+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Children use disagreement to infer what happened
+Jamie Amemiya, Gail D. Heyman & Tobias Gerstenberg
+April 09, 2024
+ +
+
+
+1 Libraries
+library("lme4") # for linear mixed effects models
+library("rsample") # for bootstrapping
+library("xtable") # for latex tables
+library("kableExtra") # for rmarkdown
+library("knitr") # for rmarkdown
+library("car") # for hypothesis test
+library("Metrics") # for rmse
+library("scales") # for percentage plots
+library("broom.mixed") # for model summaries
+library("grateful") # for package citations
+library("ggeffects") # for marginal predictions
+library("scales") # for percentage scales
+library("Hmisc") # for bootstrapped means
+library("ggtext") # for colored text in ggplot
+library("tidyverse") # for everything else
+
+
+2 Helper functions
+# set classic theme
+theme_set(theme_classic() +
+ theme(text = element_text(size = 16)))
+
+# function for printing out html or latex tables
+print_table = function(data, format = "html", digits = 2){
+ if(format == "html"){
+ data %>%
+ kable(digits = digits) %>%
+ kable_styling()
+ }else if(format == "latex"){
+ data %>%
+ xtable(digits = digits,
+ caption = "Caption",
+ label = "tab:table") %>%
+ print(include.rownames = F,
+ booktabs = T,
+ sanitize.colnames.function = identity,
+ caption.placement = "top")
+ }
+}
+
+# suppress grouping warning
+options(dplyr.summarise.inform = F)
+
+# show figures at the end of code chunks
+opts_chunk$set(comment = "",
+ fig.show = "hold")
+
+# regression function
+fun.regression = function(formula, data){
+ results = glmer(formula = formula,
+ family = binomial,
+ data = data)
+ print(results)
+ return(results)
+}
+
+# results table
+fun.table = function(results, type = "exploratory"){
+ table = results %>%
+ tidy(conf.int = T) %>%
+ filter(effect == "fixed") %>%
+ select(-group)
+
+ if (type == "exploratory"){
+ table = table %>%
+ select(-c(p.value))
+ }
+ table %>%
+ print_table()
+}
+
+# colors
+l.color = list(agreement = "#89fa50",
+ disagreement = "#ff968c",
+ ambiguous = "#d38950",
+ unambiguous = "#96d5d6")
+
+3 EXPERIMENT 1
+
+
+3.1 DATA
+
+
+
+3.1.1 Read in data
+# fixed rounding issue; one participant was actually 11 and turned 12 the next day
+# participant reported they were 9 despite birth year indicating they were 8;
+# recoded to 9.69 given reported age likely more reliable
+
+df.exp1 = read_csv("../data/data1_infer.csv") %>%
+ rename(trial_order = trial_order_dada) %>%
+ mutate(age_continuous = ifelse(age_continuous == 12, 11.99,
+ ifelse(age_continuous == 8.69, 9.69,
+ age_continuous)))
+
+3.2 STATS
+
+
+3.2.1 Counterbalancing
+-
+
- check if counterbalanced factors moderate the effect of trial type +
+
+
+
+
+
+3.2.1.1 Story order
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * story_order_wagon + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree * story_order_wagon +
+ (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 243.5716 260.2593 -116.7858 233.5716 203
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.424
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.8473 1.8559
+ story_order_wagon condition_disagree:story_order_wagon
+ 0.5618 -0.2295 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.85 + | ++0.40 + | ++-4.61 + | ++-2.63 + | ++-1.06 + | +
| +fixed + | ++condition_disagree + | ++1.86 + | ++0.42 + | ++4.46 + | ++1.04 + | ++2.67 + | +
| +fixed + | ++story_order_wagon + | ++0.56 + | ++0.36 + | ++1.56 + | ++-0.15 + | ++1.27 + | +
| +fixed + | ++condition_disagree:story_order_wagon + | ++-0.23 + | ++0.38 + | ++-0.61 + | ++-0.97 + | ++0.51 + | +
+
+
+
+
+
+3.2.1.2 Trial order
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * trial_order + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula:
+ambiguous_yes ~ 1 + condition_disagree * trial_order + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 241.2821 257.9698 -115.6410 231.2821 203
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.477
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.84349 1.83408
+ trial_order condition_disagree:trial_order
+ -0.02668 -0.64759 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.84 + | ++0.40 + | ++-4.57 + | ++-2.63 + | ++-1.05 + | +
| +fixed + | ++condition_disagree + | ++1.83 + | ++0.42 + | ++4.34 + | ++1.01 + | ++2.66 + | +
| +fixed + | ++trial_order + | ++-0.03 + | ++0.35 + | ++-0.08 + | ++-0.72 + | ++0.67 + | +
| +fixed + | ++condition_disagree:trial_order + | ++-0.65 + | ++0.39 + | ++-1.67 + | ++-1.41 + | ++0.11 + | +
+
+
+
+
+
+3.2.1.3 Valence
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * valence_neg + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula:
+ambiguous_yes ~ 1 + condition_disagree * valence_neg + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 244.9991 261.6868 -117.4996 234.9991 203
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.481
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.844699 1.867938
+ valence_neg condition_disagree:valence_neg
+ -0.004625 0.350825 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.84 + | ++0.40 + | ++-4.57 + | ++-2.64 + | ++-1.05 + | +
| +fixed + | ++condition_disagree + | ++1.87 + | ++0.42 + | ++4.44 + | ++1.04 + | ++2.69 + | +
| +fixed + | ++valence_neg + | ++0.00 + | ++0.36 + | ++-0.01 + | ++-0.70 + | ++0.69 + | +
| +fixed + | ++condition_disagree:valence_neg + | ++0.35 + | ++0.38 + | ++0.92 + | ++-0.40 + | ++1.10 + | +
+
+3.2.2 Confirmatory analysis
+
+
+
+
+
+
+3.2.2.1 Trial type effect
+Choose ambiguous statement more in disagreement than agreement trials.
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 242.4062 252.4188 -118.2031 236.4062 205
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.455
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.828 1.828 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.83 + | ++0.40 + | ++-4.62 + | ++0 + | ++-2.60 + | ++-1.05 + | +
| +fixed + | ++condition_disagree + | ++1.83 + | ++0.41 + | ++4.47 + | ++0 + | ++1.03 + | ++2.63 + | +
+
+
+
+
+
+
+3.2.2.2 Inferences above chance
+Choose unambiguous in agreement trials above chance (log odds = -.69; 33%).
