.719e+02 29.527373
## Subj.4 An:Bn1 1.457e-06 0.001207
## Subj.5 An:Bn2 7.297e+02 27.013691
## Residual 1.866e+03 43.201253
## Number of obs: 360, groups: Subj, 60
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 2.000e+02 4.770e+00 5.900e+01 41.933 <2e-16 ***
## An -4.353e-16 2.483e+00 5.900e+01 0.000 1.0000
## Bn1 -2.893e-15 6.903e+00 7.393e+01 0.000 1.0000
## Bn2 -8.422e-15 6.756e+00 7.393e+01 0.000 1.0000
## An:Bn1 -1.000e+01 5.577e+00 5.528e+01 -1.793 0.0784 .
## An:Bn2 -1.000e+01 6.578e+00 8.341e+01 -1.520 0.1322
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) An Bn1 Bn2 An:Bn1
## An 0.000
## Bn1 0.000 0.000
## Bn2 0.000 0.000 -0.334
## An:Bn1 0.000 0.000 0.000 0.000
## An:Bn2 0.000 0.000 0.000 0.000 -0.424
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')## Anova Table (Type 3 tests)
##
## Response: ysim
## Effect df MSE F ges p.value
## 1 A 1, 59 2219.64 0.00 <.001 >.999
## 2 B 2.00, 117.87 3212.66 0.00 <.001 >.999
## 3 A:B 1.82, 107.29 2424.09 5.44 ** .019 .007
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
##
## Sphericity correction method: GGNow, that we saw that our model specification works, we can perform a power analysis. Let’s assume we are mainly interested in the interaction A:B. Because this involves two fixed effects, we perform a model comparison to evaluate their common effect. If you are interested in the specific effect of each of these terms, or in the main effects, similar analyses are possible.
nsubj <- seq(10,100,1) # again, we run from 10 to 100 subjects
nsim <- length(nsubj)
COF <- COFaov <- data.frame()
for (i in 1:nsim) { # i <- 1
#print(paste0(i,"/",nsim))
# create experimental design
design <-
fixed.factor("A", levels=c("A1", "A2")) +
fixed.factor("B", levels=c("B1", "B2", "B3")) +
random.factor("Subj", instances=nsubj[i]) # set number of subjects
dat <- design.codes(design)
nsj <- length(unique(dat$Subj))
# set contrasts & compute numeric predictor variables
contrasts(dat$A) <- c(-1, +1)
contrasts(dat$B) <- contr.sdif(3)
dat$An <- model.matrix(~A,dat)[,2]
dat[,c("Bn1","Bn2")] <- model.matrix(~B,dat)[,2:3]
# compute factor F
dat$F <- factor(paste0(dat$A, ".", dat$B))
for (j in 1:4) { # j <- 1 # run 4 models for each number of subjects
# simulate data
dat$ysim <- simLMM(form, data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Res), CP=0.85, empirical=FALSE, verbose=FALSE)
# ANOVA
AOV <- aov_car(ysim ~ 1 + Error(Subj/(A*B)), data=dat)
AOVcof <- summary(AOV)$univariate.tests[4,]
COFaov <- rbind(COFaov,c(nsj,AOVcof))
# LMM: perform model comparison
LMM1 <- lmer(ysim ~ An*(Bn1+Bn2) + (An*(Bn1+Bn2) || Subj), data=dat, REML=FALSE, control=lmerControl(calc.derivs=FALSE))
LMM0 <- lmer(ysim ~ An+(Bn1+Bn2) + (An*(Bn1+Bn2) || Subj), data=dat, REML=FALSE, control=lmerControl(calc.derivs=FALSE))
LMMcof <- c(coef(summary(LMM1))[5:6,5],
as.numeric(data.frame(anova(LMM1,LMM0))[2,6:8]))
COF <- rbind(COF,c(nsj,LMMcof,isSingular(LMM1) & isSingular(LMM0)))
}
}
# results from LMMs
names(COF) <- c("nsj","p_An:Bn1","p_An:Bn2","Chisq","Df","p","singular")
COF$sign <- as.numeric(COF$p < .05)
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF, "nsjF", transform, nsjFL=mean(nsj))$nsjFL
# results for ANOVA
names(COFaov) <- c("nsj","SS","numDF","ErrorSS","denDF","F","p")
COFaov$sign <- as.numeric(COFaov$p < .05)
COFaov$nsjF <- gtools::quantcut(COFaov$nsj, q=seq(0,1,length=10))
COFaov$nsjFL <- plyr::ddply(COFaov,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
#save(COF, COFaov, file="data/Power1.rda")Note that we here compute power only for the interaction term A:B. We actually assumed in the simulations that the main effects are exactly zero. However, equivalent power analyses are also possible for main effects.
