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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)<!-- -->