Metabolite composition

Last updated: 2022-02-24

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Load packages

# it was slightly 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(tidyverse)
library(GGally)
library(gridExtra)
library(ggridges)
library(brms)
library(tidybayes)
library(DT)
library(kableExtra)
library(knitrhooks) # install with devtools::install_github("nathaneastwood/knitrhooks")
library(showtext)
output_max_height() # a knitrhook option

# set up nice font for figure
nice_font <- "Lora"
font_add_google(name = nice_font, family = nice_font, regular.wt = 400, bold.wt = 700)
showtext_auto()

options(stringsAsFactors = FALSE)

Load metabolite composition data

This analysis set out to test whether sexual selection treatment had an effect on macro-metabolite composition of flies. We measured fresh and dry fly weight in milligrams, plus the concentrations of five metabolites. These are:

Lipid_conc (i.e. the weight of the hexane fraction, divided by the full dry weight),
Carbohydrate_conc (i.e. the weight of the aqueous fraction, divided by the full dry weight),
Protein_conc (i.e. \(\mu\)g of protein per milligram as measured by the bicinchoninic acid protein assay),
Glycogen_conc (i.e. \(\mu\)g of glycogen per milligram as measured by the hexokinase assay), and
Chitin_conc (estimated as the difference between the initial and final dry weights)

We expect body weight to vary between the sexes and potentially between treatments. In turn, we expect body weight might co-vary with the five metabolite concentrations, e.g. because flies might become heavier by sequestering proportionally more of particular metabolites. There might also be weight-independent effects of sex and selection treatment on metabolite composition.

metabolites <- read_csv('data/4.metabolite_data.csv') %>%
  mutate(sex = ifelse(sex == "m", "Male", "Female"),
         line = paste(treatment, line, sep = ""),
         treatment = ifelse(treatment == "M", "Monogamy", "Polyandry")) %>%
  # log transform glycogen since it shows a long tail (others are reasonably normal-looking)
  mutate(Glycogen_ug_mg = log(Glycogen_ug_mg)) %>%
  # There was a technical error with flies collected on day 1, 
  # so they are excluded from the whole paper. All the measurements analysed are of 3d-old flies
  filter(time == '2') %>%
  select(-time)

scaled_metabolites <- metabolites %>% 
  # Find proportional metabolites as a proportion of total dry weight
  mutate(
    Dry_weight = dwt_mg,
    Lipid_conc = Hex_frac / Dry_weight,
    Carbohydrate_conc = Aq_frac / Dry_weight,
    Protein_conc = Protein_ug_mg,
    Glycogen_conc = Glycogen_ug_mg,
    Chitin_conc = Chitin_mg_mg) %>% 
  select(sex, treatment, line, Dry_weight, ends_with("conc")) %>%
  mutate_at(vars(ends_with("conc")), ~ as.numeric(scale(.x))) %>%
  mutate(Dry_weight = as.numeric(scale(Dry_weight))) %>%
  mutate(sextreat = paste(sex, treatment),
         sextreat = replace(sextreat, sextreat == "Male Monogamy", "M males"),
         sextreat = replace(sextreat, sextreat == "Male Polyandry", "E males"),
         sextreat = replace(sextreat, sextreat == "Female Monogamy", "M females"),
         sextreat = replace(sextreat, sextreat == "Female Polyandry", "E females"),
         sextreat = factor(sextreat, c("M males", "E males", "M females", "E females")))

Inspect the raw data

Raw numbers

All variables are shown in standard units (i.e. mean = 0, SD = 1).

my_data_table <- function(df){
  datatable(
    df, rownames=FALSE,
    autoHideNavigation = TRUE,
    extensions = c("Scroller",  "Buttons"),
    options = list(
      dom = 'Bfrtip',
      deferRender=TRUE,
      scrollX=TRUE, scrollY=400,
      scrollCollapse=TRUE,
      buttons = 
        list('csv', list(
          extend = 'pdf',
          pageSize = 'A4',
          orientation = 'landscape',
          filename = 'Dpseudo_metabolites')),
      pageLength = 50
    )
  )
}

scaled_metabolites %>%
  select(-sextreat) %>%
  mutate_if(is.numeric, ~ format(round(.x, 3), nsmall = 3)) %>%
  my_data_table()

Simple plots

The following plot shows how each metabolite varies between sexes and treatments, and how the concentration of each metabolite co-varies with dry weight across individuals.