Again, we first check how many fits were singular:
## [1] 0.6703297Here, we see that there is a large number of singular fits: roughly 70% of all model fits were singular suggesting that the random effects structure used in the model was not fully supported by the data. Opinions differ on whether this is a problem and how to deal with this. This paper, for example argues that singular fits can be ignored (https://debruine.github.io/lmem_sim/articles/paper.html). However, we would argue that sometimes singular fits can be problematic (in particular when one is interested in the random effects). Of course, convergence problems should also be monitored in addition.
p1 <- ggplot(data=COF)+
geom_smooth(aes(x=nsj, y=sign))+
geom_point( stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line( stat="summary", aes(x=nsjFL, y=sign))
p2 <- ggplot(data=COFaov)+
geom_smooth(aes(x=nsj, y=sign))+
geom_point( stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line( stat="summary", aes(x=nsjFL, y=sign))
gridExtra::grid.arrange(p1,p2, nrow=1)
Determine number of subjects needed for a power of 80%.
## [1] 67Let’s examine a design with only one factor X, but with crossed random effects for subjects and items, where factor X is a within-subject and between-item factor.
The computations are quite straightforward. Note that we now assume 3 random effects terms, one for subjects, one for items, and one residual noise term.
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=15) +
random.factor("Item", groups="X", instances=2)
dat <- design.codes(design)
(contrasts(dat$X) <- c(-1, +1))## [1] -1 1dat$Xn <- model.matrix(~X,dat)[,2]
# simulate data
fix <- c(200, 5) # set fixed effects
sd_Subj <- c(30, 10) # set random effects standard deviations for subjects
sd_Item <- c(10) # set random effects standard deviations for items
sd_Res <- 50 # set residual standard deviation
dat$ysim <- simLMM(form=~ X + (X | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3,
empirical=TRUE)## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + X1 + ( 1 + X1 | Subj ) + ( 1 | Item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## Subj (Intercept) 30.0
## X1 10.0 0.30
## Item (Intercept) 10.0
## Residual 50.0
## Number of obs: 60, groups: Subj, 15; Item, 4
## Fixed effects:
## (Intercept) X1
## 200 5## # A tibble: 2 × 2
## X M
## <fct> <dbl>
## 1 X1 195
## 2 X2 205# check fixed effects + random effects
summary(lmer(ysim ~ Xn + (Xn || Subj) + (1 | Item), data=dat, control=lmerControl(calc.derivs=FALSE)))## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: ysim ~ Xn + (Xn || Subj) + (1 | Item)
## Data: dat
## Control: lmerControl(calc.derivs = FALSE)
##
## REML criterion at convergence: 644.3
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.4539 -0.6250 -0.1323 0.6764 1.8460
##
## Random effects:
## Groups Name Variance Std.Dev.
## Subj (Intercept) 1171.4 34.23
## Subj.1 Xn 246.3 15.69
## Item (Intercept) 0.0 0.00
## Residual 2406.2 49.05
## Number of obs: 60, groups: Subj, 15; Item, 4
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 200.000 10.872 14.000 18.396 3.33e-11 ***
## Xn 5.000 7.518 14.000 0.665 0.517
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr)
## Xn 0.000
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')In the power analysis, we again continuously increase the number of subjects. An alternative or additional analysis could involve continuously increasing the number of items and testing how many items are needed to achieve a certain power.
nsubj <- seq(10,100,4)
nsimSj <- length(nsubj)
nitem <- seq(2,30,4)
nsimIt <- length(nitem)
COF <- data.frame()
for (i in 1:nsimSj) { # i <- 1
#print(paste0(i,"/",nsimSj))
for (j in 1:nsimIt) { # j <- 1
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=nsubj[i]) +
random.factor("Item", groups="X", instances=nitem[j])
dat <- design.codes(design)
nsj <- length(unique(dat$Subj))
nit <- length(unique(dat$Item))
contrasts(dat$X) <- c(-1, +1)
dat$Xn <- model.matrix(~X,dat)[,2]
for (j in 1:5) { # j <- 1
dat$ysim <- simLMM(~ X + (X | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3,
empirical=FALSE, verbose=FALSE)
LMM <- lmer(ysim ~ Xn + (Xn || Subj) + (1 | Item), data=dat, REML=FALSE,
control=lmerControl(calc.derivs=FALSE))
LMMcof <- coef(summary(LMM))[2,]
COF <- rbind(COF,c(nsj,nit,LMMcof, isSingular(LMM)))
}
}
}
# results for LMM
names(COF)<- c("nsj","nit","Estimate","SE","df","t","p","singular")
COF$sign <- as.numeric(COF$p < .05 & COF$Estimate>0)
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
COF$nitF <- gtools::quantcut(COF$nit, q=seq(0,1,length=5))
COF$nitFL <- plyr::ddply(COF,"nitF",transform,nitFL=mean(nsj))$nitFL
#save(COF, file="data/Power2.rda")Again, we plot the power as a function of the number of subjects. We see that power increases with increasing numbers of subjects and with increasing the numbers of items.
