Structural skew

Study 1a children

Overview

This study investigated whether 4- to 8-year-olds are sensitive to sampling information in making inferences about a social group, i.e., whether they can adjust their inferences after seeing a skewed sample of group members.

We preregistered that children would be sensitive, i.e., be more likely to report that a novel group population is taller than the sample observed, after seeing a sample selected for being short.

In a deviation from our preregistration, we included children who failed one of the warmup questions (“the same” warmup), since it proved more difficult than expected. The results are qualitatively the same if we exclude these participants.

We confirmed our prediction: participants gave different responses in the skewed and not skewed conditions. Specifically, they were more likely to choose the population as taller in the skewed condition than the not skewed condition.

Next steps

Next, we will investigate the development of this ability to adjust inferences based on sampling information, by powering for a condition by age interaction..?

We will also use a different control condition to make sure that children are not just generalizing directly off the sample by more heavily weighting the taller scrunching Zarpies, since the scrunching Zarpies only appear in the skewed condition in this study.

Also, for the warmup, we will use a different “the same” warmup (red vs white duck), add correct responses to the warmup, and include participants from warmups.

Methods

The study was preregistered on OSF.

Power analysis

See 1a_children_power_analysis.html.

Participants

223 children participated via PANDA in November 6-7, 2025.

Participants’ families were paid $10 for an estimated 10-15 minute task. Children’s parent/guardian provided consent and children provided assent.

After applying exclusion criteria, the final sample included 176 children (n = 87-88 in each of the 2 conditions).

boarding	participants
not skewed	88
skewed	87

Exclusion criteria

A total of 47 participants (21.1% of all participants) were excluded for meeting at least 1 of the following exclusion criteria:

• participating in a pilot version of this study (n = 2)

• not passing video validation, e.g., no parent or child in frame for entire duration of video (n = 9)

• video loss (n = 1)

• parental interference (n = TBD)

• failing the sound check at the start of the study (n = 4)

• failing the taller warmup during the warmup phase (n = 2)

• failing the memory check during the boat training phase (n = 28)

• failing the comprehension check during the boarding phase (n = 5)

In a deviation from our preregistration, where we intended to exclude either failure, participants were not excluded for failing the same warmup, as this question was unexpectedly challenging (n = 85 participants failed).

Warmup questions

We asked participants questions designed to elicit a “taller” and a “the same” response to get participants comfortable with answering either option.

The order of these two questions was counterbalanced. The order of options within each question was fixed, and not counterbalanced.

Taller warmup

Taller warmup

Participants mostly answered the taller warmup correctly.

Participants who made incorrect responses were excluded.

The same warmup

The same warmup

Unexpectedly, performance overall was not very good on “the same” control question - a small minority of children thought the duck was taller, perhaps due to the chicken’s comb - so participants were included if they failed this question, in a deviation from the preregistration.

Memory check

Participants mostly passed the memory check for the Quaffa boarding sequence, i.e., “no”, not all the Quaffas made it onto the boat.

Participants who made incorrect responses were excluded.

# did memory check performance vary by age
memory_check_age <-
  glm(memory_check_pass ~ age_exact,
      family = "binomial",
      data = data_all)

memory_check_age %>% 
  Anova()

There was a significant effect of age (exact) on memory check performance: younger children were more likely to fail (likelihood ratio test: \(\chi^2\)(1) = 5.62, p = 0.018).

Comprehension check

Participants overwhelmingly passed the comprehension check for the Zarpie boarding sequence. Note the correct answer to this question depends on condition:

• In the skewed condition, the correct answer is “no”, not all of the Zarpies made it onto the boat.

• In the not skewed condition, the correct answer is “yes”, all of the Zarpies made it onto the boat.

Demographics

Of included participants.

age
mean	sd	n
6.61	1.41	176

age groups
age_cat	n
4	29
5	38
6	36
7	33
8	40

age groups by condition
not skewed	skewed
4
19	10
5
24	14
6
16	20
7
16	17
8
14	26

gender	n	prop
female	93	52.8%
male	83	47.2%

race	n	prop
Caucasian	100	56.8%
Asian	25	14.2%
Asian, Caucasian	19	10.8%
African American	8	4.5%
Caucasian, Hispanic	8	4.5%
Hispanic	7	4.0%
African American, Caucasian	4	2.3%
Asian, Hispanic	2	1.1%
Caucasian, Hawaiian Pacific	1	0.6%
Caucasian, Hispanic, Colombian	1	0.6%
South American	1	0.6%