+results = fun.regression(
+ formula = "unambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: unambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 242.1031 252.1157 -118.0515 236.1031 205
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.475
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ 1.841 -1.893 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++1.84 + | ++0.40 + | ++4.61 + | ++0 + | ++1.06 + | ++2.62 + | +
| +fixed + | ++condition_disagree + | ++-1.89 + | ++0.41 + | ++-4.57 + | ++0 + | ++-2.71 + | ++-1.08 + | +
Linear hypothesis test
+
+Hypothesis:
+(Intercept) = - 0.69
+
+Model 1: restricted model
+Model 2: unambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+
+ Df Chisq Pr(>Chisq)
+1
+2 1 40.144 2.359e-10 ***
+---
+Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+
+
+
+
+
+3.2.2.3 Ambiguous choice
+Choose ambiguous in disagreement trials above chance (log odds = -.69; 33%).
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_agree + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_agree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 242.4062 252.4188 -118.2031 236.4062 205
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.455
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_agree
+ -0.0003634 -1.8277981 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++0.00 + | ++0.31 + | ++0.00 + | ++1 + | ++-0.61 + | ++0.61 + | +
| +fixed + | ++condition_agree + | ++-1.83 + | ++0.41 + | ++-4.47 + | ++0 + | ++-2.63 + | ++-1.03 + | +
Linear hypothesis test
+
+Hypothesis:
+(Intercept) = - 0.69
+
+Model 1: restricted model
+Model 2: ambiguous_yes ~ 1 + condition_agree + (1 | participant)
+
+ Df Chisq Pr(>Chisq)
+1
+2 1 4.8736 0.02727 *
+---
+Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+3.2.3 Exploratory analysis
+
+
+
+
+
+
+3.2.3.1 Trial type by age interaction
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * age_continuous + (1 | participant)",
+ data = df.exp1)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree * age_continuous + (1 |
+ participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 237.4681 254.1558 -113.7340 227.4681 203
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.546
+Number of obs: 208, groups: participant, 52
+Fixed Effects:
+ (Intercept) condition_disagree
+ 2.5127 -6.0120
+ age_continuous condition_disagree:age_continuous
+ -0.4749 0.8474 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++2.51 + | ++2.55 + | ++0.98 + | ++-2.49 + | ++7.52 + | +
| +fixed + | ++condition_disagree + | ++-6.01 + | ++2.74 + | ++-2.20 + | ++-11.38 + | ++-0.65 + | +
| +fixed + | ++age_continuous + | ++-0.47 + | ++0.28 + | ++-1.71 + | ++-1.02 + | ++0.07 + | +
| +fixed + | ++condition_disagree:age_continuous + | ++0.85 + | ++0.30 + | ++2.83 + | ++0.26 + | ++1.43 + | +
+
+
+3.2.3.2 Moderation by age
+# from 7 to 11 years
+for(i in 7:11){
+ cat(str_c("Age = ", i, "\n\n"))
+ results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp1 %>%
+ filter(age_group == i))
+}Age = 7
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 50.2215 55.2881 -22.1107 44.2215 37
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 2.049
+Number of obs: 40, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.879 1.489
+Age = 8
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 61.2404 66.8540 -27.6202 55.2404 45
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.994
+Number of obs: 48, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.1496 0.5988
+Age = 9
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 54.6152 59.6818 -24.3076 48.6152 37
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.8046
+Number of obs: 40, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.2513 0.7889
+Age = 10
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 43.7876 48.8542 -18.8938 37.7876 37
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.833
+Number of obs: 40, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -3.272 3.225
+Age = 11
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 40.0251 45.0917 -17.0125 34.0251 37
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.543
+Number of obs: 40, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.956 4.529
+
+3.3 PLOTS
+
+
+
+3.3.1 Inference
+set.seed(1)
+
+df.plot.individual = df.exp1 %>%
+ mutate(condition_disagree = as.character(condition_disagree)) %>%
+ group_by(participant, age_continuous, condition_disagree) %>%
+ summarize(pct_amb = sum(ambiguous_yes)/n())
+
+df.age.means = df.plot.individual %>%
+ distinct(participant, age_continuous) %>%
+ mutate(age_group = floor(age_continuous)) %>%
+ group_by(age_group) %>%
+ summarize(age_mean = mean(age_continuous),
+ n = str_c("n = ", n())) %>%
+ ungroup()
+
+df.plot.means = df.exp1 %>%
+ mutate(condition_disagree = as.character(condition_disagree)) %>%
+ group_by(participant, age_group, condition_disagree) %>%
+ summarize(pct_amb = sum(ambiguous_yes)/n()) %>%
+ group_by(age_group, condition_disagree) %>%
+ reframe(response = smean.cl.boot(pct_amb),
+ name = c("mean", "low", "high")) %>%
+ left_join(df.age.means,
+ by = "age_group") %>%
+ pivot_wider(names_from = name,
+ values_from = response) %>%
+ mutate(age_mean = ifelse(condition_disagree == 0, age_mean - 0.05, age_mean + 0.05))
+
+df.plot.text = df.plot.means %>%
+ distinct(age_group, n)
+
+ggplot() +
+ geom_hline(yintercept = 1/3,
+ linetype = 2,
+ alpha = 0.1) +
+ geom_point(data = df.plot.individual,
+ mapping = aes(x = age_continuous,
+ y = pct_amb,
+ color = condition_disagree),
+ alpha = 0.5,
+ show.legend = T,
+ shape = 16,
+ size = 1.5) +
+ geom_linerange(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ ymin = low,
+ ymax = high),
+ color = "gray40") +
+ geom_point(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ fill = condition_disagree),
+ shape = 21,
+ size = 3,
+ show.legend = T) +
+ geom_text(data = df.plot.text,
+ mapping = aes(x = age_group + 0.5,
+ y = 1.05,
+ label = n),
+ hjust = 0.5) +
+ scale_y_continuous(labels = percent) +
+ labs(x = "Age (in years)",
+ y = "% Infer Ambiguous Utterance",
+ title = "Experiment 1: Inference") +
+ coord_cartesian(xlim = c(7, 12),
+ ylim = c(0, 1),
+ clip = "off") +
+ scale_color_manual(name = "Trial Type",
+ labels = c("Agreement", "Disagreement"),
+ values = c(l.color$agreement, l.color$disagreement),
+ guide = guide_legend(reverse = T)) +
+ scale_fill_manual(name = "Trial Type",
+ labels = c("Agreement", "Disagreement"),
+ values = c(l.color$agreement, l.color$disagreement),
