From e2b548c00cd850d1fa26c53d2a4c9ba5df0ab5c4 Mon Sep 17 00:00:00 2001 From: nwickel Date: Fri, 9 Jan 2026 16:55:02 +0100 Subject: [PATCH] Initialize repository --- 01_intro/exercises.md | 51 + 01_intro/powersim-intro.pdf | Bin 0 -> 750187 bytes 02_ttest/exercises-ttest1.md | 59 + 02_ttest/exercises-ttest2.md | 31 + 02_ttest/powersim-ttest.R | 90 + 02_ttest/powersim-ttest.md | 99 + .../figure-gfm/unnamed-chunk-3-1.png | Bin 0 -> 8012 bytes .../figure-gfm/unnamed-chunk-4-1.png | Bin 0 -> 3375 bytes .../figure-gfm/unnamed-chunk-5-1.png | Bin 0 -> 4051 bytes .../figure-gfm/unnamed-chunk-5-2.png | Bin 0 -> 3990 bytes 03_anova/exercises-anova.md | 62 + 03_anova/powersim-anova.R | 93 + 03_anova/powersim-anova.md | 82 + .../figure-gfm/unnamed-chunk-1-1.png | Bin 0 -> 8615 bytes .../figure-gfm/unnamed-chunk-2-1.png | Bin 0 -> 5600 bytes .../figure-gfm/unnamed-chunk-3-1.png | Bin 0 -> 5241 bytes 04_ancova/exercises-ancova1.md | 33 + 04_ancova/exercises-ancova2.md | 45 + 04_ancova/intro-ancova.pdf | Bin 0 -> 198693 bytes 05_mixed1/exercises-pwrlmm.md | 50 + 05_mixed1/intro-lmm.pdf | Bin 0 -> 252642 bytes 05_mixed1/powersim-lmm.R | 185 ++ 05_mixed1/powersim-lmm.md | 340 +++ .../figure-gfm/unnamed-chunk-2-1.png | Bin 0 -> 16602 bytes .../figure-gfm/unnamed-chunk-5-1.png | Bin 0 -> 30757 bytes .../figure-gfm/unnamed-chunk-7-1.png | Bin 0 -> 4661 bytes 06_mixed2/exercises-datsimlmm.md | 81 + 06_mixed2/exercises.md | 67 + 06_mixed2/intro-datsimlmm.pdf | Bin 0 -> 175851 bytes README.md | 51 + data/healing.RData | Bin 0 -> 12947 bytes data/kleinhenz.txt | 57 + data/moeller.csv | 2010 +++++++++++++++++ data/reisby.txt | 399 ++++ 34 files changed, 3885 insertions(+) create mode 100644 01_intro/exercises.md create mode 100644 01_intro/powersim-intro.pdf create mode 100644 02_ttest/exercises-ttest1.md create mode 100644 02_ttest/exercises-ttest2.md create mode 100644 02_ttest/powersim-ttest.R create mode 100644 02_ttest/powersim-ttest.md create mode 100644 02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-3-1.png create mode 100644 02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-4-1.png create mode 100644 02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-5-1.png create mode 100644 02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-5-2.png create mode 100644 03_anova/exercises-anova.md create mode 100644 03_anova/powersim-anova.R create mode 100644 03_anova/powersim-anova.md create mode 100644 03_anova/powersim-anova_files/figure-gfm/unnamed-chunk-1-1.png create mode 100644 03_anova/powersim-anova_files/figure-gfm/unnamed-chunk-2-1.png create mode 100644 03_anova/powersim-anova_files/figure-gfm/unnamed-chunk-3-1.png create mode 100644 04_ancova/exercises-ancova1.md create mode 100644 04_ancova/exercises-ancova2.md create mode 100644 04_ancova/intro-ancova.pdf create mode 100644 05_mixed1/exercises-pwrlmm.md create mode 100644 05_mixed1/intro-lmm.pdf create mode 100644 05_mixed1/powersim-lmm.R create mode 100644 05_mixed1/powersim-lmm.md create mode 100644 05_mixed1/powersim-lmm_files/figure-gfm/unnamed-chunk-2-1.png create mode 100644 05_mixed1/powersim-lmm_files/figure-gfm/unnamed-chunk-5-1.png