education	n	prop
High school/GED	4	2.3%
Some college	20	11.4%
Bachelor's (B.A., B.S.)	60	34.1%
Master's (M.A., M.S.)	54	30.7%
Doctoral (Ph.D., J.D., M.D.)	26	14.8%
NA	12	6.8%

country	n
Canada	1
UK	1
US	174

state	n
AL	2
CA	11
CO	2
FL	6
GA	5
IA	5
ID	2
IL	6
IN	6
KY	1
LA	1
MA	7
MD	10
ME	3
MI	4
MO	3
NC	5
NH	2
NJ	9
NV	3
NY	20
OH	13
OK	4
OR	1
PA	11
SC	1
TN	1
TX	8
UT	3
VA	6
VT	1
WA	5
WI	7
NA	2

Almost all participants (n = 174) were based in the United States, from across 33 states. We also had a few international participants (n = 1 from UK, n = 1 from Canada).

median household income of zipcode
in dollars, US participants only
avg	sd
101650.2	36623.4

Procedure

This study was administered as a Qualtrics survey, and approved by the NYU IRB (IRB-FY2024-9169).

After providing their consent, participants completed a captcha and sound check, and were asked to watch videos sound on. Participants then watched the following videos in order:

0. In the warmup phase, participants were familiarized with answering questions about height in terms of who is taller or whether they are the same height. Participants saw a duck and a chicken appear on screen against a grid, who were the same in height, and were asked who is taller: the duck, the chicken, or are they the same. A same question was asked about a giraffe and a bunny, where the giraffe is in fact taller. The order of these two questions was counterbalanced.

1. In the prior setting and familiarization phase, participants saw a photorealistic picture of 5 human adults and then another picture of a different 5 adults appear on screen against a grid. These adults were all 10 gridline units tall.

Prior setting and familiarization.

2. In the boat training phase, participants were shown a parade of fictional animals attempting to board the boat, to illustrate how the boat works. In the skewed condition, the boat was 6 units tall. In the not skewed condiiton, the boat was 10 units tall.

   • The boat height was specified to be accidental (“When the boat builders were building the boat, they started building the boat from the bottom, but ran out of the special wood they needed for the boat! So the boat ended up being this tall. It might be hard for anyone who is taller than the boat to get on the boat.”), to avoid any justificatory reasoning about the height of the boat being informative about the height of Zarpies or vice versa.

   • To communicate how the boat functions to exclude those shorter than the boat, participants then watched a parade of 20 fictional animals (Quaffas, taken from Foster-Hanson et al., 2019) attempt to board the boat, one at a time, from shortest to tallest.

   • The height of animals were scaled to the height of the boat, such that 10 animals were always shorter than the boat (these animals boarded successfully) and 10 animals were always taller than the boat (all but one were unable to board; the third quaffa successfully boards by bending its head).

   

   Quaffas in the skewed condition. Note the Quaffas are short, since the skewed condition involves a short boat.
   • Participants were asked a memory check: “Did all of the animals board the boat?” (yes/no), and received an affirmation (if they said “no”) or correction (if they said “yes”).

3. In the boat boarding phase, participants learned that Zarpies live on Zarpie island, and saw an island with many Zarpies overhead. Participants learned that all the grownup Zarpies’ names were put into a hat, and some of their names “were drawn out of a hat to try and visit us”. Participants saw then saw a parade of Zarpies attempt to board the boat to visit us, one at a time. Participants were told that they were all grown-up Zarpies. The boarding phase was occluded: i.e., the heights of Zarpies were hidden behind a curtain that showed only their feet.

   • In the skewed condition, the boat is 6 units tall. 20 Zarpies attempt to board, 6 of whom successfully make it on (6 out of 16 successful = 30% successful). Of the 6 who make it on, 2 had to stoop to board.

   • In the not skewed condition, the boat is 10 units tall. 6 Zarpies attempt to board, all of whom successfully make it on (6 out of 6 successful = 100% successful). Of the 6 who board, none had to stoop to board.

Boarding in not skewed condition.

4. After the boat boarding phase, participants were asked a comprehension check: “Did all of the Zarpies board the boat?” (yes/no), and received either an affirmation (if they said “no” in the skewed condition, or “yes” in the not skewed condition) or correction (if they said “yes” in the skewed condition, or “no” in the not skewed condition).