load(file=system.file("power-analyses", "Power2.rda", package = "designr"))
ggplot(data=COF) +
geom_smooth(aes(x=nsj, y=sign, colour=factor(nitF)), se=FALSE)+
geom_point( stat="summary", aes(x=nsjFL, y=sign, colour=factor(nitF)))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign, colour=factor(nitF)))+
geom_line( stat="summary", aes(x=nsjFL, y=sign, colour=factor(nitF)))
# determine number of subjects needed for a power of 60%
idx <- which(COF$nit>46)
m0 <- loess(sign ~ nsj, data=COF[idx,])
COF$pred <- NA
COF$pred[idx] <- predict(m0)
min(COF$nsj[COF$pred>0.5], na.rm=TRUE)## [1] 38One important aspect in power analyses is that we have to assume the true effect is known. However, we are usually unsure about its true value. Therefore, it can make sense to assume different values for the effect size and to look at how this impacts on power.
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=15) +
random.factor("Item", groups="X", instances=4)
dat <- design.codes(design)
length(unique(dat$Subj))## [1] 15## [1] 8## [1] -1 1dat$Xn <- model.matrix(~X,dat)[,2]
fix <- c(200,10)
sd_Subj <- c(30, 10)
sd_Item <- c(10)
sd_Res <- 50
dat$ysim <- simLMM(~ X + (X | Subj) + (1 | Item), data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3, empirical=TRUE)## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + X1 + ( 1 + X1 | Subj ) + ( 1 | Item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## Subj (Intercept) 30.0
## X1 10.0 0.30
## Item (Intercept) 10.0
## Residual 50.0
## Number of obs: 120, groups: Subj, 15; Item, 8
## Fixed effects:
## (Intercept) X1
## 200 10## # A tibble: 2 × 2
## X M
## <fct> <dbl>
## 1 X1 190
## 2 X2 210## Linear mixed model fit by REML ['lmerModLmerTest']
## Formula: ysim ~ Xn + (Xn || Subj) + (1 | Item)
## Data: dat
## REML criterion at convergence: 1288.898
## Random effects:
## Groups Name Std.Dev.
## Subj (Intercept) 26.37
## Subj.1 Xn 16.81
## Item (Intercept) 14.08
## Residual 47.78
## Number of obs: 120, groups: Subj, 15; Item, 8
## Fixed Effects:
## (Intercept) Xn
## 200 10The data simulations look good. We next go to the power simulations.
nsubj <- seq(10,100,1)
nsim <- length(nsubj)
COF <- data.frame()
for (i in 1:nsim) {
#print(paste0(i,"/",nsim))
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=nsubj[i]) +
random.factor("Item", groups="X", instances=4)
dat <- design.codes(design)
nsj <- length(unique(dat$Subj))
contrasts(dat$X) <- c(-1, +1)
dat$Xn <- model.matrix(~X,dat)[,2]
for (j in 1:18) {
# assume three different effect sizes for the fixed effect: 5, 10, or 15
FIX <- c(5,10,15)
fix <- c(200,FIX[(j%%3)+1])
dat$ysim <- simLMM(~ X + (X | Subj) + (1 | Item), data=dat, Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3, empirical=FALSE, verbose=FALSE)
LMM <- lmer(ysim ~ Xn + (Xn || Subj) + (1 | Item), data=dat, control=lmerControl(calc.derivs=FALSE))
LMMcof <- coef(summary(LMM))[2,]
COF <- rbind(COF,c(nsj,fix[2],LMMcof)) # store effect size
}
}
# results for LMM
names(COF) <- c("nsj","EffectSize","Estimate","SE","df","t","p")
COF$sign <- as.numeric(COF$p < .05 & COF$Estimate>0)
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
#save(COF, file="data/Power3.rda")load(file=system.file("power-analyses", "Power3.rda", package = "designr"))
ggplot(data=COF) +
geom_smooth(aes(x=nsj, y=sign, colour=factor(EffectSize)))+
geom_point(stat="summary", aes(x=nsjFL, y=sign, colour=factor(EffectSize)))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign, colour=factor(EffectSize)))+
geom_line(stat="summary", aes(x=nsjFL, y=sign, colour=factor(EffectSize)))
The results show that power strongly varies as a function of the assumed effect size. This is something important to consider: maybe we are not really sure about the power of our experiment.