create mode 100644 05_mixed1/powersim-lmm_files/figure-gfm/unnamed-chunk-7-1.png create mode 100644 06_mixed2/exercises-datsimlmm.md create mode 100644 06_mixed2/exercises.md create mode 100644 06_mixed2/intro-datsimlmm.pdf create mode 100644 README.md create mode 100644 data/healing.RData create mode 100644 data/kleinhenz.txt create mode 100644 data/moeller.csv create mode 100644 data/reisby.txt diff --git a/01_intro/exercises.md b/01_intro/exercises.md new file mode 100644 index 0000000..8b4fd1c --- /dev/null +++ b/01_intro/exercises.md @@ -0,0 +1,51 @@ +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 + - + - Literature cited there (e.g., Davis, Gottlieb, and Stampnitzky 1998; + Mathews and Hamilton 2005) + +### References + +
+ +
+ +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. +. + +
+ +
+ +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. +. + +
+ +
+ +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. + +
+ +
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WJt~Y7-}R9wJ)|yBS=r3a9QbdUAl4=T literal 0 HcmV?d00001 diff --git a/02_ttest/exercises-ttest1.md b/02_ttest/exercises-ttest1.md new file mode 100644 index 0000000..520eb83 --- /dev/null +++ b/02_ttest/exercises-ttest1.md @@ -0,0 +1,59 @@ +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 + +

+ +
+ +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. +. + +
+ +
diff --git a/02_ttest/exercises-ttest2.md b/02_ttest/exercises-ttest2.md new file mode 100644 index 0000000..e7aa9cb --- /dev/null +++ b/02_ttest/exercises-ttest2.md @@ -0,0 +1,31 @@ +Exercise: Power simulation and power curves for t-test +================ + +## Directed reading activities + +Plan a replication of the “directed reading activities” study on + + +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. diff --git a/02_ttest/powersim-ttest.R b/02_ttest/powersim-ttest.R new file mode 100644 index 0000000..8344d51 --- /dev/null +++ b/02_ttest/powersim-ttest.R @@ -0,0 +1,90 @@ +#' --- +#' 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)) + diff --git a/02_ttest/powersim-ttest.md b/02_ttest/powersim-ttest.md new file mode 100644 index 0000000..de3c29c --- /dev/null +++ b/02_ttest/powersim-ttest.md @@ -0,0 +1,99 @@ +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) diff --git a/02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-3-1.png b/02_ttest/powersim-ttest_files/figure-gfm/unnamed-chunk-3-1.png new file mode 100644 index 0000000000000000000000000000000000000000..bfc8331434a3829a5f05612ddc62c86519b70ca4 GIT binary patch literal 8012 zcmch6c|4Tw_xC+$44E-)vM-;67DY42V2mY8VN{Z>qGZcfb_S!dl#&r;2{RHSO3Id{ zknJNevWCW*eM!g~e)m0nf6rgf@A>C>J90N4op(UG z7dVCiFc@+@$N`Sw00)Q#`cZKX4mfJRP3D*bThal5-W^JI$QXcJKSsp?R5lZp>Ok$^ z$+Q77$Hqz=fD$UMq@P-{v(t4I=%?cPOG^58cE&Po#_GvqWA$TWW8*Ynr^I2WpSrUH za)E<83eM+iL6|(A29{m`fCB%cYGI8YIrab`ZfvNlW#*qcm$o}-wET8tS4&gVS?871 zqT3&0tW{XuO&5Ag9v@%Y*(usvX^?QQ8Az3akF08J&bqiI(e(T^-lMscmAH;{%dCo! zC24@%3oBs%1$lt?r2r6NwwtV-yNe*z#X+aKz)TH_|9=k`W9f?_BPfKOAvciA1f&dX z3}Pf)E{DN@mPs=DsW(91;6vZ7i|Aa-@I4?1v^`|A4wROMN#?!-gpc30{oOo5Mq7vi 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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 +- Fix the parameters of the ANOVA model + +### Some background + +- Open anchoring quest (Röseler et al. 2022), + +### 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 + +
