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01_intro/exercises.md

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+Exercise: Binomial test power simulation
+================
+
+## Birth rates
+
+Kanazawa (2007) claims that beautiful parents have more daughters
+
+- Plan a study and calculate the sample size necessary to
+  - detect a deviation from the global 106:100 male-female sex ratio
+  - with about 80% power
+- Wanted: Substance-matter knowledge
+  - What would be a minimum relevant deviation (effect)?
+  - Considering the literature on birth rates, what would be a realistic
+    deviation?
+- Some background
+  - <https://en.wikipedia.org/wiki/Human_sex_ratio>
+  - Literature cited there (e.g., Davis, Gottlieb, and Stampnitzky 1998;
+    Mathews and Hamilton 2005)
+
+### References
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-DavisGottlieb98" class="csl-entry">
+
+Davis, D. L., M. B. Gottlieb, and J. R. Stampnitzky. 1998. “Reduced
+Ratio of Male to Female Births in Several Industrial Countries: A
+Sentinel Health Indicator?” *Journal of the American Medical
+Association* 279 (13): 1018–23.
+<https://doi.org/10.1001/jama.279.13.1018>.
+
+</div>
+
+<div id="ref-Kanazawa07" class="csl-entry">
+
+Kanazawa, S. 2007. “Beautiful Parents Have More Daughters: A Further
+Implication of the Generalized Trivers–Willard Hypothesis (gTWH).”
+*Journal of Theoretical Biology* 244 (1): 133–40.
+<https://doi.org/10.1016/j.jtbi.2006.07.017>.
+
+</div>
+
+<div id="ref-MathewsHamilton05" class="csl-entry">
+
+Mathews, T. J., and B. E. Hamilton. 2005. “Trend Analysis of the Sex
+Ratio at Birth in the United States.” *National Vital Statistics
+Reports* 53 (20): 1–20.
+
+</div>
+
+</div>

02_ttest/exercises-ttest1.md

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+Exercise: Power simulation and power curves for t-test
+================
+
+## Temporal value asymmetry
+
+“participants … were asked to imagine that they had agreed to spend 5 hr
+entering data into a computer and to indicate how much money it would be
+fair for them to receive. Some participants imagined that they had
+completed the work 1 month previously, and others imagined that they
+would complete the work 1 month in the future . . . Participants
+believed that they should receive 101% more money for work they would do
+1 month later ($M = \$125.04$) than for identical work that they had
+done 1 month previously ($M = \$62.20$), $t(119) = 2.22$, $p = .03$,
+$d = 0.41$” (Caruso, Gilbert, and Wilson 2008, 797)
+
+### Plan a direct replication
+
+1.  What is a plausible standard deviation? Hint: $d = (M1 − M2)/SD$
+2.  What is an interesting minimal effect size (in \$, Euro, or min)?
+3.  Simulate responses for 120 participants in both the *past* and the
+    *future* condition, assuming normal distributions with the same
+    variance. Use the standard deviation and the minimal effect size
+    from 1. and 2.
+4.  Parameter recovery: Repeat the simulation from 3. 2000 times to
+    re-estimate the parameters ($\mu_1, \mu_2, \sigma$) from the
+    simulated responses. Visualize the recovered parameters in box
+    plots.  
+    Hint: $SE = 2/\sqrt{n} \cdot SD$, where $n$ is the total sample
+    size.
+
+``` r
+t <- t.test(x, y, mu = 0, var.equal = TRUE)
+c(t$estimate, sd.pool = sqrt(n) / 2 * t$stderr)
+```
+
+5.  Power simulation: Increase the total sample size to find out the `n`
+    necessary for 80% power for the t-test.
+6.  Power curves:
+    - Write an `R` function that takes sample size `n`, minimal effect
+      `d`, standard deviation `sd`, and number of replications `nrep` as
+      arguments. It should return the simulated power.
+    - Use this function to simulate the power for each combination of 4
+      different standard deviations and 4 sample sizes.
+    - Visualize these power curves in a single plot.
+
+### References
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-CarusoGilbert08" class="csl-entry">
+
+Caruso, E. M., D. T. Gilbert, and T. D. Wilson. 2008. “A Wrinkle in
+Time: Asymmetric Valuation of Past and Future Events.” *Psychological
+Science* 19 (8): 796–801.
+<https://doi.org/10.1111/j.1467-9280.2008.02159.x>.
+
+</div>
+
+</div>

02_ttest/exercises-ttest2.md

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+Exercise: Power simulation and power curves for t-test
+================
+
+## Directed reading activities
+
+Plan a replication of the “directed reading activities” study on
+<https://jasp-stats.github.io/jasp-data-library/myChapters/chapter_2.html>
+
+1.  Which parameter value represents the minimum relevant effect that
+    the test should be able to detect? How large should this effect be?
+    Briefly say what it means in the context of the study.
+
+2.  What would be a plausible value for the standard deviation?
+
+3.  Simulate and draw power curves that depend on the effect size and
+    $n$. Use at least five different effect sizes and at least four
+    different sample sizes. The minimum relevant effect size should be
+    among the simulation conditions.
+
+4.  From inspecting the power curves, what $n$ in each group is required
+    to detect the minimum relevant effect with at least 80% power?
+
+5.  Create a renderable R script or an R Markdown file that includes
+
+    - a header with title, author, date
+    - at least one section head line
+    - the homework questions and your answers
+    - the simulation code and output
+    - the plot of the power curves.
+
+    Render the R or Rmd file to HTML.

