Analysis of Experiment 2
Last updated: 2020-04-25
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Knit directory: social_immunity/
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Load data and R packages
# All but 1 of these packages can be easily installed from CRAN.
# However it was harder to install the showtext package. On Mac, I did this:
# installed 'homebrew' using Terminal: ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"
# installed 'libpng' using Terminal: brew install libpng
# installed 'showtext' in R using: devtools::install_github("yixuan/showtext")
library(showtext)
library(brms)
library(bayesplot)
library(tidyverse)
library(gridExtra)
library(kableExtra)
library(bayestestR)
library(tidybayes)
library(cowplot)
source("code/helper_functions.R")
# set up nice font for figure
nice_font <- "PT Serif"
font_add_google(name = nice_font, family = nice_font, regular.wt = 400, bold.wt = 700)
showtext_auto()
exp2_treatments <- c("Ringers", "LPS")
durations <- read_csv("data/data_collection_sheets/experiment_durations.csv") %>%
filter(experiment == 2) %>% select(-experiment)
outcome_tally <- read_csv(file = "data/clean_data/experiment_2_outcome_tally.csv") %>%
mutate(
outcome = str_replace_all(outcome, "Stayed inside the hive", "Stayed inside"),
outcome = str_replace_all(outcome, "Left of own volition", "Left voluntarily"),
outcome = factor(outcome, levels = c("Stayed inside", "Left voluntarily", "Forced out")),
treatment = str_replace_all(treatment, "Ringer CHC", "Ringers"),
treatment = str_replace_all(treatment, "LPS CHC", "LPS"),
treatment = factor(treatment, levels = exp2_treatments))
# Re-formatted version of the same data, where each row is an individual bee. We need this format to run the brms model.
data_for_categorical_model <- outcome_tally %>%
mutate(id = 1:n()) %>%
split(.$id) %>%
map(function(x){
if(x$n[1] == 0) return(NULL)
data.frame(
treatment = x$treatment[1],
hive = x$hive[1],
colour = x$colour[1],
outcome = rep(x$outcome[1], x$n))
}) %>% do.call("rbind", .) %>% as_tibble() %>%
arrange(hive, treatment) %>%
mutate(outcome_numeric = as.numeric(outcome),
hive = as.character(hive),
treatment = factor(treatment, levels = exp2_treatments)) %>%
left_join(durations, by = "hive") %>%
mutate(hive = C(factor(hive), sum)) # use "sum coding" for hive, since there is no obvious reference levelInspect the raw data
Sample sizes by treatment
sample_sizes <- data_for_categorical_model %>%
group_by(treatment) %>%
summarise(n = n())
sample_sizes %>%
kable() %>% kable_styling(full_width = FALSE)| treatment | n |
|---|---|
| Ringers | 294 |
| LPS | 291 |
Sample sizes by treatment and hive
data_for_categorical_model %>%
group_by(hive, treatment) %>%
summarise(n = n()) %>%
kable() %>% kable_styling(full_width = FALSE)| hive | treatment | n |
|---|---|---|
| Arts | Ringers | 70 |
| Arts | LPS | 68 |
| Garden | Ringers | 75 |
| Garden | LPS | 75 |
| Skylab | Ringers | 99 |
| Skylab | LPS | 100 |
| Zoology | Ringers | 50 |
| Zoology | LPS | 48 |
Inspect the results
outcome_tally %>%
select(-colour) %>%
kable(digits = 3) %>% kable_styling(full_width = FALSE) %>%
scroll_box(height = "380px")| hive | treatment | outcome | n |
|---|---|---|---|
| Arts | LPS | Stayed inside | 56 |
| Arts | LPS | Left voluntarily | 5 |
| Arts | LPS | Forced out | 7 |
| Arts | Ringers | Stayed inside | 64 |
| Arts | Ringers | Left voluntarily | 5 |
| Arts | Ringers | Forced out | 1 |
| Garden | LPS | Stayed inside | 70 |
| Garden | LPS | Left voluntarily | 2 |
| Garden | LPS | Forced out | 3 |
| Garden | Ringers | Stayed inside | 73 |
| Garden | Ringers | Left voluntarily | 2 |
| Garden | Ringers | Forced out | 0 |
| Skylab | LPS | Stayed inside | 93 |
| Skylab | LPS | Left voluntarily | 2 |
| Skylab | LPS | Forced out | 5 |
| Skylab | Ringers | Stayed inside | 97 |
| Skylab | Ringers | Left voluntarily | 1 |
