Fitness consequences of the selfish supergene Segregation Distorter

Load R packages

packages <- c("dplyr", "brms", "ggplot2", "reshape2", "Cairo", "knitr", "pander", "lazerhawk",
              "grid", "gridExtra", "ggthemes", "readr", "tibble", "stringr")
shh <- suppressMessages(lapply(packages, library, character.only = TRUE, quietly = TRUE))
nCores <- 1

summarise_brms <- function(brmsfit){
  lazerhawk::brms_SummaryTable(brmsfit, astrology = TRUE) %>%
  mutate(Covariate = str_replace_all(Covariate, "SDM", "SD-"),
         Covariate = str_replace_all(Covariate, "SD-other", "SDMother")) %>% 
  pander(split.cell = 40, split.table = Inf)
} 

Load and clean the data

This section also saves a copy of the cleaned data in the directory data/cleaned_data. If you wish to re-use the data, I suggest you use these cleaned files; the factor and level names are more intuitive than in the original data sheets (however, the actual data are identical).

load_and_clean_data <- function(path){
  
   if(path == "data/messy_data/experiment2.csv"){
   
    expt2 <- read_csv(path) %>%
      mutate(prop_SD_females = female_SD / (female_SD + female_cyo),
             prop_SD_males = male_SD / (male_SD + male_cyo),
             prop_males = (male_SD + male_cyo) / (male_SD + male_cyo + female_SD + female_cyo),
             starting_number_focal_sex = num_embryos,
             SD_offspring = female_SD,
             CyO_offspring = female_cyo) 
    
    expt2$starting_number_focal_sex[expt2$sex_of_embryos == "male"] <- 
      expt2$starting_number_focal_sex[expt2$sex_of_embryos == "male"] -
      expt2$female_SD[expt2$sex_of_embryos == "male"] - expt2$female_cyo[expt2$sex_of_embryos == "male"]
    
    expt2$starting_number_focal_sex[expt2$sex_of_embryos == "female"] <- 
      expt2$starting_number_focal_sex[expt2$sex_of_embryos == "female"] -
      expt2$male_SD[expt2$sex_of_embryos == "female"] - expt2$male_cyo[expt2$sex_of_embryos == "female"]
    
    expt2$SD_offspring[expt2$sex_of_embryos == "male"] <- expt2$male_SD[expt2$sex_of_embryos == "male"]
    expt2$CyO_offspring[expt2$sex_of_embryos == "male"] <- expt2$male_cyo[expt2$sex_of_embryos == "male"]
    
    expt2$starting_number_focal_sex[expt2$starting_number_focal_sex < expt2$SD_focal_sex] <-
      expt2$SD_focal_sex[expt2$starting_number_focal_sex < expt2$SD_focal_sex]
    
    expt2$Dead_offspring <- expt2$starting_number_focal_sex - expt2$SD_offspring - expt2$CyO_offspring
    expt2$Dead_offspring[expt2$Dead_offspring < 0] <- 0
    
    expt2$starting_number_focal_sex <- with(expt2, SD_offspring + CyO_offspring + Dead_offspring)
    
    expt2 <- expt2 %>% 
      select(block, SD_parent, sex_of_embryos, SD, starting_number_focal_sex, SD_offspring, CyO_offspring, Dead_offspring) %>%
      mutate(vial = 1:n(), 
             SD_embryos = "to be estimated",
             CyO_embryos = "to be estimated",
             block = as.numeric(as.factor(block))) %>%
      filter(!(SD %in% c("w1118", "LHm")))
    write_csv(expt2, path = str_replace(path, "messy", "clean")) 
    return(expt2)
  }
  
  # For experiment 1....
  dat <- read_csv(path) %>% 
    mutate(block = as.character(block), 
           copies_of_SD = substr(genotype, 2, 2),
           parent_with_SD = substr(genotype, 1, 1),
           parent_with_SD = replace(parent_with_SD, 
                                    parent_with_SD == "M", "Mother"),
           parent_with_SD = replace(parent_with_SD, 
                                    parent_with_SD == "P", "Father"),
           parent_with_SD = replace(parent_with_SD, 
                                    parent_with_SD == "A", "Both parents"),
           parent_with_SD = replace(parent_with_SD, 
                                    parent_with_SD == "N", "Neither parent"),
           parent_with_SD =  factor(parent_with_SD, 
                                    levels = c("Neither parent", 
                                               "Father", "Mother", 
                                               "Both parents")),
           SD = paste("SD-", SD, sep = ""),
           SD = replace(SD, SD == "SD-N", "No SD chromosome"),
  
           MotherHasSD = ifelse(parent_with_SD != "Father", 1, 0),
           FatherHasSD = ifelse(parent_with_SD != "Mother", 1, 0),
           MotherHasSD = replace(MotherHasSD, SD == "No SD chromosome", 0),
           FatherHasSD = replace(FatherHasSD, SD == "No SD chromosome", 0),
           SD_from_mum = ifelse(MotherHasSD==1 & copies_of_SD=="1", "Yes", "No"),
           SD_from_mum = replace(SD_from_mum, FatherHasSD==1 & MotherHasSD==1 & copies_of_SD=="2", "Yes"),
           SD_from_dad = ifelse(FatherHasSD == 1 & copies_of_SD == "1", "Yes", "No"),
           SD_from_dad = replace(SD_from_dad, FatherHasSD==1 & MotherHasSD==1 & copies_of_SD=="2", "Yes"),
           mum_genotype = relevel(as.factor(ifelse(MotherHasSD==1, SD, "wild_type")), ref = "wild_type"),
           dad_genotype = relevel(as.factor(ifelse(FatherHasSD==1, SD, "wild_type")), ref = "wild_type"))
  
  if(path == "data/messy_data/male_fitness.csv"){
    dat <- dat %>% 
      mutate(focal.male.fitness = focal_daughters + focal_sons,
             rival.male.fitness = rival_daughters + rival_sons,
             total = focal.male.fitness + rival.male.fitness,
             vial = 1:n())
  }
  
  if(path == "data/messy_data/larval_fitness.csv"){
    
    # Note that for 'Mother has SD' crosses (Cross 2), the 'initial_number' of larvae is the maximum possible.
    # The real initial number of non-recombinants is probably a bit lower, since the survival rate of the recombinants is probably not 100%
    # This means the survival rate in Cross 2 is an under-estimate of the true survival rate.
    # For SD-5, it is a very slight underestimate, since SD-5 does not recombine much.
    # For the other two SDs, it is a larger underestimate. 
    