levels <- c("Carbohydrate", "Chitin", "Glycogen", "Lipid", "Protein", "Dry weight")

cols <- c("M females" = "pink", 
          "E females" = "red", 
          "M males" = "skyblue", 
          "E males" = "blue")

grid.arrange(
  scaled_metabolites %>% 
    rename_all(~ str_remove_all(.x, "_conc")) %>%
    rename(`Dry weight` = Dry_weight) %>%
    mutate(sex = factor(sex, c("Male", "Female"))) %>%
    reshape2::melt(id.vars = c('sex', 'treatment', 'sextreat', 'line')) %>% 
    mutate(variable = factor(variable, levels)) %>%
    ggplot(aes(x = sex, y = value,  fill  = sextreat)) +
    geom_hline(yintercept = 0, linetype = 2) + 
    geom_boxplot() + 
    facet_grid( ~ variable) +
    theme_bw() +
    xlab("Sex") + ylab("Concentration") +
    theme(legend.position = 'top',
          text = element_text(family = nice_font)) + 
    scale_fill_manual(values = cols, name = ""),
  
  arrangeGrob(
    scaled_metabolites %>% 
      rename_all(~ str_remove_all(.x, "_conc")) %>%
      reshape2::melt(id.vars = c('sex', 'treatment', 'sextreat', 'line', 'Dry_weight')) %>% 
      mutate(variable = factor(variable, levels)) %>%
      ggplot(aes(x = Dry_weight, y = value, colour = sextreat, fill = sextreat)) +
      geom_smooth(method = 'lm', se = TRUE, aes(colour = NULL, fill = NULL), colour = "grey20", size = .4) +
      geom_point(pch = 21, colour = "grey20") +
      facet_grid( ~ variable) +
      theme_bw() +
      xlab("Dry weight") + ylab("Concentration") +
      theme(legend.position = 'none',
            text = element_text(family = nice_font),) + 
      scale_colour_manual(values = cols, name = "") + 
      scale_fill_manual(values = cols, name = ""),
    scaled_metabolites %>% 
      rename_all(~ str_remove_all(.x, "_conc")) %>%
      reshape2::melt(id.vars = c('sex', 'treatment', 'sextreat', 'line', 'Dry_weight')) %>% 
      mutate(variable = factor(variable, levels)) %>%
      ggplot(aes(x = Dry_weight, y = value, colour = sextreat, fill = sextreat)) +
      theme_void() + ylab(NULL), nrow = 1, widths = c(0.84, 0.16)),
  heights = c(0.55, 0.45)
)

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
034431b	lukeholman	2020-12-18
49e7792	lukeholman	2020-12-11
deb7183	lukeholman	2020-12-09
43cc270	lukeholman	2020-12-09
4f5ee28	lukeholman	2020-12-04

Plot of correlations between variables

Some of the metabolites, especially lipid concentration, are correlated with dry weight. There is also a large difference in dry weight between sexes (and treatments, to a lesser extent), and sex and treatment effects are evident for some of the metabolites in the raw data. Some of the metabolites are weakly correlated with other metabolites, e.g. lipid and glycogen concentration.

modified_densityDiag <- function(data, mapping, ...) {
  ggally_densityDiag(data, mapping, colour = "grey10", ...) + 
    scale_fill_manual(values = cols) + 
  scale_x_continuous(guide = guide_axis(check.overlap = TRUE))
}

modified_points <- function(data, mapping, ...) {
  ggally_points(data, mapping, pch = 21, colour = "grey10", ...) +
    scale_fill_manual(values = cols) + 
    scale_x_continuous(guide = guide_axis(check.overlap = TRUE))
}

modified_facetdensity <- function(data, mapping, ...) {
  ggally_facetdensity(data, mapping, ...) + 
    scale_colour_manual(values = cols)
}

modified_box_no_facet <- function(data, mapping, ...) {
  ggally_box_no_facet(data, mapping, colour = "grey10", ...) +
    scale_fill_manual(values = cols)
}

metabolite_pairs_plot <- scaled_metabolites %>% 
  arrange(sex, treatment) %>%
  select(-line, -sex, -treatment) %>%
  rename(`Sex and treatment` = sextreat) %>%
  rename_all(~ str_replace_all(.x, "_", " ")) %>%
  
  ggpairs(aes(colour = `Sex and treatment`, fill = `Sex and treatment`),
          diag = list(continuous = wrap(modified_densityDiag, alpha = 0.7),
                      discrete = wrap("blank")),
          lower = list(continuous = wrap(modified_points, alpha = 0.7, size = 1.1), 
                       discrete = wrap("blank"),
                       combo = wrap(modified_box_no_facet, alpha = 0.7)),
          upper = list(continuous = wrap(modified_points, alpha = 0.7, size = 1.1),
                       discrete = wrap("blank"),
                       combo = wrap(modified_box_no_facet, alpha = 0.7, size = 0.5))) 