+ +
+ +Jacowitz, K. E., and D. Kahneman. 1995. “Measures of Anchoring in +Estimation Tasks.” *Personality and Social Psychology Bulletin* 21 (11): +1161–66. . + +
+ +
+ +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.” . + +
+ +
diff --git a/03_anova/powersim-anova.R b/03_anova/powersim-anova.R new file mode 100644 index 0000000..7f4ef02 --- /dev/null +++ b/03_anova/powersim-anova.R @@ -0,0 +1,93 @@ +#' --- +#' 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) + diff --git a/03_anova/powersim-anova.md b/03_anova/powersim-anova.md new file mode 100644 index 0000000..5e5761d --- /dev/null +++ b/03_anova/powersim-anova.md @@ -0,0 +1,82 @@ +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]]$"Pr(>F)"[3] # test of interaction +}) +mean(pval < 0.05) +``` + + ## [1] 0.817 diff --git a/03_anova/powersim-anova_files/figure-gfm/unnamed-chunk-1-1.png b/03_anova/powersim-anova_files/figure-gfm/unnamed-chunk-1-1.png new file mode 100644 index 0000000000000000000000000000000000000000..81c32ffecded92e2117c1b53c3cb230fa4c59b18 GIT binary patch literal 8615 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Reanalyze the original data + - Re-estimate the ANCOVA model for the Kleinhenz et al. (1999) + [data](../data/kleinhenz.txt) +2. Run a power simulation for a replication study + 1. Draw plausible pre-CMS values + 2. Specify the minimum relevant average treatment effect (ATE) + 3. Set the remaining parameters to plausible values + 4. What is the sample size required for the test to detect the + effect with 80% power? + 5. How robust is the power simulation when you repeat it with a new + set of pre-CMS values? Try it! + 6. Recover the parameters of the ANCOVA model + +### References + +
+ +
+ +Kleinhenz, J., K. Streitberger, J. Windeler, A. Güßbacher, G. Mavridis, +and E. Martin. 1999. “Randomised Clinical Trial Comparing the Effects of +Acupuncture and a Newly Designed Placebo Needle in Rotator Cuff +Tendinitis.” *Pain* 83 (2): 235–41. + +
+ +
diff --git a/04_ancova/exercises-ancova2.md b/04_ancova/exercises-ancova2.md new file mode 100644 index 0000000..8a26156 --- /dev/null +++ b/04_ancova/exercises-ancova2.md @@ -0,0 +1,45 @@ +Exercise: Analysis and power simulation for baseline/follow-up +measurements +================ + +## MASS anorexia data + +1. Analyze the original data: + - In R, see ?MASS::anorexia + - Data preparation + +``` r +data(anorexia, package = "MASS") +dat <- + subset(anorexia, Treat != "Cont") |> # exclude control group + droplevels() # drop empty factor levels +lbs2kg <- 0.4535924 +dat$Prewt <- lbs2kg * dat$Prewt # to kg +dat$Postwt <- lbs2kg * dat$Postwt +lattice::xyplot(Postwt ~ Prewt, dat, groups = Treat, + type = c("g", "r", "p"), auto.key = TRUE) +``` + +- Estimate the average treatment effect (ATE) for FT relative to CBT. +- What is the 95% CI for the ATE? +- What are the pre- and post-weight