| Skylab | Ringers | Forced out | 1 |
| Zoology | LPS | Stayed inside | 38 |
| Zoology | LPS | Left voluntarily | 4 |
| Zoology | LPS | Forced out | 6 |
| Zoology | Ringers | Stayed inside | 42 |
| Zoology | Ringers | Left voluntarily | 2 |
| Zoology | Ringers | Forced out | 6 |
Raw results as a bar chart
pd <- position_dodge(.3)
outcome_tally %>%
group_by(treatment, outcome) %>%
summarise(n = sum(n)) %>% mutate() %>%
group_by(treatment) %>%
mutate(total_n = sum(n),
percent = 100 * n / sum(n),
SE = sqrt(total_n * (percent/100) * (1-(percent/100)))) %>%
ungroup() %>%
mutate(lowerCI = map_dbl(1:n(), ~ 100 * binom.test(n[.x], total_n[.x])$conf.int[1]),
upperCI = map_dbl(1:n(), ~ 100 * binom.test(n[.x], total_n[.x])$conf.int[2])) %>%
filter(outcome != "Stayed inside") %>%
ggplot(aes(treatment, percent, fill = outcome)) +
geom_errorbar(aes(ymin=lowerCI, ymax=upperCI), position = pd, width = 0) +
geom_point(stat = "identity", position = pd, colour = "grey15", pch = 21, size = 4) +
scale_fill_brewer(palette = "Pastel1", name = "Outcome", direction = -1) +
xlab("Treatment") + ylab("% bees (+95% CIs)") +
theme(legend.position = "top") +
coord_flip()
| Version | Author | Date |
|---|---|---|
| 1ce9e19 | lukeholman | 2020-04-21 |
Multinomial model of outcome
Run the models
Fit a multinomial logisitic model, with 3 possible outcomes describing what happened to each bee introduced to the hive: stayed inside, left of its own volition, or forced out by the other workers. To assess the effects of our predictor variables, we compare 5 models with different fixed factors, ranking them by posterior model probability.
if(!file.exists("output/exp2_model.rds")){
exp2_model_v1 <- brm(
outcome_numeric ~ treatment * hive + observation_time_minutes,
data = data_for_categorical_model,
prior = c(set_prior("normal(0, 3)", class = "b", dpar = "mu2"),
set_prior("normal(0, 3)", class = "b", dpar = "mu3")),
family = "categorical", save_all_pars = TRUE, sample_prior = TRUE,
chains = 4, cores = 1, iter = 5000, seed = 1)
exp2_model_v2 <- brm(
outcome_numeric ~ treatment + hive + observation_time_minutes,
data = data_for_categorical_model,
prior = c(set_prior("normal(0, 3)", class = "b", dpar = "mu2"),
set_prior("normal(0, 3)", class = "b", dpar = "mu3")),
family = "categorical", save_all_pars = TRUE, sample_prior = TRUE,
chains = 4, cores = 1, iter = 5000, seed = 1)
exp2_model_v3 <- brm(
outcome_numeric ~ hive + observation_time_minutes,
data = data_for_categorical_model,
prior = c(set_prior("normal(0, 3)", class = "b", dpar = "mu2"),
set_prior("normal(0, 3)", class = "b", dpar = "mu3")),
family = "categorical", save_all_pars = TRUE, sample_prior = TRUE,
chains = 4, cores = 1, iter = 5000, seed = 1)
posterior_model_probabilities <- tibble(
Model = c("treatment * hive + observation_time_minutes",
"treatment + hive + observation_time_minutes",
"hive + observation_time_minutes"),
post_prob = as.numeric(post_prob(exp2_model_v1,
exp2_model_v2,
exp2_model_v3))) %>%
arrange(-post_prob)
saveRDS(exp2_model_v2, "output/exp2_model.rds") # save the top model, treatment + hive
saveRDS(posterior_model_probabilities, "output/exp2_post_prob.rds")
}
exp2_model <- readRDS("output/exp2_model.rds")
posterior_model_probabilities <- readRDS("output/exp2_post_prob.rds")Posterior model probabilites
posterior_model_probabilities %>%
kable(digits = 3) %>% kable_styling()| Model | post_prob |
|---|---|
| hive + observation_time_minutes | 0.530 |
| treatment + hive + observation_time_minutes | 0.466 |
| treatment * hive + observation_time_minutes | 0.003 |
Fixed effects from the top model
The code chunk below wrangles the raw output of the summary() functions for brms models into a more readable table of results, and also adds ‘Bayesian p-values’ (i.e. the posterior probability that the true effect size has the same sign as the reported effect).
