Henrik Singmann

@singmann.bsky.social

Associate Professor at UCL Experimental Psychology; math psych & cognitive psychology; statistical and cognitive modelling in R; German migrant worker in UK

Figure 8: 2M Forced-Choice Task Results, consisting of two panels.
The figure shows the average accuracy results in the 2M-MIN task (left panel) and the 2M-MAX task (right panel) as a function of set size. The error bars correspond to 95% bootstrap based confidence intervals.
In the left panel (2M-min task) accuracy is constant with M. In the right panel (2M-max task) accuracy increases with M.

Figure B1: Example of a 2M-MIN Test Trial
Study description: The experiment began with a study phase in which participants were presented with a list of 100 common nouns taken from Kellen et al. (2021), each presented for 2,000 ms, with a 400 ms interval between each word. After the study phase, participants initiated the test phase, which was comprised of 40 test trials, with 10 trials per M ∈ {1, 2, 3, 4}. Two hundred fifty-two and 253 participants engaged in the 2M-MIN and 2M-MAX tasks, respectively. During the test phase, words were displayed adjacent to one another, arranged in an M × 2 matrix. In the 2M-MIN task, participants were instructed to “click the item you are most certain is new,” whereas in the 2M-MAX task, they were told to “click the item you are most certain is old.” Participants selected the word of their choice by simply clicking on it, which led to the next test trial. Figure B1 illustrates a test trial.

Figure 7: Illustration of the Gumbel-min Signal Detection Model
Figure consists of three panels.
Bottom-left panel: The Gumbel-min latent-strength distributions associated with SIGNAL (old) and NOISE (new) items. Top-left panel: The log likelihood ratio (log-LR) for the two latent-strength distributions. Right panel: The receiver operating characteristic function produced by the two latent-strength distributions. pH = hit probabilities; pFA = false-alarm probabilities.

Picture of the Bayesian Workflow book by Gelman and colleagues

Fig. 4 from the paper showing argument quality ratings as a function of belief consistency for each argument in Experiment 1. The overall pattern is shown for each argument shown.
Note. Results of Experiment 1 conditional on the topic and the level of argument support. Each line and colour in each panel shows responses to exactly one argument (i.e., there is no aggregation across items within a panel). The dots show individual responses and the curved lines show predictions from the linear mixed model. Blue dots represent argument quality ratings to good arguments in the data, orange dots represent argument quality ratings to bad arguments in the data, and the size of the dots represents the number of argument quality rating responses for the corresponding belief rating. Data points are dodged so that responses for good and bad arguments do not overlap. Model predictions are based on the fixed effects of the final model and the random effects of the by-topic grouping factor. Ext. = extremely.

$R code and output showing the new functionality: ``` r ## pak::pkg_install("quentingronau/bridgesampling#44") ## see: https://cran.r-project.org/web/packages/bridgesampling/vignettes/bridgesampling_example_stan.html library(bridgesampling) ### generate data ### set.seed(12345) mu <- 0 tau2 <- 0.5 sigma2 <- 1 n <- 20 theta <- rnorm(n, mu, sqrt(tau2)) y <- rnorm(n, theta, sqrt(sigma2)) ### set prior parameters ### mu0 <- 0 tau20 <- 1 alpha <- 1 beta <- 1 stancodeH0 <- 'data { int<lower=1> n; // number of observations vector[n] y; // observations real<lower=0> alpha; real<lower=0> beta; real<lower=0> sigma2; } parameters { real<lower=0> tau2; // group-level variance vector[n] theta; // participant effects } model { target += inv_gamma_lpdf(tau2 | alpha, beta); target += normal_lpdf(theta | 0, sqrt(tau2)); target += normal_lpdf(y | theta, sqrt(sigma2)); } ' tf <- withr::local_tempfile(fileext = ".stan") writeLines(stancodeH0, tf) mod <- cmdstanr::cmdstan_model(tf, quiet = TRUE, force_recompile = TRUE) fitH0 <- mod$sample( data = list(y = y, n = n, alpha = alpha, beta = beta, sigma2 = sigma2), seed = 202, chains = 4, parallel_chains = 4, iter_warmup = 1000, iter_sampling = 50000, refresh = 0 ) #> Running MCMC with 4 parallel chains... #> #> Chain 3 finished in 0.8 seconds. #> Chain 2 finished in 0.8 seconds. #> Chain 4 finished in 0.8 seconds. #> Chain 1 finished in 1.1 seconds. #> #> All 4 chains finished successfully. #> Mean chain execution time: 0.9 seconds. #> Total execution time: 1.2 seconds. H0.bridge <- bridge_sampler(fitH0, silent = TRUE) print(H0.bridge) #> Bridge sampling estimate of the log marginal likelihood: -37.73301 #> Estimate obtained in 8 iteration(s) via method "normal". #### Expected output: ## Bridge sampling estimate of the log marginal likelihood: -37.53183 ## Estimate obtained in 5 iteration(s) via method "normal". ```$