means for the two groups? +- What are the baseline-adjusted means for the two groups? + +2. Run a power simulation for a replication study: + - Draw plausible pre-weights. + - Specify the minimum relevant effect. + - Set the remaining parameters to plausible values. + - What is the sample size required for the test to detect the effect + with 80% power? + - How robust is the power simulation when you repeat it with a new + set of pre-weights? Try it! + - Recover the parameters of the ANCOVA model. +3. Create a renderable R script or an R Markdown file that includes + - a header with title, author, date + - at least one section head line + - the homework questions and your answers + - the R code, output, and plots (if any) + + Render the R or Rmd file to HTML. + +### Reference diff --git a/04_ancova/intro-ancova.pdf b/04_ancova/intro-ancova.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c9da59d5d5021e651820dfcc5d2da124af70d085 GIT binary patch literal 198693 zcmeFZWpE_Pk}fK#TT+XeTFlJM%*@Qp%+xJrW@ct)wV0WinVF@ho|)Y{cX!S9JbUk) zU#HBfGRi5-BQqk>-MLiLe_Sc;M9D4)-E{ezvXFZ{&;JGQ~&1tp+yT$Evf5ZXZk5Y`zi6s z|NZ$XA!lfB?P#ZOXpcktn|laJ%ZT&W>(h*>g@d6T4z-Ylu7ja~p}w_& zAvhNoxV?j&p{^yk%km7&xD3hoCJ+GV8{j3lp_Rd3oX^(p-%d;aY58B|1j9eg3C4e& z6HNa!Cz$_rPO$v5oS^&H-A;7>wA+d9_ci|4j0w8m7yX|yu<2O;Wlqrlp0xfcPSDf+ z>zw#~=l)ZiVEDTef6d@@%*_Ar41T7-DcwT@yLLlvRR`MY+@Q~HL5UX+iwy*ftP_M- zSQ5t+g?RN)t*GdrqdUaIzBX_Zd-`x5Ca1bbku|IG$d9T-`vT>nVZ;LHs{{OXdTh+&E?;oTkE3*(hgJ^s)-DBE-6Q 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[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 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- 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$sigma)) + +beta +Lt <- chol(Su) +t(Lt)[lower.tri(Lt, diag = TRUE)] / se +se + +#' ### References + diff --git a/05_mixed1/powersim-lmm.md b/05_mixed1/powersim-lmm.md new file mode 100644 index 0000000..a0fc3b9 --- /dev/null +++ b/05_mixed1/powersim-lmm.md @@ -0,0 +1,340 @@ +Power for mixed-effects models +================ +Last modified: 2026-01-09 + +``` r +library(lattice) +library(lme4) +``` + +# Reanalysis + +## Application context: Depression and type of diagnosis + +- Reisby et al. (1977) studied the effect of Imipramin on 66 inpatients + treated for depression +- Depression was measured with the Hamilton depression rating scale +- Patients were classified into endogenous and non-endogenous depressed +- Depression was measured weekly for 6 time points + +Data: [reisby.txt](../data/reisby.txt) + +``` r +dat <- read.table("../data/reisby.txt", header = TRUE) +dat$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 + +
+ +
+ +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. +. + +
+ +
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100644 index 0000000..aa653f4 --- /dev/null +++ b/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$rt2 <- X %*% beta + Z %*% u + e + +#--------------- (5) Visualize simulated data --------------------------------- +xyplot(rt ~ soa | subject, datsim, group = item, type = "b", layout = c(3, 1)) +``` + +### Reference + +
+ +