02_ttest/powersim-ttest.R

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+#' ---
+#' title: "Independent t-test: Power simulation and power curves"
+#' author: ""
+#' date: "Last modified: 2026-01-09"
+#' ---
+
+
+#' ## Application context
+
+#'
+#' Listening experiment
+#'
+#' - Task of each participant is to repeatedly adjust the frequency of a
+#'   comparison tone to sound equal in pitch to a 1000-Hz standard tone
+#' - Mean adjustment estimates the point of subjective equality $\mu$
+#' - Two participants will take part in the experiment providing adjustments
+#'   $X$ and $Y$
+#' - Goal is to detect a difference between their points of subjective
+#'   equality $\mu_x$ and $\mu_y$ of 4 Hz
+#'
+
+#' ## Model
+
+#'
+#' Assumptions
+#' 
+#' - $X_1, \ldots, X_n \sim N(\mu_x, \sigma_x^2)$ i.i.d.
+#' - $Y_1, \ldots, Y_m \sim N(\mu_y, \sigma_y^2)$ i.i.d.
+#' - both samples independent
+#' - $\sigma_x^2 = \sigma_y^2$ but unknown
+#' 
+#' Hypothesis
+#' 
+#' - H$_0\colon~ \mu_x - \mu_y = \delta = 0$
+#'
+
+#' ## Power simulation
+#+ cache = TRUE
+
+n <- 110; m <- 110
+pval <- replicate(2000, {
+  x <- rnorm(n, mean = 1000 + 4, sd = 10)         # Participant 1 responses
+  y <- rnorm(m, mean = 1000,     sd = 10)         # Participant 2 responses
+  t.test(x, y, mu = 0, var.equal = TRUE)$p.value
+})
+mean(pval < 0.05)
+
+#' ## Power curves
+#'
+#' Turn into a function of n and effect size
+
+pwrFun <- function(n = 30, d = 4, sd = 10, nrep = 50) {
+  n <- n; m <- n
+  pval <- replicate(nrep, {
+    x <- rnorm(n, mean = 1000 + d, sd = sd)
+    y <- rnorm(m, mean = 1000,     sd = sd)
+    t.test(x, y, mu = 0, var.equal = TRUE)$p.value
+  })
+  mean(pval < 0.05)
+}
+
+#'
+#' Set up conditions and call power function
+#+ cache = TRUE
+
+cond <- expand.grid(d = 0:5,
+                    n = c(50, 75, 100, 125))
+cond$pwr <- mapply(pwrFun, n = cond$n, d = cond$d, MoreArgs = list(nrep = 500))
+
+## Plot results
+lattice::xyplot(pwr ~ d, cond, groups = n, type = c("g", "b"),
+                auto.key = list(corner = c(0, 1)))
+
+#' ## Parameter recovery
+
+n <- 100
+
+out <- replicate(2000, {
+  x <- rnorm(n, mean = 1000 + 4, sd = 10)
+  y <- rnorm(n, mean = 1000,     sd = 10)
+  t <- t.test(x, y, mu = 0, var.equal = TRUE)
+  c(
+    effect = -as.numeric(diff(t$estimate)),
+    sd.pool = t$stderr * sqrt(2*n) / 2
+  )
+})
+
+rowMeans(out)
+boxplot(t(out))
+

02_ttest/powersim-ttest.md

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+Independent t-test: Power simulation and power curves
+================
+Last modified: 2026-01-09
+
+## Application context
+
+Listening experiment
+
+- Task of each participant is to repeatedly adjust the frequency of a
+  comparison tone to sound equal in pitch to a 1000-Hz standard tone
+- Mean adjustment estimates the point of subjective equality $\mu$
+- Two participants will take part in the experiment providing
+  adjustments $X$ and $Y$
+- Goal is to detect a difference between their points of subjective
+  equality $\mu_x$ and $\mu_y$ of 4 Hz
+
+## Model
+
+Assumptions
+
+- $X_1, \ldots, X_n \sim N(\mu_x, \sigma_x^2)$ i.i.d.
+- $Y_1, \ldots, Y_m \sim N(\mu_y, \sigma_y^2)$ i.i.d.
+- both samples independent
+- $\sigma_x^2 = \sigma_y^2$ but unknown
+
+Hypothesis
+
+- H$_0\colon~ \mu_x - \mu_y = \delta = 0$
+
+## Power simulation
+
+``` r
+n <- 110; m <- 110
+pval <- replicate(2000, {
+  x <- rnorm(n, mean = 1000 + 4, sd = 10)         # Participant 1 responses
+  y <- rnorm(m, mean = 1000,     sd = 10)         # Participant 2 responses
+  t.test(x, y, mu = 0, var.equal = TRUE)$p.value
+})
+mean(pval < 0.05)
+```
+
+    ## [1] 0.831
+
+## Power curves
+
+Turn into a function of n and effect size
+
+``` r
+pwrFun <- function(n = 30, d = 4, sd = 10, nrep = 50) {
+  n <- n; m <- n
+  pval <- replicate(nrep, {
+    x <- rnorm(n, mean = 1000 + d, sd = sd)
+    y <- rnorm(m, mean = 1000,     sd = sd)
+    t.test(x, y, mu = 0, var.equal = TRUE)$p.value
+  })
+  mean(pval < 0.05)
+}
+```
+
+Set up conditions and call power function
+
+``` r
+cond <- expand.grid(d = 0:5,
+                    n = c(50, 75, 100, 125))
+cond$pwr <- mapply(pwrFun, n = cond$n, d = cond$d, MoreArgs = list(nrep = 500))
+
+## Plot results
+lattice::xyplot(pwr ~ d, cond, groups = n, type = c("g", "b"),
+                auto.key = list(corner = c(0, 1)))
+```
+
+![](powersim-ttest_files/figure-gfm/unnamed-chunk-3-1.png)<!-- -->
+
+## Parameter recovery
+
+``` r
+n <- 100
+
+out <- replicate(2000, {
+  x <- rnorm(n, mean = 1000 + 4, sd = 10)
+  y <- rnorm(n, mean = 1000,     sd = 10)
+  t <- t.test(x, y, mu = 0, var.equal = TRUE)
+  c(
+    effect = -as.numeric(diff(t$estimate)),
+    sd.pool = t$stderr * sqrt(2*n) / 2
+  )
+})
+
+rowMeans(out)
+```
+
+    ##   effect  sd.pool 
+    ## 3.978400 9.993142
+
+``` r
+boxplot(t(out))
+```
+
+![](powersim-ttest_files/figure-gfm/unnamed-chunk-4-1.png)<!-- -->

03_anova/exercises-anova.md

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+Exercise: Testing the interaction in two-by-two ANOVA
+================
+
+## Anchoring and adjustment
+
+Items and anchor values (Jacowitz and Kahneman 1995)
+
+- How tall is the largest coast redwood in the world? \[20, 168m\]
+- How many member states belong to the United Nations? \[14, 127
+  members\]
+- How much km/h is the maximum speed of a house cat? \[11, 48km/h\]
+
+### Research question
+
+- Does time pressure (respond within 7s) increase the anchor effect?
+
+### Suggest a minimum relevant effect
+
+- Go to <https://apps.mathpsy.uni-tuebingen.de/fw/pars2eta/>
+- Fix the parameters of the ANOVA model
+
+### Some background
+
+- Open anchoring quest (Röseler et al. 2022), <https://osf.io/ygnvb/>
+
+### Plan the study
+
+- Pick one of the three items
+- Parameter recovery
+  - Make a data frame for the two-by-two design
+  - With the parameter values determined before, simulate responses
+  - Re-estimate the parameters
+- Power simulation
+  - Calculate the sample size necessary to detect the time-pressure
+    effect
+
+### Bonus task: Verify the plausibility of your model
+
+- Download the raw data from the open anchoring quest project
+- Estimate $\sigma$ and compare it to your value
+
+### References
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-JacowitzKahneman95" class="csl-entry">
+
+Jacowitz, K. E., and D. Kahneman. 1995. “Measures of Anchoring in
+Estimation Tasks.” *Personality and Social Psychology Bulletin* 21 (11):
+1161–66. <https://doi.org/10.1177/01461672952111004>.
+
+</div>
+
+<div id="ref-RoeselerWeber22" class="csl-entry">
+
+Röseler, L., L. Weber, K. A. C. Helgerth, E. Stich, M. Günther, P.
+Tegethoff, F. S. Wagner, et al. 2022. “OpAQ: Open Anchoring Quest,
+Version 1.1.50.97.” <https://doi.org/10.17605/OSF.IO/YGNVB>.
+
+</div>
+
+</div>