#metabolite_pairs_plot %>% ggsave(filename = "figures/metabolite_pairs_plot.pdf", height = 10, width = 10)
metabolite_pairs_plot

Version	Author	Date
21567bb	MartinGarlovsky	2021-03-12
034431b	lukeholman	2020-12-18
43cc270	lukeholman	2020-12-09

Mean dry weight

se <- function(x) sd(x) / sqrt(length(x))
metabolites %>% 
  group_by(sex, treatment) %>% 
  summarise(mean_dwt = mean(dwt_mg), 
            SE = se(dwt_mg), 
            n = n()) %>% 
  kable(digits = 3) %>% kable_styling(full_width = FALSE)

sex	treatment	mean_dwt	SE	n
Female	Monogamy	0.562	0.019	12
Female	Polyandry	0.644	0.017	12
Male	Monogamy	0.330	0.009	12
Male	Polyandry	0.353	0.009	12

Directed acyclic graph (DAG)

This directed acyclic graph (DAG) illustrates the causal pathways that we observed between the experimental or measured variables (square boxes) and latent variables (ovals). We hypothesise that sex and mating system potentially influence dry weight as well as the metabolite composition (which we assessed by estimating the concentrations of carbohydrates, chitin, glycogen, lipids and protein). Additionally, dry weight is likely correlated with metabolite composition, and so dry weight acts as a ‘mediator variable’ between metabolite composition, and sex and treatment. The structural equation model below is built with this DAG in mind.

DiagrammeR::grViz('digraph {

graph [layout = dot, rankdir = LR]

# define the global styles of the nodes. We can override these in box if we wish
node [shape = rectangle, style = filled, fillcolor = Linen]

"Metabolite\ncomposition" [shape = oval, fillcolor = Beige]

# edge definitions with the node IDs
"Mating system\ntreatment (E vs M)" -> {"Dry weight"}
"Mating system\ntreatment (E vs M)" -> {"Metabolite\ncomposition"} 

"Sex\n(Female vs Male)" -> {"Dry weight"} -> {"Metabolite\ncomposition"}
"Sex\n(Female vs Male)" -> {"Metabolite\ncomposition"}

{"Metabolite\ncomposition"} -> "Carbohydrates"
{"Metabolite\ncomposition"} -> "Chitin"
{"Metabolite\ncomposition"} -> "Glycogen"
{"Metabolite\ncomposition"} -> "Lipids"
{"Metabolite\ncomposition"} -> "Protein"
}')

Fit `brms` structural equation model

Here we fit a model of the five metabolites, which includes dry body weight as a mediator variable. That is, our model estimates the effect of treatment, sex and line (and all the 2- and 3-way interactions) on dry weight, and then estimates the effect of those some predictors (plus dry weight) on the five metabolites. The model assumes that although the different sexes, treatment groups, and lines may differ in their dry weight, the relationship between dry weight and the metabolites does not vary by sex/treatment/line. This assumption was made to constrain the number of parameters in the model, and to reflect out prior beliefs about allometric scaling of metabolites.

Define Priors

We set fairly tight Normal priors on all fixed effect parameters, which ‘regularises’ the estimates towards zero – this is conservative (because it ensures that a stronger signal in the data is needed to produce a given posterior effect size estimate), and it also helps the model to converge. Similarly, we set a somewhat conservative half-cauchy prior (mean 0, scale 0.01) on the random effects for line (i.e. we consider large differences between lines – in terms of means and treatment effects – to be possible but improbable). We leave all other priors at the defaults used by brms. Note that the Normal priors are slightly wider in the model of dry weight, because we expect larger effect sizes of sex and treatment on dry weight than on the metabolite composition.

prior1 <- c(set_prior("normal(0, 0.5)", class = "b", resp = 'Lipid'),
            set_prior("normal(0, 0.5)", class = "b", resp = 'Carbohydrate'),
            set_prior("normal(0, 0.5)", class = "b", resp = 'Protein'),
            set_prior("normal(0, 0.5)", class = "b", resp = 'Glycogen'),
            set_prior("normal(0, 0.5)", class = "b", resp = 'Chitin'),
            set_prior("normal(0, 1)", class = "b", resp = 'Dryweight'),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Lipid', group = "line"),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Carbohydrate', group = "line"),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Protein', group = "line"),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Glycogen', group = "line"),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Chitin', group = "line"),
            set_prior("cauchy(0, 0.01)", class = "sd", resp = 'Dryweight', group = "line"))