+ guide = guide_legend(reverse = T)) +
+ theme(plot.title = element_text(hjust = 0.5,
+ vjust = 2,
+ size = 18,
+ face = "bold"),
+ axis.title.y = element_markdown(color = l.color$ambiguous),
+ legend.position = "right")
+
+ggsave(filename = "../figures/plots/exp1_inference.pdf",
+ width = 8,
+ height = 4)
+
+4 EXPERIMENT 2
+
+
+4.1 DATA
+ +
+
+4.2 STATS
+
+
+4.2.1 Counterbalancing
+
+
+4.2.1.1 Prediction condition
+
+
+
+
+
+
+4.2.1.1.1 Story order
+results = fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb * story_order_wagon + (1 | participant)",
+ data = df.exp2.predict)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb * story_order_wagon + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 533.8373 554.1795 -261.9187 523.8373 427
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.5109
+Number of obs: 432, groups: participant, 54
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.2683 1.5824
+ story_order_wagon condition_amb:story_order_wagon
+ -0.1105 0.2620 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.27 + | ++0.19 + | ++-6.85 + | ++-1.63 + | ++-0.91 + | +
| +fixed + | ++condition_amb + | ++1.58 + | ++0.23 + | ++7.00 + | ++1.14 + | ++2.03 + | +
| +fixed + | ++story_order_wagon + | ++-0.11 + | ++0.18 + | ++-0.62 + | ++-0.46 + | ++0.24 + | +
| +fixed + | ++condition_amb:story_order_wagon + | ++0.26 + | ++0.22 + | ++1.20 + | ++-0.17 + | ++0.69 + | +
+
+
+
+
+
+4.2.1.1.2 Trial order
+results = fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb*trial_order_auau + (1 | participant)",
+ data = df.exp2.predict)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb * trial_order_auau + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 533.6323 553.9745 -261.8162 523.6323 427
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.473
+Number of obs: 432, groups: participant, 54
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.25973 1.58341
+ trial_order_auau condition_amb:trial_order_auau
+ 0.15624 0.01714 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.26 + | ++0.18 + | ++-6.90 + | ++-1.62 + | ++-0.90 + | +
| +fixed + | ++condition_amb + | ++1.58 + | ++0.23 + | ++7.03 + | ++1.14 + | ++2.02 + | +
| +fixed + | ++trial_order_auau + | ++0.16 + | ++0.18 + | ++0.88 + | ++-0.19 + | ++0.50 + | +
| +fixed + | ++condition_amb:trial_order_auau + | ++0.02 + | ++0.22 + | ++0.08 + | ++-0.41 + | ++0.44 + | +
+
+
+
+
+
+4.2.1.1.3 Valence
+results = fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb * valence_neg + (1 | participant)",
+ data = df.exp2.predict)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb * valence_neg + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 531.5686 551.9108 -260.7843 521.5686 427
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.4787
+Number of obs: 432, groups: participant, 54
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.26341 1.59408
+ valence_neg condition_amb:valence_neg
+ -0.05144 0.34376 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.26 + | ++0.18 + | ++-6.92 + | ++-1.62 + | ++-0.91 + | +
| +fixed + | ++condition_amb + | ++1.59 + | ++0.23 + | ++7.08 + | ++1.15 + | ++2.04 + | +
| +fixed + | ++valence_neg + | ++-0.05 + | ++0.18 + | ++-0.29 + | ++-0.40 + | ++0.30 + | +
| +fixed + | ++condition_amb:valence_neg + | ++0.34 + | ++0.22 + | ++1.57 + | ++-0.08 + | ++0.77 + | +
+
+4.2.1.2 Inference condition
+
+
+
+
+
+
+4.2.1.2.1 Story order
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * story_order_wagon + (1 | participant)",
+ data = df.exp2.infer)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree * story_order_wagon +
+ (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 398.6856 419.1984 -194.3428 388.6856 442
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.009
+Number of obs: 447, groups: participant, 56
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.687817 3.783142
+ story_order_wagon condition_disagree:story_order_wagon
+ 0.005322 -0.262981 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-2.69 + | ++0.31 + | ++-8.70 + | ++-3.29 + | ++-2.08 + | +
| +fixed + | ++condition_disagree + | ++3.78 + | ++0.35 + | ++10.69 + | ++3.09 + | ++4.48 + | +
| +fixed + | ++story_order_wagon + | ++0.01 + | ++0.28 + | ++0.02 + | ++-0.55 + | ++0.56 + | +
| +fixed + | ++condition_disagree:story_order_wagon + | ++-0.26 + | ++0.31 + | ++-0.86 + | ++-0.86 + | ++0.33 + | +
+
+
+
+
+
+4.2.1.2.2 Trial order
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * trial_order_dada + (1 | participant)",
+ data = df.exp2.infer)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree * trial_order_dada + (1 |
+ participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 400.1794 420.6922 -195.0897 390.1794 442
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.032
+Number of obs: 447, groups: participant, 56
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.70545 3.78219
+ trial_order_dada condition_disagree:trial_order_dada
+ -0.06539 0.08335 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-2.71 + | ++0.31 + | ++-8.67 + | ++-3.32 + | ++-2.09 + | +
| +fixed + | ++condition_disagree + | ++3.78 + | ++0.36 + | ++10.62 + | ++3.08 + | ++4.48 + | +
| +fixed + | ++trial_order_dada + | ++-0.07 + | ++0.29 + | ++-0.23 + | ++-0.63 + | ++0.49 + | +
| +fixed + | ++condition_disagree:trial_order_dada + | ++0.08 + | ++0.31 + | ++0.27 + | ++-0.52 + | ++0.68 + | +
+
+
+
+
+
+4.2.1.2.3 Valence
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * valence_neg + (1 | participant)",
+ data = df.exp2.infer)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula:
+ambiguous_yes ~ 1 + condition_disagree * valence_neg + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 396.7389 417.2517 -193.3695 386.7389 442
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.9986
+Number of obs: 447, groups: participant, 56
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.6816 3.7756
+ valence_neg condition_disagree:valence_neg
+ -0.0941 -0.3050 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-2.68 + | ++0.31 + | ++-8.73 + | ++-3.28 + | ++-2.08 + | +
| +fixed + | ++condition_disagree + | ++3.78 + | ++0.35 + | ++10.65 + | ++3.08 + | ++4.47 + | +
| +fixed + | ++valence_neg + | ++-0.09 + | ++0.28 + | ++-0.33 + | ++-0.65 + | ++0.46 + | +
| +fixed + | ++condition_disagree:valence_neg + | ++-0.31 + | ++0.31 + | ++-0.99 + | ++-0.91 + | ++0.30 + | +
+
+4.2.2 Confirmatory analyses
+
+
+4.2.2.1 Trial type effect
+
+
+
+
+
+
+4.2.2.1.1 Prediction condition
+Predict disagreement more in ambiguous than unambiguous trials.