Determine number of subjects needed for a power of 50% under the assumption of an effect size of middle size (i.e., fixed effect of 10).
m0 <- loess(sign ~ nsj, data=subset(COF,EffectSize==10))
COF$pred <- NA
COF$pred[COF$EffectSize==10] <- predict(m0)
idx <- COF$pred>0.5
min(COF$nsj[idx], na.rm=TRUE) # 70## [1] 88Now let’s consider the case that we want to use results from a prior experiment as the assumption about effect sizes in our power analysis. First we load the data from the prior study.
# load empirical data
data(gibsonwu2013) # the Gibson & Wu (2013) dataset is included in the designr package
gw1 <- subset(gibsonwu2013, region=="headnoun") # subset critical region
gw1$so <- ifelse(gw1$type %in% c("subj-ext"),-1,1) # sum-coding for predictor
dat2 <- gw1[,c("subj","item","so","type2","rt")] # extract experimental design
datE <- dplyr::arrange(dat2, subj, item)
length(unique(datE$subj)) # 37## [1] 37## [1] 15# we re-create the empirical design from the experiment using designr
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("subj", instances=18) +
random.factor("item", instances= 8) +
random.factor(c("subj", "item"), groups=c("X"))
dat <- dplyr::arrange(design.codes(design), subj, item)
(contrasts(dat$X) <- c(+1, -1))## [1] 1 -1## [1] 36## [1] 16## subj item so type2 rt
## 2001 1 1 1 object relative 246
## 1192 1 2 -1 subject relative 542
## 1591 1 3 1 object relative 406
## 753 1 4 -1 subject relative 286
## 341 1 5 1 object relative 582
## 221 1 6 -1 subject relative 959
## 1471 1 7 1 object relative 270
## 922 1 8 -1 subject relative 438
## 461 1 9 1 object relative 294
## 1054 1 10 -1 subject relative 278
## 1342 1 11 1 object relative 494
## 94 1 13 1 object relative 1561
## 621 1 14 -1 subject relative 438
## 1891 1 15 1 object relative 286
## 1761 1 16 -1 subject relative 374
## 20371 2 1 -1 subject relative 286
## 19801 2 2 1 object relative 278
## 19501 2 3 -1 subject relative 254
## 20521 2 4 1 object relative 734
## 18991 2 5 -1 subject relative 390## subj item X so
## 1 subj01 item01 X1 1
## 2 subj01 item02 X2 -1
## 3 subj01 item03 X1 1
## 4 subj01 item04 X2 -1
## 5 subj01 item05 X1 1
## 6 subj01 item06 X2 -1
## 7 subj01 item07 X1 1
## 8 subj01 item08 X2 -1
## 9 subj01 item09 X1 1
## 10 subj01 item10 X2 -1
## 11 subj01 item11 X1 1
## 12 subj01 item12 X2 -1
## 13 subj01 item13 X1 1
## 14 subj01 item14 X2 -1
## 15 subj01 item15 X1 1
## 16 subj01 item16 X2 -1
## 17 subj02 item01 X2 -1
## 18 subj02 item02 X1 1
## 19 subj02 item03 X2 -1
## 20 subj02 item04 X1 1## subj item so type2 rt
## 59661 39 11 1 object relative 2053
## 59151 39 13 1 object relative 445
## 60061 39 14 -1 subject relative 222
## 59951 39 15 1 object relative 430
## 58901 39 16 -1 subject relative 302
## 63041 40 1 -1 subject relative 422
## 64031 40 2 1 object relative 318
## 63191 40 3 -1 subject relative 421
## 62871 40 4 1 object relative 462
## 64301 40 5 -1 subject relative 1262
## 64551 40 6 1 object relative 438
## 64181 40 7 -1 subject relative 356
## 63761 40 8 1 object relative 1509
## 63471 40 9 -1 subject relative 390
## 63891 40 10 1 object relative 358
## 64801 40 11 -1 subject relative 390
## 64421 40 13 -1 subject relative 318
## 64671 40 14 1 object relative 278
## 63361 40 15 -1 subject relative 461
## 63631 40 16 1 object relative 294## subj item X so
## 557 subj35 item13 X1 1
## 558 subj35 item14 X2 -1
## 559 subj35 item15 X1 1
## 560 subj35 item16 X2 -1
## 561 subj36 item01 X2 -1
## 562 subj36 item02 X1 1
## 563 subj36 item03 X2 -1
## 564 subj36 item04 X1 1
## 565 subj36 item05 X2 -1
## 566 subj36 item06 X1 1
## 567 subj36 item07 X2 -1
## 568 subj36 item08 X1 1
## 569 subj36 item09 X2 -1
## 570 subj36 item10 X1 1
## 571 subj36 item11 X2 -1
## 572 subj36 item12 X1 1
## 573 subj36 item13 X2 -1
## 574 subj36 item14 X1 1
## 575 subj36 item15 X2 -1
## 576 subj36 item16 X1 1
# run LMM on speed variable
lmm <- lmer(speed ~ so + (so|subj) + (so|item), data=datE, control=lmerControl(calc.derivs=FALSE))## boundary (singular) fit: see help('isSingular')## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: speed ~ so + (so | subj) + (so | item)