prior1

           prior class coef group         resp dpar nlpar bound source
  normal(0, 0.5)     b                   Lipid                    user
  normal(0, 0.5)     b            Carbohydrate                    user
  normal(0, 0.5)     b                 Protein                    user
  normal(0, 0.5)     b                Glycogen                    user
  normal(0, 0.5)     b                  Chitin                    user
    normal(0, 1)     b               Dryweight                    user
 cauchy(0, 0.01)    sd       line        Lipid                    user
 cauchy(0, 0.01)    sd       line Carbohydrate                    user
 cauchy(0, 0.01)    sd       line      Protein                    user
 cauchy(0, 0.01)    sd       line     Glycogen                    user
 cauchy(0, 0.01)    sd       line       Chitin                    user
 cauchy(0, 0.01)    sd       line    Dryweight                    user

Define the SEM’s sub-models

The fixed effects formula is sex * treatment + Dryweight (or sex * treatment in the case of the model of dry weight). The random effects part of the formula indicates that the 8 independent selection lines may differ in their means, and that the treatment effect may vary in sign/magnitude between lines. The notation | p | means that the model estimates the correlations in line effects (both slopes and intercepts) between the 6 response variables. Finally, the notation set_rescor(TRUE) means that the model should estimate the residual correlations between the response variables.

brms_formula <- 
  
  # Sub-models of the 5 metabolites
  bf(mvbind(Lipid, Carbohydrate, Protein, Glycogen, Chitin) ~ 
       sex * treatment + Dryweight + (1 | p | line)) +
  
  # dry weight sub-model
   bf(Dryweight ~ sex * treatment + (1 | p | line)) +
  
  # Allow for (and estimate) covariance between the residuals of the difference response variables
  set_rescor(TRUE)

# # specify models with random slopes terms for sex and treatment
# brms_formula <- 
#   
#   # Sub-models of the 5 metabolites
#   bf(mvbind(Lipid, Carbohydrate, Protein, Glycogen, Chitin) ~ 
#        sex*treatment + Dryweight + (sex + treatment| p | line)) +
#   
#   # dry weight sub-model
#    bf(Dryweight ~ sex * treatment + (sex + treatment| p | line)) +
#   
#   # Allow for (and estimate) covariance between the residuals of the difference response variables
#   set_rescor(TRUE)

# # specify models with random slopes terms for treatment
# brms_formula <- 
#   
#   # Sub-models of the 5 metabolites
#   bf(mvbind(Lipid, Carbohydrate, Protein, Glycogen, Chitin) ~ 
#        sex*treatment + Dryweight + (treatment | p | line)) +
#   
#   # dry weight sub-model
#    bf(Dryweight ~ sex * treatment + (treatment | p | line)) +
#   
#   # Allow for (and estimate) covariance between the residuals of the difference response variables
#   set_rescor(TRUE)

brms_formula

Lipid ~ sex * treatment + Dryweight + (1 | p | line) 
Carbohydrate ~ sex * treatment + Dryweight + (1 | p | line) 
Protein ~ sex * treatment + Dryweight + (1 | p | line) 
Glycogen ~ sex * treatment + Dryweight + (1 | p | line) 
Chitin ~ sex * treatment + Dryweight + (1 | p | line) 
Dryweight ~ sex * treatment + (1 | p | line)

Running the model

The model is run over 4 chains with 5000 iterations each (with the first 2500 discarded as burn-in), for a total of 2500*4 = 10,000 posterior samples.

if(!file.exists("output/brms_metabolite_SEM_noslope.rds")){
  
  brms_metabolite_SEM <- brm(
    brms_formula,
    data = scaled_metabolites %>% # brms does not like underscores in variable names
      rename(Dryweight = Dry_weight) %>%
      rename_all(~ gsub("_conc", "", .x)),
    iter = 5000, chains = 4, cores = 1,
    prior = prior1,
    control = list(max_treedepth = 20,
                   adapt_delta = 0.99)
  )
  
  saveRDS(brms_metabolite_SEM, "output/brms_metabolite_SEM_noslope.rds")
  #saveRDS(brms_metabolite_SEM, "output/brms_metabolite_SEM.rds") # save with treatment random slope 
  #saveRDS(brms_metabolite_SEM, "output/brms_metabolite_SEM_sextreatslope.rds") # save with sex and treatment random slopes
} else {
  brms_metabolite_SEM <- readRDS('output/brms_metabolite_SEM_noslope.rds')
}

Posterior predictive check of model fit

The plot below shows that the fitted model is able to produce posterior predictions that have a similar distribution to the original data, for each of the response variables, which is a necessary condition for the model to be used for statistical inference.