5. In the sample observation phase, all participants saw the Zarpies who successfully boarded the boat get off the boat to visit us. The Zarpies got off one at a time, and each waved/descrunched if relevant. The height of this observed sample (4, 5, 6, 6, 7, 8) was held constant across conditions.

   • To emphasize the height of the Zarpies relative to the boat, participants watched Zarpies deboard the boat, wave, reboard the boat (with any Zarpies taller than the boat stooping down again to board again), and deboard again (with any Zarpies taller than the boat straightening up again).

Observed sample in skewed condition. Note the observed sample is the same, but the height of the boat is short in the skewed condition, vs tall in the not skewed condition.

Participants were asked a single DV:

6. Participants were asked an explicit comparison question asking them to who is taller: Zarpies who visited, Zarpies on Zarpie island, or are they the same (see [explicit comparison]). The order of the first two options were counterbalanced (“the same” always came last).

Finally, participants’ parent or guardian were asked for any problems or confusion they had and demographic information.

Pre-registered analyses

Explicit comparison question in skewed condition.

Participants were explicitly asked to compare the population and the sample: “Who is taller? The Zarpies on Zarpie island, the Zarpies who visited, or are they the same?” The order of the first two options was counterbalanced across participants.

Responses by condition

We pre-registered that if children do adjust, they should be more likely to say “Zarpies on Zarpie island” in the skewed condition than the not skewed condition.

Main sample

# fisher's exact
dv_comp_table <- data %>%
  group_by(boarding, dv_comp) %>%
  summarize(n = n()) %>%
  pivot_wider(names_from = dv_comp,
              values_from = n,
              values_fill = 0) %>% 
  column_to_rownames(var = "boarding") 

fisher_test <- 
  fisher.test(dv_comp_table)
fisher_test

As predicted, overall, participants provided different responses to the question “Who’s taller?” in the skewed condition versus not skewed condition (Fisher’s test, p = 0.008).

## 
## Call:
## lm(formula = dv_comp_pop ~ boarding, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.5058 -0.2809 -0.2809  0.4943  0.7191 
## 
## Coefficients:
##                Estimate Std. Error t value    Pr(>|t|)    
## (Intercept)     0.28090    0.05065   5.546 0.000000107 ***
## boardingskewed  0.22485    0.07204   3.121     0.00211 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4778 on 174 degrees of freedom
## Multiple R-squared:  0.05302,    Adjusted R-squared:  0.04758 
## F-statistic: 9.742 on 1 and 174 DF,  p-value: 0.002109

As predicted, specifically, participants in the skewed condition were more likely to say that Zarpies on Zarpie island (the population) is taller in the skewed condition, compared to the not skewed condition (t(174) = 3.12, p = 0.002).

Excluding those who failed “the same” warmup

# fisher's exact
dv_comp_table <- data_exclude_on_both_warmups %>%
  group_by(boarding, dv_comp) %>%
  summarize(n = n()) %>%
  pivot_wider(names_from = dv_comp,
              values_from = n,
              values_fill = 0) %>% 
  column_to_rownames(var = "boarding") 

fisher_test <- 
  fisher.test(dv_comp_table)
fisher_test

As predicted, overall, participants provided different responses to the question “Who’s taller?” in the skewed condition versus not skewed condition (Fisher’s test, p = 0.006).

dv_comp_pop <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_exclude_on_both_warmups)

dv_comp_pop %>% 
  summary()

As predicted, specifically, participants in the skewed condition were more likely to say that Zarpies on Zarpie island (the population) is taller in the skewed condition, compared to the not skewed condition (z(109) = 3.08, p = 0.002).

Exploratory analyses

The following analyses include those who failed “the same” warmup.

Correct responses by condition

Was one condition “harder” than the other?

dv_comp_correct <-
  glm(dv_comp_correct ~ boarding,
      family = "binomial",
      data = data)

dv_comp_correct %>% 
  summary()

No, there were no differences between conditions in how likely participants were to give the hypothesized/correct response (z(174) = 0.3, p = 0.764).

Responses vs chance within condition

Within each condition, are responses different from chance?

# chi sq: not skewed condition, vs chance
dv_comp_not_skewed_counts <- data %>%
  filter(boarding == "not skewed") %>%
  count(dv_comp) 

dv_comp_not_skewed_chisq <- 
  chisq.test(
    dv_comp_not_skewed_counts$n,
    p = rep(1/3, 3) # chance
    )

Yes, responses in the not skewed condition are significantly different from chance (\(\chi^2\)(2) = 9.26, p = 0.01).