## Data: datE
## Control: lmerControl(calc.derivs = FALSE)
##
## REML criterion at convergence: -5932.5
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.2501 -0.5996 0.1237 0.6430 2.5441
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subj (Intercept) 3.712e-07 6.093e-04
## so 1.331e-08 1.154e-04 -0.51
## item (Intercept) 1.100e-07 3.317e-04
## so 2.305e-09 4.801e-05 1.00
## Residual 8.916e-07 9.442e-04
## Number of obs: 547, groups: subj, 37; item, 15
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 2.672e-03 1.379e-04 3.806e+01 19.369 <2e-16 ***
## so 3.879e-05 4.644e-05 2.966e+01 0.835 0.41
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr)
## so 0.012
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')# the random effects correlation for items is estimated as 1, indicating a singular fit
lmm <- lmer(speed ~ so + (so|subj) + (so||item), data=datE, control=lmerControl(calc.derivs=FALSE))## boundary (singular) fit: see help('isSingular')## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: speed ~ so + (so | subj) + (so || item)
## Data: datE
## Control: lmerControl(calc.derivs = FALSE)
##
## REML criterion at convergence: -5931.3
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.1531 -0.5908 0.1328 0.6442 2.5066
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subj (Intercept) 3.718e-07 6.098e-04
## so 1.315e-08 1.147e-04 -0.51
## item (Intercept) 1.098e-07 3.314e-04
## item.1 so 1.809e-15 4.253e-08
## Residual 8.941e-07 9.456e-04
## Number of obs: 547, groups: subj, 37; item, 15
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 2.672e-03 1.379e-04 3.807e+01 19.371 <2e-16 ***
## so 3.810e-05 4.477e-05 3.553e+01 0.851 0.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr)
## so -0.159
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')Based on these analyses, we take the last fitted lmm model, and use
this to simulate new data. This is possible with the simple function
call simLMM(LMM=lmm). This uses the data that the model was
fit on. However, we can also use the fit lmm object to simulate data for
a new data set (i.e., here the experimental design we created we created
using designr), by simply adding the new data frame:
simLMM(data=dat, LMM=lmm). Importantly, the new data frame
needs to have the variables that are used in the model.
## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + so + ( 1 + so | subj ) + ( 1 | item ) + ( 0 + so | item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## subj (Intercept) 0.0006098
## so 0.0001147 -0.51
## item (Intercept) 0.0003314
## item so 0.0000000
## Residual 0.0009456
## Number of obs: 547, groups: subj, 37; item, 15
## Fixed effects:
## (Intercept) so
## 2.672153e-03 3.809511e-05## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + so + ( 1 + so | subj ) + ( 1 | item ) + ( 0 + so | item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev. Corr
## subj (Intercept) 0.0006098
## so 0.0001147 -0.51
## item (Intercept) 0.0003314
## item so 0.0000000
## Residual 0.0009456
## Number of obs: 576, groups: subj, 36; item, 16
## Fixed effects:
## (Intercept) so
## 2.672153e-03 3.809511e-05## # A tibble: 2 × 2
## so M
## <dbl> <dbl>
## 1 -1 0.00263
## 2 1 0.00271## boundary (singular) fit: see help('isSingular')## Linear mixed model fit by REML ['lmerModLmerTest']
## Formula: speed ~ so + (so || subj) + (so || item)
## Data: dat
## REML criterion at convergence: -6250.4
## Random effects:
## Groups Name Std.Dev.
## subj (Intercept) 5.353e-04
## subj.1 so 3.669e-05
## item (Intercept) 3.383e-04
## item.1 so 0.000e+00
## Residual 9.565e-04
## Number of obs: 576, groups: subj, 36; item, 16
## Fixed Effects:
## (Intercept) so
## 0.0026722 0.0000381
## optimizer (nloptwrap) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warningsThe results show that we can exactly recover the parameters from the previous model in the new simulated data set.