+results = fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb + (1 | participant)",
+ data = df.exp2.predict)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 531.3732 543.5785 -262.6866 525.3732 429
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.5009
+Number of obs: 432, groups: participant, 54
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.267 1.584
+ 0 1
+ 0 0.7685185 0.2314815
+ 1 0.4259259 0.5740741| +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.27 + | ++0.18 + | ++-6.88 + | ++0 + | ++-1.63 + | ++-0.91 + | +
| +fixed + | ++condition_amb + | ++1.58 + | ++0.23 + | ++7.03 + | ++0 + | ++1.14 + | ++2.02 + | +
+
+
+
+
+
+4.2.2.1.2 Inference condition
+Choose ambiguous statement more in disagreement than agreement trials.
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp2.infer)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 396.2555 408.5632 -195.1278 390.2555 444
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.028
+Number of obs: 447, groups: participant, 56
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.697 3.771
+ 0 1
+ 0 0.91071429 0.08928571
+ 1 0.29147982 0.70852018| +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-2.70 + | ++0.31 + | ++-8.72 + | ++0 + | ++-3.30 + | ++-2.09 + | +
| +fixed + | ++condition_disagree + | ++3.77 + | ++0.35 + | ++10.69 + | ++0 + | ++3.08 + | ++4.46 + | +
+
+4.2.3 Exploratory analysis
+
+
+4.2.3.1 Trial type by age interaction
+
+
+
+
+
+
+4.2.3.1.1 Prediction
+results = fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb * age_continuous + (1 | participant)",
+ data = df.exp2.predict)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb * age_continuous + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 533.1134 553.4555 -261.5567 523.1134 427
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.4888
+Number of obs: 432, groups: participant, 54
+Fixed Effects:
+ (Intercept) condition_amb
+ 0.3225 -0.3813
+ age_continuous condition_amb:age_continuous
+ -0.1702 0.2100 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++0.32 + | ++1.16 + | ++0.28 + | ++-1.95 + | ++2.60 + | +
| +fixed + | ++condition_amb + | ++-0.38 + | ++1.43 + | ++-0.27 + | ++-3.18 + | ++2.41 + | +
| +fixed + | ++age_continuous + | ++-0.17 + | ++0.12 + | ++-1.37 + | ++-0.41 + | ++0.07 + | +
| +fixed + | ++condition_amb:age_continuous + | ++0.21 + | ++0.15 + | ++1.39 + | ++-0.09 + | ++0.51 + | +
+
+
+
+
+
+4.2.3.1.2 Inference
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree * age_continuous + (1 | participant)",
+ data = df.exp2.infer)Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree * age_continuous + (1 |
+ participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 370.9069 391.4197 -180.4534 360.9069 442
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.188
+Number of obs: 447, groups: participant, 56
+Fixed Effects:
+ (Intercept) condition_disagree
+ 4.0689 -7.5859
+ age_continuous condition_disagree:age_continuous
+ -0.7699 1.2725 | +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++4.07 + | ++2.20 + | ++1.85 + | ++-0.24 + | ++8.38 + | +
| +fixed + | ++condition_disagree + | ++-7.59 + | ++2.30 + | ++-3.30 + | ++-12.09 + | ++-3.08 + | +
| +fixed + | ++age_continuous + | ++-0.77 + | ++0.25 + | ++-3.05 + | ++-1.26 + | ++-0.27 + | +
| +fixed + | ++condition_disagree:age_continuous + | ++1.27 + | ++0.27 + | ++4.70 + | ++0.74 + | ++1.80 + | +
+
+4.2.3.2 Moderation by age
+
+
+
+4.2.3.2.1 Prediction condition
+# from 7 to 11 years
+for(i in 7:11){
+ cat(str_c("Age = ", i, "\n\n"))
+ fun.regression(
+ formula = "dis_yes ~ 1 + condition_amb + (1 | participant)",
+ data = df.exp2.predict %>%
+ filter(age_group == i))
+}Age = 7
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+124.0071 131.7002 -59.0036 118.0071 93
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.161
+Number of obs: 96, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_amb
+ -0.9926 1.2093
+Age = 8boundary (singular) fit: see help('isSingular')Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 93.6577 100.8037 -43.8288 87.6577 77
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 3.525e-08
+Number of obs: 80, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.735 2.140
+optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
+Age = 9
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+130.8159 138.5089 -62.4079 124.8159 93
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.1557
+Number of obs: 96, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_amb
+ -0.793 1.132
+Age = 10
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 94.7171 101.8631 -44.3585 88.7171 77
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.3675
+Number of obs: 80, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.782 1.989
+Age = 11boundary (singular) fit: see help('isSingular')Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: dis_yes ~ 1 + condition_amb + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 99.8731 107.0192 -46.9366 93.8731 77
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.804e-07
+Number of obs: 80, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_amb
+ -1.386 1.792
+optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
+
+
+4.2.3.2.2 Inference condition
+# from 7 to 11 years
+for(i in 7:11){
+ cat(str_c("Age = ", i, "\n\n"))
+ fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp2.infer %>%
+ filter(age_group == i))
+}Age = 7
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+120.7673 128.4603 -57.3836 114.7673 93
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.8776
+Number of obs: 96, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.410 1.197
+Age = 8
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 88.856 96.549 -41.428 82.856 93
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 1.184
+Number of obs: 96, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.911 3.917
+Age = 9
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 52.2821 59.9752 -23.1411 46.2821 93
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 3.168
+Number of obs: 96, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -5.704 8.442
+Age = 10boundary (singular) fit: see help('isSingular')Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 43.0340 50.1423 -18.5170 37.0340 76
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0
+Number of obs: 79, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -2.944 5.429
+optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
+Age = 11
+
+Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 65.8653 73.0114 -29.9327 59.8653 77
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0.843
+Number of obs: 80, groups: participant, 10
+Fixed Effects:
+ (Intercept) condition_disagree
+ -3.241 4.510
+
+
+
+
+
+
+
+
+
+
+
+4.2.3.2.3 Inference condition: First story only
+Examine story 1 (trials 1 and 2) and story 4 (trials 7 and 8) among 7-year-olds.