tableS3 <- get_fixed_effects_with_p_values(exp2_model) %>%
mutate(mu = map_chr(str_extract_all(Parameter, "mu[:digit:]"), ~ .x[1]),
Parameter = str_remove_all(Parameter, "mu[:digit:]_"),
Parameter = str_replace_all(Parameter, "treatment", "Treatment: "),
Parameter = str_replace_all(Parameter, "observation_time_minutes", "Observation duration (minutes)")) %>%
arrange(mu) %>%
select(-mu, -Rhat, -Bulk_ESS, -Tail_ESS)
names(tableS3)[3:5] <- c("Est. Error", "Lower 95% CI", "Upper 95% CI")
saveRDS(tableS3, file = "figures/tableS3.rds")
tableS3 %>%
kable(digits = 3) %>%
kable_styling(full_width = FALSE) %>%
pack_rows("% bees leaving voluntarily", 1, 6) %>%
pack_rows("% bees forced out", 7, 12)| Parameter | Estimate | Est. Error | Lower 95% CI | Upper 95% CI | p | |
|---|---|---|---|---|---|---|
| % bees leaving voluntarily | ||||||
| Intercept | -6.368 | 6.669 | -19.689 | 6.679 | 0.168 | |
| Treatment: LPS | 0.377 | 0.439 | -0.480 | 1.245 | 0.197 | |
| hive1 | 0.111 | 1.522 | -2.868 | 3.075 | 0.466 | |
| hive2 | -0.183 | 0.679 | -1.513 | 1.160 | 0.390 | |
| hive3 | 0.082 | 2.531 | -4.835 | 5.113 | 0.491 | |
| Observation duration (minutes) | 0.029 | 0.068 | -0.103 | 0.165 | 0.337 | |
| % bees forced out | ||||||
| Intercept | -5.179 | 6.708 | -18.321 | 7.894 | 0.222 | |
| Treatment: LPS | 1.105 | 0.433 | 0.291 | 1.989 | 0.004 | ** |
| hive1 | -0.030 | 1.544 | -3.059 | 2.971 | 0.490 | |
| hive2 | -0.847 | 0.705 | -2.266 | 0.490 | 0.112 | |
| hive3 | 0.049 | 2.554 | -4.892 | 5.089 | 0.493 | |
| Observation duration (minutes) | 0.015 | 0.069 | -0.118 | 0.149 | 0.414 | |
Table S3: Table summarising the posterior estimates of each fixed effect in the best-fitting model of Experiment 2. This was a multinomial model with three possible outcomes (stay inside, leave voluntarily, be forced out), and so there are two parameter estimates for the intercept and for each predictor in the model. ‘Treatment’ is a fixed factor with two levels, and the effect of LPS shown here is expressed relative to the ‘Ringers’ treatment. ‘Hive’ was a fixed factor with four levels; unlike for treatment, we modelled hive using deviation coding, such that the intercept term represents the mean across all hives (in the Ringers treatment), and the three hive terms represent the deviation from this mean for three of the four hives. Lastly, observation duration was a continuous variable expressed to the nearest minute. The \(p\) column gives the posterior probability that the true effect size is opposite in sign to what is reported in the Estimate column, similarly to a \(p\)-value.