    dat <- dat %>%
      mutate(surviving_females = nonrecombinant_females,
             surviving_males = nonrecombinant_males,
             survivors = nonrecombinant_females + nonrecombinant_males,
             initial_number = number_larvae_in_vial - recombinant_females - recombinant_males,
             number_died = initial_number - survivors,
             larval_density = as.numeric(scale(number_larvae_in_vial)), # mean-centre the larval density covariate
             vial = 1:n(),
             total = survivors + number_died) %>%
      select(-number_larvae_in_vial, -nonrecombinant_females, -nonrecombinant_males, -recombinant_females, -recombinant_males)
      
    # We never put more than 100 larvae in one vial, so for cases where the initial number was 200, it was 2 vials of 100 each, etc.
    dat$larval_density[dat$larval_density > 100] <- 100
    
    # For Cross 1, where both parents have SD, we cannot discriminate larvae with 0 or 1 copies of SD 
    # (until they become adults and develop eyes)
    dat$copies_of_SD[dat$genotype == "A0+A1"] <- "0 or 1"
  }
  
  if(path == "data/messy_data/female_fitness.csv"){
    dat$vial <- 1:nrow(dat)
  }
  
  # 'genotype' variable is a composite of SD genotype and parental origin, used during data collection. Not needed for analysis
  dat <- dat %>% select(-genotype) 
  
  # save the clean data
  write_csv(dat, path = str_replace(path, "messy", "clean")) 
  dat 
}

larvae  <- load_and_clean_data("data/messy_data/larval_fitness.csv") 
males   <- load_and_clean_data("data/messy_data/male_fitness.csv") 
females <- load_and_clean_data("data/messy_data/female_fitness.csv") 
expt2   <- load_and_clean_data("data/messy_data/experiment2.csv") 

Larval survival data

Table S1: Number and percentage of L1 larvae surviving to adulthood for each SD genotype and cross type.

larvae %>% 
  group_by(SD, copies_of_SD, parent_with_SD) %>% 
  summarise(number_larvae_counted = sum(survivors + number_died),
            number_of_survivors = sum(survivors),
            percent_surviving = 100 * (number_of_survivors / number_larvae_counted)) %>%
    mutate(percent_surviving = format(round(percent_surviving, 1), nsmall = 1)) %>%
  as.data.frame() %>% pander(split.cell = 40, split.table = Inf)

Adult sex ratio

Table S2: Number and percentage of male and female adults emerging from the juvenile fitness assay vials.

larvae %>% group_by(SD, copies_of_SD, parent_with_SD) %>% 
  summarise(n.males = sum(surviving_males),
            n.females = sum(surviving_females),
            n.total = sum(survivors), 
            percent.male = format(round(100 * sum(surviving_males) / n.total, 1), nsmall = 1)) %>%
  as.data.frame() %>% pander(split.cell = 40, split.table = Inf)

Male fitness data

Table S3: Average relative fitness of adult males for each SD genotype and cross type, expressed as the average proportion of offspring sired. The last two columns give the sample size in terms of number of vials (each of which contained 5 focal males), and number of males.

SE <- function(x) sd(x) / sqrt(length(x))
males %>% 
  group_by(SD, copies_of_SD, parent_with_SD) %>% 
  summarise(average_relative_fitness = mean(focal.male.fitness / (focal.male.fitness + rival.male.fitness)),
            SE = SE(focal.male.fitness / (focal.male.fitness + rival.male.fitness)),
            number_of_vials = n(),
            number_of_males = number_of_vials * 5) %>%
  mutate(average_relative_fitness = format(round(average_relative_fitness, 2), nsmall = 2),
         SE = format(round(SE, 3), nsmall = 3)) %>%
  as.data.frame() %>% pander(split.cell = 40, split.table = Inf)

Female fitness data

Table S4: Average fecundity of adult females for each SD genotype and cross type. The last two columns give the sample size in terms of number of oviposition vials (each of which contained up to 5 focal females), and number of males.

females %>% 
  mutate(larvae_per_female = larvae_produced / number_of_laying_females) %>%
  group_by(SD, copies_of_SD, parent_with_SD) %>% 
  summarise(average_fecundity = mean(larvae_per_female),
            SE = SE(larvae_per_female),
            number_of_vials = n(), 
            number_of_females = sum(number_of_laying_females)) %>%
  mutate(average_fecundity = format(round(average_fecundity, 2), nsmall = 2),
         SE = format(round(SE, 3), nsmall = 3)) %>%
  as.data.frame() %>% pander(split.cell = 40, split.table = Inf)

Experiment 2 larval survival data

Table S5: Number and percentage of L1 larvae surviving to adulthood in Experiment 2, for each SD genotype, cross type, and offspring sex.

set.seed(1)
expt2 %>%
  rename(parent_with_SD = SD_parent, offspring_sex = sex_of_embryos) %>%
  mutate(SD_embryos = SD_offspring + rbinom(n(), Dead_offspring, 0.5),
         CyO_embryos = starting_number_focal_sex - SD_embryos,
         parent_with_SD = ifelse(parent_with_SD == "dad", "Father", "Mother"),
         offspring_sex = ifelse(offspring_sex == "female", "Female", "Male")) %>%
  group_by(SD, parent_with_SD, offspring_sex) %>% 
  summarise(percent_surviving_SD_larvae = mean(100 * SD_offspring / SD_embryos),
            percent_surviving_CyO_larvae = mean(100 * CyO_offspring / CyO_embryos),
            n_larvae_counted = sum(starting_number_focal_sex),
            n_crosses = n()) %>%
  mutate(percent_surviving_SD_larvae = format(round(percent_surviving_SD_larvae, 1), nsmall = 1),
         percent_surviving_CyO_larvae = format(round(percent_surviving_CyO_larvae, 1), nsmall = 1)) %>% 
  pander(split.cell = 40, split.table = Inf)

Run a Bayesian binomial generalised linear mixed model

if(!file.exists("data/model_output/larvae_brms.rds")){
  
  larvae_brms <-  brm(survivors | trials(total) ~ larval_density + SD * copies_of_SD * parent_with_SD + (1 | vial), 
                      family = binomial,
                      data = larvae, 
                      iter = 4000, chains = 4, cores = nCores,
                      control = list(adapt_delta = 0.99, max_treedepth = 15),
                      prior = prior(normal(0, 5), class = "b"))
  
  saveRDS(larvae_brms, "data/model_output/larvae_brms.rds")
} else larvae_brms <- readRDS("data/model_output/larvae_brms.rds")