+ +Baayen, R. H., D. J. Davidson, and D. M. Bates. 2008. “Mixed-Effects +Modeling with Crossed Random Effects for Subjects and Items.” *Journal +of Memory and Language* 59 (4): 390–412. +. + +
+ +
diff --git a/06_mixed2/exercises.md b/06_mixed2/exercises.md new file mode 100644 index 0000000..d3fd939 --- /dev/null +++ b/06_mixed2/exercises.md @@ -0,0 +1,67 @@ +Exercise: Power simulation for LMM +================ + +## Physical healing as a function of perceived time + +Aungle and Langer (2023) investigate how perceived time influences +physical healing + +- They used cupping to induce bruises on 33 subjects, then took a + picture, waited for 28 min and took another picture +- Subjective time was manipulated to feel like 14, 28, or 56 min +- The pre and post pictures were presented to 25 raters who rated the + amount of healing on a 10-point-scale with 0 = not at all healed, 5 = + somewhat healed, 10 = completely healed +- Subjects participated in all three conditions over a two week period + +Data: [healing.RData](../data/healing.RData) + +``` r +load("../data/healing.RData") + +str(dat) + +# Subject ID +dat$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 + +
+ +
+ +Aungle, P., and E. Langer. 2023. “Physical Healing as a Function of +Perceived Time.” *Scientific Reports* 13 (1): 22432. +. + +
+ +
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+ +
+ +Wickelmaier, F. 2022. “Simulating the Power of Statistical Tests: A +Collection of R Examples.” *ArXiv*. +. + +
+ +
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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 + diff --git a/data/moeller.csv b/data/moeller.csv new file mode 100644 index 0000000..99a5695 --- /dev/null +++ b/data/moeller.csv @@ -0,0 +1,2010 @@ +# Data from Moeller et al., International Journal of +# Neuropsychopharmacology (2008), 11, 985-997 +id,treat,week,pans +1,risp,0,85 +1,risp,1,91 +1,risp,2,86 +1,risp,3,77 +1,risp,4,69 +1,risp,5,59 +1,risp,6,69 +1,risp,7,60 +1,risp,8,64 +2,risp,0,77 +2,risp,1,69 +2,risp,2,59 +2,risp,3,52 +2,risp,4,67 +2,risp,5,59 +2,risp,6,56 +2,risp,7,56 +2,risp,8,61 +3,halo,0,66 +3,halo,1,59 +3,halo,2,66 +3,halo,3,55 +3,halo,4,66 +3,halo,5,61 +3,halo,6,48 +3,halo,7,57 +3,halo,8,57 +4,risp,0,99 +4,risp,1,86 +4,risp,2,66 +4,risp,3,58 +4,risp,4,57 +4,risp,5,57 +4,risp,6,57 +4,risp,7,57 +4,risp,8,57 +5,halo,0,64 +5,halo,1,57 +5,halo,2,46 +5,halo,3,50 +5,halo,4,52 +5,halo,5,52 +5,halo,6,52 +5,halo,7,52 +5,halo,8,52 +6,risp,0,88 +6,risp,1,81 +6,risp,2,83 +6,risp,3,71 +6,risp,4,49 +6,risp,5,72 +6,risp,6,57 +6,risp,7,74 +6,risp,8,74 +7,halo,0,112 +7,halo,1,79 +7,halo,2,93 +7,halo,3,93 +7,halo,4,93 +7,halo,5,93 +7,halo,6,93 +7,halo,7,93 +7,halo,8,93 +8,halo,0,68 +8,halo,1,30 +8,halo,2,82 +8,halo,3,74 +8,halo,4,103 +8,halo,5,103 +8,halo,6,103 +8,halo,7,103 +8,halo,8,103 +9,risp,0,105 +9,risp,1,126 +9,risp,2,126 +9,risp,3,126 +9,risp,4,126 +9,risp,5,126 +9,risp,6,126 +9,risp,7,126 +9,risp,8,126 +10,halo,0,84 +10,halo,1,56 +10,halo,2,64 +10,halo,3,55 +10,halo,4,53 +10,halo,5,44 +10,halo,6,44 +10,halo,7,44 +10,halo,8,44 +11,risp,0,88 +11,risp,1,88 +11,risp,2,93 +11,risp,3,94 +11,risp,4,97 +11,risp,5,99 +11,risp,6,99 +11,risp,7,99 +11,risp,8,99 +12,halo,0,121 +12,halo,1,109 +12,halo,2,105 +12,halo,3,82 +12,halo,4,75 +12,halo,5,75 +12,halo,6,75 +12,halo,7,73 +12,halo,8,73 +13,risp,0,111 +13,risp,2,52 +13,risp,3,45 +13,risp,4,45 +13,risp,5,45 +13,risp,6,45 +13,risp,7,45 +13,risp,8,57 +14,risp,0,118 +14,risp,1,106 +14,risp,2,106 +14,risp,3,106 +14,risp,4,106 +14,risp,5,106 +14,risp,6,106 +14,risp,7,106 +14,risp,8,106 +15,halo,0,102 +15,halo,1,94 +15,halo,2,90 +15,halo,3,89 +15,halo,4,96 +15,halo,5,96 +15,halo,6,96 +15,halo,7,96 +15,halo,8,96 +16,halo,0,96 +16,halo,1,63 +16,halo,2,50 +16,halo,3,73 +16,halo,4,81 +16,halo,5,92 +16,halo,6,92 +16,halo,7,92 +16,halo,8,92 +17,halo,0,73 +17,halo,1,41 +17,halo,2,41 +17,halo,3,41 +17,halo,4,41 +17,halo,5,41 +17,halo,6,41 +17,halo,7,41 +17,halo,8,41 +18,halo,0,84 +18,halo,1,40 +18,halo,2,45 +18,halo,3,35 +18,halo,4,34 +18,halo,5,37 +18,halo,6,32 +18,halo,7,32 +18,halo,8,32 +19,halo,0,74 +19,halo,1,65 +19,halo,2,38 +19,halo,3,31 +19,halo,4,33 +19,halo,5,33 +19,halo,6,33 +19,halo,7,33 +19,halo,8,33 +20,risp,0,64 +20,risp,1,30 +20,risp,2,36 +20,risp,3,34 +20,risp,4,39 +20,risp,5,45 +20,risp,6,41 +20,risp,7,45 +20,risp,8,41 +21,halo,0,52 +21,halo,1,38 +21,halo,2,42 +21,halo,3,41 +21,halo,4,36 +21,halo,5,36 +21,halo,6,43 +21,halo,7,41 +21,halo,8,41 +22,risp,0,94 +22,risp,1,72 +22,risp,2,73 +22,risp,3,72 +22,risp,4,69 +22,risp,5,74 +22,risp,6,81 +22,risp,7,89 +22,risp,8,87 +23,risp,0,89 +23,risp,1,35 +23,risp,2,34 +23,risp,3,52 +23,risp,4,58 +23,risp,5,54 +23,risp,6,54 +23,risp,7,54 +23,risp,8,54 +24,halo,0,72 +24,halo,1,64 +24,halo,2,63 +24,halo,3,65 +24,halo,4,71 +24,halo,5,73 +24,halo,6,62 +24,halo,7,62 +24,halo,8,62 +25,halo,0,87 +25,halo,1,95 +25,halo,2,77 +25,halo,3,88 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+221,halo,6,35 +221,halo,7,32 +221,halo,8,32 +222,risp,0,65 +222,risp,1,62 +222,risp,2,60 +222,risp,3,60 +222,risp,4,56 +222,risp,5,56 +222,risp,6,59 +222,risp,7,59 +222,risp,8,59 +223,halo,0,69 +223,halo,1,46 +223,halo,2,46 +223,halo,3,67 +223,halo,4,46 +223,halo,5,53 +223,halo,6,53 +223,halo,7,53 +223,halo,8,53 +224,risp,0,96 +224,risp,1,68 +224,risp,2,68 +224,risp,3,68 +224,risp,4,68 +224,risp,5,68 +224,risp,6,68 +224,risp,7,68 +224,risp,8,68 diff --git a/data/reisby.txt b/data/reisby.txt new file mode 100644 index 0000000..60f81d1 --- /dev/null +++ b/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 +101 4 4 nonen 0 +101 3 5 nonen 0 +103 33 0 nonen 0 +103 24 1 nonen 0 +103 15 2 nonen 0 +103 24 3 nonen 0 +103 15 4 nonen 0 +103 13 5 nonen 0 +104 29 0 endog 0 +104 22 1 endog 1 +104 18 2 endog 2 +104 13 3 endog 3 +104 19 4 endog 4 +104 0 5 endog 5 +105 22 0 nonen 0 +105 12 1 nonen 0 +105 16 2 nonen 0 +105 16 3 nonen 0 +105 13 4 nonen 0 +105 9 5 nonen 0 +106 21 0 endog 0 +106 25 1 endog 1 +106 23 2 endog 2 +106 18 3 endog 3 +106 20 4 endog 4 +106 NA 5 endog 5 +107 21 0 endog 0 +107 21 1 endog 1 +107 16 2 endog 2 +107 19 3 endog 3 +107 NA 4 endog 4 +107 6 5 endog 5 +108 21 0 endog 0 +108 22 1 endog 1 +108 11 2 endog 2 +108 9 3 endog 3 +108 9 4 endog 4 +108 7 5 endog 5 +113 21 0 nonen 0 +113 23 1 nonen 0 +113 19 2 nonen 0 +113 23 3 nonen 0 +113 