grid.arrange(
  pp_check(brms_metabolite_SEM, resp = "Dryweight") + 
    ggtitle("Dry weight") + theme(legend.position = "none"),
  pp_check(brms_metabolite_SEM, resp = "Lipid") + 
    ggtitle("Lipid") + theme(legend.position = "none"),
  pp_check(brms_metabolite_SEM, resp = "Carbohydrate") + 
    ggtitle("Carbohydrate") + theme(legend.position = "none"),
  pp_check(brms_metabolite_SEM, resp = "Protein") + 
    ggtitle("Protein") + theme(legend.position = "none"),
  pp_check(brms_metabolite_SEM, resp = "Glycogen") + 
    ggtitle("Glycogen") + theme(legend.position = "none"),
  pp_check(brms_metabolite_SEM, resp = "Chitin") + 
    ggtitle("Chitin") + theme(legend.position = "none"),
  nrow = 2
)

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
ffb09dd	MartinGarlovsky	2021-02-08
034431b	lukeholman	2020-12-18
bb96acf	lukeholman	2020-12-11
e5c580f	lukeholman	2020-12-11
31ed22a	lukeholman	2020-12-10
f7c88a2	lukeholman	2020-12-10

Table of model parameter estimates

Formatted table

This tables shows the fixed effects estimates for treatment, sex, their interaction, as well as the slope associated with dry weight (where relevant), for each of the six response variables. The p column shows 1 - minus the “probability of direction”, i.e. the posterior probability that the reported sign of the estimate is correct given the data and the prior; subtracting this value from one gives a Bayesian equivalent of a one-sided p-value. For brevity, we have omitted all the parameter estimates involving the predictor variable line, as well as the estimates of residual (co)variance. Click the next tab to see a complete summary of the model and its output.

vars <- c("Lipid", "Carbohydrate", "Glycogen", "Protein", "Chitin")
tests <- c('_Dryweight', '_sexMale',
           '_sexMale:treatmentPolyandry',
           '_treatmentPolyandry')

hypSEM <- data.frame(expand_grid(vars, tests) %>% 
                       mutate(est = NA,
                              err = NA,
                              lwr = NA,
                              upr = NA) %>% 
                       # bind body weight on the end
                       rbind(data.frame(
                         vars = rep('Dryweight', 3),
                         tests = c('_sexMale', 
                                   '_treatmentPolyandry', 
                                   '_sexMale:treatmentPolyandry'),
                         est = NA,
                         err = NA,
                         lwr = NA,
                         upr = NA)))

for(i in 1:nrow(hypSEM)) {
  
  result = hypothesis(brms_metabolite_SEM, 
                      paste0(hypSEM[i, 1], hypSEM[i, 2], ' = 0'))$hypothesis

  hypSEM[i, 3] = round(result$Estimate, 3)
  hypSEM[i, 4] = round(result$Est.Error, 3)
  hypSEM[i, 5] = round(result$CI.Lower, 3)
  hypSEM[i, 6] = round(result$CI.Upper, 3)

}

pvals <- bayestestR::p_direction(brms_metabolite_SEM) %>% 
  as.data.frame() %>%
  mutate(vars = map_chr(str_split(Parameter, "_"), ~ .x[2]),
         tests = map_chr(str_split(Parameter, "_"), ~ .x[3]),
         tests = str_c("_", str_remove_all(tests, "[.]")),
         tests = replace(tests, tests == "_sexMaletreatmentPolyandry", "_sexMale:treatmentPolyandry")) %>%
  filter(!str_detect(tests, "line")) %>%
  mutate(p_val = 1 - pd, star = ifelse(p_val < 0.05, "\\*", "")) %>%
  select(vars, tests, p_val, star)


hypSEM <- hypSEM %>% left_join(pvals, by = c("vars", "tests"))

hypSEM %>% 
  mutate(Parameter = c(rep(c('Dry weight', 'Sex (M)', 
                             'Sex (M) x Treatment (E)', 
                             'Treatment (E)'), 5), 
                       'Sex (M)', 'Treatment (E)', 'Sex (M) x Treatment (E)'))  %>% 
  mutate(Parameter = factor(Parameter, c("Dry weight", "Sex (M)", "Treatment (E)", "Sex (M) x Treatment (E)")),
         vars = factor(vars, c("Carbohydrate", "Chitin", "Glycogen", "Lipid", "Protein", "Dryweight"))) %>%
  arrange(vars, Parameter) %>%
  select(Parameter, Estimate = est, `Est. error` = err, 
         `CI lower` = lwr, `CI upper` = upr, `p` = p_val, star) %>% 
  rename(` ` = star) %>%
  #write_csv('output/metabolite_SEM_table.csv')
  mutate(p = ifelse(p > 0.001, round(p, 3), '< 0.001')) %>% 
  #write_csv('output/metabolite_SEM_slopes_table.csv')
  kable() %>% 
  kable_styling(full_width = FALSE) %>%
  group_rows("Carbohydrates", 1, 4) %>%
  group_rows("Chitin", 5, 8) %>%
  group_rows("Glycogen", 9, 12) %>%
  group_rows("Lipids", 13, 16) %>% 
  group_rows("Protein", 17, 20) %>% 
  group_rows("Dry weight", 21, 23)