Next, we can turn to perform power analyses.
# perform power analysis based on fitted model
nsubj <- seq(10,100,1)
nsim <- length(nsubj)
COF <- data.frame()
for (i in 1:nsim) { # i <- 1
#print(paste0(i,"/",nsim))
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("subj", instances=nsubj[i]) +
random.factor("item", instances= 8) +
random.factor(c("subj", "item"), groups=c("X"))
dat <- dplyr::arrange(design.codes(design), subj, item)
contrasts(dat$X) <- c(+1, -1)
dat$so <- model.matrix(~X,dat)[,2]
nsj <- length(unique(dat$subj))
for (j in 1:10) { # j <- 1
dat$speed <- simLMM(data=dat, LMM=lmm, empirical=FALSE, verbose=FALSE)
LMMcof <- coef(summary(lmer(speed ~ so + (so||subj) + (so||item), data=dat, control=lmerControl(calc.derivs=FALSE))))[2,]
COF <- rbind(COF,c(nsj,LMMcof))
}
}
names(COF) <- c("nsj","Estimate","SE","df","t","p")
COF$sign <- as.numeric(COF$p < .05 & COF$Estimate>0)
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
#save(COF, file="data/Power4.rda")load(file=system.file("power-analyses", "Power4.rda", package = "designr"))
ggplot(data=COF) +
geom_smooth(aes(x=nsj, y=sign))+
geom_point(stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line(stat="summary", aes(x=nsjFL, y=sign))
## (Intercept) so
## 2.672153e-03 3.809511e-05## [1] 3.859278e-05# determine number of subjects needed for a power of 40%
m0 <- loess(sign ~ nsj, data=COF)
COF$pred <- predict(m0)
idx <- COF$pred>0.4
min(COF$nsj[idx]) # 164## [1] 174We can see that the power for this study is quite low. Notice that we are using a realistic effect size estimate from a previous study. Yet, even 200 subjects are not sufficient to obtain a power of 80% to detect the true effect.
Next, we treat the case of a continuous covariate. Interestingly, we
can use simLMM to simulate not only a dependent variable,
but also a covariate.
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=15) +
random.factor("Item", groups="X", instances=10)
dat <- design.codes(design)
(contrasts(dat$X) <- c(-1, +1))## [1] -1 1simLMM has a further argument: family. This
can be set to “lp”, which stands for “linear predictor”. This means that
simLMM does not add any residual noise, but puts out the
best prediction for each data point. Using this functionality, we can
generate covariates that only differ by subject (i.e., age in the
example below), or covariates that only differ by item (i.e., word
frequency in the example below). However, we can also simulate a
covariate that varies across subjects, items, and across individual
trials. For example, when modeling fixation durations during reading, we
might include the previous fixation duration as a predictor (covariate).
When modeling ratings, we might have a task where each trial has two
ratings, and we want to predict the second rating using the first as
predictor (covariate). However, these differences are mainly important
for simulating the covariate.
# simulate covariate
dat$age <- round( simLMM(form=~ 1 + (1 | Subj), data=dat,
Fixef=30, VC_sd=list(5),
empirical=TRUE, family="lp") )## Data simulation from a linear mixed-effects model (LMM): linear predictor
## Formula: lp ~ 1 + ( 1 | Subj )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev.Corr
## Subj (Intercept) 5.0
## Number of obs: 300, groups: Subj, 15
## Fixed effects:
## (Intercept)
## 30
dat$wordFrequency <- round( simLMM(form=~ 1 + (1 | Item), data=dat,
Fixef=4, VC_sd=list(1),
empirical=TRUE, family="lp") )## Data simulation from a linear mixed-effects model (LMM): linear predictor
## Formula: lp ~ 1 + ( 1 | Item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev.Corr
## Item (Intercept) 1.0
## Number of obs: 300, groups: Item, 20
## Fixed effects:
## (Intercept)
## 4
fix <- c(5) # set fixed effects
sd_Subj <- c(1) # set random effects standard deviations for subjects
sd_Item <- c(1) # set random effects standard deviations for items
sd_Res <- 1 # set residual standard deviation
dat$rating <- round( simLMM(form=~ 1 + (1 | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res),
empirical=TRUE) )## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + ( 1 | Subj ) + ( 1 | Item )
## empirical = TRUE
## Random effects:
## Groups Name Std.Dev.Corr
## Subj (Intercept) 1.0
## Item (Intercept) 1.0
## Residual 1.0
## Number of obs: 300, groups: Subj, 15; Item, 20
## Fixed effects:
## (Intercept)
## 5
Note that we mean-center the covariate.