23 4 nonen 0 +113 NA 5 nonen 0 +114 NA 0 nonen 0 +114 17 1 nonen 0 +114 11 2 nonen 0 +114 13 3 nonen 0 +114 7 4 nonen 0 +114 7 5 nonen 0 +115 NA 0 endog 0 +115 16 1 endog 1 +115 16 2 endog 2 +115 16 3 endog 3 +115 16 4 endog 4 +115 11 5 endog 5 +117 19 0 endog 0 +117 16 1 endog 1 +117 13 2 endog 2 +117 12 3 endog 3 +117 7 4 endog 4 +117 6 5 endog 5 +118 NA 0 endog 0 +118 26 1 endog 1 +118 18 2 endog 2 +118 18 3 endog 3 +118 14 4 endog 4 +118 11 5 endog 5 +120 20 0 nonen 0 +120 19 1 nonen 0 +120 17 2 nonen 0 +120 18 3 nonen 0 +120 16 4 nonen 0 +120 17 5 nonen 0 +121 20 0 nonen 0 +121 22 1 nonen 0 +121 19 2 nonen 0 +121 19 3 nonen 0 +121 12 4 nonen 0 +121 14 5 nonen 0 +123 15 0 nonen 0 +123 15 1 nonen 0 +123 15 2 nonen 0 +123 13 3 nonen 0 +123 5 4 nonen 0 +123 5 5 nonen 0 +501 29 0 endog 0 +501 30 1 endog 1 +501 26 2 endog 2 +501 22 3 endog 3 +501 19 4 endog 4 +501 24 5 endog 5 +502 21 0 endog 0 +502 22 1 endog 1 +502 13 2 endog 2 +502 11 3 endog 3 +502 2 4 endog 4 +502 1 5 endog 5 +504 19 0 nonen 0 +504 17 1 nonen 0 +504 15 2 nonen 0 +504 16 3 nonen 0 +504 12 4 nonen 0 +504 12 5 nonen 0 +505 21 0 nonen 0 +505 11 1 nonen 0 +505 18 2 nonen 0 +505 0 3 nonen 0 +505 0 4 nonen 0 +505 4 5 nonen 0 +507 27 0 endog 0 +507 26 1 endog 1 +507 26 2 endog 2 +507 25 3 endog 3 +507 24 4 endog 4 +507 19 5 endog 5 +603 28 0 nonen 0 +603 22 1 nonen 0 +603 18 2 nonen 0 +603 20 3 nonen 0 +603 11 4 nonen 0 +603 13 5 nonen 0 +604 27 0 nonen 0 +604 27 1 nonen 0 +604 13 2 nonen 0 +604 5 3 nonen 0 +604 7 4 nonen 0 +604 NA 5 nonen 0 +606 19 0 endog 0 +606 33 1 endog 1 +606 12 2 endog 2 +606 12 3 endog 3 +606 3 4 endog 4 +606 1 5 endog 5 +607 30 0 endog 0 +607 39 1 endog 1 +607 30 2 endog 2 +607 27 3 endog 3 +607 20 4 endog 4 +607 4 5 endog 5 +608 24 0 nonen 0 +608 19 1 nonen 0 +608 14 2 nonen 0 +608 12 3 nonen 0 +608 3 4 nonen 0 +608 4 5 nonen 0 +609 NA 0 endog 1 +609 25 1 endog 1 +609 22 2 endog 2 +609 14 3 endog 3 +609 15 4 endog 4 +609 2 5 endog 5 +610 34 0 endog 0 +610 NA 1 endog 1 +610 33 2 endog 2 +610 23 3 endog 3 +610 NA 4 endog 4 +610 11 5 endog 5 +302 18 0 endog 0 +302 22 1 endog 1 +302 16 2 endog 2 +302 8 3 endog 3 +302 9 4 endog 4 +302 12 5 endog 5 +303 21 0 nonen 0 +303 21 1 nonen 0 +303 13 2 nonen 0 +303 14 3 nonen 0 +303 10 4 nonen 0 +303 5 5 nonen 0 +304 21 0 endog 0 +304 27 1 endog 1 +304 29 2 endog 2 +304 NA 3 endog 3 +304 12 4 endog 4 +304 24 5 endog 5 +305 19 0 nonen 0 +305 17 1 nonen 0 +305 15 2 nonen 0 +305 11 3 nonen 0 +305 5 4 nonen 0 +305 1 5 nonen 0 +308 22 0 nonen 0 +308 21 1 nonen 0 +308 18 2 nonen 0 +308 17 3 nonen 0 +308 12 4 nonen 0 +308 11 5 nonen 0 +309 22 0 nonen 0 +309 22 1 nonen 0 +309 16 2 nonen 0 +309 19 3 nonen 0 +309 20 4 nonen 0 +309 11 5 nonen 0 +310 24 0 endog 0 +310 19 1 endog 1 +310 11 2 endog 2 +310 7 3 endog 3 +310 6 4 endog 4 +310 NA 5 endog 5 +311 20 0 endog 0 +311 16 1 endog 1 +311 21 2 endog 2 +311 17 3 endog 3 +311 NA 4 endog 4 +311 15 5 endog 5 +312 17 0 endog 0 +312 NA 1 endog 1 +312 18 2 endog 2 +312 17 3 endog 3 +312 17 4 endog 4 +312 6 5 endog 5 +313 21 0 nonen 0 +313 19 1 nonen 0 +313 10 2 nonen 0 +313 11 3 nonen 0 +313 11 4 nonen 0 +313 8 5 