# chi sq: skewed condition, vs chance
dv_comp_skewed_counts <- data %>%
  filter(boarding == "skewed") %>%
  count(dv_comp) 

dv_comp_skewed_chisq <- 
  chisq.test(
    dv_comp_skewed_counts$n,
    p = rep(1/3, 3) # chance
    )

Yes, responses in the skewed condition are significantly different from chance (\(\chi^2\)(2) = 17.86, p < .001).

Age effects

Responses

We suspected but did not preregister an interaction between age and condition (see 1a power analysis).

Condition by age histogram

Main sample

# all responses
dv_comp_cond_age <- 
  multinom(dv_comp ~ boarding * age_exact,
           data = data)

dv_comp_cond_age %>% 
  Anova()

When predicting responses in general, there was not a statistically significant interaction between condition and age (exact) in a multinomial model of responses with age, condition, and their interaction as predictors (\(\chi^2\)(2) = 0.29, p = 0.865).

# population responses only
dv_comp_pop_cond_age <- 
  glm(dv_comp_pop ~ boarding * age_exact,
      family = "binomial",
      data = data)

dv_comp_pop_cond_age %>% 
  Anova()

When predicting responses of “Zarpies on Zarpie island”, there was not a statistically significant interaction between condition and age (exact) in a binomial model of responses with age, condition, and their interaction as predictors (\(\chi^2\)(1) = 0.26, p = 0.61).

# all responses
dv_comp_cond_age_brm <-
  brm(dv_comp ~ boarding * age_exact,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muZarpiesonZarpieisland")),
      data = data)

# vs null
dv_comp_cond_age_brm_null <-
  brm(dv_comp ~ boarding + age_exact,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muZarpiesonZarpieisland")),
      data = data)

bayes_factor(dv_comp_cond_age_brm, dv_comp_cond_age_brm_null)

A Bayesian analysis revealed strong evidence against an interaction between condition and age (exact) on responses overall (BF = 0.03), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

# population responses only
dv_comp_pop_cond_age_brm <-
  brm(dv_comp_pop ~ boarding * age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data)

dv_comp_pop_cond_age_brm %>% 
  hypothesis("boardingskewed:age_exact > 0") 

A Bayesian analysis revealed moderate evidence for an interaction between condition and age (exact) on responses of “Zarpies on Zarpie island” (BF = 5.83), in a Bernoulli model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

Including memory check failures

dv_comp_cond_age <- 
  multinom(dv_comp ~ boarding * age_exact,
           data = data_incl_memory_check)

dv_comp_cond_age %>% 
  Anova()

When predicting responses in general, there was no interaction between condition and age (exact) in a multinomial model of responses with age, condition, and their interaction as predictors (\(\chi^2\)(2) = 1.7, p = 0.427).

# binomial model on population responses
dv_comp_pop_cond_age <- 
  glm(dv_comp_pop ~ boarding * age_exact,
      family = "binomial",
      data = data_incl_memory_check)

dv_comp_pop_cond_age %>% 
  Anova()

When predicting responses of “Zarpies on Zarpie island”, there was no interaction between condition and age (exact) in a binomial model of responses with age, condition, and their interaction as predictors (\(\chi^2\)(1) = 1.48, p = 0.224).

# all responses
dv_comp_cond_age_brm <-
  brm(dv_comp ~ boarding * age_exact,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check)

# vs null
dv_comp_cond_age_brm_null <-
  brm(dv_comp ~ boarding + age_exact,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b",
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check)

bayes_factor(dv_comp_cond_age_brm, dv_comp_cond_age_brm_null)

# population responses only
dv_comp_pop_cond_age_brm <-
  brm(dv_comp_pop ~ boarding * age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data_incl_memory_check)

dv_comp_pop_cond_age_brm %>% 
  hypothesis("boardingskewed:age_exact > 0") 

A Bayesian analysis revealed strong evidence against an interaction between condition and age (exact) on responses overall (BF = 0.06), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

A Bayesian analysis revealed strong evidence for an interaction between condition and age (exact) on responses of “Zarpies on Zarpie island” (BF = 12.03), in a Bernoulli model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

Condition by age group

Main sample

4yo

dv_comp_4 <- 
  multinom(dv_comp ~ boarding,
           data = data %>% 
             filter(age_cat == 4))

dv_comp_4 %>% 
  Anova()

dv_comp_pop_4 <- 
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_cat == 4))

dv_comp_4 %>% 
  Anova()

