library(lmerTest)
library(simr)
data(sleepstudy)
S <- sleepstudyTechnical Details- Sleepstudy
- Finger Tapping
- Power Simulations
Sleepstudy
Reaction ~ Days
. . .
Model Illustration
Model Equation
fit <- lmer(Reaction ~ Days + (Days|Subject), S). . .
equatiomatic::extract_eq(fit)\[ \begin{aligned} \operatorname{Reaction}_{i} &\sim N \left(\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{Days}), \sigma^2 \right) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{j}} \\ &\mu_{\beta_{1j}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\ \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} \end{array} \right) \right) \text{, for Subject j = 1,} \dots \text{,J} \end{aligned} \]
Test
LMM
coef(summary(fit))["Days", , drop=F] Estimate Std. Error df t value Pr(>|t|)
Days 10.47 1.546 17 6.771 3.264e-06. . .
Aggregate + t-Test
COEFS <- coef(lmList(Reaction ~ Days | Subject, S))
head(COEFS) (Intercept) Days
308 244.2 21.765
309 205.1 2.262
310 203.5 6.115
330 289.7 3.008
331 285.7 5.266
332 264.3 9.567. . .
t.test(COEFS$Days)$p.value[1] 3.264e-06Finger Tapping
Tap_Speed ~ Fatigued + Caffein + Physical_Activity + Time
. . .
LMM
D <- readRDS("./FingerTapping.rds")
anova(RSLP_fit <-
lmer(Tap_Speed_t0 ~
Fatigued + Caffein + Physical_Activity + Time +
(Fatigued + Caffein + Physical_Activity + Time || PersonID), D
))Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
Fatigued 1.118 1.118 1 20.3 16.52 0.00059 ***
Caffein 0.710 0.710 1 16.7 10.48 0.00492 **
Physical_Activity 0.134 0.134 1 14.2 1.97 0.18150
Time 0.843 0.843 1 27.5 12.45 0.00149 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Aggregate + t-Test
lmper person- Extract coefficients
- t-test
. . .
COEFS <- coef(lmList(Tap_Speed_t0 ~
Fatigued + Caffein + Physical_Activity + Time| PersonID, D))
head(COEFS) (Intercept) Fatigued Caffein Physical_Activity Time
1 5.537 -0.005006 0.11454 0.01294 0.010271
2 5.954 0.008884 0.06979 0.24502 -0.009993
4 6.089 -0.091132 0.24615 NA -0.006202
6 5.690 -0.002435 0.09211 NA -0.025624
7 6.197 -0.139987 NA NA 0.034949
8 6.288 -0.042551 NA 0.45775 0.010620. . .
t.test(COEFS$Fatigued)$p.value[1] 0.006832Comparison
LMM
- Challenging numerics
- Crossed random effects
aggregate + t-Test
- KISS
- Fewer assumptions
Type I Error & Power
Simulation Code
# Setup auxiliary functions
get_type1_CI <- function(P.vals){
C.int <- apply(P.vals < 0.05, 1, function(x)
prop.test(sum(x), length(x), p = 0.05)$conf.int |> round(4))
cbind(
`Power` = rowMeans(P.vals < 0.05),
lwr = C.int[1, ],
upr = C.int[2, ]
) |> as.data.frame()
}
get_p_vals <- function(Y){
D$Y <- Y
c(
LMM = lmer(Y ~
Fatigued + Caffein + Physical_Activity + Time +
(Fatigued + Caffein + Physical_Activity + Time || PersonID), D) |>
summary() |> coef() |> _["Fatigued", "Pr(>|t|)"],
`t-Test` = lmList(Y ~
Fatigued + Caffein + Physical_Activity + Time | PersonID, D
) |> coef() |> _$Fatigued |> t.test() |> _$p.value
)
}
# Simulate H0 ===========
xfun::cache_rds({
RSLP_fit <- lmer(Tap_Speed_t0 ~
Fatigued + Caffein + Physical_Activity + Time +
(Fatigued + Caffein + Physical_Activity + Time | PersonID), D
)
fixef(RSLP_fit) <- c(0,0,0,0,0)
simulate(
RSLP_fit,
nsim = 10000,
seed=123) |>
parallel::mclapply(get_p_vals)
},
file = "cache_sim3.1.rds",
dir = "cache/",
hash = list(D, as.character(body(get_p_vals)))
) |> simplify2array() |> get_type1_CI() -> H0_CI
# Simulate H1 ===========
xfun::cache_rds({
RSLP_fit <- lmer(Tap_Speed_t0 ~
Fatigued + Caffein + Physical_Activity + Time +
(Fatigued + Caffein + Physical_Activity + Time | PersonID), D
)
fixef(RSLP_fit) <- c(0,0.02,0,0,0)
simulate(
RSLP_fit,
nsim = 10000,
seed=123) |>
parallel::mclapply(get_p_vals)
},
file = "cache_sim3.2.rds",
dir = "cache/",
hash = list(D, as.character(body(get_p_vals)))
) |> simplify2array() |> get_type1_CI() -> H1_CI
# Plot results ============
H0_CI$Hypothesis <- "H0: Fatigue = 0"
H1_CI$Hypothesis <- "H1: Fatigue = 0.02"
H_all <- rbind(H0_CI, H1_CI)
H_all$test <- rownames(H_all)
ggplot(H_all, aes(x = test, y = `Power`, ymin = lwr, ymax = upr)) +
geom_pointrange() +
geom_hline(yintercept = 0.05, linetype = 2) +
facet_wrap(~Hypothesis, scales = "free") +
theme_minimal() +
xlab("")
Power-Simulations
Using simr
powerSim(RSLP_fit). . .
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Simulate with rnorm() & co.
lme4::simulate.merMod
- Existing model
- Modify existing model (
simr)
. . .
fixef<-coef<-VarCorr<-sigma<-scale<-
. . .
fixef(RSLP_fit) <- c(0,0,0,0,0)
simulate(RSLP_fit, nsim = 100)No existing model?
simulate(Y ~ ..., newparams = list(theta = ???, ...))1
. . .
simr::makeLmer(
Y ~ Fatigued + Caffein + Physical_Activity + Time +
(Fatigued + Caffein + Physical_Activity + Time | PersonID),
fixef = c(0,0,0,0,0),
VarCorr = diag(5),
data = D,
sigma = 1
)Linear mixed model fit by REML ['lmerMod']
Formula: Y ~ Fatigued + Caffein + Physical_Activity + Time + (Fatigued +
Caffein + Physical_Activity + Time | PersonID)
Data: D
REML criterion at convergence: 2033
Random effects:
Groups Name Std.Dev. Corr
PersonID (Intercept) 1
Fatigued 1 0.00
Caffein 1 0.00 0.00
Physical_Activity 1 0.00 0.00 0.00
Time 1 0.00 0.00 0.00 0.00
Residual 1
Number of obs: 567, groups: PersonID, 27
Fixed Effects:
(Intercept) Fatigued Caffein Physical_Activity
0 0 0 0
Time
0 Caching: xfun::cache_rds
xfun::cache_rds({
expression
},
hash = list_with_relevant_objects
)Have Fun Simulating!
- Aggregate per Subject \(\rightarrow\) t-Test
equatiomatic::extract_eq()lme4::simulate.merMod()fixef<-& co.simr::makeLmer()
xfun::cache_rds()
Footnotes
theta: cholesky factor of the normalized random effects covariance matrix.↩︎