nonen 0 +315 27 0 endog 0 +315 21 1 endog 1 +315 17 2 endog 2 +315 13 3 endog 3 +315 5 4 endog 4 +315 NA 5 endog 5 +316 32 0 endog 0 +316 26 1 endog 1 +316 23 2 endog 2 +316 26 3 endog 3 +316 23 4 endog 4 +316 24 5 endog 5 +318 17 0 endog 0 +318 18 1 endog 1 +318 19 2 endog 2 +318 21 3 endog 3 +318 17 4 endog 4 +318 11 5 endog 5 +319 24 0 endog 0 +319 18 1 endog 1 +319 10 2 endog 2 +319 14 3 endog 3 +319 13 4 endog 4 +319 12 5 endog 5 +322 28 0 endog 0 +322 21 1 endog 1 +322 25 2 endog 2 +322 32 3 endog 3 +322 34 4 endog 4 +322 NA 5 endog 5 +327 17 0 nonen 0 +327 18 1 nonen 0 +327 15 2 nonen 0 +327 8 3 nonen 0 +327 19 4 nonen 0 +327 17 5 nonen 0 +328 22 0 nonen 0 +328 24 1 nonen 0 +328 28 2 nonen 0 +328 26 3 nonen 0 +328 28 4 nonen 0 +328 29 5 nonen 0 +331 19 0 nonen 0 +331 21 1 nonen 0 +331 18 2 nonen 0 +331 16 3 nonen 0 +331 14 4 nonen 0 +331 10 5 nonen 0 +333 23 0 nonen 0 +333 20 1 nonen 0 +333 21 2 nonen 0 +333 20 3 nonen 0 +333 24 4 nonen 0 +333 14 5 nonen 0 +334 31 0 nonen 0 +334 25 1 nonen 0 +334 NA 2 nonen 0 +334 7 3 nonen 0 +334 8 4 nonen 0 +334 11 5 nonen 0 +335 21 0 nonen 0 +335 21 1 nonen 0 +335 18 2 nonen 0 +335 15 3 nonen 0 +335 12 4 nonen 0 +335 10 5 nonen 0 +337 27 0 nonen 0 +337 22 1 nonen 0 +337 23 2 nonen 0 +337 21 3 nonen 0 +337 12 4 nonen 0 +337 13 5 nonen 0 +338 22 0 nonen 0 +338 20 1 nonen 0 +338 22 2 nonen 0 +338 23 3 nonen 0 +338 19 4 nonen 0 +338 18 5 nonen 0 +339 27 0 endog 0 +339 NA 1 endog 1 +339 14 2 endog 2 +339 12 3 endog 3 +339 11 4 endog 4 +339 12 5 endog 5 +344 NA 0 endog 0 +344 21 1 endog 1 +344 12 2 endog 2 +344 13 3 endog 3 +344 13 4 endog 4 +344 18 5 endog 5 +345 29 0 nonen 0 +345 27 1 nonen 0 +345 27 2 nonen 0 +345 22 3 nonen 0 +345 22 4 nonen 0 +345 23 5 nonen 0 +346 25 0 endog 0 +346 24 1 endog 1 +346 19 2 endog 2 +346 23 3 endog 3 +346 14 4 endog 4 +346 21 5 endog 5 +347 18 0 endog 0 +347 15 1 endog 1 +347 14 2 endog 2 +347 10 3 endog 3 +347 8 4 endog 4 +347 NA 5 endog 5 +348 24 0 nonen 0 +348 21 1 nonen 0 +348 12 2 nonen 0 +348 13 3 nonen 0 +348 12 4 nonen 0 +348 5 5 nonen 0 +349 17 0 endog 0 +349 19 1 endog 1 +349 15 2 endog 2 +349 12 3 endog 3 +349 9 4 endog 4 +349 13 5 endog 5 +350 22 0 nonen 0 +350 25 1 nonen 0 +350 12 2 nonen 0 +350 16 3 nonen 0 +350 10 4 nonen 0 +350 16 5 nonen 0 +351 30 0 endog 0 +351 27 1 endog 1 +351 23 2 endog 2 +351 20 3 endog 3 +351 12 4 endog 4 +351 11 5 endog 5 +352 21 0 endog 0 +352 19 1 endog 1 +352 18 2 endog 2 +352 15 3 endog 3 +352 18 4 endog 4 +352 19 5 endog 5 +353 27 0 endog 0 +353 21 1 endog 1 +353 24 2 endog 2 +353 22 3 endog 3 +353 16 4 endog 4 +353 11 5 endog 5 +354 28 0 endog 0 +354 27 1 endog 1 +354 27 2 endog 2 +354 26 3 endog 3 +354 23 4 endog 4 +354 NA 5 endog 5 +355 22 0 endog 0 +355 26 1 endog 1 +355 20 2 endog 2 +355 13 3 endog 3 +355 10 4 endog 4 +355 7 5 endog 5 +357 27 0 endog 0 +357 22 1 endog 1 +357 24 2 endog 2 +357 25 3 endog 3 +357 19 4 endog 4 +357 19 5 endog 5 +360 21 0 endog 0 +360 28 1 endog 1 +360 27 2 endog 2 +360 29 3 endog 3 +360 28 4 endog 4 +360 33 5 endog 5 +361 30 0 endog 0 +361 22 1 endog 1 +361 11 2 endog 2 +361 8 3 endog 3 +361 7 4 endog 4 +361 19 5 endog 5 +