Importantly, when we use a covariate (i.e., continuous variable),
then it is not possible to use empirical=TRUE. We have to
switch to empirical=FALSE.
# simulate data
fix <- c(200,20, 7, 5) # set fixed effects
sd_Subj <- c(30, 10, 5, 5) # set random effects standard deviations for subjects
sd_Item <- c(10) # set random effects standard deviations for items
sd_Res <- 50 # set residual standard deviation
dat$ysim <- simLMM(form=~ X*rating.c + (X*rating.c | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3,
empirical=FALSE)## Data simulation from a linear mixed-effects model (LMM)
## Formula: gaussian ~ 1 + X1 + rating.c + X1:rating.c + ( 1 + X1 + rating.c + X1:rating.c | Subj ) + ( 1 | Item )
## empirical = FALSE
## Random effects:
## Groups Name Std.Dev. Corr
## Subj (Intercept) 30.0
## X1 10.0 0.30
## rating.c 5.0 0.30 0.30
## X1:rating.c 5.0 0.30 0.30 0.30
## Item (Intercept) 10.0
## Residual 50.0
## Number of obs: 300, groups: Subj, 15; Item, 20
## Fixed effects:
## (Intercept) X1 rating.c X1:rating.c
## 200 20 7 5## # A tibble: 2 × 2
## X M
## <fct> <dbl>
## 1 X1 174.
## 2 X2 228.# check fixed effects + random effects
summary(lmer(ysim ~ Xn*rating.c + (Xn*rating.c || Subj) + (1 | Item), data=dat, control=lmerControl(calc.derivs=FALSE)))## boundary (singular) fit: see help('isSingular')## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: ysim ~ Xn * rating.c + (Xn * rating.c || Subj) + (1 | Item)
## Data: dat
## Control: lmerControl(calc.derivs = FALSE)
##
## REML criterion at convergence: 3224.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.95068 -0.61584 -0.03999 0.65256 2.74467
##
## Random effects:
## Groups Name Variance Std.Dev.
## Item (Intercept) 173.637 13.177
## Subj Xn:rating.c 4.919 2.218
## Subj.1 rating.c 0.000 0.000
## Subj.2 Xn 0.000 0.000
## Subj.3 (Intercept) 655.056 25.594
## Residual 2545.620 50.454
## Number of obs: 300, groups: Item, 20; Subj, 15
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 201.946 7.808 16.951 25.863 4.63e-15 ***
## Xn 29.072 4.195 16.404 6.930 2.96e-06 ***
## rating.c 9.183 2.346 132.621 3.915 0.000144 ***
## Xn:rating.c 6.205 2.039 12.925 3.043 0.009480 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) Xn rtng.c
## Xn 0.000
## rating.c 0.001 0.095
## Xn:rating.c 0.041 0.007 0.014
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')nsubj <- seq(10,100,1)
nsim <- length(nsubj)
COF <- data.frame()
for (i in 1:nsim) { # i <- 1
#print(paste0(i,"/",nsim))
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=nsubj[i]) +
random.factor("Item", groups="X", instances=2)
dat <- design.codes(design)
nsj <- length(unique(dat$Subj))
contrasts(dat$X) <- c(-1, +1)
dat$Xn <- model.matrix(~X,dat)[,2]
for (j in 1:10) { # j <- 1
dat$rating <- round( simLMM(form=~ 1 + (1 | Subj) + (1 | Item), data=dat,
Fixef=5, VC_sd=list(1, 1, 1), empirical=FALSE, verbose=FALSE) )
dat$rating.c <- dat$rating - mean(dat$rating)
fix <- c(200,20, 7, 5) # set fixed effects
sd_Subj <- c(30, 10, 5, 5) # set random effects standard deviations for subjects
sd_Item <- c(10) # set random effects standard deviations for items
sd_Res <- 50 # set residual standard deviation
dat$ysim <- simLMM(form=~ X*rating.c + (X*rating.c | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item, sd_Res), CP=0.3,
empirical=FALSE, verbose=FALSE)
LMM <- lmer(ysim ~ Xn*rating.c + (Xn*rating.c || Subj) + (1 | Item), data=dat,
control=lmerControl(calc.derivs=FALSE))
LMMcof <- coef(summary(LMM))[4,] # extract the stats for the interaction
COF <- rbind(COF,c(nsj,LMMcof))
}
}
names(COF) <- c("nsj","Estimate","SE","df","t","p")
COF$sign <- as.numeric(COF$p < .05 & COF$Estimate>0)
COF$nsjF <- gtools::quantcut(COF$nsj, q=seq(0,1,length=10))
COF$nsjFL <- plyr::ddply(COF,"nsjF",transform,nsjFL=mean(nsj))$nsjFL
#save(COF, file="data/Power5.rda")We are here only interested in the interaction between the covariate and the factor, i.e., whether the slopes differ between conditions. Of course, one could also study the main effects.