+# story 1, 7 year olds
+df.exp2.infer.7.1 = df.exp2.infer %>%
+ filter(age_group == 7 &
+ (trial == "trial 1" |trial == "trial 2"))
+
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp2.infer.7.1)boundary (singular) fit: see help('isSingular')Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 34.7724 38.3065 -14.3862 28.7724 21
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0
+Number of obs: 24, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -0.6931 -0.4055
+optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings prop.table(table(df.exp2.infer.7.1$condition_disagree, df.exp2.infer.7.1$ambiguous_yes),
+ margin = 1)
+ 0 1
+ 0 0.6666667 0.3333333
+ 1 0.7500000 0.2500000| +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-0.69 + | ++0.61 + | ++-1.13 + | ++0.26 + | ++-1.89 + | ++0.51 + | +
| +fixed + | ++condition_disagree + | ++-0.41 + | ++0.91 + | ++-0.45 + | ++0.65 + | ++-2.18 + | ++1.37 + | +
# story 4, 7 year olds
+df.exp2.infer.7.4 = df.exp2.infer %>%
+ filter(age_group == 7 &
+ (trial == "trial 7" |trial == "trial 8"))
+
+results = fun.regression(
+ formula = "ambiguous_yes ~ 1 + condition_disagree + (1 | participant)",
+ data = df.exp2.infer.7.4)boundary (singular) fit: see help('isSingular')Generalized linear mixed model fit by maximum likelihood (Laplace
+ Approximation) [glmerMod]
+ Family: binomial ( logit )
+Formula: ambiguous_yes ~ 1 + condition_disagree + (1 | participant)
+ Data: data
+ AIC BIC logLik deviance df.resid
+ 35.7967 39.3308 -14.8983 29.7967 21
+Random effects:
+ Groups Name Std.Dev.
+ participant (Intercept) 0
+Number of obs: 24, groups: participant, 12
+Fixed Effects:
+ (Intercept) condition_disagree
+ -1.099 1.435
+optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings prop.table(table(df.exp2.infer.7.4$condition_disagree, df.exp2.infer.7.4$ambiguous_yes),
+ margin = 1)
+ 0 1
+ 0 0.7500000 0.2500000
+ 1 0.4166667 0.5833333| +effect + | ++term + | ++estimate + | ++std.error + | ++statistic + | ++p.value + | ++conf.low + | ++conf.high + | +
|---|---|---|---|---|---|---|---|
| +fixed + | ++(Intercept) + | ++-1.10 + | ++0.67 + | ++-1.65 + | ++0.10 + | ++-2.41 + | ++0.21 + | +
| +fixed + | ++condition_disagree + | ++1.44 + | ++0.89 + | ++1.62 + | ++0.11 + | ++-0.30 + | ++3.17 + | +
+
+4.2.4 Bayesian model
+
+
+
+4.2.4.1 Prediction data
+df.exp2.predict.prob = df.exp2.predict %>%
+ count(age_group, condition_amb_c, dis_yes) %>%
+ group_by(age_group, condition_amb_c) %>%
+ mutate(probability = n/sum(n)) %>%
+ ungroup() %>%
+ mutate(utterance = str_remove_all(condition_amb_c, " Trials"),
+ utterance = factor(utterance,
+ levels = c("Unambiguous", "Ambiguous")),
+ agreement = factor(dis_yes,
+ levels = c(0, 1),
+ labels = c("agree", "disagree"))) %>%
+ select(-c(condition_amb_c, dis_yes, n)) %>%
+ relocate(probability, .after = last_col()) %>%
+ arrange(age_group, utterance, agreement)
+
+
+4.2.4.2 Without softmax
+utterance_prior = c(0.5, 0.5)
+
+df.inference = df.exp2.predict.prob %>%
+ group_by(agreement, age_group) %>%
+ mutate(prior = utterance_prior) %>%
+ mutate(posterior = probability * prior /
+ sum(probability * prior)) %>%
+ ungroup()
+
+df.model.posterior = df.inference %>%
+ rename(condition = agreement) %>%
+ mutate(condition = factor(condition,
+ levels = c("agree", "disagree"),
+ labels = c("Agreement Trials", "Disagreement Trials"))) %>%
+ filter(utterance == "Ambiguous")
+
+
+4.2.4.3 One temperature parameter
+age = 7:11
+
+softmax = function(vec, temp = 3) {
+ out = exp(vec*temp) / sum(exp(vec*temp))
+ return(out)
+}
+
+df.data = df.exp2.infer %>%
+ count(age_group, condition_disagree_c, ambiguous_yes) %>%
+ group_by(age_group, condition_disagree_c) %>%
+ reframe(p = n/sum(n)) %>%
+ filter(row_number() %% 2 == 0) %>%
+ rename(agreement = condition_disagree_c) %>%
+ mutate(agreement = ifelse(agreement == "Agreement Trials", "agree", "disagree"))
+
+fit_softmax = function(beta){
+ df.prediction = df.inference %>%
+ filter(age_group %in% age) %>%
+ select(age_group, utterance, agreement, posterior) %>%
+ pivot_wider(names_from = utterance,
+ values_from = posterior) %>%
+ rowwise() %>%
+ mutate(Unambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[1],
+ Ambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[2]) %>%
+ select(age_group, agreement, prediction = Ambiguous_soft)
+
+ # compute loss as squared error
+ loss = df.data %>%
+ filter(age_group %in% age) %>%
+ left_join(df.prediction) %>%
+ mutate(loss = (p-prediction)^2) %>%
+ pull(loss) %>%
+ sum()
+
+ return(loss)
+}
+
+# find best fitting softmax parameter
+fit = optim(par = 0,
+ fn = fit_softmax)
+
+# use the best parameter
+beta = fit[[1]]
+
+# model with softmax
+df.model.softmax = df.inference %>%
+ select(age_group, utterance, agreement, posterior) %>%
+ pivot_wider(names_from = utterance,
+ values_from = posterior) %>%
+ rowwise() %>%
+ mutate(Unambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[1],
+ Ambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[2]) %>%
+ select(age_group, condition = agreement, posterior = Ambiguous_soft) %>%
+ mutate(condition = factor(condition,
+ levels = c("agree", "disagree"),
+ labels = c("Agreement Trials", "Disagreement Trials")))
+
+
+4.2.4.4 Linear increase of temperature parameter
+-
+
- fit linear model of softmax temperature as a function of age +
# rm(beta)
+
+fit_softmax_age = function(par){
+ df.prediction = df.inference %>%
+ select(age_group, utterance, agreement, posterior) %>%
+ mutate(beta = par[1] + par[2] * age_group) %>%
+ pivot_wider(names_from = utterance,
+ values_from = posterior) %>%
+ rowwise() %>%
+ mutate(Unambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[1],
+ Ambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[2]) %>%
+ select(age_group, agreement, prediction = Ambiguous_soft)
+
+ # compute loss as squared error
+ loss = df.data %>%
+ filter(age_group %in% age) %>%
+ left_join(df.prediction,
+ by = c("age_group", "agreement")) %>%
+ mutate(loss = (p-prediction)^2) %>%
+ pull(loss) %>%
+ sum()
+
+ return(loss)
+}
+
+# find best fitting softmax parameter
+fit = optim(par = c(0, 0),
+ fn = fit_softmax_age)
+
+df.model.softmax.linear = df.inference %>%
+ select(age_group, utterance, agreement, posterior) %>%
+ pivot_wider(names_from = utterance,
+ values_from = posterior) %>%
+ mutate(beta = fit$par[1] + fit$par[2] * age_group) %>%
+ rowwise() %>%
+ mutate(Unambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[1],
+ Ambiguous_soft = softmax(c(Unambiguous, Ambiguous),
+ temp = beta)[2]) %>%
+ select(age_group, condition = agreement, posterior = Ambiguous_soft) %>%
+ mutate(condition = factor(condition,
+ levels = c("agree", "disagree"),
+ labels = c("Agreement Trials", "Disagreement Trials")))
+
+
+
+
+
+4.2.4.5 Model comparison
+df.model.posterior %>%
+ mutate(name = "posterior") %>%
+ select(-c(utterance, probability, prior)) %>%
+ bind_rows(df.model.softmax %>%
+ mutate(name = "softmax")) %>%
+ bind_rows(df.model.softmax.linear %>%
+ mutate(name = "softmax increase")) %>%
+ pivot_wider(names_from = name,
+ values_from = posterior) %>%
+ left_join(df.data %>%
+ mutate(condition = factor(agreement,
+ levels = c("agree", "disagree"),
+ labels = c("Agreement Trials",
+ "Disagreement Trials"))) %>%
+ select(-agreement),
+ by = c("age_group", "condition")) %>%
+ summarize(
+ r_posterior = cor(p, posterior),
+ r_softmax = cor(p, softmax),
+ r_softmaxincrease = cor(p, `softmax increase`),
+ rmse_posterior = rmse(p, posterior),
+ rmse_softmax = rmse(p, softmax),
+ rmse_softmaxincrease = rmse(p, `softmax increase`)) %>%
+ pivot_longer(cols = everything(),
+ names_to = c("index", "name"),
+ names_sep = "_") %>%
+ pivot_wider(names_from = index,
+ values_from = value) %>%
+ print_table()| +name + | ++r + | ++rmse + | +
|---|---|---|
| +posterior + | ++0.96 + | ++0.21 + | +
| +softmax + | ++0.96 + | ++0.17 + | +
| +softmaxincrease + | ++0.98 + | ++0.15 + | +
+
+4.3 PLOTS
+
+
+
+4.3.1 Prediction
+set.seed(1)
+
+df.plot.individual = df.exp2.predict %>%
+ mutate(condition_amb = as.character(condition_amb)) %>%
+ group_by(participant, age_continuous, condition_amb) %>%
+ summarize(pct_dis = sum(dis_yes)/n())
+
+df.age.means = df.plot.individual %>%
+ distinct(participant, age_continuous) %>%
+ mutate(age_group = floor(age_continuous)) %>%
+ group_by(age_group) %>%
+ summarize(age_mean = mean(age_continuous),
+ n = str_c("n = ", n())) %>%
+ ungroup()
+
+df.plot.means = df.exp2.predict %>%
+ mutate(condition_amb = as.character(condition_amb)) %>%
+ group_by(participant, age_group, condition_amb) %>%
+ summarize(pct_dis = sum(dis_yes)/n()) %>%
+ group_by(age_group, condition_amb) %>%
+ reframe(response = smean.cl.boot(pct_dis),
+ name = c("mean", "low", "high")) %>%
+ left_join(df.age.means,
+ by = "age_group") %>%
+ pivot_wider(names_from = name,
+ values_from = response) %>%
+ mutate(age_mean = ifelse(condition_amb == 0, age_mean - 0.05, age_mean + 0.05))
+
+df.plot.text = df.plot.means %>%
+ distinct(age_group, n)
+
+
+ggplot() +
+ geom_hline(yintercept = 0.5,
+ linetype = 2,
+ alpha = 0.1) +
+ geom_point(data = df.plot.individual,
+ mapping = aes(x = age_continuous,
+ y = pct_dis,
+ color = condition_amb),
+ alpha = 0.5,
+ show.legend = T,
+ shape = 16,
+ size = 1.5) +
+ geom_linerange(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ ymin = low,
+ ymax = high),
+ color = "gray40") +
+ geom_point(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ fill = condition_amb),
+ shape = 21,
+ size = 3,
+ show.legend = T) +
+ geom_text(data = df.plot.text,
+ mapping = aes(x = age_group + 0.5,
+ y = 1.05,
+ label = n),
+ hjust = 0.5) +
+ scale_y_continuous(labels = percent) +
+ labs(x = "Age (in years)",
+ y = "% Predict Disagreement",
+ title = "Experiment 2: Prediction") +
+ coord_cartesian(xlim = c(7, 12),
+ ylim = c(0, 1),
+ clip = "off") +
+ scale_color_manual(name = "Trial Type",