Check a diagnostic plot of the model

This plot shows the frequency distribution of the original data (black line), along with predicted data that were computed using 10 random samples from the posterior of the model parameters. The posterior predicted values closely match the originals, suggesting that the model is approximating the data reasonably well.

pp_check(larvae_brms)

Past versions of unnamed-chunk-9-1.png

Inspect the model summary

This summary shows the median, error, and 95% CIs for the posterior for all the fixed effects. This table is difficult to interpret and not very informative, but is presented here for completeness.

summarise_brms(larvae_brms)

Hypothesis testing

Table S6: The results of hypothesis tests computed using the model of larval survival in Experiment 1. Each row gives the posterior estimate of a difference in means, such that the estimate is positive if mean 1 is larger than mean 2, and negative otherwise (expressed in % larval survival). The mean 1 and mean 2 columns list the parent which had SD (mother, father, or both), followed by the number of SD alleles present in the offspring (0, 1 or 2). The Posterior probability column gives the probability that the mean with the smaller point estimate is actually larger than the other mean, analagously to a one-tailed p-value. The Evidence ratio (ER) column gives the ratio of evidence, such that ER = 5 means that it is 5 times more likely that the mean with the smaller point estimate really is the smaller one. Asterisks highlight rows where the posterior probability is less than 0.05.

# Function to compute posterior difference in means, with the posterior_probability of the smaller mean actually being the larger one (plus the larger evidence ratio, e.g. ER = 10 means it's 10x more likely the smaller mean really is smaller not larger, ER=1 means it's equally likely the smaller one is actually larger)
compare <- function(mean1, mean2){
  difference <- mean1 - mean2
  prob_less <- hypothesis(data.frame(x = difference), "x < 0")$hypothesis
  prob_more <- hypothesis(data.frame(x = difference), "x > 0")$hypothesis
  hypothesis(data.frame(x = difference), "x = 0")$hypothesis %>% 
    mutate(Posterior_probability = min(c(prob_less$Post.Prob, prob_more$Post.Prob)),
           Evidence_ratio = max(c(prob_less$Evid.Ratio, prob_more$Evid.Ratio)))
}

hypothesis_tests <- function(model, dat, SR = FALSE, divisor = NULL){
  
  SDs <- c("SD-5", "SD-72", "SD-Mad")
  
  new <- dat %>%
    select(SD, copies_of_SD, MotherHasSD, FatherHasSD, SD_from_mum, 
           SD_from_dad, mum_genotype, dad_genotype, parent_with_SD) %>%
    distinct() %>%
    arrange(SD, copies_of_SD, MotherHasSD, FatherHasSD, SD_from_mum, 
            SD_from_dad, mum_genotype, dad_genotype) %>%
    mutate(total = 100, 
           number_of_laying_females = 5, 
           larval_density = 0,
           copies_of_SD = as.character(copies_of_SD),
           i = 1:n(),
           parent = ifelse(FatherHasSD == 1, "B", "M"),
           parent = replace(parent, MotherHasSD == 0, "F"))
  
  
  SD5 <- new %>% filter(SD == "SD-5")
  SD72 <- new %>% filter(SD == "SD-72")
  SDMad <- new %>% filter(SD == "SD-Mad")
  
  pred <- fitted(model, re_formula = NA,  
                 newdata = new, 
                 summary = FALSE) 
  
  if(!is.null(divisor)) pred <- pred / divisor
  
  control_0SD_neither <- pred[, new %>% filter(SD == "No SD chromosome") %>% pull(i)]
  
  SD5_0SD_mother <- pred[, SD5 %>% filter(copies_of_SD == "0" & parent == "M") %>% pull(i)]
  SD5_1SD_mother <- pred[, SD5 %>% filter(copies_of_SD == "1" & parent == "M") %>% pull(i)]
  SD5_0SD_father <- pred[, SD5 %>% filter(copies_of_SD == "0" & parent == "F") %>% pull(i)]
  SD5_1SD_father <- pred[, SD5 %>% filter(copies_of_SD == "1" & parent == "F") %>% pull(i)]
  
  SD72_0SD_mother <- pred[, SD72 %>% filter(copies_of_SD == "0" & parent == "M") %>% pull(i)]
  SD72_1SD_mother <- pred[, SD72 %>% filter(copies_of_SD == "1" & parent == "M") %>% pull(i)]
  SD72_0SD_father <- pred[, SD72 %>% filter(copies_of_SD == "0" & parent == "F") %>% pull(i)]
  SD72_1SD_father <- pred[, SD72 %>% filter(copies_of_SD == "1" & parent == "F") %>% pull(i)]
  
  SDMad_0SD_mother <- pred[, SDMad %>% filter(copies_of_SD == "0" & parent == "M") %>% pull(i)]
  SDMad_1SD_mother <- pred[, SDMad %>% filter(copies_of_SD == "1" & parent == "M") %>% pull(i)]
  SDMad_0SD_father <- pred[, SDMad %>% filter(copies_of_SD == "0" & parent == "F") %>% pull(i)]
  SDMad_1SD_father <- pred[, SDMad %>% filter(copies_of_SD == "1" & parent == "F") %>% pull(i)]
  
  SDMad_1SD_both <- pred[, SDMad %>% filter(copies_of_SD %in% c("0 or 1", "1") & parent == "B") %>% pull(i)]
  SDMad_2 <- pred[, SDMad %>% filter(copies_of_SD == "2") %>% pull(i)]
  
  # For larvae data only, we also have observations of SD-5 and SD-Mad homozygotes
  if(sum(new$copies_of_SD == "2") > 1){ 
    SD5_1SD_both  <- pred[, SD5  %>% filter(copies_of_SD %in% c("0 or 1", "1") & parent == "B") %>% pull(i)]
    SD72_1SD_both <- pred[, SD72 %>% filter(copies_of_SD %in% c("0 or 1", "1") & parent == "B") %>% pull(i)]
    SD5_2  <- pred[, SD5  %>% filter(copies_of_SD == "2") %>% pull(i)]
    SD72_2 <- pred[, SD72 %>% filter(copies_of_SD == "2") %>% pull(i)]
    
    two_compare <- data.frame(
      SD = SDs,
      Mean_1 = rep(paste("Both parents,", ifelse("0 or 1" %in% SD5$copies_of_SD, "0 or 1", "1")), 3),
      Mean_2 = rep("Both parents, 2", 3),
      rbind(
        compare(SD5_1SD_both, SD5_2),
        compare(SD72_1SD_both, SD72_2),
        compare(SDMad_1SD_both, SDMad_2)
      ) 
    )
    