Plotting estimates from the model
Derive prediction from the posterior
get_posterior_preds <- function(hive){
new <- expand.grid(treatment = levels(data_for_categorical_model$treatment),
hive = "Zoology",
observation_time_minutes = 120)
preds <- fitted(exp2_model, newdata = new, summary = FALSE)
dimnames(preds) <- list(NULL, paste(new$treatment, new$hive, sep = "~"), NULL)
rbind(
as.data.frame(preds[,, 1]) %>% mutate(outcome = "Stayed inside", posterior_sample = 1:n()),
as.data.frame(preds[,, 2]) %>% mutate(outcome = "Left voluntarily", posterior_sample = 1:n()),
as.data.frame(preds[,, 3]) %>% mutate(outcome = "Forced out", posterior_sample = 1:n())) %>%
gather(treatment, prop, contains("~")) %>%
mutate(treatment = strsplit(treatment, split = "~"),
hive = map_chr(treatment, ~ .x[2]),
treatment = map_chr(treatment, ~ .x[1]),
treatment = factor(treatment, c("Ringers", "LPS")),
outcome = factor(outcome, c("Stayed inside", "Left voluntarily", "Forced out"))) %>%
arrange(treatment, outcome) %>% as_tibble() %>% select(-hive)
}
# plotting data for panel A: one specific hive
plotting_data <- get_posterior_preds(hive = "Zoology")
# stats data: for comparing means across all hives
stats_data <- get_posterior_preds(hive = NA)Make Figure 2
cols <- c("#34558b", "#4ec5a5", "#ffaf12")
dot_plot <- plotting_data %>%
left_join(sample_sizes, by = "treatment") %>%
arrange(treatment) %>%
mutate(treatment = factor(paste(treatment, "\n(n = ", n, ")", sep = ""),
unique(paste(treatment, "\n(n = ", n, ")", sep = "")))) %>%
ggplot(aes(100 * prop, treatment)) +
stat_dotsh(quantiles = 100, fill = "grey40", colour = "grey40") +
stat_pointintervalh(aes(colour = outcome, fill = outcome),
.width = c(0.5, 0.95),
position = position_nudge(y = -0.07), point_colour = "grey26", pch = 21, stroke = 0.4) +
scale_colour_manual(values = cols) +
scale_fill_manual(values = cols) +
facet_wrap( ~ outcome, scales = "free_x") +
xlab("% bees (posterior estimate)") + ylab("Treatment") +
theme_bw() +
coord_cartesian(ylim=c(1.4, 2.2)) +
theme(
text = element_text(family = nice_font),
strip.background = element_rect(fill = "#eff0f1"),
panel.grid.major.y = element_blank(),
legend.position = "none"
)
get_log_odds <- function(trt1, trt2){ # positive effect = odds of this outcome are higher for trt2 than trt1 (put control as trt1)
log((trt2 / (1 - trt2) / (trt1 / (1 - trt1))))
}
LOR <- plotting_data %>%
spread(treatment, prop) %>%
mutate(LOR = get_log_odds(Ringers, LPS)) %>%
select(posterior_sample, outcome, LOR)
LOR_plot <- LOR %>%
ggplot(aes(LOR, outcome, colour = outcome)) +
geom_vline(xintercept = 0, linetype = 2) +
geom_vline(xintercept = log(2), linetype = 2, colour = "pink") +