• no main effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 2.87, p = 0.238)

• no main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 0.01, p = 0.93)

# all responses
dv_comp_4_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 4))

# vs null
dv_comp_4_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 4))

bayes_factor(dv_comp_4_brm, dv_comp_4_brm_null)

dv_comp_pop_4_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 4))

hypothesis(dv_comp_pop_4_brm, "boardingskewed > 0")

• anecdotal evidence against a difference between conditions on responses overall (BF = 0.61), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• anecdotal evidence against a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 0.42), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

5yo

dv_comp_5 <-
  multinom(dv_comp ~ boarding,
           data = data %>% 
             filter(age_cat == 6))

dv_comp_5 %>% 
  Anova()

dv_comp_pop_5 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_cat == 6))

dv_comp_pop_5 %>% 
  Anova()

• marginal effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 5.83, p = 0.054)

• marginal main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 2.86, p = 0.091)

# all responses
dv_comp_5_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 5))

# vs null
dv_comp_5_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 5))

bayes_factor(dv_comp_5_brm, dv_comp_5_brm_null)

dv_comp_pop_5_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data %>% 
        filter(age_cat == 5))

hypothesis(dv_comp_pop_5_brm, "boardingskewed > 0")

• anecdotal evidence for a difference between conditions on responses overall (BF = 2.68), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• strong evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 19.73), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

6yo

dv_comp_6 <-
  multinom(dv_comp ~ boarding,
           data = data %>% 
             filter(age_cat == 6))

dv_comp_6 %>% 
  Anova()

dv_comp_pop_6 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_cat == 6))

dv_comp_pop_6 %>% 
  Anova()

• marginal effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 5.83, p = 0.054)

• marginal main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 2.86, p = 0.091)

# all responses
dv_comp_6_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 6))

# vs null
dv_comp_6_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 6))

bayes_factor(dv_comp_6_brm, dv_comp_6_brm_null)

dv_comp_pop_6_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data %>% 
        filter(age_cat == 6))

hypothesis(dv_comp_pop_6_brm, "boardingskewed > 0")

• anecdotal evidence for a difference between conditions on responses overall (BF = 1.22), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• strong evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 10.73), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

7yo

dv_comp_7 <-
  multinom(dv_comp ~ boarding,
           data = data %>% 
             filter(age_cat == 7))

dv_comp_7 %>% 
  Anova()

dv_comp_pop_7 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_cat == 7))

dv_comp_pop_7 %>% 
  Anova()

• no main effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 1.75, p = 0.417)

• no main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 1.51, p = 0.219)

# all responses
dv_comp_7_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 7))

# vs null
dv_comp_7_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 7))

bayes_factor(dv_comp_7_brm, dv_comp_7_brm_null)

dv_comp_pop_7_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data %>% 
        filter(age_cat == 7))

hypothesis(dv_comp_pop_7_brm, "boardingskewed > 0")

• anecdotal evidence against a difference between conditions on responses overall (BF = 0.76), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• moderate evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 5.5), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

8yo

dv_comp_8 <-
  multinom(dv_comp ~ boarding,
           data = data %>% 
             filter(age_cat == 8))

dv_comp_8 %>% 
  Anova()

dv_comp_pop_8 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_cat == 8))

dv_comp_pop_8 %>% 
  Anova()

• no main effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 2.49, p = 0.288)

• no main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 2.41, p = 0.121)

# all responses
dv_comp_8_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 8))

# vs null
dv_comp_8_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data %>% 
        filter(age_cat == 8))

bayes_factor(dv_comp_8_brm, dv_comp_8_brm_null)

dv_comp_pop_8_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data %>% 
        filter(age_cat == 8))

hypothesis(dv_comp_pop_8_brm, "boardingskewed > 0")

• anecdotal evidence for a difference between conditions on responses overall (BF = 1.05), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• moderate evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 9.96), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

Including memory check failures

4yo

dv_comp_4 <- 
  multinom(dv_comp ~ boarding,
           data = data_incl_memory_check %>% 
             filter(age_cat == 4))

dv_comp_4 %>% 
  Anova()

dv_comp_pop_4 <-  
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_cat == 4))

dv_comp_4 %>% 
  Anova()

• no main effect of condition on responses overall (likelihood ratio \(\chi^2\)(2) = 3.59, p = 0.166)