  } else { # for non-larva data
    two_compare <- data.frame(SD = "SD-Mad", 
                              Mean_1 = "Both parents, 1",
                              Mean_2 = "Both parents, 2", 
                              compare(SDMad_1SD_both, SDMad_2)) # CONTROLS HERE
  }
  
  if("0 or 1" %in% new$copies_of_SD){  
    SD5_0or1   <- pred[, SD5 %>% filter(copies_of_SD == "0 or 1") %>% pull(i)]
    SD72_0or1  <- pred[, SD72 %>% filter(copies_of_SD == "0 or 1") %>% pull(i)]
    SDMad_0or1 <- pred[, SDMad %>% filter(copies_of_SD == "0 or 1") %>% pull(i)]
  }
  
  rbind(
    data.frame(
      SD = rep(SDs, 6),
      Mean_1 = c(rep("Neither, 0", 6), rep("Mother, 0", 3), rep("Mother, 1", 3), rep("Mother, 0", 3), rep("Father, 0", 3)),
      Mean_2 = c(rep("Father, 0", 3), rep("Mother, 0", 3),  
                 rep("Father, 0", 3), rep("Father, 1", 3), rep("Mother, 1", 3), rep("Father, 1", 3)),
      rbind(
        
        compare(control_0SD_neither, SD5_0SD_father), # controls vs 0 SD individuals
        compare(control_0SD_neither, SD72_0SD_father),
        compare(control_0SD_neither, SDMad_0SD_father),
        compare(control_0SD_neither, SD5_0SD_mother),
        compare(control_0SD_neither, SD72_0SD_mother),
        compare(control_0SD_neither, SDMad_0SD_mother),
        
        compare(SD5_0SD_mother, SD5_0SD_father), # parental effects on 0 SD individuals
        compare(SD72_0SD_mother, SD72_0SD_father),
        compare(SDMad_0SD_mother, SDMad_0SD_father),
        
        compare(SD5_1SD_mother, SD5_1SD_father), # parental effects on 1 SD individuals
        compare(SD72_1SD_mother, SD72_1SD_father),
        compare(SDMad_1SD_mother, SDMad_1SD_father),
        
        compare(SD5_0SD_mother, SD5_1SD_mother), # effect of inheriting SD, from the mother
        compare(SD72_0SD_mother, SD72_1SD_mother),
        compare(SDMad_0SD_mother, SDMad_1SD_mother),
        
        compare(SD5_0SD_father, SD5_1SD_father), # effect of inheriting SD, from the father
        compare(SD72_0SD_father, SD72_1SD_father),
        compare(SDMad_0SD_father, SDMad_1SD_father)
        
      )), 
    two_compare) %>% 
    select(-Hypothesis, -Evid.Ratio, -Post.Prob, -Star) %>%
    mutate(Notable = ifelse(Posterior_probability < 0.05, "*", "")) 
}

clean_table <- function(tabl){
  cols <- c("Trait", "SD", "Comparison", "Difference", "Error", "Posterior_probability")
  if(!("Trait" %in% names(tabl))){
    cols <- cols[cols != "Trait"]
  }
  
  tabl <- mutate(tabl, 
                 Estimate = format(round(Estimate, 1), nsmall = 1),  
                 Est.Error = format(round(Est.Error, 1), nsmall = 1),
                 CI.Lower = format(round(CI.Lower, 1), nsmall = 1),
                 CI.Upper = format(round(CI.Upper, 1), nsmall = 1), 
                 Evidence_ratio = format(round(Evidence_ratio, 1), nsmall = 1), 
                 Difference = paste(Estimate, " (", CI.Lower, " to ", CI.Upper, ")", sep = ""),
                 Comparison = paste(Mean_1, "-", Mean_2),
                 Error = Est.Error) %>%
    select(!! cols)
  names(tabl) <- str_replace_all(names(tabl), "_", " ")
  tabl
}

tests1 <- hypothesis_tests(larvae_brms, larvae)
tests1 %>% clean_table() %>% pander(split.cell = 40, split.table = Inf)

Run a Bayesian binomial generalised linear mixed model

SR_data <- larvae %>% filter(survivors > 0) %>% 
  mutate(Males = surviving_males, 
         total = surviving_males + surviving_females)

if(!file.exists("data/model_output/SR_brms.rds")){
  SR_brms <-  brm(Males | trials(total) ~ SD * copies_of_SD * parent_with_SD + (1 | vial), 
                  family = binomial,
                  data = SR_data, iter = 4000, chains = 4, cores = nCores,
                  control = list(adapt_delta = 0.99, max_treedepth = 15), 
                  prior = prior(normal(0, 5), class = "b"))
  
  saveRDS(SR_brms, "data/model_output/SR_brms.rds")
} else SR_brms <- readRDS("data/model_output/SR_brms.rds")

Check a diagnostic plot of the model

This plot shows the frequency distribution of the original data (black line), along with predicted data that were computed using 10 random samples from the posterior of the model parameters. The posterior predicted values closely match the originals, suggesting that the model is approximating the data reasonably well.

pp_check(SR_brms)

Past versions of unnamed-chunk-13-1.png

Inspect the model summary

This summary shows the median, error, and 95% CIs for the posterior for all the fixed effects. This table is difficult to interpret and not very informative, but is presented here for completeness.

summarise_brms(SR_brms)

Hypothesis testing

Table S7: The results of hypothesis tests computed using the model of adult sex ratio in Experiment 1. Each row gives the posterior estimate of a difference in means, such that the estimate is positive if mean 1 is larger than mean 2, and negative otherwise (expressed in % males). The mean 1 and mean 2 columns list the parent which had SD (mother, father, or both), followed by the number of SD alleles present in the offspring (0, 1 or 2). The Posterior probability column gives the probability that the mean with the smaller point estimate is actually larger than the other mean, analagously to a one-tailed p-value. The Evidence ratio (ER) column gives the ratio of evidence, such that ER = 5 means that it is 5 times more likely that the mean with the smaller point estimate really is the smaller one. Asterisks highlight rows where the posterior probability is less than 0.05.

tests2 <- hypothesis_tests(SR_brms, SR_data)
tests2 %>% clean_table() %>% pander(split.cell = 40, split.table = Inf)

