Power Analysis for Longitudinal Multilevel/Linear Mixed-Effects Models.

The purpose of `powerlmm`

is to help design longitudinal treatment studies (parallel groups), with or without higher-level clustering (e.g. longitudinally clustered by therapists, groups, or physician), and missing data. The main features of the package are:

- Longitudinal two- and three-level (nested) linear mixed-effects models, and partially nested designs.
- Random slopes at the subject- and cluster-level.
- Missing data.
- Unbalanced designs (both unequal cluster sizes, and treatment groups).
- Design effect, and estimated type I error when the third-level is ignored.
- Fast analytical power calculations for all designs.
- Power for small samples sizes using Satterthwaite’s degrees of freedom approximation.
- Explore bias, Type I errors, model misspecification, and LRT model selection using convenient simulation methods.

`powerlmm`

can be installed from CRAN and GitHub.

```
# CRAN, version 0.3.0
install.packages("powerlmm")
# GitHub, dev version
devtools::install_github("rpsychologist/powerlmm")
```

This is an example of setting up a three-level longitudinal model with random slopes at both the subject- and cluster-level, with different missing data patterns per treatment arm. Relative standardized inputs are used, but unstandardized raw parameters values can also be used.

```
library(powerlmm)
d <- per_treatment(control = dropout_weibull(0.3, 2),
treatment = dropout_weibull(0.2, 2))
p <- study_parameters(n1 = 11,
n2 = 10,
n3 = 5,
icc_pre_subject = 0.5,
icc_pre_cluster = 0,
icc_slope = 0.05,
var_ratio = 0.02,
dropout = d,
effect_size = cohend(-0.8,
standardizer = "pretest_SD"))
p
#>
#> Study setup (three-level)
#>
#> n1 = 11
#> n2 = 10 x 5 (treatment)
#> 10 x 5 (control)
#> n3 = 5 (treatment)
#> 5 (control)
#> 10 (total)
#> total_n = 50 (treatment)
#> 50 (control)
#> 100 (total)
#> dropout = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (time)
#> 0, 0, 1, 3, 6, 9, 12, 16, 20, 25, 30 (%, control)
#> 0, 0, 1, 2, 4, 5, 8, 10, 13, 17, 20 (%, treatment)
#> icc_pre_subjects = 0.5
#> icc_pre_clusters = 0
#> icc_slope = 0.05
#> var_ratio = 0.02
#> effect_size = -0.8 (Cohen's d [SD: pretest_SD])
```

```
get_power(p, df = "satterthwaite")
#>
#> Power Analyis for Longitudinal Linear Mixed-Effects Models (three-level)
#> with missing data and unbalanced designs
#>
#> n1 = 11
#> n2 = 10 x 5 (treatment)
#> 10 x 5 (control)
#> n3 = 5 (treatment)
#> 5 (control)
#> 10 (total)
#> total_n = 50 (control)
#> 50 (treatment)
#> 100 (total)
#> dropout = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (time)
#> 0, 0, 1, 3, 6, 9, 12, 16, 20, 25, 30 (%, control)
#> 0, 0, 1, 2, 4, 5, 8, 10, 13, 17, 20 (%, treatment)
#> icc_pre_subjects = 0.5
#> icc_pre_clusters = 0
#> icc_slope = 0.05
#> var_ratio = 0.02
#> effect_size = -0.8 (Cohen's d [SD: pretest_SD])
#> df = 7.953218
#> alpha = 0.05
#> power = 68%
```

Unequal cluster sizes is also supported, the cluster sizes can either be random (sampled), or the marginal distribution can be specified.

```
p <- study_parameters(n1 = 11,
n2 = unequal_clusters(2, 3, 5, 20),
icc_pre_subject = 0.5,
icc_pre_cluster = 0,
icc_slope = 0.05,
var_ratio = 0.02,
effect_size = cohend(-0.8,
standardizer = "pretest_SD"))
get_power(p)
#>
#> Power Analyis for Longitudinal Linear Mixed-Effects Models (three-level)
#> with missing data and unbalanced designs
#>
#> n1 = 11
#> n2 = 2, 3, 5, 20 (treatment)
#> 2, 3, 5, 20 (control)
#> n3 = 4 (treatment)
#> 4 (control)
#> 8 (total)
#> total_n = 30 (control)
#> 30 (treatment)
#> 60 (total)
#> dropout = No missing data
#> icc_pre_subjects = 0.5
#> icc_pre_clusters = 0
#> icc_slope = 0.05
#> var_ratio = 0.02
#> effect_size = -0.8 (Cohen's d [SD: pretest_SD])
#> df = 6
#> alpha = 0.05
#> power = 44%
```

