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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