Run a Bayesian negative binomial generalised linear mixed model

if(!file.exists("data/model_output/female_brms.rds")){
  female_brms <-  brm(larvae_produced ~ number_of_laying_females + SD * copies_of_SD * parent_with_SD, 
                    family = negbinomial,
                    data = females, iter = 4000, chains = 4, cores = nCores,
                    control = list(adapt_delta = 0.99, max_treedepth = 15),
                    prior = prior(normal(0, 5), class = "b"))
  
  saveRDS(female_brms, "data/model_output/female_brms.rds")
} else female_brms <- readRDS("data/model_output/female_brms.rds")

Check a diagnostic plot of the model

This plot shows the frequency distribution of the original data (black line), along with predicted data that were computed using 10 random samples from the posterior of the model parameters. The posterior predicted values closely match the originals, suggesting that the model is approximating the data reasonably well.

pp_check(female_brms)

Past versions of unnamed-chunk-17-1.png

Inspect the model summary

This summary shows the median, error, and 95% CIs for the posterior for all the fixed effects. This table is difficult to interpret and not very informative, but is presented here for completeness.

summarise_brms(female_brms)

Hypothesis testing

Table S8: The results of hypothesis tests computed using the model of female fitness in Experiment 1. Each row gives the posterior estimate of a difference in means, such that the estimate is positive if mean 1 is larger than mean 2, and negative otherwise (expressed as the number of offspring produced). The mean 1 and mean 2 columns list the parent which had SD (mother, father, or both), followed by the number of SD alleles present in the offspring (0, 1 or 2). The Posterior probability column gives the probability that the mean with the smaller point estimate is actually larger than the other mean, analagously to a one-tailed p-value. The Evidence ratio (ER) column gives the ratio of evidence, such that ER = 5 means that it is 5 times more likely that the mean with the smaller point estimate really is the smaller one. Asterisks highlight rows where the posterior probability is less than 0.05.

tests3 <- hypothesis_tests(female_brms, females, divisor = 5)
tests3 %>% clean_table() %>% pander(split.cell = 40, split.table = Inf)

Run a Bayesian binomial generalised linear mixed model

if(!file.exists("data/model_output/male_brms.rds")){
  male_brms <-  brm(focal.male.fitness | trials(total) ~ SD * copies_of_SD * parent_with_SD + (1 | vial), 
                    family = binomial,
                    data = males, iter = 4000, chains = 4, cores = nCores,
                    control = list(adapt_delta = 0.99, max_treedepth = 15),
                    prior = prior(normal(0, 5), class = "b"))
  
  saveRDS(male_brms, "data/model_output/male_brms.rds")
} else male_brms <- readRDS("data/model_output/male_brms.rds")

Check a diagnostic plot of the model

This plot shows the frequency distribution of the original data (black line), along with predicted data that were computed using 10 random samples from the posterior of the model parameters. The posterior predicted values closely match the originals, suggesting that the model is approximating the data reasonably well.

pp_check(male_brms)

Past versions of unnamed-chunk-21-1.png

Inspect the model summary

This summary shows the median, error, and 95% CIs for the posterior for all the fixed effects. This table is difficult to interpret and not very informative, but is presented here for completeness.

summarise_brms(male_brms)

Hypothesis testing

Table S9: The results of hypothesis tests computed using the model of male fitness in Experiment 1. Each row gives the posterior estimate of a difference in means, such that the estimate is positive if mean 1 is larger than mean 2, and negative otherwise (expressed in % offspring sired). The mean 1 and mean 2 columns list the parent which had SD (mother, father, or both), followed by the number of SD alleles present in the offspring (0, 1 or 2). The Posterior probability column gives the probability that the mean with the smaller point estimate is actually larger than the other mean, analagously to a one-tailed p-value. The Evidence ratio (ER) column gives the ratio of evidence, such that ER = 5 means that it is 5 times more likely that the mean with the smaller point estimate really is the smaller one. Asterisks highlight rows where the posterior probability is less than 0.05.

tests4 <- hypothesis_tests(male_brms, males)
tests4 %>% clean_table() %>% pander(split.cell = 40, split.table = Inf)

Run a Bayesian binomial generalised linear mixed model

# Function to simular the missing genotypes of embryos that died, assuming fair meiosis in SD mothers and k = k in SD fathers
simulate_expt2_data <- function(expt2_dataset, k){
  
  have_SD_mum <- which(expt2_dataset$SD_parent == "mum")
  have_SD_dad <- which(expt2_dataset$SD_parent == "dad")
  
  expt2_dataset$SD_embryos[have_SD_mum] <- with(expt2_dataset[have_SD_mum, ], 
                                                SD_offspring + rbinom(length(SD_embryos), Dead_offspring, 0.5))
  
  expt2_dataset$SD_embryos[have_SD_dad] <- with(expt2_dataset[have_SD_dad, ], 
                                                SD_offspring + rbinom(length(SD_embryos), Dead_offspring, k))
  
  expt2_dataset <- expt2_dataset %>%
    mutate(SD_embryos = as.numeric(SD_embryos),
           CyO_embryos = starting_number_focal_sex - SD_embryos) %>% 
    select(-starting_number_focal_sex, -Dead_offspring) 
  
  expt2_dataset %>% 
    select(sex_of_embryos, SD, SD_parent, vial, block, SD_offspring, CyO_offspring) %>%
    tidyr::gather(genotype, num_survivors, SD_offspring, CyO_offspring) %>%
    mutate(genotype = gsub("_offspring", "", genotype)) %>%
    left_join(expt2_dataset %>% 
                select(sex_of_embryos, SD, SD_parent, vial, block, SD_embryos, CyO_embryos) %>%
                tidyr::gather(genotype, total_number, SD_embryos, CyO_embryos) %>%
                mutate(genotype = gsub("_embryos", "", genotype))) %>%
    arrange(vial)
}

# Simulate some data and run the Bayesian GLMM
model_simulated_data <- function(expt2_dataset, k){
  
  # First re-format teh data and simulate the missing embryo genotypes using stated value of k
  dat <- simulate_expt2_data(expt2_dataset, k)

  # Now run the model
  brm(num_survivors  | trials(total_number) ~ genotype * sex_of_embryos * SD * SD_parent + (1 | vial), 
      data = dat, family = "binomial",
      chains = 4, cores = nCores, seed = 1,
      prior = prior(normal(0, 5), class = "b"),
      control = list(adapt_delta = 0.9999, max_treedepth = 15))  
  
