# Appendix A — Reference points, categorization, and discrimination

## A.1 R packages used

As indicated in the Methods section, for all our analyses we used R (Version 4.0.4; R Core Team, 2021) and the R-packages *brms* (Version 2.16.1; Bürkner, 2017, 2018), *cowplot* (Version 1.1.1; Wilke, 2020), *dplyr* (Version 1.0.10; Wickham, François, et al., 2022), *forcats* (Version 0.5.2; Wickham, 2022a), *ggimage* (Version 0.3.1; Yu, 2022), *ggiraph* (Version 0.7.10; Gohel & Skintzos, 2021), *ggplot2* (Version 3.3.6; Wickham, 2016), *ggthemes* (Version 4.2.4; Arnold, 2021), *glue* (Version 1.6.2; Hester & Bryan, 2022), *here* (Version 1.0.1; Müller, 2020), *lubridate* (Version 1.8.0; Grolemund & Wickham, 2011), *magick* (Version 2.7.3; Ooms, 2021), *modelr* (Version 0.1.9; Wickham, 2022b), *papaja* (Version 0.1.1; Aust & Barth, 2022), *patchwork* (Version 1.1.2; Pedersen, 2022), *purrr* (Version 0.3.4; Henry & Wickham, 2020), *Rcpp* (Eddelbuettel & Balamuta, 2018; Version 1.0.9; Eddelbuettel & François, 2011), *readr* (Version 2.1.2; Wickham, Hester, et al., 2022), *readxl* (Version 1.4.1; Wickham & Bryan, 2022), *rstan* (Version 2.21.2; Stan Development Team, 2020a), *StanHeaders* (Version 2.21.0.7; Stan Development Team, 2020b), *stringr* (Version 1.4.0; Wickham, 2019), *tibble* (Version 3.1.8; Müller & Wickham, 2022), *tidybayes* (Version 3.0.1; Kay, 2021), *tidyr* (Version 1.2.1; Wickham & Girlich, 2022), *tidyverse* (Version 1.3.2; Wickham et al., 2019), and *tinylabels* (Version 0.2.3; Barth, 2022).

## A.2 Bayesian hierarchical model implementation details

### A.2.1 Categorization responses

We fitted a hierarchical Bayesian binomial logistic regression model to the categorization response data with morph level as fixed effect, for each morph series separately, and participant ID within each morph series as random effect for both intercept and slope: \[ freqB \; | \; trials(n) \sim a + b * morph\_level\] \[ a \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] \[ b \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] As priors, we specified a normal distribution with a mean of zero and a standard deviation of 1.5 for the intercept and a normal distribution with a mean of zero and a standard deviation of 0.5 for the slope. For the other model parameters, default priors were used. We used 4 chains consisting of 8000 iterations, with 4000 warmup iterations per chain.

### A.2.2 Categorization response times

We fitted a hierarchical Bayesian lognormal regression model to the categorization response time data with morph level and morph level squared as fixed effects, for each morph series separately, and participant ID within each morph series as random effect for both intercept and slopes: \[ catRT \sim a + b * morph\_level + c * morph\_level^2\] \[ a \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] \[ b \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] \[ c \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] As priors, we specified a normal distribution with a mean of -1 and a standard deviation of 0.5 for the intercept and a normal distribution with a mean of zero and a standard deviation of 0.3 for the slopes. The prior for sigma was specified as a normal distribution with mean 0.4 and standard deviation 0.3, and the priors for the standard deviations of intercept and slopes were specified as a normal distribution with mean 0.3 and standard deviation 0.1. We used 4 chains consisting of 8000 iterations, with 4000 warmup iterations per chain.

### A.2.3 Discrimination responses

To investigate the presence of differences in discrimination sensitivity across stimulus pairs, we fitted a hierarchical Bayesian binomial logistic regression model to the discrimination response data with stepsize as fixed effect, for each morph series separately, and with trial stimuli and participant ID within each morph series as random effects for intercept and participant ID within each morph series as random effect for the slope: \[ freqdiff \; | \; trials(n) \sim a + b * stepsize\] \[ \begin{aligned} a \sim 0 + morph\_series + (1 \; | \; q \; | \; morph\_series:trial\_stimuli) \\ + (1 \; | \; p \; | \; morph\_series:pp\_id) \end{aligned} \] \[ b \sim 0 + morph\_series + (1 \; | \; p \; | \; morph\_series:pp\_id)\] As priors, we specified a normal distribution with a mean of zero and a standard deviation of 1.5 for the intercept and a normal distribution with a mean of zero and a standard deviation of 0.5 for the slope. We used 4 chains consisting of 8000 iterations, with 4000 warmup iterations per chain.

Figure A.1 shows the empirical proportions and posterior predictive distributions for responding ‘different’ in the successive discrimination task, per stepsize, trial type, and morph series, averaged across participants. Stepsize indicates the absolute difference in morph level between the two morph stimuli presented in a trial, with a minimum of zero (for same trials) and a maximum of eleven (when both extremes of the morph series are presented). In this supplementary figure, all stepsizes are shown.

Figure A.2 shows the posterior distributions for the intercept (A) and the effect of stepsize (B) on the probability of responding ‘different’ in the successive discrimination task, for each morph series separately.

Figure A.3 shows the estimated pairwise differences between the posterior distributions for the effect of stepsize on responding ‘different’ in the successive discrimination task, for each of the different recognizable and non-recognizable morph series combinations.

To investigate the presence of an overall category boundary effect, we fitted a hierarchical Bayesian binomial logistic regression model to the discrimination response data with stepsize as fixed effect, for each morph series and trial type separately, and with trial stimuli and participant ID within each morph series as random effects for intercept and participant ID within each morph series as random effect for the slope: \[ freqdiff \; | \; trials(n) \sim a + b * stepsize\] \[ \begin{aligned} a \sim 0 + morph\_series * between\_category + (1 \; | \; q \; | \; morph\_series:trial\_stimuli) \\+ (1 \; | \; p \; | \; morph\_series:pp\_id) \end{aligned} \] \[ b \sim 0 + morph\_series * between\_category + (1 \; | \; p \; | \; morph\_series:pp\_id)\] As priors, we specified a normal distribution with a mean of zero and a standard deviation of 1.5 for the intercept and a normal distribution with a mean of zero and a standard deviation of 0.5 for the slope. We used 4 chains consisting of 8000 iterations, with 4000 warmup iterations per chain.

Figure A.4 shows the posterior distributions for the intercept (A), effect of stepsize (B), effect of trial type (C), and interaction between stepsize and trial type (D) on the probability of responding ‘different’ in the successive discrimination task, for each morph series separately.