+ labels = c("Unambiguous", "Ambiguous"),
+ values = c(l.color$unambiguous, l.color$ambiguous),
+ guide = guide_legend(reverse = T)) +
+ scale_fill_manual(name = "Trial Type",
+ labels = c("Unambiguous", "Ambiguous"),
+ values = c(l.color$unambiguous, l.color$ambiguous),
+ guide = guide_legend(reverse = T)) +
+ theme(plot.title = element_text(hjust = 0.5,
+ vjust = 2,
+ size = 18,
+ face = "bold"),
+ axis.title.y = element_markdown(color = l.color$disagreement),
+ legend.position = "right")
+
+ggsave(filename = "../figures/plots/exp2_prediction.pdf",
+ width = 8,
+ height = 4)
+
+
+4.3.2 Inference
+set.seed(1)
+
+df.plot.individual = df.exp2.infer %>%
+ mutate(condition_disagree = as.character(condition_disagree)) %>%
+ group_by(participant, age_continuous, condition_disagree) %>%
+ summarize(pct_amb = sum(ambiguous_yes)/n())
+
+df.age.means = df.plot.individual %>%
+ distinct(participant, age_continuous) %>%
+ mutate(age_group = floor(age_continuous)) %>%
+ group_by(age_group) %>%
+ summarize(age_mean = mean(age_continuous),
+ n = str_c("n = ", n())) %>%
+ ungroup()
+
+df.plot.means = df.exp2.infer %>%
+ mutate(condition_disagree = as.character(condition_disagree)) %>%
+ group_by(participant, age_group, condition_disagree) %>%
+ summarize(pct_amb = sum(ambiguous_yes)/n()) %>%
+ group_by(age_group, condition_disagree) %>%
+ reframe(response = smean.cl.boot(pct_amb),
+ name = c("mean", "low", "high")) %>%
+ left_join(df.age.means,
+ by = "age_group") %>%
+ pivot_wider(names_from = name,
+ values_from = response) %>%
+ mutate(age_mean = ifelse(condition_disagree == 0, age_mean - 0.05, age_mean + 0.05))
+
+df.plot.text = df.plot.means %>%
+ distinct(age_group, n)
+
+df.model = df.model.posterior %>%
+ mutate(name = "posterior") %>%
+ select(-c(utterance, probability, prior)) %>%
+ bind_rows(df.model.softmax %>%
+ mutate(name = "softmax")) %>%
+ bind_rows(df.model.softmax.linear %>%
+ mutate(name = "softmax increase")) %>%
+ mutate(condition_disagree = factor(condition,
+ levels = c("Agreement Trials",
+ "Disagreement Trials"),
+ labels = c(0,
+ 1))) %>%
+ left_join(df.age.means %>%
+ select(-n),
+ by = "age_group") %>%
+ mutate(age_mean = ifelse(condition_disagree == 0,
+ age_mean - 0.05,
+ age_mean + 0.05))
+
+ggplot() +
+ geom_hline(yintercept = 0.5,
+ linetype = 2,
+ alpha = 0.1) +
+ geom_point(data = df.plot.individual,
+ mapping = aes(x = age_continuous,
+ y = pct_amb,
+ color = condition_disagree),
+ alpha = 0.5,
+ show.legend = T,
+ shape = 16,
+ size = 1.5) +
+ geom_linerange(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ ymin = low,
+ ymax = high),
+ color = "gray40",
+ show.legend = F) +
+ geom_point(data = df.plot.means,
+ mapping = aes(x = age_mean,
+ y = mean,
+ fill = condition_disagree),
+ shape = 21,
+ size = 3,
+ show.legend = F) +
+ geom_point(data = df.model,
+ mapping = aes(x = age_mean,
+ y = posterior,
+ shape = name,
+ fill = condition_disagree),
+ size = 1.5,
+ alpha = 0.5,
+ show.legend = T) +
+ geom_text(data = df.plot.text,
+ mapping = aes(x = age_group + 0.5,
+ y = 1.05,
+ label = n),
+ hjust = 0.5) +
+ scale_y_continuous(labels = percent) +
+ labs(x = "Age (in years)",
+ y = "% Infer Ambiguous Utterance",
+ title = "Experiment 2: Inference") +
+ coord_cartesian(xlim = c(7, 12),
+ ylim = c(0, 1),
+ clip = "off") +
+ scale_color_manual(name = "Trial Type",
+ labels = c("Agreement", "Disagreement"),
+ values = c(l.color$agreement, l.color$disagreement)) +
+ scale_fill_manual(name = "Trial Type",
+ labels = c("Agreement", "Disagreement"),
+ values = c(l.color$agreement, l.color$disagreement)) +
+ scale_shape_manual(name = "Model",
+ labels = c("posterior", "softmax", "softmax increase"),
+ values = c(21, 22, 23)) +
+ theme(plot.title = element_text(hjust = 0.5,
+ vjust = 2,
+ size = 18,
+ face = "bold"),
+ axis.title.y = element_markdown(color = l.color$ambiguous),
+ legend.position = "right") +
+ guides(fill = guide_legend(override.aes = list(shape = 21,
+ size = 3,
+ alpha = 1),
+ reverse = T,
+ order = 1),
+ shape = guide_legend(override.aes = list(fill = "white",
+ alpha = 1)),
+ color = "none")
+
+ggsave(filename = "../figures/plots/exp2_inference.pdf",
+ width = 8,
+ height = 4)
+
+
+
+
+5 Session info
+ +We used R version 4.3.2 (R Core Team 2023) and the following R packages: bookdown v. 0.37 (Xie 2016, 2023a), broom.mixed v. 0.2.9.4 (Bolker and Robinson 2022), car v. 3.1.2 (Fox and Weisberg 2019), ggeffects v. 1.3.4 (Lüdecke 2018), ggtext v. 0.1.2 (Wilke and Wiernik 2022), Hmisc v. 5.1.1 (Harrell Jr 2023), kableExtra v. 1.3.4 (Zhu 2021), knitr v. 1.45 (Xie 2014, 2015, 2023b), lme4 v. 1.1.35.1 (Bates et al. 2015), Metrics v. 0.1.4 (Hamner and Frasco 2018), rmarkdown v. 2.25 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2023), rsample v. 1.2.0 (Frick et al. 2023), scales v. 1.3.0 (Wickham, Pedersen, and Seidel 2023), tidyverse v. 2.0.0 (Wickham et al. 2019), xtable v. 1.8.4 (Dahl et al. 2019).