03_anova/powersim-anova.R

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+#' ---
+#' title: "Two-by-two ANOVA: Power simulation of the interaction test"
+#' author: ""
+#' date: "Last modified: 2026-01-09"
+#' ---
+
+#' ## Application context
+
+#'
+#' Effect of fertilizers
+#'
+#' In an experiment, two fertilizers (A and B, each either low or high dose)
+#' will be combined and the yield of peas (Y) in kg be observed. Goal is to
+#' detect an increase of the Fertilizer-A effect by an additional 12 kg when
+#' combined with a high dose of Fertilizer B (interaction effect).
+#'
+
+#+ echo = FALSE
+dat <- data.frame(
+  A = rep(1:2, each = 2),
+  B = rep(1:2, times = 2),
+  y = c(30, 30 + 5, 30 + 30, 30 + 30 + 5 + 12)
+)
+par(mai = c(.6, .6, .1, .1), mgp = c(2, .7, 0))
+plot(y ~ A, dat, type = "n", xlim = c(0.8, 2.2), ylim = c(20, 80),
+     xlab = "Fertilizer A", ylab = "Yield (kg)", xaxt = "n")
+lines(y ~ A, dat[dat$B == 1, ], col = "darkblue")
+lines(y ~ A, dat[dat$B == 2, ], col = "darkblue")
+lines(1:2, c(30 + 5, 30 + 30 + 5), lty = 2, col = "darkblue")
+axis(1, 1:2, c("low", "high"))
+arrows(2, 30 + 30 + 6, 2, 30 + 30 + 5 + 11, code = 3, length = 0.1,
+       col = "darkgray")
+text(c(1, 2, 1, 2, 2.07), c(27, 55, 40, 80, 65 + 6),
+     c(expression(mu), expression(mu + alpha[2]),
+       expression(mu + beta[2]),
+       expression(mu + alpha[2] + beta[2] + (alpha * beta)[22]),
+       "12 kg")
+)
+text(1.5, 27 + 30/2, "Fertilizer B: low", srt = 21, col = "darkgray")
+text(1.5, 38 + (30 + 12)/2, "Fertilizer B: high", srt = 32, col = "darkgray")
+
+#' ## Model
+
+#'
+#' Assumptions
+#' 
+#' - $Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} +
+#'              \varepsilon_{ijk}$
+#' - $\varepsilon_{ijk} \sim N(0, \sigma^2) \text{ i.i.d.}$
+#' - $i = 1, \dots, I$; $j = 1, \dots, J$; $k = 1, \dots, K$
+#' - $\alpha_1 = \beta_1 := 0$
+#' 
+#' Hypothesis
+#' 
+#' - H$_0^{AB}\colon~ (\alpha\beta)_{ij} = 0 \text{ for all } i,j$
+#' 
+
+#' ## Setup
+
+set.seed(1704)
+n <- 96
+dat <- data.frame(
+  A = factor(rep(1:2, each = n/2), labels = c("low", "high")),
+  B = factor(rep(rep(1:2, each = n/4), 2), labels = c("low", "high"))
+)
+X <- model.matrix(~ A*B, dat)
+unique(X)
+beta <- c(mu = 30, a2 = 30, b2 = 5, ab22 = 12)
+means <- X %*% beta
+
+lattice::xyplot(I(means + rnorm(n, sd = 10)) ~ A, dat, groups = B,
+                type = c("g", "p", "a"), auto.key = TRUE, ylab = "Yield (kg)")
+
+#' ## Parameter recovery
+#+ cache = TRUE
+
+out <- replicate(2000, {
+  y <- means + rnorm(n, sd = 10)   # y = mu + a + b + ab + e
+  m <- aov(y ~ A * B, dat)
+  c(coef(m), sigma = sigma(m))
+})
+boxplot(t(out))
+
+#' ## Power simulation
+#+ cache = TRUE
+
+pval <- replicate(2000, {
+  y <- means + rnorm(n, sd = 10)
+  m <- aov(y ~ A*B, dat)
+  summary(m)[[1]]$"Pr(>F)"[3]      # test of interaction
+})
+mean(pval < 0.05)
+

03_anova/powersim-anova.md

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+Two-by-two ANOVA: Power simulation of the interaction test
+================
+Last modified: 2026-01-09
+
+## Application context
+
+Effect of fertilizers
+
+In an experiment, two fertilizers (A and B, each either low or high
+dose) will be combined and the yield of peas (Y) in kg be observed. Goal
+is to detect an increase of the Fertilizer-A effect by an additional 12
+kg when combined with a high dose of Fertilizer B (interaction effect).
+
+![](powersim-anova_files/figure-gfm/unnamed-chunk-1-1.png)<!-- -->
+
+## Model
+
+Assumptions
+
+- $Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} +  \varepsilon_{ijk}$
+- $\varepsilon_{ijk} \sim N(0, \sigma^2) \text{ i.i.d.}$
+- $i = 1, \dots, I$; $j = 1, \dots, J$; $k = 1, \dots, K$
+- $\alpha_1 = \beta_1 := 0$
+
+Hypothesis
+
+- H$_0^{AB}\colon~ (\alpha\beta)_{ij} = 0 \text{ for all } i,j$
+
+## Setup
+
+``` r
+set.seed(1704)
+n <- 96
+dat <- data.frame(
+  A = factor(rep(1:2, each = n/2), labels = c("low", "high")),
+  B = factor(rep(rep(1:2, each = n/4), 2), labels = c("low", "high"))
+)
+X <- model.matrix(~ A*B, dat)
+unique(X)
+```
+
+    ##    (Intercept) Ahigh Bhigh Ahigh:Bhigh
+    ## 1            1     0     0           0
+    ## 25           1     0     1           0
+    ## 49           1     1     0           0
+    ## 73           1     1     1           1
+
+``` r
+beta <- c(mu = 30, a2 = 30, b2 = 5, ab22 = 12)
+means <- X %*% beta
+
+lattice::xyplot(I(means + rnorm(n, sd = 10)) ~ A, dat, groups = B,
+                type = c("g", "p", "a"), auto.key = TRUE, ylab = "Yield (kg)")
+```
+
+![](powersim-anova_files/figure-gfm/unnamed-chunk-2-1.png)<!-- -->
+
+## Parameter recovery
+
+``` r
+out <- replicate(2000, {
+  y <- means + rnorm(n, sd = 10)   # y = mu + a + b + ab + e
+  m <- aov(y ~ A * B, dat)
+  c(coef(m), sigma = sigma(m))
+})
+boxplot(t(out))
+```
+
+![](powersim-anova_files/figure-gfm/unnamed-chunk-3-1.png)<!-- -->
+
+## Power simulation
+
+``` r
+pval <- replicate(2000, {
+  y <- means + rnorm(n, sd = 10)
+  m <- aov(y ~ A*B, dat)
+  summary(m)[[1]]$"Pr(>F)"[3]      # test of interaction
+})
+mean(pval < 0.05)
+```
+
+    ## [1] 0.817