Cluster sizes follow a Poisson distribution

```
p <- study_parameters(n1 = 11,
n2 = unequal_clusters(func = rpois(5, 5)), # sample from Poisson
icc_pre_subject = 0.5,
icc_pre_cluster = 0,
icc_slope = 0.05,
var_ratio = 0.02,
effect_size = cohend(-0.8,
standardizer = "pretest_SD"))
get_power(p, R = 100, progress = FALSE) # expected power by averaging over R realizations
#>
#> Power Analyis for Longitudinal Linear Mixed-Effects Models (three-level)
#> with missing data and unbalanced designs
#>
#> n1 = 11
#> n2 = rpois(5, 5) (treatment)
#> - (control)
#> n3 = 5 (treatment)
#> 5 (control)
#> 10 (total)
#> total_n = 26 (control)
#> 26 (treatment)
#> 52 (total)
#> dropout = No missing data
#> icc_pre_subjects = 0.5
#> icc_pre_clusters = 0
#> icc_slope = 0.05
#> var_ratio = 0.02
#> effect_size = -0.8 (Cohen's d [SD: pretest_SD])
#> df = 8
#> alpha = 0.05
#> power = 49% (MCSE: 1%)
#>
#> NOTE: n2 is randomly sampled. Values are the mean from R = 100 realizations.
```

Several convenience functions are also included, e.g. for creating power curves.

```
x <- get_power_table(p,
n2 = 5:10,
n3 = c(4, 8, 12),
effect_size = cohend(c(0.5, 0.8), standardizer = "pretest_SD"))
```

The package includes a flexible simulation method that makes it easy to investigate the performance of different models. As an example, let’s compare the power difference between the 2-level LMM with 11 repeated measures, to doing an ANCOVA at posttest. Using `sim_formula`

different models can be fit to the same data set during the simulation.

```
p <- study_parameters(n1 = 11,
n2 = 40,
icc_pre_subject = 0.5,
cor_subject = -0.4,
var_ratio = 0.02,
effect_size = cohend(-0.8,
standardizer = "pretest_SD"))
# 2-lvl LMM
f0 <- sim_formula("y ~ time + time:treatment + (1 + time | subject)")
# ANCOVA, formulas with no random effects are with using lm()
f1 <- sim_formula("y ~ treatment + pretest",
data_transform = transform_to_posttest,
test = "treatment")
f <- sim_formula_compare("LMM" = f0,
"ANCOVA" = f1)
res <- simulate(p,
nsim = 2000,
formula = f,
cores = parallel::detectCores(logical = FALSE))
```

We then summarize the results using `summary`

. Let’s look specifically at the treatment effects.

```
summary(res, para = list("LMM" = "time:treatment",
"ANCOVA" = "treatment"))
#> Model: summary
#>
#> Fixed effects: 'time:treatment', 'treatment'
#>
#> model M_est theta M_se SD_est Power Power_bw Power_satt
#> LMM -1.1 -1.1 0.32 0.31 0.94 0.93 .
#> ANCOVA -11.0 0.0 3.70 3.70 0.85 0.85 0.85
#> ---
#> Number of simulations: 2000 | alpha: 0.05
#> Time points (n1): 11
#> Subjects per cluster (n2 x n3): 40 (treatment)
#> 40 (control)
#> Total number of subjects: 80
#> ---
#> At least one of the models applied a data transformation during simulation,
#> summaries that depend on the true parameter values will no longer be correct,
#> see 'help(summary.plcp_sim)'
```

We can also look at a specific model, here’s the results for the 2-lvl LMM.

```
summary(res, model = "LMM")
#> Model: LMM
#>
#> Random effects
#>
#> parameter M_est theta est_rel_bias prop_zero is_NA
#> subject_intercept 100.00 100.0 0.00600 0 0
#> subject_slope 2.00 2.0 0.00630 0 0
#> error 100.00 100.0 -0.00049 0 0
#> cor_subject -0.39 -0.4 -0.01400 0 0
#>
#> Fixed effects
#>
#> parameter M_est theta M_se SD_est Power Power_bw Power_satt
#> (Intercept) 0.0150 0.0 1.30 1.30 0.046 . .
#> time 0.0013 0.0 0.25 0.25 0.055 . .
#> time:treatment -1.1000 -1.1 0.32 0.31 0.940 0.93 .
#> ---
#> Number of simulations: 2000 | alpha: 0.05
#> Time points (n1): 11
#> Subjects per cluster (n2 x n3): 40 (treatment)
#> 40 (control)
#> Total number of subjects: 80
#> ---
#> At least one of the models applied a data transformation during simulation,
#> summaries that depend on the true parameter values will no longer be correct,
#> see 'help(summary.plcp_sim)'
```

The package’s basic functionality is also implemented in a Shiny web application, aimed at users that are less familiar with R. Launch the application by typing