+ +R version 4.3.2 (2023-10-31)
+Platform: aarch64-apple-darwin20 (64-bit)
+Running under: macOS Sonoma 14.4.1
+
+Matrix products: default
+BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
+LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
+
+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/Los_Angeles
+tzcode source: internal
+
+attached base packages:
+[1] stats graphics grDevices utils datasets methods base
+
+other attached packages:
+ [1] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
+ [4] dplyr_1.1.4 purrr_1.0.2 readr_2.1.4
+ [7] tidyr_1.3.0 tibble_3.2.1 ggplot2_3.4.4
+[10] tidyverse_2.0.0 ggtext_0.1.2 Hmisc_5.1-1
+[13] ggeffects_1.3.4 grateful_0.2.4 broom.mixed_0.2.9.4
+[16] scales_1.3.0 Metrics_0.1.4 car_3.1-2
+[19] carData_3.0-5 knitr_1.45 kableExtra_1.3.4
+[22] xtable_1.8-4 rsample_1.2.0 lme4_1.1-35.1
+[25] Matrix_1.6-4
+
+loaded via a namespace (and not attached):
+ [1] gridExtra_2.3 rlang_1.1.3 magrittr_2.0.3 furrr_0.3.1
+ [5] compiler_4.3.2 systemfonts_1.0.5 vctrs_0.6.5 rvest_1.0.3
+ [9] pkgconfig_2.0.3 crayon_1.5.2 fastmap_1.1.1 backports_1.4.1
+[13] labeling_0.4.3 utf8_1.2.4 rmarkdown_2.25 markdown_1.12
+[17] tzdb_0.4.0 nloptr_2.0.3 ragg_1.2.7 bit_4.0.5
+[21] xfun_0.41 cachem_1.0.8 jsonlite_1.8.8 highr_0.10
+[25] broom_1.0.5 parallel_4.3.2 cluster_2.1.6 R6_2.5.1
+[29] bslib_0.6.1 stringi_1.8.3 parallelly_1.37.0 boot_1.3-28.1
+[33] rpart_4.1.23 jquerylib_0.1.4 Rcpp_1.0.12 bookdown_0.37
+[37] base64enc_0.1-3 splines_4.3.2 nnet_7.3-19 timechange_0.2.0
+[41] tidyselect_1.2.0 rstudioapi_0.15.0 abind_1.4-5 yaml_2.3.8
+[45] codetools_0.2-19 listenv_0.9.1 lattice_0.22-5 withr_3.0.0
+[49] evaluate_0.23 foreign_0.8-86 future_1.33.1 xml2_1.3.6
+[53] pillar_1.9.0 renv_1.0.3 checkmate_2.3.1 generics_0.1.3
+[57] vroom_1.6.5 hms_1.1.3 commonmark_1.9.0 munsell_0.5.0
+[61] minqa_1.2.6 globals_0.16.2 glue_1.7.0 tools_4.3.2
+[65] data.table_1.14.10 webshot_0.5.5 grid_4.3.2 colorspace_2.1-0
+[69] nlme_3.1-164 htmlTable_2.4.2 Formula_1.2-5 cli_3.6.2
+[73] textshaping_0.3.7 fansi_1.0.6 viridisLite_0.4.2 svglite_2.1.3
+[77] gtable_0.3.4 sass_0.4.8 digest_0.6.34 htmlwidgets_1.6.4
+[81] farver_2.1.1 htmltools_0.5.7 lifecycle_1.0.4 httr_1.4.7
+[85] gridtext_0.1.5 bit64_4.0.5 MASS_7.3-60
+
+
+Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2023. rmarkdown: Dynamic Documents for r. https://github.com/rstudio/rmarkdown.
+
+
+Bates, Douglas, Martin Mächler, Ben Bolker, and Steve Walker. 2015. “Fitting Linear Mixed-Effects Models Using lme4.” Journal of Statistical Software 67 (1): 1–48. https://doi.org/10.18637/jss.v067.i01.
+
+
+Bolker, Ben, and David Robinson. 2022. broom.mixed: Tidying Methods for Mixed Models. https://CRAN.R-project.org/package=broom.mixed.
+
+
+Dahl, David B., David Scott, Charles Roosen, Arni Magnusson, and Jonathan Swinton. 2019. xtable: Export Tables to LaTeX or HTML. https://CRAN.R-project.org/package=xtable.
+
+
+Fox, John, and Sanford Weisberg. 2019. An R Companion to Applied Regression. Third. Thousand Oaks CA: Sage. https://socialsciences.mcmaster.ca/jfox/Books/Companion/.
+
+
+Frick, Hannah, Fanny Chow, Max Kuhn, Michael Mahoney, Julia Silge, and Hadley Wickham. 2023. rsample: General Resampling Infrastructure. https://CRAN.R-project.org/package=rsample.
+
+
+Hamner, Ben, and Michael Frasco. 2018. Metrics: Evaluation Metrics for Machine Learning. https://CRAN.R-project.org/package=Metrics.
+
+
+Harrell Jr, Frank E. 2023. Hmisc: Harrell Miscellaneous. https://CRAN.R-project.org/package=Hmisc.
+
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+Lüdecke, Daniel. 2018. “ggeffects: Tidy Data Frames of Marginal Effects from Regression Models.” Journal of Open Source Software 3 (26): 772. https://doi.org/10.21105/joss.00772.
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+R Core Team. 2023. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
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+Wickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy D’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019. “Welcome to the tidyverse.” Journal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.
+
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+Wickham, Hadley, Thomas Lin Pedersen, and Dana Seidel. 2023. scales: Scale Functions for Visualization. https://CRAN.R-project.org/package=scales.
+
+
+Wilke, Claus O., and Brenton M. Wiernik. 2022. ggtext: Improved Text Rendering Support for “ggplot2”. https://CRAN.R-project.org/package=ggtext.
+
+
+Xie, Yihui. 2014. “knitr: A Comprehensive Tool for Reproducible Research in R.” In Implementing Reproducible Computational Research, edited by Victoria Stodden, Friedrich Leisch, and Roger D. Peng. Chapman; Hall/CRC.
+
+
+———. 2015. Dynamic Documents with R and Knitr. 2nd ed. Boca Raton, Florida: Chapman; Hall/CRC. https://yihui.org/knitr/.
+
+
+———. 2016. bookdown: Authoring Books and Technical Documents with R Markdown. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/bookdown.
+
+
+———. 2023a. bookdown: Authoring Books and Technical Documents with r Markdown. https://github.com/rstudio/bookdown.
+
+
+———. 2023b. knitr: A General-Purpose Package for Dynamic Report Generation in r. https://yihui.org/knitr/.
+
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+Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018. R Markdown: The Definitive Guide. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown.
+
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+Xie, Yihui, Christophe Dervieux, and Emily Riederer. 2020. R Markdown Cookbook. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown-cookbook.
+
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+Zhu, Hao. 2021. kableExtra: Construct Complex Table with “kable” and Pipe Syntax. https://CRAN.R-project.org/package=kableExtra.
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+