04_ancova/exercises-ancova1.md

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+Exercise: Analysis and power simulation for baseline/follow-up
+measurements
+================
+
+## Shoulder pain and acupuncture
+
+1.  Reanalyze the original data
+    - Re-estimate the ANCOVA model for the Kleinhenz et al. (1999)
+      [data](../data/kleinhenz.txt)
+2.  Run a power simulation for a replication study
+    1.  Draw plausible pre-CMS values
+    2.  Specify the minimum relevant average treatment effect (ATE)
+    3.  Set the remaining parameters to plausible values
+    4.  What is the sample size required for the test to detect the
+        effect with 80% power?
+    5.  How robust is the power simulation when you repeat it with a new
+        set of pre-CMS values? Try it!
+    6.  Recover the parameters of the ANCOVA model
+
+### References
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-KleinhenzStreitberger99" class="csl-entry">
+
+Kleinhenz, J., K. Streitberger, J. Windeler, A. Güßbacher, G. Mavridis,
+and E. Martin. 1999. “Randomised Clinical Trial Comparing the Effects of
+Acupuncture and a Newly Designed Placebo Needle in Rotator Cuff
+Tendinitis.” *Pain* 83 (2): 235–41.
+
+</div>
+
+</div>

04_ancova/exercises-ancova2.md

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+Exercise: Analysis and power simulation for baseline/follow-up
+measurements
+================
+
+## MASS anorexia data
+
+1.  Analyze the original data:
+    - In R, see ?MASS::anorexia
+    - Data preparation
+
+``` r
+data(anorexia, package = "MASS")
+dat <-
+  subset(anorexia, Treat != "Cont") |>  # exclude control group
+  droplevels()                          # drop empty factor levels
+lbs2kg <- 0.4535924
+dat$Prewt  <- lbs2kg * dat$Prewt        # to kg
+dat$Postwt <- lbs2kg * dat$Postwt
+lattice::xyplot(Postwt ~ Prewt, dat, groups = Treat,
+                type = c("g", "r", "p"), auto.key = TRUE)
+```
+
+- Estimate the average treatment effect (ATE) for FT relative to CBT.
+- What is the 95% CI for the ATE?
+- What are the pre- and post-weight means for the two groups?
+- What are the baseline-adjusted means for the two groups?
+
+2.  Run a power simulation for a replication study:
+    - Draw plausible pre-weights.
+    - Specify the minimum relevant effect.
+    - Set the remaining parameters to plausible values.
+    - What is the sample size required for the test to detect the effect
+      with 80% power?
+    - How robust is the power simulation when you repeat it with a new
+      set of pre-weights? Try it!
+    - Recover the parameters of the ANCOVA model.
+3.  Create a renderable R script or an R Markdown file that includes
+    - a header with title, author, date
+    - at least one section head line
+    - the homework questions and your answers
+    - the R code, output, and plots (if any)
+
+    Render the R or Rmd file to HTML.
+
+### Reference

05_mixed1/exercises-pwrlmm.md

@@ -0,0 +1,50 @@
+Exercise: Power simulation for longitudinal data
+================
+
+## Risperidone vs. haloperidol and schizophrenia
+
+``` r
+dat       <- read.table("../data/moeller.csv", header = TRUE, sep = ",")
+dat$id    <- factor(dat$id)
+dat$treat <- factor(dat$treat, levels = c("risp", "halo"))
+
+lattice::xyplot(pans ~ week, data = dat, groups = treat, type = c("g", "p", "a"), auto.key = TRUE)
+```
+
+1)  Analyze the original data from [moeller.csv](../data/moeller.csv):
+    - `pans`: Positive and Negative Symptom Scale for schizophrenia
+
+    - `treat`: medication group
+
+      - `risp`: atypical neuroleptic risperidone
+      - `halo`: conventional neuroleptic haloperidol
+
+    - What is the sample size in each treatment group?
+
+    - Estimate the by-group random-slope model.
+
+    - What are the estimates for the fixed effects and variance
+      components?
+
+    - Interpret the interaction effect.
+
+    - Test the interaction effect.
+2)  Run a power simulation for a replication study:
+    - Set up a data frame containing the study design and sample size.
+
+    - Specify the minimum relevant effect.
+
+    - Set the fixed effects and variance components to plausible values.
+
+    - How many participants are required for the test of the interaction
+      to detect the specified effect with a power of 80%?
+
+    - Recover the parameters of the by-group random-slope model for one
+      simulated data set.
+3)  Create a renderable R script or an R Markdown file that includes
+    - a header with title, author, date
+    - at least one section head line
+    - the questions from above and your answers
+    - the R code, output, and plots (if any)
+
+    Render the R or Rmd file to HTML.