}

# Do the same model for k = 0.5, 0.6 and 0.7
if(!file.exists("data/model_output/expt2_brms.rds")){
  model_list <- list(
    model_simulated_data(expt2, 0.5),
    model_simulated_data(expt2, 0.6),
    model_simulated_data(expt2, 0.7))
  
  saveRDS(model_list, "data/model_output/expt2_brms.rds")
} else model_list <- readRDS("data/model_output/expt2_brms.rds")

Check a diagnostic plot of the model

This plot shows the frequency distribution of the original data (black line), along with predicted data that were computed using 10 random samples from the posterior of the model parameters. The posterior predicted values closely match the originals, suggesting that the model is approximating the data reasonably well. There are two plots for this model because we used a multivariate model that jointly modelled two response variables: the survival of SD larvae and of CyO larvae.

grid.arrange(
  pp_check(model_list[[1]], resp = "SDoffspring")  + labs(title = "Number of SD offspring"),
  pp_check(model_list[[1]], resp = "CyOoffspring") + labs(title = "Number of CyO offspring")
)

Past versions of unnamed-chunk-27-1.png

Inspect the model summary

This summary shows the median, error, and 95% CIs for the posterior for all the fixed effects. This table is difficult to interpret and not very informative, but is presented here for completeness.

summarise_brms(model_list[[1]])

Plot the posterior predictions of the group means

new_data <- simulate_expt2_data(expt2, 0.5) %>% 
  select(sex_of_embryos, SD, SD_parent, genotype) %>% 
  distinct() %>% mutate(total_number = 100) %>%
  arrange(desc(SD_parent), SD, desc(sex_of_embryos))

plot_expt2 <- function(model){

  pd <- position_dodge(0.4)
  
  predictions <- fitted(model, 
                        summary = TRUE, 
                        newdata = new_data, re_formula = NA)
  
  inner_bar <- fitted(model, 
                      summary = TRUE, 
                      newdata = new_data, re_formula = NA,
                      probs = c(0.25, 0.75))
  
  process_preds <- function(preds){
    data.frame(new_data, preds) %>%
      rename(Offspring_type = genotype) %>%
      mutate(sex_of_embryos  = str_replace_all(sex_of_embryos, "female", "daughters"),
             sex_of_embryos  = str_replace_all(sex_of_embryos, "male", "sons"),
             Offspring_type = paste(Offspring_type, sex_of_embryos),
             SD_parent = str_replace_all(SD_parent, "dad", "SD/CyO father"),
             SD_parent = str_replace_all(SD_parent, "mum", "SD/CyO mother"))
  }

  left_join(process_preds(predictions), 
            process_preds(inner_bar)) %>% 
    ggplot(aes(Offspring_type, Estimate, colour = SD_parent, fill = SD_parent)) +
    geom_errorbar(aes(ymin = Q2.5, ymax = Q97.5), width = 0, position = pd, size = 1.3) + 
    geom_errorbar(aes(ymin = Q25, ymax = Q75), width = 0, position = pd, size = 3) + 
    geom_point(position = pd, size = 2, colour = "black",  pch = 21) +
    facet_grid(~SD) + 
    scale_colour_brewer(name = "Cross", palette = "Pastel1", direction = -1, aesthetics = c("colour", "fill")) +
    theme_hc() + 
    theme(axis.text.x = element_text(angle = 45, vjust = 1, hjust = 1),
          legend.position = "top", axis.ticks.y = element_blank()) +
    ylab("Juvenile fitness") + xlab("Genotype and sex of offspring")
}

fig2 <- plot_expt2(model_list[[1]])
ggsave(fig2, file = "figures/fig2.pdf", height = 3.9, width = 7)
fig2

Past versions of unnamed-chunk-29-1.png

Figure 2: Posterior estimates of % L1 larva-to-adult survival in Experiment 2 for each combination of offspring sex and genotype (x-axis), SD variant (panels), and cross (colours). Error bars show the 95% quantiles of the posterior. See Tables 2 and S10 for associated hypothesis tests. This plot was generated assuming fair meiosis (\(k\) = 0.5) in SD/CyO males; see Figure S1 for equivalent plots made using different assumed values of \(k\).

Check the effect of our assumption that meiosis is fair in SD/CyO males

make_expt2_multiplot <- function(...) {
    plots <- list(...)
    g <- ggplotGrob(plots[[1]] + theme(legend.position="bottom"))$grobs
    legend <- g[[which(sapply(g, function(x) x$name) == "guide-box")]]
    lheight <- sum(legend$height)
    plots <- lapply(plots, function(x) ggplotGrob(x + theme(legend.position = "none")))
    
    p <- gtable_rbind(plots[[1]], plots[[2]], plots[[3]])
    
    grid.arrange(p, legend, 
                 nrow = 2, 
                 heights = unit.c(unit(1, "npc") - lheight, lheight))
  }

make_expt2_multiplot(
  plot_expt2(model_list[[1]]) + theme(axis.text.x = element_blank(), axis.ticks.x = element_blank()) + xlab(NULL) + ylab(NULL),
  plot_expt2(model_list[[2]]) + theme(axis.text.x = element_blank(), axis.ticks.x = element_blank()) + xlab(NULL),
  plot_expt2(model_list[[3]]) + ylab(NULL))

Past versions of unnamed-chunk-30-1.png

Figure S1: Equivalent plots to Figure 2, under the assumption that meiosis is fair (\(k\) = 0.5, top row, same as Figure 2), slightly biased (\(k\) = 0.6, middle row), and more strongly biased (\(k\) = 0.7, bottom row). Note that the significant results for Figure 2 mostly stay the same or increase in magnitude, suggesting that they are genuine and are not sensitive to our assumptions about the data.