geom_vline(xintercept = -log(2), linetype = 2, colour = "pink") +
stat_pointintervalh(aes(colour = outcome, fill = outcome),
position = position_dodge(0.4),
.width = c(0.5, 0.95),
point_colour = "grey26", pch = 21, stroke = 0.4) +
scale_colour_manual(values = cols) +
scale_fill_manual(values = cols) +
xlab("Effect size of LPS\n(log odds ratio)") + ylab("Mode of exit") +
theme_bw() +
theme(
text = element_text(family = nice_font),
panel.grid.major.y = element_blank(),
legend.position = "none"
)
diff_in_forced_out_plot <- plotting_data %>%
spread(outcome, prop) %>%
mutate(prop_leavers_that_were_forced_out = `Forced out` / (`Forced out` + `Left voluntarily`)) %>%
select(posterior_sample, treatment, prop_leavers_that_were_forced_out) %>%
spread(treatment, prop_leavers_that_were_forced_out) %>%
mutate(difference_prop_forced_out_LOR = get_log_odds(Ringers, LPS)) %>%
ggplot(aes(difference_prop_forced_out_LOR, y =1)) +
geom_vline(xintercept = 0, linetype = 2) +
stat_dotsh(quantiles = 100, fill = "grey40", colour = "grey40") +
xlab("Effect of LPS on proportion\nbees leaving by force\n(log odds ratio)") +
ylab("Posterior density") +
theme_bw() +
theme(
text = element_text(family = nice_font),
panel.grid.major.y = element_blank(),
legend.position = "none"
)
bottom_row <- cowplot::plot_grid(LOR_plot, diff_in_forced_out_plot,
labels = c("B", "C"),
nrow = 1, align = 'hv', axis = 'l', rel_heights = c(1.6, 1))
top_row <- cowplot::plot_grid(dot_plot, labels = "A")
p <- cowplot::plot_grid(top_row, bottom_row,
nrow = 2, align = 'v', axis = 'l', rel_heights = c(1.4, 1))
ggsave(plot = p, filename = "figures/fig2.pdf", height = 6, width = 6)
p
| Version | Author | Date |
|---|---|---|
| 1ce9e19 | lukeholman | 2020-04-21 |
The dashed lines mark \(LOR = 0\), indicating no effect, and \(LOR = \pm log(2)\), i.e. the point at which the odds are twice as high in one treatment as the other.
Hypothesis testing and effect sizes
This section calculates the posterior difference in treatment group means, in order to perform some null hypothesis testing, calculate effect size, and calculate the 95% credible intervals on the effect size. In all cases, the effect size expresses the effect of the “LPS-treated bee CHCs” treatment relative to the “Ringer-treated bee CHCs” control.
Stats associated with Figure 2B
Making Table S4
my_summary <- function(df, columns, outcome) {
lapply(columns, function(x){
p <- 1 - (df %>% pull(!! x) %>%
bayestestR::p_direction() %>% as.numeric())
df %>% pull(!! x) %>% posterior_summary() %>% as_tibble() %>%
mutate(p=p, Outcome = outcome, Metric = x) %>%
select(Outcome, Metric, everything())
}) %>% do.call("rbind", .)