05_mixed1/powersim-lmm.R

@@ -0,0 +1,185 @@
+#' ---
+#' title: "Power for mixed-effects models"
+#' author: ""
+#' date: "Last modified: 2026-01-09"
+#' bibliography: ../lit.bib
+#' ---
+
+library(lattice)
+library(lme4)
+
+#' # Reanalysis
+
+#' ## Application context: Depression and type of diagnosis
+#'
+#' - @ReisbyGram77 studied the effect of Imipramin on 66 inpatients
+#'   treated for depression
+#' - Depression was measured with the Hamilton depression rating scale
+#' - Patients were classified into endogenous and non-endogenous depressed
+#' - Depression was measured weekly for 6 time points
+#'
+#' Data: [reisby.txt](../data/reisby.txt)
+
+dat      <- read.table("../data/reisby.txt", header = TRUE)
+dat$id   <- factor(dat$id)
+dat$diag <- factor(dat$diag, levels = c("nonen", "endog"))
+dat      <- na.omit(dat)     # drop missing values
+head(dat, n = 13)
+
+xyplot(hamd ~ week | id, data = dat, type=c("g", "r", "p"),
+  pch = 16, layout = c(11, 6), ylab = "HDRS score", xlab = "Time (week)")
+
+
+#' ## Random-intercept model
+#'
+#' $$
+#' \begin{aligned}
+#' Y_{ij} &= \beta_0 + \beta_1 \, \mathtt{week}_{ij}
+#'         + \upsilon_{0i}
+#'         + \varepsilon_{ij} \\
+#' \upsilon_{0i} &\sim N(0, \sigma^2_{\upsilon_0}) \text{ i.i.d.} \\
+#' \mathbf{\varepsilon}_i &\sim N(0, \, \sigma^2)  \text{ i.i.d.} \\
+#' i &= 1, \ldots, I, \quad j = 1, \ldots n_i
+#' \end{aligned}
+#' $$
+
+m1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE)
+summary(m1)
+
+#' ## Random-slope model
+#'
+#' $$
+#' \begin{aligned}
+#' Y_{ij} &= \beta_0 + \beta_1 \, \mathtt{week}_{ij}
+#'         + \upsilon_{0i} + \upsilon_{1i}\, \mathtt{week}_{ij}
+#'         + \varepsilon_{ij} \\
+#' \begin{pmatrix} \upsilon_{0i}\\ \upsilon_{1i} \end{pmatrix} &\sim
+#'    N \left(\begin{pmatrix} 0\\ 0 \end{pmatrix}, \,
+#'            \mathbf{\Sigma}_\upsilon =
+#' \begin{pmatrix}
+#'           \sigma^2_{\upsilon_0} & \sigma_{\upsilon_0 \upsilon_1} \\
+#'           \sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} \\
+#' \end{pmatrix} \right)
+#'       \text{ i.i.d.} \\
+#' \mathbf{\varepsilon}_i &\sim N(\mathbf{0}, \, \sigma^2 \mathbf{I}_{n_i})
+#'       \text{ i.i.d.} \\
+#' i &= 1, \ldots, I, \quad j = 1, \ldots n_i
+#' \end{aligned}
+#' $$
+
+m2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE)
+summary(m2)
+
+#' ## Partial pooling
+
+#+ fig.height = 10, fig.width = 6.5, fig.aling = "center"
+indiv <- unlist(
+  sapply(unique(dat$id),
+         function(i) predict(lm(hamd ~ week, dat[dat$id == i, ])))
+)
+
+xyplot(hamd + predict(m2, re.form = ~ 0) + predict(m2) + indiv ~ week | id,
+  data = dat, type = c("p", "l", "l", "l"), pch = 16, grid = TRUE,
+  distribute.type = TRUE, layout = c(11, 6), ylab = "HDRS score",
+  xlab = "Time (week)",
+  # customize colors
+  col = c("#434F4F", "#3CB4DC", "#FF6900", "#78004B"),
+  # add legend
+  key = list(space = "top", columns = 3,
+             text = list(c("Population", "Mixed model", "Within-subject")),
+             lines = list(col = c("#3CB4DC", "#FF6900", "#78004B")))
+  )
+
+#' ## By-group random-slope model
+
+m3 <- lmer(hamd ~ week + diag + (week | id), data = dat, REML = FALSE)
+m4 <- lmer(hamd ~ week * diag + (week | id), data = dat, REML = FALSE)
+anova(m3, m4)
+
+#' ## Means and predicted HDRS score by group
+
+dat2 <- aggregate(hamd ~ week + diag, dat, mean)
+dat2$m4 <- predict(m4, newdata = dat2, re.form = ~ 0)
+
+plot(m4 ~ week, dat2[dat2$diag == "endog", ], type = "l",
+     ylim=c(0, 28), xlab="Week", ylab = "HDRS score")
+lines(m4 ~ week, dat2[dat2$diag == "nonen", ], lty = 2)
+points(hamd ~ week, dat2[dat2$diag == "endog", ], pch = 16)
+points(hamd ~ week, dat2[dat2$diag == "nonen", ], pch = 21, bg = "white")
+legend("topright", c("Endogenous", "Non endogenous"),
+       lty = 1:2, pch = c(16, 21), pt.bg = "white", bty = "n")
+
+#' ## Gory details
+
+fixef(m4)
+getME(m4, "theta")
+t(chol(VarCorr(m4)$id))[lower.tri(diag(2), diag = TRUE)] / sigma(m4)
+
+#' # Power simulation
+
+#' ## Setup
+
+## Study design and sample sizes
+n_week <- 6
+n_subj <- 80
+n <- n_week * n_subj
+
+dat <- data.frame(
+     id = factor(rep(seq_len(n_subj), each = n_week)),
+   week = rep(0:(n_week - 1), times = n_subj),
+  treat = factor(rep(0:1, each = n/2), labels = c("ctr", "trt"))
+)
+
+## Fixed effects and variance components
+beta <- c(22, -2, 0, -1)
+Su <- matrix(c(12, -1.5, -1.5, 2), nrow = 2)
+se <- 3.5
+
+#' ## Power
+
+#+ cache = TRUE, warning = FALSE
+pval <- replicate(200, {
+
+  # Data generation
+  means <- model.matrix( ~ week * treat, dat) %*% beta
+  ranu <- MASS::mvrnorm(n_subj, mu = c(0, 0), Sigma = Su)
+  e <- rnorm(n_subj * n_week, mean = 0, sd = se)
+  
+  y <- means + ranu[dat$id, 1] + ranu[dat$id, 2] * dat$week + e
+  
+  # Fitting model to test H0
+  m0 <- lmer(y ~ week + treat + (1 + week | id), data = dat, REML = FALSE)
+  m1 <- lmer(y ~ week * treat + (1 + week | id), data = dat, REML = FALSE)
+  anova(m0, m1)["m1", "Pr(>Chisq)"]
+  }
+)
+
+mean(pval < 0.05)
+
+#' ## Parameter recovery
+
+#+ cache = TRUE, warning = FALSE
+par <- replicate(200, {
+  
+  means <- model.matrix( ~ week * treat, dat) %*% beta
+  ranu <- MASS::mvrnorm(n_subj, mu = c(0, 0), Sigma = Su)
+  e <- rnorm(n_subj * n_week, mean = 0, sd = se)
+  
+  y <- means + ranu[dat$id, 1] + ranu[dat$id, 2] * dat$week + e
+  
+  m1 <- lmer(y ~ week * treat + (1 + week | id), data = dat, REML = FALSE)
+  list(fixef = fixef(m1), theta = getME(m1, "theta"), sigma = sigma(m1))
+  }, simplify = FALSE
+)
+
+rowMeans(sapply(par, function(x) x$fixef))
+rowMeans(sapply(par, function(x) x$theta))
+mean(sapply(par, function(x) x$sigma))
+
+beta
+Lt <- chol(Su)
+t(Lt)[lower.tri(Lt, diag = TRUE)] / se
+se
+
+#' ### References
+