load(file=system.file("power-analyses", "Power5.rda", package = "designr"))
ggplot(data=COF) +
geom_smooth(aes(x=nsj, y=sign))+
geom_point( stat="summary", aes(x=nsjFL, y=sign))+
geom_errorbar(stat="summary", aes(x=nsjFL, y=sign))+
geom_line( stat="summary", aes(x=nsjFL, y=sign))
# determine number of subjects needed for a power of 50%
m0 <- loess(sign ~ nsj, data=COF)
COF$pred <- predict(m0)
idx <- COF$pred>0.5
min(COF$nsj[idx]) # 84## [1] 56We can also simulate data from a logistic GLMM. This can be
implemented by setting the argument family="binomial".
Note, however, that the argument empirical=TRUE does not
work for family="binomial". To be precise, the random
effects for subjects or items can still be sampled using
empirical=TRUE; however, the residual Bernoulli noise is
not simulated exactly.
design <-
fixed.factor("X", levels=c("X1", "X2")) +
random.factor("Subj", instances=50) +
random.factor("Item", groups="X", instances=10)
dat <- dplyr::arrange(design.codes(design), Subj, Item)[c(3, 1, 2)]
(contrasts(dat$X) <- c(-1, +1))## [1] -1 1dat$Xn <- model.matrix(~X,dat)[,2]
fix <- c(0.5,2) # specify fixed-effects
sd_Subj <- c(3 ,1) # specify subject random effects (standard deviation)
sd_Item <- c(1) # specify item random effects (standard deviation)
dat$ysim <- simLMM(form=~ X + (X | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item), CP=0.3,
empirical=FALSE, family="binomial")## Data simulation from a logistic generalized linear mixed-effects model (GLMM)
## Formula: binomial ~ 1 + X1 + ( 1 + X1 | Subj ) + ( 1 | Item )
## empirical = FALSE
## Random effects:
## Groups Name Std.Dev.Corr
## Subj (Intercept) 3.0
## X1 1.0 0.30
## Item (Intercept) 1.0
## Number of obs: 1000, groups: Subj, 50; Item, 20
## Fixed effects:
## (Intercept) X1
## 0.5 2.0## # A tibble: 2 × 2
## X M
## <fct> <dbl>
## 1 X1 0.322
## 2 X2 0.788glmer(ysim ~ X + (X | Subj) + (1 | Item), data=dat, family="binomial", control=lmerControl(calc.derivs=FALSE))## Warning in glmer(ysim ~ X + (X | Subj) + (1 | Item), data = dat, family =
## "binomial", : please use glmerControl() instead of lmerControl()## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: ysim ~ X + (X | Subj) + (1 | Item)
## Data: dat
## AIC BIC logLik deviance df.resid
## 696.9263 726.3729 -342.4632 684.9263 994
## Random effects:
## Groups Name Std.Dev. Corr
## Subj (Intercept) 3.631
## X1 1.251 0.23
## Item (Intercept) 1.450
## Number of obs: 1000, groups: Subj, 50; Item, 20
## Fixed Effects:
## (Intercept) X1
## 0.8764 2.8495We don’t execute the power simulation here, since this would take a long time. But it should be a straightforward extension from the other analyses shown in detail.
Maybe you want to simulate data from a GLMM with a different
probability density function (PDF) or with a different link function.
Currently, such further family options are not implemented in
simLMM (except for family="lognormal").
However, using the argument family="lp", it is possible to
extract the linear predictor from the GLMM, and to compute the link
function and desired probability distribution manually.
# provide the linear predictor
# i.e., predictions of the LMM before passing through the link function / probability density function (PDF)
# this allows using custom links / PDFs
dat$lp <- simLMM(form=~ X + (X | Subj) + (1 | Item), data=dat,
Fixef=fix, VC_sd=list(sd_Subj, sd_Item), CP=0.3,
empirical=FALSE, family="lp")## Data simulation from a linear mixed-effects model (LMM): linear predictor
## Formula: lp ~ 1 + X1 + ( 1 + X1 | Subj ) + ( 1 | Item )
## empirical = FALSE
## Random effects:
## Groups Name Std.Dev.Corr
## Subj (Intercept) 3.0
## X1 1.0 0.30
## Item (Intercept) 1.0
## Number of obs: 1000, groups: Subj, 50; Item, 20
## Fixed effects:
## (Intercept) X1
## 0.5 2.0