Parameter	Estimate	Est. error	CI lower	CI upper	p
Carbohydrates
Dry weight	0.087	0.265	-0.441	0.600	0.364
Sex (M)	0.004	0.419	-0.804	0.844	0.499
Treatment (E)	-0.245	0.303	-0.838	0.353	0.205
Sex (M) x Treatment (E)	-0.418	0.348	-1.101	0.253	0.117
Chitin
Dry weight	-0.484	0.268	-1.014	0.041	0.036	*
Sex (M)	0.398	0.430	-0.455	1.244	0.176
Treatment (E)	-0.120	0.282	-0.658	0.446	0.326
Sex (M) x Treatment (E)	-0.313	0.325	-0.962	0.328	0.161
Glycogen
Dry weight	0.339	0.263	-0.177	0.856	0.101
Sex (M)	-0.268	0.419	-1.094	0.550	0.259
Treatment (E)	0.216	0.297	-0.359	0.803	0.233
Sex (M) x Treatment (E)	0.394	0.346	-0.292	1.059	0.126
Lipids
Dry weight	0.537	0.255	0.026	1.028	0.019	*
Sex (M)	-0.120	0.415	-0.930	0.680	0.393
Treatment (E)	0.392	0.271	-0.152	0.918	0.075
Sex (M) x Treatment (E)	-0.053	0.314	-0.681	0.547	0.431
Protein
Dry weight	-0.211	0.271	-0.734	0.331	0.214
Sex (M)	-0.009	0.427	-0.831	0.828	0.491
Treatment (E)	-0.211	0.312	-0.826	0.402	0.251
Sex (M) x Treatment (E)	0.343	0.357	-0.366	1.045	0.166
Dry weight
Sex (M)	-1.616	0.140	-1.894	-1.347	< 0.001	*
Treatment (E)	0.522	0.156	0.211	0.821	0.002	*
Sex (M) x Treatment (E)	-0.352	0.196	-0.741	0.043	0.038	*

Complete output from `summary.brmsfit()`

‘Group-Level Effects’ (also called random effects): This shows the (co)variances associated with the line-specific intercepts (which have names like sd(Lipid_Intercept)) and slopes (e.g. sd(Dryweight_treatmentPolyandry)), as well as the correlations between these effects (e.g. cor(Lipid_Intercept,Protein_Intercept) is the correlation in line effects on lipids and proteins)
‘Population-Level Effects:’ (also called fixed effects): These give the estimates of the intercept (i.e. for female M flies) and the effects of treatment, sex, dry weight, and the treatment \(\times\) sex interaction, for each response variable.
‘Family Specific Parameters’: This is the parameter sigma for the residual variance for each response variable
‘Residual Correlations:’ This give the correlations between the residuals for each pairs of response variables.

Note that the model has converged (Rhat = 1) and the posterior is adequately samples (high ESS values).

brms_metabolite_SEM

 Family: MV(gaussian, gaussian, gaussian, gaussian, gaussian, gaussian) 
  Links: mu = identity; sigma = identity
         mu = identity; sigma = identity
         mu = identity; sigma = identity
         mu = identity; sigma = identity
         mu = identity; sigma = identity
         mu = identity; sigma = identity 
Formula: Lipid ~ sex * treatment + Dryweight + (1 | p | line) 
         Carbohydrate ~ sex * treatment + Dryweight + (1 | p | line) 
         Protein ~ sex * treatment + Dryweight + (1 | p | line) 
         Glycogen ~ sex * treatment + Dryweight + (1 | p | line) 
         Chitin ~ sex * treatment + Dryweight + (1 | p | line) 
         Dryweight ~ sex * treatment + (1 | p | line) 
   Data: scaled_metabolites %>% rename(Dryweight = Dry_weig (Number of observations: 48) 
  Draws: 4 chains, each with iter = 5000; warmup = 2500; thin = 1;
         total post-warmup draws = 10000