}
stats_table <- rbind(
plotting_data %>%
filter(outcome == "Stayed inside") %>%
spread(treatment, prop) %>%
mutate(`Absolute difference in % bees staying inside` = 100 * (LPS - Ringers),
`Log odds ratio` = get_log_odds(Ringers, LPS)) %>%
my_summary(c("Absolute difference in % bees staying inside",
"Log odds ratio"),
outcome = "Stayed inside") %>%
mutate(p = c(" ", format(round(p[2], 4), nsmall = 4))),
plotting_data %>%
filter(outcome == "Left voluntarily") %>%
spread(treatment, prop) %>%
mutate(`Absolute difference in % bees leaving voluntarily` = 100 * (LPS - Ringers),
`Log odds ratio` = get_log_odds(Ringers, LPS)) %>%
my_summary(c("Absolute difference in % bees leaving voluntarily",
"Log odds ratio"),
outcome = "Left voluntarily") %>%
mutate(p = c(" ", format(round(p[2], 4), nsmall = 4))),
plotting_data %>%
filter(outcome == "Forced out") %>%
spread(treatment, prop) %>%
mutate(`Absolute difference in % bees forced out` = 100 * (LPS - Ringers),
`Log odds ratio` = get_log_odds(Ringers, LPS)) %>%
my_summary(c("Absolute difference in % bees forced out",
"Log odds ratio"),
outcome = "Forced out") %>%
mutate(p = c(" ", format(round(p[2], 4), nsmall = 4)))
) %>%
mutate(` ` = ifelse(p < 0.05, "\\*", ""),
` ` = replace(` `, p < 0.01, "**"),
` ` = replace(` `, p < 0.001, "***"),
` ` = replace(` `, p == " ", ""))
stats_table[c(2,4,6), 1] <- " "
stats_table %>%
select(-Outcome) %>%
kable(digits = 3) %>% kable_styling(full_width = FALSE) %>%
row_spec(c(0,2,4,6), extra_css = "border-bottom: solid;") %>%
pack_rows("% bees staying inside", 1, 2) %>%
pack_rows("% bees leaving voluntarily", 3, 4) %>%
pack_rows("% bees forced out", 5, 6)| Metric | Estimate | Est.Error | Q2.5 | Q97.5 | p | |
|---|---|---|---|---|---|---|
| % bees staying inside | ||||||
| Absolute difference in % bees staying inside | -11.949 | 4.942 | -22.208 | -2.753 | ||
| Log odds ratio | -0.836 | 0.331 | -1.520 | -0.199 | 0.0048 | ** |
| % bees leaving voluntarily | ||||||
| Absolute difference in % bees leaving voluntarily | 1.335 | 2.623 | -3.720 | 7.086 | ||
| Log odds ratio | 0.242 | 0.440 | -0.621 | 1.106 | 0.2881 | |
| % bees forced out | ||||||
| Absolute difference in % bees forced out | 10.614 | 4.627 | 2.538 | 20.464 | ||
| Log odds ratio | 1.079 | 0.433 | 0.270 | 1.967 | 0.0047 | ** |
Table S4: This table gives statistics associated with each of the contrasts plotted in Figure 2B. Each pair of rows gives the absolute and standardised effect size (as log odds ratio; LOR) for the LPS treatment, relative to the Ringers treatment, for one of the three possible outcomes (stayed inside, left voluntarily, or forced out). A LOR of \(|log(x)|\) indicates that the outcome is \(x\) times more frequent in one treatment compared to the other, e.g. \(log(2) = 0.69\) indicates a two-fold difference in frequency. The \(p\) column gives the posterior probability that the true effect size has the same sign as is shown in the Estimate column; this metric has a similar interpretation to a one-tailed \(p\) value in frequentist statistics.
Evaluating evidence for the null hypothesis
Here, we derive the result present in prose in the Results, that the true effect size
LOR_left <- LOR %>%