• no main effect of condition on “Zarpies on Zarpie island” responses (binomial model: likelihood ratio \(\chi^2\)(1) = 0.44, p = 0.509)

# all responses
dv_comp_4_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 4))

# vs null
dv_comp_4_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 4))

bayes_factor(dv_comp_4_brm, dv_comp_4_brm_null)

dv_comp_pop_4_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 4))

hypothesis(dv_comp_pop_4_brm, "boardingskewed > 0")

• anecdotal evidence against a difference between conditions on responses overall (BF = 0.79), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

• anecdotal evidence against a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 0.4), in a Bernoulli model of population responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

5yo

dv_comp_5 <-
  multinom(dv_comp ~ boarding,
           data = data_incl_memory_check %>% 
             filter(age_cat == 6))

dv_comp_5 %>% 
  Anova()

dv_comp_pop_5 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_cat == 6))

dv_comp_pop_5 %>% 
  Anova()

• marginal effect of condition on responses overall (\(\chi^2\)(2) = 7.76, p = 0.021)

• marginal effect of condition on “Zarpies on Zarpie island” responses (\(\chi^2\)(1) = 3.51, p = 0.061)

6yo

dv_comp_6 <-
  multinom(dv_comp ~ boarding,
           data = data_incl_memory_check %>% 
             filter(age_cat == 6))

dv_comp_6 %>% 
  Anova()

dv_comp_pop_6 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_cat == 6))

dv_comp_pop_6 %>% 
  Anova()

• marginal effect of condition on responses overall (\(\chi^2\)(2) = 7.76, p = 0.021)

• marginal effect of condition on “Zarpies on Zarpie island” responses (\(\chi^2\)(1) = 3.51, p = 0.061)

7yo

dv_comp_7 <-
  multinom(dv_comp ~ boarding,
           data = data_incl_memory_check %>% 
             filter(age_cat == 7))

dv_comp_7 %>% 
  Anova()

dv_comp_pop_7 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_cat == 7))

dv_comp_pop_7 %>% 
  Anova()

• no effect of condition on responses overall (\(\chi^2\)(2) = 3.22, p = 0.2)

• no effect of condition on “Zarpies on Zarpie island” responses (\(\chi^2\)(1) = 2.62, p = 0.106)

# all responses
dv_comp_7_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 7))

# vs null
dv_comp_7_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 7))

bayes_factor(dv_comp_7_brm, dv_comp_7_brm_null)

• anecdotal evidence for a difference between conditions on responses overall (BF = 1.25), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

dv_comp_pop_7_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 7))

hypothesis(dv_comp_pop_7_brm, "boardingskewed > 0")

• strong evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 10.73), in a Bernoulli model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

8yo

dv_comp_8 <-
  multinom(dv_comp ~ boarding,
           data = data_incl_memory_check %>% 
             filter(age_cat == 8))

dv_comp_8 %>% 
  Anova()

dv_comp_pop_8 <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_cat == 8))

dv_comp_pop_8 %>% 
  Anova()

• no effect of condition on responses overall (\(\chi^2\)(2) = 3.51, p = 0.173)

• no effect of condition on “Zarpies on Zarpie island” responses (\(\chi^2\)(1) = 3.35, p = 0.067)

# all responses
dv_comp_8_brm <-
  brm(dv_comp ~ boarding,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept", 
                          dpar = "muZarpiesonZarpieisland"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "b", 
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 8))

# vs null
dv_comp_8_brm_null <-
  brm(dv_comp ~ 1,
      family = categorical,
      prior = c(set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muthesame"),
                set_prior("normal(0,1)", class = "Intercept",
                          dpar = "muZarpiesonZarpieisland")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 8))

bayes_factor(dv_comp_8_brm, dv_comp_8_brm_null)

• anecdotal evidence for a difference between conditions on responses overall (BF = 1.6), in a categorical model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

dv_comp_pop_8_brm <-
  brm(dv_comp_pop ~ boarding,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b", coef = "boardingskewed")),
      data = data_incl_memory_check %>% 
        filter(age_cat == 8))

hypothesis(dv_comp_pop_8_brm, "boardingskewed > 0")

• strong evidence for a difference between conditions on responses of “Zarpies on Zarpie island” (BF = 13.39), in a Bernoulli model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

Condition by age

Main sample

# pop responses, median split
dv_comp_pop_younger <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_exact < data$age_exact %>% median()))

dv_comp_pop_younger %>% 
  Anova()

dv_comp_pop_older <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data %>% 
        filter(age_exact >= data$age_exact %>% median()))

dv_comp_pop_older %>% 
  Anova()

After doing a median split by age, the younger half of children showed a marginal effect of condition (\(\chi^2\)(1) = 3.82, p = 0.051), while the older half of children showed a significant effect of condition (\(\chi^2\)(1) = 4.79, p = 0.029).