05_mixed1/powersim-lmm.md

@@ -0,0 +1,340 @@
+Power for mixed-effects models
+================
+Last modified: 2026-01-09
+
+``` r
+library(lattice)
+library(lme4)
+```
+
+# Reanalysis
+
+## Application context: Depression and type of diagnosis
+
+- Reisby et al. (1977) studied the effect of Imipramin on 66 inpatients
+  treated for depression
+- Depression was measured with the Hamilton depression rating scale
+- Patients were classified into endogenous and non-endogenous depressed
+- Depression was measured weekly for 6 time points
+
+Data: [reisby.txt](../data/reisby.txt)
+
+``` r
+dat      <- read.table("../data/reisby.txt", header = TRUE)
+dat$id   <- factor(dat$id)
+dat$diag <- factor(dat$diag, levels = c("nonen", "endog"))
+dat      <- na.omit(dat)     # drop missing values
+head(dat, n = 13)
+```
+
+    ##     id hamd week  diag endweek
+    ## 1  101   26    0 nonen       0
+    ## 2  101   22    1 nonen       0
+    ## 3  101   18    2 nonen       0
+    ## 4  101    7    3 nonen       0
+    ## 5  101    4    4 nonen       0
+    ## 6  101    3    5 nonen       0
+    ## 7  103   33    0 nonen       0
+    ## 8  103   24    1 nonen       0
+    ## 9  103   15    2 nonen       0
+    ## 10 103   24    3 nonen       0
+    ## 11 103   15    4 nonen       0
+    ## 12 103   13    5 nonen       0
+    ## 13 104   29    0 endog       0
+
+``` r
+xyplot(hamd ~ week | id, data = dat, type=c("g", "r", "p"),
+  pch = 16, layout = c(11, 6), ylab = "HDRS score", xlab = "Time (week)")
+```
+
+![](powersim-lmm_files/figure-gfm/unnamed-chunk-2-1.png)<!-- -->
+
+## Random-intercept model
+
+$$
+\begin{aligned}
+Y_{ij} &= \beta_0 + \beta_1 \, \mathtt{week}_{ij}
+        + \upsilon_{0i}
+        + \varepsilon_{ij} \\
+\upsilon_{0i} &\sim N(0, \sigma^2_{\upsilon_0}) \text{ i.i.d.} \\
+\mathbf{\varepsilon}_i &\sim N(0, \, \sigma^2)  \text{ i.i.d.} \\
+i &= 1, \ldots, I, \quad j = 1, \ldots n_i
+\end{aligned}
+$$
+
+``` r
+m1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE)
+summary(m1)
+```
+
+    ## Linear mixed model fit by maximum likelihood  ['lmerMod']
+    ## Formula: hamd ~ week + (1 | id)
+    ##    Data: dat
+    ## 
+    ##       AIC       BIC    logLik -2*log(L)  df.resid 
+    ##    2293.2    2308.9   -1142.6    2285.2       371 
+    ## 
+    ## Scaled residuals: 
+    ##     Min      1Q  Median      3Q     Max 
+    ## -3.1739 -0.5876 -0.0342  0.5465  3.5297 
+    ## 
+    ## Random effects:
+    ##  Groups   Name        Variance Std.Dev.
+    ##  id       (Intercept) 16.16    4.019   
+    ##  Residual             19.04    4.363   
+    ## Number of obs: 375, groups:  id, 66
+    ## 
+    ## Fixed effects:
+    ##             Estimate Std. Error t value
+    ## (Intercept)  23.5518     0.6385   36.88
+    ## week         -2.3757     0.1350  -17.60
+    ## 
+    ## Correlation of Fixed Effects:
+    ##      (Intr)
+    ## week -0.524
+
+## Random-slope model
+
+$$
+\begin{aligned}
+Y_{ij} &= \beta_0 + \beta_1 \, \mathtt{week}_{ij}
+        + \upsilon_{0i} + \upsilon_{1i}\, \mathtt{week}_{ij}
+        + \varepsilon_{ij} \\
+\begin{pmatrix} \upsilon_{0i}\\ \upsilon_{1i} \end{pmatrix} &\sim
+   N \left(\begin{pmatrix} 0\\ 0 \end{pmatrix}, \,
+           \mathbf{\Sigma}_\upsilon =
+\begin{pmatrix}
+          \sigma^2_{\upsilon_0} & \sigma_{\upsilon_0 \upsilon_1} \\
+          \sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} \\
+\end{pmatrix} \right)
+      \text{ i.i.d.} \\
+\mathbf{\varepsilon}_i &\sim N(\mathbf{0}, \, \sigma^2 \mathbf{I}_{n_i})
+      \text{ i.i.d.} \\
+i &= 1, \ldots, I, \quad j = 1, \ldots n_i
+\end{aligned}
+$$
+
+``` r
+m2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE)
+summary(m2)
+```
+
+    ## Linear mixed model fit by maximum likelihood  ['lmerMod']
+    ## Formula: hamd ~ week + (week | id)
+    ##    Data: dat
+    ## 
+    ##       AIC       BIC    logLik -2*log(L)  df.resid 
+    ##    2231.0    2254.6   -1109.5    2219.0       369 
+    ## 
+    ## Scaled residuals: 
+    ##     Min      1Q  Median      3Q     Max 
+    ## -2.7460 -0.5016  0.0332  0.5177  3.6834 
+    ## 
+    ## Random effects:
+    ##  Groups   Name        Variance Std.Dev. Corr 
+    ##  id       (Intercept) 12.631   3.554         
+    ##           week         2.079   1.442    -0.28
+    ##  Residual             12.216   3.495         
+    ## Number of obs: 375, groups:  id, 66
+    ## 
+    ## Fixed effects:
+    ##             Estimate Std. Error t value
+    ## (Intercept)  23.5769     0.5456   43.22
+    ## week         -2.3771     0.2087  -11.39
+    ## 
+    ## Correlation of Fixed Effects:
+    ##      (Intr)
+    ## week -0.449
+
+## Partial pooling
+
+``` r
+indiv <- unlist(
+  sapply(unique(dat$id),
+         function(i) predict(lm(hamd ~ week, dat[dat$id == i, ])))
+)
+
+xyplot(hamd + predict(m2, re.form = ~ 0) + predict(m2) + indiv ~ week | id,
+  data = dat, type = c("p", "l", "l", "l"), pch = 16, grid = TRUE,
+  distribute.type = TRUE, layout = c(11, 6), ylab = "HDRS score",
+  xlab = "Time (week)",
+  # customize colors
+  col = c("#434F4F", "#3CB4DC", "#FF6900", "#78004B"),
+  # add legend
+  key = list(space = "top", columns = 3,
+             text = list(c("Population", "Mixed model", "Within-subject")),
+             lines = list(col = c("#3CB4DC", "#FF6900", "#78004B")))
+  )
+```
+
+![](powersim-lmm_files/figure-gfm/unnamed-chunk-5-1.png)<!-- -->
+
+## By-group random-slope model
+
+``` r
+m3 <- lmer(hamd ~ week + diag + (week | id), data = dat, REML = FALSE)
+m4 <- lmer(hamd ~ week * diag + (week | id), data = dat, REML = FALSE)
+anova(m3, m4)
+```
+
+    ## Data: dat
+    ## Models:
+    ## m3: hamd ~ week + diag + (week | id)
+    ## m4: hamd ~ week * diag + (week | id)
+    ##    npar    AIC    BIC  logLik -2*log(L)  Chisq Df Pr(>Chisq)
+    ## m3    7 2228.9 2256.4 -1107.5    2214.9                     
+    ## m4    8 2230.9 2262.3 -1107.5    2214.9 0.0042  1     0.9486
+
+## Means and predicted HDRS score by group
+
+``` r
+dat2 <- aggregate(hamd ~ week + diag, dat, mean)
+dat2$m4 <- predict(m4, newdata = dat2, re.form = ~ 0)
+
+plot(m4 ~ week, dat2[dat2$diag == "endog", ], type = "l",
+     ylim=c(0, 28), xlab="Week", ylab = "HDRS score")
+lines(m4 ~ week, dat2[dat2$diag == "nonen", ], lty = 2)
+points(hamd ~ week, dat2[dat2$diag == "endog", ], pch = 16)
+points(hamd ~ week, dat2[dat2$diag == "nonen", ], pch = 21, bg = "white")
+legend("topright", c("Endogenous", "Non endogenous"),
+       lty = 1:2, pch = c(16, 21), pt.bg = "white", bty = "n")
+```
+
+![](powersim-lmm_files/figure-gfm/unnamed-chunk-7-1.png)<!-- -->
+
+## Gory details
+
+``` r
+fixef(m4)
+```
+
+    ##    (Intercept)           week      diagendog week:diagendog 
+    ##    22.47626332    -2.36568746     1.98802087    -0.02705576
+
+``` r
+getME(m4, "theta")
+```
+
+    ##      id.(Intercept) id.week.(Intercept)             id.week 
+    ##           0.9760823          -0.1175194           0.3951992
+
+``` r
+t(chol(VarCorr(m4)$id))[lower.tri(diag(2), diag = TRUE)] / sigma(m4)
+```
+
+    ## [1]  0.9760823 -0.1175194  0.3951992
+
+# Power simulation
+
+## Setup
+
+``` r
+## Study design and sample sizes
+n_week <- 6
+n_subj <- 80
+n <- n_week * n_subj
+
+dat <- data.frame(
+     id = factor(rep(seq_len(n_subj), each = n_week)),
+   week = rep(0:(n_week - 1), times = n_subj),
+  treat = factor(rep(0:1, each = n/2), labels = c("ctr", "trt"))
+)
+
+## Fixed effects and variance components
+beta <- c(22, -2, 0, -1)
+Su <- matrix(c(12, -1.5, -1.5, 2), nrow = 2)
+se <- 3.5
+```
+
+## Power
+
+``` r
+pval <- replicate(200, {
+
+  # Data generation
+  means <- model.matrix( ~ week * treat, dat) %*% beta
+  ranu <- MASS::mvrnorm(n_subj, mu = c(0, 0), Sigma = Su)
+  e <- rnorm(n_subj * n_week, mean = 0, sd = se)
+  
+  y <- means + ranu[dat$id, 1] + ranu[dat$id, 2] * dat$week + e
+  
+  # Fitting model to test H0
+  m0 <- lmer(y ~ week + treat + (1 + week | id), data = dat, REML = FALSE)
+  m1 <- lmer(y ~ week * treat + (1 + week | id), data = dat, REML = FALSE)
+  anova(m0, m1)["m1", "Pr(>Chisq)"]
+  }
+)
+
+mean(pval < 0.05)
+```
+
+    ## [1] 0.805
+
+## Parameter recovery
+
+``` r
+par <- replicate(200, {
+  
+  means <- model.matrix( ~ week * treat, dat) %*% beta
+  ranu <- MASS::mvrnorm(n_subj, mu = c(0, 0), Sigma = Su)
+  e <- rnorm(n_subj * n_week, mean = 0, sd = se)
+  
+  y <- means + ranu[dat$id, 1] + ranu[dat$id, 2] * dat$week + e
+  
+  m1 <- lmer(y ~ week * treat + (1 + week | id), data = dat, REML = FALSE)
+  list(fixef = fixef(m1), theta = getME(m1, "theta"), sigma = sigma(m1))
+  }, simplify = FALSE
+)
+
+rowMeans(sapply(par, function(x) x$fixef))
+```
+
+    ##   (Intercept)          week      treattrt week:treattrt 
+    ##   22.00069056   -1.98276116    0.03001846   -1.03298775
+
+``` r
+rowMeans(sapply(par, function(x) x$theta))
+```
+
+    ##      id.(Intercept) id.week.(Intercept)             id.week 
+    ##           0.9767601          -0.1180799           0.3763657
+
+``` r
+mean(sapply(par, function(x) x$sigma))
+```
+
+    ## [1] 3.479607
+
+``` r
+beta
+```
+
+    ## [1] 22 -2  0 -1
+
+``` r
+Lt <- chol(Su)
+t(Lt)[lower.tri(Lt, diag = TRUE)] / se
+```
+
+    ## [1]  0.9897433 -0.1237179  0.3846546
+
+``` r
+se
+```
+
+    ## [1] 3.5
+
+### References
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-ReisbyGram77" class="csl-entry">
+
+Reisby, N., L. F. Gram, P. Bech, A. Nagy, G. O. Petersen, J. Ortmann, I.
+Ibsen, et al. 1977. “Imipramine: Clinical Effects and Pharmacokinetic
+Variability.” *Psychopharmacology* 54: 263–72.
+<https://doi.org/10.1007/BF00426574>.
+
+</div>
+
+</div>