Hypothesis testing

hypothesis_tests_expt2 <- function(model){
  
  compare <- function(mean1, mean2, SD, Mean_1, Mean_2){
  difference <- mean1 - mean2
  prob_less <- hypothesis(data.frame(x = difference), "x < 0")$hypothesis
  prob_more <- hypothesis(data.frame(x = difference), "x > 0")$hypothesis
  output <- data.frame(
    SD, Mean_1, Mean_2,
    hypothesis(data.frame(x = difference), "x = 0")$hypothesis %>% 
    mutate(Posterior_probability = min(c(prob_less$Post.Prob, prob_more$Post.Prob)),
           Evidence_ratio = max(c(prob_less$Evid.Ratio, prob_more$Evid.Ratio)))
  )
}
  preds <- fitted(model, 
                  summary = FALSE, 
                  newdata = new_data, re_formula = NA)
  
  
  SD5_sons_mum_SD <- preds[,1]
  SD5_sons_mum_CyO <- preds[,2]
  SD5_daughters_mum_SD <- preds[,3]
  SD5_daughters_mum_CyO <- preds[,4]
  SD72_sons_mum_SD <- preds[,5]
  SD72_sons_mum_CyO <- preds[,6]
  SD72_daughters_mum_SD <- preds[,7]
  SD72_daughters_mum_CyO <- preds[,8]
  SDMad_sons_mum_SD <- preds[,9]
  SDMad_sons_mum_CyO <- preds[,10]
  SDMad_daughters_mum_SD <- preds[,11]
  SDMad_daughters_mum_CyO <- preds[,12]
  SD5_sons_dad_SD <- preds[,13]
  SD5_sons_dad_CyO <- preds[,14]
  SD5_daughters_dad_SD <- preds[,15]
  SD5_daughters_dad_CyO <- preds[,16]
  SD72_sons_dad_SD <- preds[,17]
  SD72_sons_dad_CyO <- preds[,18]
  SD72_daughters_dad_SD <- preds[,19]
  SD72_daughters_dad_CyO <- preds[,20]
  SDMad_sons_dad_SD <- preds[,21]
  SDMad_sons_dad_CyO <- preds[,22]
  SDMad_daughters_dad_SD <- preds[,23]
  SDMad_daughters_dad_CyO <- preds[,24]
  
  rbind(
    # sex effect on SD individuals when mother has SD
    compare(SD5_sons_mum_SD, SD5_daughters_mum_SD, "SD-5", "Sons, SD, mother", "Daughters, SD, mother"), 
    compare(SD72_sons_mum_SD, SD72_daughters_mum_SD, "SD-72", "Sons, SD, mother", "Daughters, SD, mother"),
    compare(SDMad_sons_mum_SD, SDMad_daughters_mum_SD, "SD-Mad", "Sons, SD, mother", "Daughters, SD, mother"),
    # sex effect on CyO individuals when mother has SD
    compare(SD5_sons_mum_CyO, SD5_daughters_mum_CyO, "SD-5", "Sons, CyO, mother", "Daughters, CyO, mother"), 
    compare(SD72_sons_mum_CyO, SD72_daughters_mum_CyO, "SD-72", "Sons, CyO, mother", "Daughters, CyO, mother"),
    compare(SDMad_sons_mum_CyO, SDMad_daughters_mum_CyO, "SD-Mad", "Sons, CyO, mother", "Daughters, CyO, mother"),
    # sex effect on SD individuals when father has SD
    compare(SD5_sons_dad_SD, SD5_daughters_dad_SD, "SD-5", "Sons, SD, father", "Daughters, SD, father"), 
    compare(SD72_sons_dad_SD, SD72_daughters_dad_SD, "SD-72", "Sons, SD, father", "Daughters, SD, father"),
    compare(SDMad_sons_dad_SD, SDMad_daughters_dad_SD, "SD-Mad", "Sons, SD, father", "Daughters, SD, father"),
    # sex effect on CyO individuals when father has SD
    compare(SD5_sons_dad_CyO, SD5_daughters_dad_CyO, "SD-5", "Sons, CyO, father", "Daughters, CyO, father"), 
    compare(SD72_sons_dad_CyO, SD72_daughters_dad_CyO, "SD-72", "Sons, CyO, father", "Daughters, CyO, father"),
    compare(SDMad_sons_dad_CyO, SDMad_daughters_dad_CyO, "SD-Mad", "Sons, CyO, father", "Daughters, CyO, father"),
    # effect of inheriting SD not CyO from mother on sons
    compare(SD5_sons_mum_CyO, SD5_sons_mum_SD, "SD-5", "Sons, CyO, mother", "Sons, SD, mother"), 
    compare(SD72_sons_mum_CyO, SD72_sons_mum_SD, "SD-72", "Sons, CyO, mother", "Sons, SD, mother"),
    compare(SDMad_sons_mum_CyO, SDMad_sons_mum_SD, "SD-Mad", "Sons, CyO, mother", "Sons, SD, mother"),
    # effect of inheriting SD not CyO from father on sons
    compare(SD5_sons_dad_CyO, SD5_sons_dad_SD, "SD-5", "Sons, CyO, father", "Sons, SD, father"),
    compare(SD72_sons_dad_CyO, SD72_sons_dad_SD, "SD-72", "Sons, CyO, father", "Sons, SD, father"),
    compare(SDMad_sons_dad_CyO, SDMad_sons_dad_SD, "SD-Mad", "Sons, CyO, father", "Sons, SD, father"),
    # effect of inheriting SD not CyO from mother on daughters
    compare(SD5_daughters_mum_CyO, SD5_daughters_mum_SD, "SD-5", "Daughters, CyO, mother", "Daughters, SD, mother"), 
    compare(SD72_daughters_mum_CyO, SD72_daughters_mum_SD, "SD-72", "Daughters, CyO, mother", "Daughters, SD, mother"),
    compare(SDMad_daughters_mum_CyO, SDMad_daughters_mum_SD, "SD-Mad", "Daughters, CyO, mother", "Daughters, SD, mother"),
    # effect of inheriting SD not CyO from father on daughters
    compare(SD5_daughters_dad_CyO, SD5_daughters_dad_SD, "SD-5", "Daughters, CyO, father", "Daughters, SD, father"), 
    compare(SD72_daughters_dad_CyO, SD72_daughters_dad_SD, "SD-72", "Daughters, CyO, father", "Daughters, SD, father"),
    compare(SDMad_daughters_dad_CyO, SDMad_daughters_dad_SD, "SD-Mad", "Daughters, CyO, father", "Daughters, SD, father"),
    # effect of cross direction for SD sons
    compare(SD5_sons_mum_SD, SD5_sons_dad_SD, "SD-5", "Sons, SD, mother", "Sons, SD, father"), 
    compare(SD72_sons_mum_SD, SD72_sons_dad_SD, "SD-72", "Sons, SD, mother", "Sons, SD, father"),
    compare(SDMad_sons_mum_SD, SDMad_sons_dad_SD, "SD-Mad", "Sons, SD, mother", "Sons, SD, father"),
    # effect of cross direction for SD daughters
    compare(SD5_daughters_mum_SD, SD5_daughters_dad_SD, "SD-5", "Daughters, SD, mother", "Daughters, SD, father"), 
    compare(SD72_daughters_mum_SD, SD72_daughters_dad_SD, "SD-72", "Daughters, SD, mother", "Daughters, SD, father"),
    compare(SDMad_daughters_mum_SD, SDMad_daughters_dad_SD, "SD-Mad", "Daughters, SD, mother", "Daughters, SD, father"),
    # effect of cross direction for CyO sons
    compare(SD5_sons_mum_CyO, SD5_sons_dad_CyO, "SD-5", "Sons, CyO, mother", "Sons, CyO, father"), 
    compare(SD72_sons_mum_CyO, SD72_sons_dad_CyO, "SD-72", "Sons, CyO, mother", "Sons, CyO, father"),
    compare(SDMad_sons_mum_CyO, SDMad_sons_dad_CyO, "SD-Mad", "Sons, CyO, mother", "Sons, CyO, father"),
    # effect of cross direction for CyO daughters
    compare(SD5_daughters_mum_CyO, SD5_daughters_dad_CyO, "SD-5", "Daughters, CyO, mother", "Daughters, CyO, father"), 
    compare(SD72_daughters_mum_CyO, SD72_daughters_dad_CyO, "SD-72", "Daughters, CyO, mother", "Daughters, CyO, father"),
    compare(SDMad_daughters_mum_CyO, SDMad_daughters_dad_CyO, "SD-Mad", "Daughters, CyO, mother", "Daughters, CyO, father")
  ) %>% 
    select(-Hypothesis, -Evid.Ratio, -Post.Prob, -Star) %>%
    mutate(Notable = ifelse(Posterior_probability < 0.05, "*", ""),
           Test = c(rep("Sex difference in survival", 12),
                    rep("Effect of offspring genotype", 12),
                    rep("Effect of parental genotype", 12))) %>% as_tibble() 
}