Group-Level Effects: 
~line (Number of levels: 8) 
                                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Lipid_Intercept)                                 0.09      0.15     0.00     0.50 1.00     2226     3040
sd(Carbohydrate_Intercept)                          0.07      0.13     0.00     0.50 1.00     3422     2717
sd(Protein_Intercept)                               0.05      0.12     0.00     0.44 1.00     4846     3055
sd(Glycogen_Intercept)                              0.03      0.07     0.00     0.26 1.00     9019     5018
sd(Chitin_Intercept)                                0.07      0.12     0.00     0.44 1.00     3130     2735
sd(Dryweight_Intercept)                             0.05      0.08     0.00     0.27 1.00     2860     4008
cor(Lipid_Intercept,Carbohydrate_Intercept)         0.02      0.38    -0.70     0.71 1.00    13199     6832
cor(Lipid_Intercept,Protein_Intercept)              0.00      0.38    -0.70     0.71 1.00    14787     6474
cor(Carbohydrate_Intercept,Protein_Intercept)       0.01      0.38    -0.70     0.72 1.00    11274     6759
cor(Lipid_Intercept,Glycogen_Intercept)             0.01      0.37    -0.69     0.71 1.00    15679     6795
cor(Carbohydrate_Intercept,Glycogen_Intercept)      0.01      0.37    -0.69     0.70 1.00    11674     7312
cor(Protein_Intercept,Glycogen_Intercept)           0.00      0.38    -0.71     0.71 1.00     9312     7759
cor(Lipid_Intercept,Chitin_Intercept)               0.00      0.38    -0.70     0.71 1.00    12344     7086
cor(Carbohydrate_Intercept,Chitin_Intercept)       -0.00      0.38    -0.71     0.71 1.00    11084     7659
cor(Protein_Intercept,Chitin_Intercept)             0.01      0.38    -0.70     0.71 1.00     8263     7879
cor(Glycogen_Intercept,Chitin_Intercept)            0.01      0.38    -0.70     0.71 1.00     7362     7966
cor(Lipid_Intercept,Dryweight_Intercept)           -0.01      0.37    -0.69     0.69 1.00    11583     7502
cor(Carbohydrate_Intercept,Dryweight_Intercept)     0.01      0.38    -0.70     0.72 1.00     9377     7379
cor(Protein_Intercept,Dryweight_Intercept)          0.01      0.38    -0.70     0.71 1.00     8299     7795
cor(Glycogen_Intercept,Dryweight_Intercept)        -0.01      0.38    -0.70     0.71 1.00     7296     8107
cor(Chitin_Intercept,Dryweight_Intercept)          -0.01      0.38    -0.71     0.70 1.00     7056     8688

Population-Level Effects: 
                                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Lipid_Intercept                            -0.12      0.24    -0.58     0.36 1.00     8283     7353
Carbohydrate_Intercept                      0.22      0.27    -0.31     0.77 1.00     9902     7985
Protein_Intercept                           0.02      0.28    -0.52     0.56 1.00    10823     8268
Glycogen_Intercept                         -0.07      0.26    -0.58     0.43 1.00    10745     8442
Chitin_Intercept                           -0.06      0.26    -0.57     0.45 1.00     8522     6813
Dryweight_Intercept                         0.63      0.11     0.42     0.85 1.00     8348     7726
Lipid_sexMale                              -0.12      0.42    -0.93     0.68 1.00     6798     7196
Lipid_treatmentPolyandry                    0.39      0.27    -0.15     0.92 1.00     8798     7658
Lipid_Dryweight                             0.54      0.25     0.03     1.03 1.00     5568     6453
Lipid_sexMale:treatmentPolyandry           -0.05      0.31    -0.68     0.55 1.00     9816     7636
Carbohydrate_sexMale                        0.00      0.42    -0.80     0.84 1.00     7962     7496
Carbohydrate_treatmentPolyandry            -0.24      0.30    -0.84     0.35 1.00    10245     7539
Carbohydrate_Dryweight                      0.09      0.26    -0.44     0.60 1.00     6631     7408
Carbohydrate_sexMale:treatmentPolyandry    -0.42      0.35    -1.10     0.25 1.00    11374     7608
Protein_sexMale                            -0.01      0.43    -0.83     0.83 1.00     8339     7487
Protein_treatmentPolyandry                 -0.21      0.31    -0.83     0.40 1.00    10374     8676
Protein_Dryweight                          -0.21      0.27    -0.73     0.33 1.00     6785     7231
Protein_sexMale:treatmentPolyandry          0.34      0.36    -0.37     1.04 1.00    10331     7256
Glycogen_sexMale                           -0.27      0.42    -1.09     0.55 1.00     7302     7487
Glycogen_treatmentPolyandry                 0.22      0.30    -0.36     0.80 1.00     9860     8345
Glycogen_Dryweight                          0.34      0.26    -0.18     0.86 1.00     6689     7518
Glycogen_sexMale:treatmentPolyandry         0.39      0.35    -0.29     1.06 1.00    11426     8369
Chitin_sexMale                              0.40      0.43    -0.45     1.24 1.00     7576     7166
Chitin_treatmentPolyandry                  -0.12      0.28    -0.66     0.45 1.00     8551     7564
Chitin_Dryweight                           -0.48      0.27    -1.01     0.04 1.00     6202     6889
Chitin_sexMale:treatmentPolyandry          -0.31      0.32    -0.96     0.33 1.00     9734     8005
Dryweight_sexMale                          -1.62      0.14    -1.89    -1.35 1.00     8443     7594
Dryweight_treatmentPolyandry                0.52      0.16     0.21     0.82 1.00     6327     7005
Dryweight_sexMale:treatmentPolyandry       -0.35      0.20    -0.74     0.04 1.00     7145     7157