06_mixed2/exercises-datsimlmm.md

@@ -0,0 +1,81 @@
+Exercises: Data simulation for crossed random-effects models
+================
+
+- Change the data simulation by Baayen, Davidson, and Bates (2008) for
+  $N = 30$ subjects instead of only 3
+- You can use the following script and adjust it accordingly
+- You can choose if you want to use model matrices or create the vectors
+  “manually”
+
+``` r
+library(lattice)
+library(lme4)
+
+#--------------- (1) Create data frame ----------------------------------------
+datsim <- expand.grid(subject = factor(c("s1" , "s2" , "s3" )),
+                      item    = factor(c("w1" , "w2" , "w3" )),
+                      soa     = factor(c("long" , "short" ))) |>
+  sort_by(~ subject)
+
+#--------------- (2) Define parameters ----------------------------------------
+beta0 <- 522.22
+beta1 <- -19
+
+sw  <- 21
+sy0 <- 24
+sy1 <- 7
+ry  <- -0.7
+se  <- 9
+
+#--------------- (3) Create vectors and simulate data -------------------------
+# Fixed effects
+b0 <- rep(beta0, 18)
+b1 <- rep(rep(c(0, beta1), each = 3), 3)
+
+# Draw random effects
+w <- rep(rnorm(3, mean = 0, sd = sw), 6)
+e <- rnorm(18, mean = 0, sd = se)
+
+# Bivariate normal distribution
+sig <- matrix(c(sy0^2, ry * sy0 * sy1, ry * sy0 * sy1, sy1^2), 2, 2)
+y01 <- MASS::mvrnorm(3, mu = c(0, 0), Sigma = sig)
+y0 <- rep(y01[,1], each = 6)
+y1 <- rep(c(0, y01[1,2],
+            0, y01[2,2],
+            0, y01[3,2]), each = 3)
+
+datsim$rt <- b0 + b1 + w + y0 + y1 + e
+
+#--------------- (4) Simulate data using model matrices -----------------------
+X <- model.matrix( ~ soa, datsim)
+Z <- model.matrix( ~ 0 + item + subject + subject:soa, datsim,
+                  contrasts.arg = list(subject = contrasts(datsim$subject,
+                                                        contrasts = FALSE)))
+
+# Fixed effects
+beta <- c(beta0, beta1)
+# Random effects
+u <- c(w = unique(w),
+       y0 = y01[,1],
+       y1 = y01[,2])
+
+datsim$rt2 <- X %*% beta + Z %*% u + e
+
+#--------------- (5) Visualize simulated data ---------------------------------
+xyplot(rt ~ soa | subject, datsim, group = item, type = "b", layout = c(3, 1))
+```
+
+### Reference
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-Baayen08" class="csl-entry">
+
+Baayen, R. H., D. J. Davidson, and D. M. Bates. 2008. “Mixed-Effects
+Modeling with Crossed Random Effects for Subjects and Items.” *Journal
+of Memory and Language* 59 (4): 390–412.
+<https://doi.org/10.1016/j.jml.2007.12.005>.
+
+</div>
+
+</div>