tests_expt2 <- hypothesis_tests_expt2(model_list[[1]]) %>% 
  mutate(Estimate = format(round(Estimate, 1), nsmall = 1),  
         Est.Error = format(round(Est.Error, 1), nsmall = 1),
         CI.Lower = format(round(CI.Lower, 1), nsmall = 1),
         CI.Upper = format(round(CI.Upper, 1), nsmall = 1), 
         Difference = paste(Estimate, " (", CI.Lower, " to ", CI.Upper, ")", sep = ""),
         Comparison = paste(Mean_1, "-", Mean_2),
         Error = Est.Error) %>%
  select(SD, Comparison, Difference, Error, Posterior_probability, Notable) %>% 
  arrange(SD, Comparison)

Table 2: List of the all the notable differences between groups in Experiment 2 (posterior probability $<$0.05; see Table S10 for the remaining results). For each group, we list the sex of the focal larvae, their genotype (SD or CyO), and the parent that carried SD (mother or father). The difference in means is expressed in % larvae surviving; other details are as in Table 1.

table2 <- tests_expt2 %>% 
  filter(Notable == "*") %>% select(-Notable) 
names(table2) <- str_replace_all(names(table2), "_", " ")

write_csv(table2 %>% rename(p = `Posterior probability`), path = "figures/Table2.csv")
table2 %>% pander(split.cell = 50, split.table = Inf)

Table S10: Complete version of Table 2, showing all the contrasts that were tested in Experiment 2.

tests_expt2 %>% pander(split.cell = 50, split.table = Inf) 

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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] grid      stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
 [1] stringr_1.3.1      tibble_2.0.99.9000 readr_1.1.1       
 [4] ggthemes_4.0.1     gridExtra_2.3      lazerhawk_0.2.4   
 [7] pander_0.6.2       Cairo_1.5-9        reshape2_1.4.3    
[10] brms_2.6.0         ggplot2_3.1.0      Rcpp_1.0.0        
[13] dplyr_0.8.0.1      knitr_1.20        

loaded via a namespace (and not attached):
 [1] nlme_3.1-137         matrixStats_0.54.0   fs_1.2.6            
 [4] xts_0.11-0           devtools_1.13.6      RColorBrewer_1.1-2  
 [7] threejs_0.3.1        rprojroot_1.3-2      rstan_2.18.2        
[10] tools_3.5.1          backports_1.1.2      R6_2.4.0            
[13] DT_0.4               lazyeval_0.2.1       colorspace_1.3-2    
[16] withr_2.1.2          tidyselect_0.2.5     prettyunits_1.0.2   
[19] processx_3.2.1       Brobdingnag_1.2-5    compiler_3.5.1      
[22] git2r_0.23.0         cli_1.0.1            shinyjs_1.0         
[25] labeling_0.3         colourpicker_1.0     scales_1.0.0        
[28] dygraphs_1.1.1.6     mvtnorm_1.0-8        ggridges_0.5.0      
[31] callr_2.0.4          digest_0.6.18        StanHeaders_2.18.0  
[34] rmarkdown_1.10       base64enc_0.1-3      pkgconfig_2.0.2     
[37] htmltools_0.3.6      htmlwidgets_1.2      rlang_0.3.1         
[40] shiny_1.2.0          zoo_1.8-3            crosstalk_1.0.0     
[43] gtools_3.8.1         inline_0.3.15        magrittr_1.5        
[46] loo_2.0.0            bayesplot_1.5.0      Matrix_1.2-14       
[49] munsell_0.5.0        abind_1.4-5          stringi_1.3.1       
[52] whisker_0.3-2        yaml_2.2.0           pkgbuild_1.0.2      
[55] plyr_1.8.4           parallel_3.5.1       promises_1.0.1      
[58] crayon_1.3.4         miniUI_0.1.1.1       lattice_0.20-35     
[61] hms_0.4.2            ps_1.3.0             pillar_1.3.1.9000   
[64] igraph_1.2.1         markdown_0.8         shinystan_2.5.0     
[67] stats4_3.5.1         rstantools_1.5.0     glue_1.3.0.9000     
[70] evaluate_0.11        httpuv_1.4.5         gtable_0.2.0        
[73] purrr_0.3.1          tidyr_0.8.2          assertthat_0.2.0    
[76] mime_0.6             xtable_1.8-3         colortools_0.1.5    
[79] coda_0.19-2          later_0.7.5          rsconnect_0.8.8     
[82] shinythemes_1.1.1    memoise_1.1.0        workflowr_1.2.0     
[85] bridgesampling_0.4-0