filter(outcome == "Left voluntarily") %>% # Forced out
pull(LOR)
sum(LOR_left > -log(1.5) & LOR_left < log(1.5)) / length(LOR_left)[1] 0.5837sum(LOR_left > -log(2) & LOR_left < log(2)) / length(LOR_left)[1] 0.8323Stats associated with Figure 2C
Here, we derive the result present in prose in the Results, that the
difference_prop_forced_out_LOR <- plotting_data %>%
spread(outcome, prop) %>%
mutate(prop_leavers_that_were_forced_out = `Forced out` / (`Forced out` + `Left voluntarily`)) %>%
select(posterior_sample, treatment, prop_leavers_that_were_forced_out) %>%
spread(treatment, prop_leavers_that_were_forced_out) %>%
mutate(difference_prop_forced_out_LOR = get_log_odds(Ringers, LPS)) %>%
pull(difference_prop_forced_out_LOR)
hypothesis(difference_prop_forced_out_LOR, "x > 0")$hypothesis Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
1 (x) > 0 0.7281842 0.5984938 -0.2336917 1.712823 8.082652 0.8899
sessionInfo()R version 3.6.3 (2020-02-29)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Catalina 10.15.4
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_AU.UTF-8/en_AU.UTF-8/en_AU.UTF-8/C/en_AU.UTF-8/en_AU.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] cowplot_1.0.0 tidybayes_2.0.1 bayestestR_0.5.1 kableExtra_1.1.0
[5] gridExtra_2.3 forcats_0.5.0 stringr_1.4.0 dplyr_0.8.5
[9] purrr_0.3.3 readr_1.3.1 tidyr_1.0.2 tibble_3.0.0
[13] ggplot2_3.3.0 tidyverse_1.3.0 bayesplot_1.7.1 brms_2.12.0
[17] Rcpp_1.0.3 showtext_0.7-1 showtextdb_2.0 sysfonts_0.8
loaded via a namespace (and not attached):
[1] colorspace_1.4-1 ellipsis_0.3.0 ggridges_0.5.2
[4] rsconnect_0.8.16 rprojroot_1.3-2 markdown_1.1
[7] base64enc_0.1-3 fs_1.3.1 rstudioapi_0.11
[10] farver_2.0.3 rstan_2.19.3 svUnit_0.7-12
[13] DT_0.13 fansi_0.4.1 mvtnorm_1.1-0
[16] lubridate_1.7.8 xml2_1.3.1 bridgesampling_1.0-0
[19] knitr_1.28 shinythemes_1.1.2 jsonlite_1.6.1
[22] workflowr_1.6.0 broom_0.5.4 dbplyr_1.4.2
[25] shiny_1.4.0 compiler_3.6.3 httr_1.4.1
[28] backports_1.1.6 assertthat_0.2.1 Matrix_1.2-18
[31] fastmap_1.0.1 cli_2.0.2 later_1.0.0
[34] htmltools_0.4.0 prettyunits_1.1.1 tools_3.6.3
[37] igraph_1.2.5 coda_0.19-3 gtable_0.3.0
[40] glue_1.4.0 reshape2_1.4.4 cellranger_1.1.0
[43] vctrs_0.2.4 nlme_3.1-147 crosstalk_1.1.0.1
[46] insight_0.8.1 xfun_0.13 ps_1.3.0
[49] rvest_0.3.5 mime_0.9 miniUI_0.1.1.1
[52] lifecycle_0.2.0 gtools_3.8.2 zoo_1.8-7
[55] scales_1.1.0 colourpicker_1.0 hms_0.5.3
[58] promises_1.1.0 Brobdingnag_1.2-6 parallel_3.6.3
[61] inline_0.3.15 RColorBrewer_1.1-2 shinystan_2.5.0
[64] curl_4.3 yaml_2.2.1 loo_2.2.0
[67] StanHeaders_2.19.2 stringi_1.4.6 highr_0.8
[70] dygraphs_1.1.1.6 pkgbuild_1.0.6 rlang_0.4.5
[73] pkgconfig_2.0.3 matrixStats_0.56.0 evaluate_0.14
[76] lattice_0.20-41 labeling_0.3 rstantools_2.0.0
[79] htmlwidgets_1.5.1 processx_3.4.2 tidyselect_1.0.0
[82] plyr_1.8.6 magrittr_1.5 R6_2.4.1
[85] generics_0.0.2 DBI_1.1.0 pillar_1.4.3
[88] haven_2.2.0 whisker_0.4 withr_2.1.2
[91] xts_0.12-0 abind_1.4-5 modelr_0.1.5
[94] crayon_1.3.4 arrayhelpers_1.1-0 rmarkdown_2.1
[97] grid_3.6.3 readxl_1.3.1 callr_3.4.3
[100] git2r_0.26.1 threejs_0.3.3 reprex_0.3.0
[103] digest_0.6.25 webshot_0.5.2 xtable_1.8-4
[106] httpuv_1.5.2 stats4_3.6.3 munsell_0.5.0
[109] viridisLite_0.3.0 shinyjs_1.1