Including memory check failures

# pop responses, median split
dv_comp_pop_younger <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_exact < data$age_exact %>% median()))

dv_comp_pop_younger %>% 
  Anova()

dv_comp_pop_older <-
  glm(dv_comp_pop ~ boarding,
      family = "binomial",
      data = data_incl_memory_check %>% 
        filter(age_exact >= data$age_exact %>% median()))

dv_comp_pop_older %>% 
  Anova()

After doing a median split by age, the younger half of children showed a marginal effect of condition (\(\chi^2\)(1) = 2.18, p = 0.14), while the older half of children showed a significant effect of condition (\(\chi^2\)(1) = 7.13, p = 0.008).

Correct responses

Collapsed across conditions

dv_comp_correct_age <- 
  glm(dv_comp_correct ~ age_exact,
      family = "binomial",
      data = data)

dv_comp_correct_age %>% 
  Anova()

When predicting correct responses (“Zarpies on Zarpie island” in the skewed condition, “the same” in the not skewed condition), there was no effect of age (exact) in a binomial model of responses with age as the sole predictor (\(\chi^2\)(1) = 0.06, p = 0.801).

# correct responses
dv_comp_correct_age_brm <-
  brm(dv_comp_correct ~ age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data)

dv_comp_correct_age_brm %>% 
  hypothesis("age_exact > 0") 

A Bayesian analysis revealed anecdotal evidence for an effect of age on correct responses (BF = 1.5), in a Bernoulli model of correct responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

By condition

# correct ~ boarding * age interaction
dv_comp_correct_cond_age <- 
  glm(dv_comp_correct ~ boarding * age_exact,
      family = "binomial",
      data = data)

dv_comp_correct_cond_age %>% 
  Anova()

When predicting correct responses (“Zarpies on Zarpie island” in the skewed condition, “the same” in the not skewed condition), there was no interaction between condition and age (exact) in a binomial model of responses with age, condition, and their interaction as predictors (\(\chi^2\)(1) = 1.67, p = 0.196).

# correct ~ boarding * age interaction
dv_comp_correct_cond_age_brm <-
  brm(dv_comp_correct ~ boarding * age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data)

# vs null (no interaction)
dv_comp_correct_cond_age_brm_null <-
  brm(dv_comp_correct ~ boarding + age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data)

bayes_factor(dv_comp_correct_cond_age_brm, dv_comp_correct_cond_age_brm_null)

A Bayesian analysis revealed moderate evidence against an interaction between condition and age (exact) on responses overall (BF = 0.18), in a Bernoulli model of responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

dv_comp_pop_skewed_age <- 
  glm(dv_comp_pop ~ age_exact,
      family = "binomial",
      data = data %>% 
        filter(boarding == "skewed"))

dv_comp_pop_skewed_age %>% 
  Anova()

Subsetting to the skewed condition, there was no effect of age (exact) in a binomial model of responses with age as the sole predictor (\(\chi^2\)(1) = 1.08, p = 0.299).

# skewed condition only
dv_comp_pop_skewed_age_brm <-
  brm(dv_comp_pop ~ age_exact,
      family = bernoulli,
      prior = c(set_prior("normal(0,1)", class = "Intercept"),
                set_prior("normal(0,1)", class = "b")),
      data = data %>% 
        filter(boarding == "skewed"))

dv_comp_pop_skewed_age_brm %>% 
  hypothesis("age_exact > 0") 

Subsetting to the skewed condition, a Bayesian analysis revealed moderate evidence for an effect of age on population responses (BF = 5.98), in a Bernoulli model of correct responses with age, condition, and their interaction as predictors with normal(0,1) as the prior on all terms.