06_mixed2/exercises.md

@@ -0,0 +1,67 @@
+Exercise: Power simulation for LMM
+================
+
+## Physical healing as a function of perceived time
+
+Aungle and Langer (2023) investigate how perceived time influences
+physical healing
+
+- They used cupping to induce bruises on 33 subjects, then took a
+  picture, waited for 28 min and took another picture
+- Subjective time was manipulated to feel like 14, 28, or 56 min
+- The pre and post pictures were presented to 25 raters who rated the
+  amount of healing on a 10-point-scale with 0 = not at all healed, 5 =
+  somewhat healed, 10 = completely healed
+- Subjects participated in all three conditions over a two week period
+
+Data: [healing.RData](../data/healing.RData)
+
+``` r
+load("../data/healing.RData")
+
+str(dat)
+
+# Subject ID
+dat$Subject <- factor(dat$Subject)
+# Rater ID
+dat$ResponseId <- factor(dat$ResponseId)
+```
+
+1.  Visualize the data.
+
+    - Aggregate the data over Raters and plot the data for each subject
+      using `lattice::xyplot()`
+    - Aggregate the data over Subjects and plot one panel for each rater
+    - How would you choose the random effects for a model testing
+      healing over the three conditions
+
+2.  Fit the model you think fits the experimental design best
+
+3.  Test the effects of condition
+
+4.  Run a power simulation for a replication study:
+
+    - Set up a data frame containing the study design and sample size.
+
+    - Specify the minimum relevant effects.
+
+    - Set the fixed effects and variance components to plausible values.
+
+    - How many participants are required to detect the specified effect
+      with a power of 80%?
+
+    - Recover the parameters of the model for one simulated data set.
+
+### Reference
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-Aungle23" class="csl-entry">
+
+Aungle, P., and E. Langer. 2023. “Physical Healing as a Function of
+Perceived Time.” *Scientific Reports* 13 (1): 22432.
+<https://doi.org/10.1038/s41598-023-50009-3>.
+
+</div>
+
+</div>

README.md

@@ -0,0 +1,51 @@
+Power simulations
+================
+Nora Wickelmaier
+January, 21-22, 2026
+
+## Schedule
+
+| Day | Time        | Topic                                                                                                              | Exercises                                                                                                                         |
+|:----|:------------|:-------------------------------------------------------------------------------------------------------------------|:----------------------------------------------------------------------------------------------------------------------------------|
+| Wed | 9:00–10:30  | [Simulation-based power analysis](01_intro/powersim-intro.pdf)                                                     | [Binomial test power simulation](01_intro/exercises.md)                                                                           |
+|     | 10:45–12:15 | [Power (curves) for t-tests](02_ttest/powersim-ttest.md)                                                           | [Temporal value asymmetry](02_ttest/exercises-ttest1.md), [Directed reading activities](02_ttest/exercises-ttest2.md)             |
+|     | 13:00–14:00 | [Power for ANOVAs](03_anova/powersim-anova.md)                                                                     | [Testing the interaction in two-by-two ANOVA](03_anova/exercises-anova.md)                                                        |
+| Thu | 9:00–10:30  | [Power for baseline/follow-up measurements](04_ancova/intro-ancova.pdf)                                            | [Shoulder pain and acupuncture](04_ancova/exercises-ancova1.md), [MASS anorexia data](04_ancova/exercises-ancova2.md)             |
+|     | 10:45–12:15 | [Introduction to LMMs](05_mixed1/intro-lmm.pdf), [Power for longitudinal data analysis](05_mixed1/powersim-lmm.md) | [Risperidone vs. haloperidol and schizophrenia](05_mixed1/exercises-pwrlmm.md)                                                    |
+|     | 13:00–14:00 | [Data simulation for crossed random-effects models](06_mixed2/intro-datsimlmm.pdf)                                 | [Data simulation for crossed random-effects models](06_mixed2/exercises-datsimlmm.md), [Physical healing](06_mixed2/exercises.md) |
+
+## Content
+
+- Power and the significance filter
+- Simulation-based power analysis with R
+- Drawing power curves
+- Power for t-tests, ANOVA, ANCOVA, and mixed-effects models
+
+## Prerequisites
+
+- Introductory knowledge of statistics
+- Basic knowledge of R
+
+## Software
+
+Participants will need to have installed:
+
+- A current R version (<https://CRAN.R-project.org/>)
+- An IDE for R (like RStudio or VSCode) or a text editor with syntax
+  highlighting (like Vim or Notepad++)
+- Additional R package:
+  - lme4: <https://CRAN.R-project.org/package=lme4>
+
+## Literature
+
+<div id="refs" class="references csl-bib-body hanging-indent">
+
+<div id="ref-Wickelmaier22" class="csl-entry">
+
+Wickelmaier, F. 2022. “Simulating the Power of Statistical Tests: A
+Collection of R Examples.” *ArXiv*.
+<https://doi.org/10.48550/arXiv.2110.09836>.
+
+</div>
+
+</div>

data/kleinhenz.txt

@@ -0,0 +1,57 @@
+# Reference: Kleinhenz et al., Pain 83 (1999) 235-241
+# Source: Vickers, A. J., & Altman, D. G., BMJ 323 (2001) 1123-1124 
+
+pre post  grp
+ 35   35 plac
+ 54   37 plac
+ 35   40 plac
+ 30   42 plac
+ 44   45 plac
+ 49   47 plac
+ 38   51 plac
+ 52   52 plac
+ 52   52 plac
+ 39   53 plac
+ 44   53 plac
+ 53   53 plac
+ 73   53 plac
+ 48   54 plac
+ 58   54 plac
+ 55   57 plac
+ 73   74 plac
+ 65   76 plac
+ 78   78 plac
+ 80   80 plac
+ 59   81 plac
+ 52   81 plac
+ 46   83 plac
+ 57   85 plac
+ 44   85 plac
+ 63   86 plac
+ 80   95 plac
+ 43   74  acu
+ 75   99  acu
+ 66   88  acu
+ 65   85  acu
+ 74  100  acu
+ 58   58  acu
+ 62   77  acu
+ 64  100  acu
+ 59   61  acu
+ 45   83  acu
+ 70   78  acu
+ 66  100  acu
+ 59  100  acu
+ 68   94  acu
+ 73   60  acu
+ 60   92  acu
+ 41   41  acu
+ 53   53  acu
+ 62   80  acu
+ 74   83  acu
+ 54   57  acu
+ 40   67  acu
+ 32   82  acu
+ 78   78  acu
+ 69  100  acu
+

data/reisby.txt

@@ -0,0 +1,399 @@
+## Data from Reisby et al., Psychopharmacology 54 (1977) 263-272
+ id hamd week  diag endweek
+101   26    0 nonen       0
+101   22    1 nonen       0
+101   18    2 nonen       0
+101    7    3 nonen       0
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