# [R-sig-ME] Minimum detectable effect size in linear mixed model

Han Zhang h@nzh @end|ng |rom um|ch@edu
Sat Jul 4 00:27:12 CEST 2020

```Hello,

I'm trying to find the minimum detectable effect size (MDES) given my
sample, alpha (.05), and desired power (90%) in a linear mixed model
setting. I'm using the simr package for a simulation-based approach. What I
did is changing the original effect size to a series of hypothetical effect
sizes and find the minimum effect size that has a 90% chance of producing a
significant result. Below is a toy code:

library(lmerTest)
library(simr)

# fit the model
model <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
summary(model)

Fixed effects:
Estimate Std. Error      df t value Pr(>|t|)
(Intercept)  251.405      6.825  17.000  36.838  < 2e-16 ***
Days          10.467      1.546  17.000   6.771 3.26e-06 ***

Here is the code for minimum detectable effect size:

pwr <- NA

# define a set of reasonable effect sizes
es <- seq(0, 10, 2)

# loop through the effect sizes
for (i in 1:length(es)) {
# replace the original effect size with new one
fixef(model)['Days'] =  es[i]
# run simulation to obtain power estimate
pwr.summary <- summary(powerSim(
model,
test = fixed('Days', "t"),
nsim = 100,
progress = T
))
# store output
pwr[i] <- as.numeric(pwr.summary)
}

# display results
cbind("Coefficient" = es,
Power = pwr)

Output:

Coefficient   Power
[1,]                                     0  0.09
[2,]                                     2  0.24
[3,]                                     4  0.60
[4,]                                     6  0.99
[5,]                                     8  1.00
[6,]                                    10  1.00

My questions:

(1) Is this the right way to find the MDES?

(2) I have some trouble making sense of the output. Can I say the
following: because the estimated power when the effect = 6 is 99%, and
because the actual model has an estimate of 10.47, then the study is
sufficiently powered? Conversely, imagine that if the actual estimate was
3.0, then can I say the study is insufficiently powered?

Thank you,
Han
--
Han Zhang, Ph.D.
Department of Psychology
University of Michigan, Ann Arbor
https://sites.lsa.umich.edu/hanzh/

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