Session info

## R version 4.5.2 (2025-10-31)
## Platform: aarch64-apple-darwin20
## Running under: macOS Sequoia 15.7.2
## 
## Matrix products: default
## BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: America/New_York
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] brms_2.23.0      Rcpp_1.1.0       effectsize_1.0.1 emmeans_2.0.0   
##  [5] nnet_7.3-20      lmerTest_3.1-3   lme4_1.1-37      Matrix_1.7-4    
##  [9] car_3.1-3        carData_3.0-5    tidycensus_1.7.3 zipcodeR_0.3.5  
## [13] ggtext_0.1.2     lubridate_1.9.4  forcats_1.0.1    stringr_1.6.0   
## [17] dplyr_1.1.4      purrr_1.2.0      readr_2.1.6      tidyr_1.3.1     
## [21] tibble_3.3.0     ggplot2_4.0.1    tidyverse_2.0.0  gt_1.1.0        
## [25] scales_1.4.0     janitor_2.2.1    here_1.0.2      
## 
## loaded via a namespace (and not attached):
##   [1] RColorBrewer_1.1-3    tensorA_0.36.2.1      rstudioapi_0.17.1    
##   [4] jsonlite_2.0.0        datawizard_1.3.0      magrittr_2.0.4       
##   [7] TH.data_1.1-5         estimability_1.5.1    farver_2.1.2         
##  [10] nloptr_2.2.1          rmarkdown_2.30        ragg_1.5.0           
##  [13] fs_1.6.6              vctrs_0.6.5           memoise_2.0.1        
##  [16] minqa_1.2.8           terra_1.8-80          htmltools_0.5.8.1    
##  [19] distributional_0.5.0  curl_7.0.0            raster_3.6-32        
##  [22] Formula_1.2-5         StanHeaders_2.32.10   sass_0.4.10          
##  [25] KernSmooth_2.23-26    bslib_0.9.0           sandwich_3.1-1       
##  [28] zoo_1.8-14            cachem_1.1.0          uuid_1.2-1           
##  [31] lifecycle_1.0.4       pkgconfig_2.0.3       R6_2.6.1             
##  [34] fastmap_1.2.0         rbibutils_2.4         snakecase_0.11.1     
##  [37] digest_0.6.38         numDeriv_2016.8-1.1   colorspace_2.1-2     
##  [40] ps_1.9.1              rprojroot_2.1.1       textshaping_1.0.4    
##  [43] RSQLite_2.4.4         labeling_0.4.3        timechange_0.3.0     
##  [46] httr_1.4.7            abind_1.4-8           compiler_4.5.2       
##  [49] proxy_0.4-27          bit64_4.6.0-1         withr_3.0.2          
##  [52] inline_0.3.21         S7_0.2.1              backports_1.5.0      
##  [55] DBI_1.2.3             QuickJSR_1.8.1        pkgbuild_1.4.8       
##  [58] MASS_7.3-65           rappdirs_0.3.3        classInt_0.4-11      
##  [61] loo_2.8.0             tools_4.5.2           units_1.0-0          
##  [64] glue_1.8.0            callr_3.7.6           nlme_3.1-168         
##  [67] gridtext_0.1.5        grid_4.5.2            sf_1.0-22            
##  [70] checkmate_2.3.3       generics_0.1.4        gtable_0.3.6         
##  [73] tzdb_0.5.0            class_7.3-23          hms_1.1.4            
##  [76] sp_2.2-0              xml2_1.5.0            pillar_1.11.1        
##  [79] vroom_1.6.6           posterior_1.6.1       splines_4.5.2        
##  [82] lattice_0.22-7        survival_3.8-3        bit_4.6.0            
##  [85] tidyselect_1.2.1      knitr_1.50            gridExtra_2.3        
##  [88] reformulas_0.4.2      V8_8.0.1              stats4_4.5.2         
##  [91] xfun_0.54             bridgesampling_1.1-2  matrixStats_1.5.0    
##  [94] rstan_2.32.7          stringi_1.8.7         yaml_2.3.10          
##  [97] boot_1.3-32           evaluate_1.0.5        codetools_0.2-20     
## [100] cli_3.6.5             RcppParallel_5.1.11-1 systemfonts_1.3.1    
## [103] xtable_1.8-4          parameters_0.28.2     Rdpack_2.6.4         
## [106] processx_3.8.6        jquerylib_0.1.4       coda_0.19-4.1        
## [109] parallel_4.5.2        rstantools_2.5.0      blob_1.2.4           
## [112] bayestestR_0.17.0     bayesplot_1.14.0      Brobdingnag_1.2-9    
## [115] ggthemes_5.1.0        mvtnorm_1.3-3         tigris_2.2.1         
## [118] e1071_1.7-16          insight_1.4.2         crayon_1.5.3         
## [121] rlang_1.1.6           rvest_1.0.5           multcomp_1.4-29