Family Specific Parameters: 
                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_Lipid            0.74      0.09     0.59     0.93 1.00     6746     7390
sigma_Carbohydrate     1.00      0.11     0.81     1.25 1.00     9278     7809
sigma_Protein          1.02      0.12     0.82     1.28 1.00     9185     7780
sigma_Glycogen         0.95      0.11     0.76     1.19 1.00    11909     7770
sigma_Chitin           0.85      0.10     0.67     1.06 1.00     7486     7023
sigma_Dryweight        0.36      0.04     0.29     0.46 1.00     8239     7406

Residual Correlations: 
                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
rescor(Lipid,Carbohydrate)        -0.32      0.14    -0.58    -0.04 1.00     5457     6371
rescor(Lipid,Protein)             -0.06      0.15    -0.34     0.24 1.00    10423     7214
rescor(Carbohydrate,Protein)       0.04      0.14    -0.25     0.31 1.00    11539     7956
rescor(Lipid,Glycogen)             0.04      0.15    -0.26     0.33 1.00     7796     6676
rescor(Carbohydrate,Glycogen)     -0.01      0.15    -0.29     0.28 1.00    11161     7736
rescor(Protein,Glycogen)          -0.13      0.14    -0.41     0.16 1.00    10372     7470
rescor(Lipid,Chitin)              -0.07      0.15    -0.35     0.22 1.00     8329     7466
rescor(Carbohydrate,Chitin)       -0.42      0.12    -0.64    -0.16 1.00     9965     8172
rescor(Protein,Chitin)             0.06      0.15    -0.22     0.35 1.00     9097     7373
rescor(Glycogen,Chitin)           -0.04      0.15    -0.32     0.25 1.00     9577     7869
rescor(Lipid,Dryweight)            0.16      0.18    -0.20     0.49 1.00     5978     6731
rescor(Carbohydrate,Dryweight)     0.02      0.17    -0.33     0.35 1.00     5275     5847
rescor(Protein,Dryweight)         -0.04      0.17    -0.38     0.30 1.00     6616     7322
rescor(Glycogen,Dryweight)         0.01      0.17    -0.33     0.35 1.00     6280     7242
rescor(Chitin,Dryweight)           0.25      0.18    -0.12     0.57 1.00     4907     5998

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

Posterior effect size of treatment on metabolite abundance, for each sex

Here, we use the model to predict the mean concentration of each metabolite (in standard units) in each treatment and sex (averaged across the eight replicate selection lines). We then calculate the effect size of treatment by subtracting the (sex-specific) mean for the M treatment from the mean for the E treatment; thus a value of 1 would mean that the E treatment has a mean that is larger by 1 standard deviation. Thus, the y-axis in the following graphs essentially shows the posterior estimate of standardised effect size (Cohen’s d), from the model shown above.

Because the model contains dry weight as a mediator variable, we created these predictions two different ways, and display the answer for both using tabs in the following figures/tables. Firstly, we predicted the means controlling for differences in dry weight between sexes and treatments; this was done by deriving the predictions with dry weight set to its global mean, for both sexes and treatments. Secondly, we derived predictions without controlling for dry weight. This was done by deriving the predictions with dry weight set to its average value for the appropriate treatment-sex combination.

By clicking the tabs and comparing, one can see that the estimates of the treatment effect hardly change when differences in dry weight are controlled for. This indicates that dry mass does not have an important role in mediating the effect of treatment on metabolite composition, even though body size differs between treatments. Thus, we conclude that the M vs E treatments caused metabolite composition to evolve, through mechanisms other than the evolution of dry weight.

Figure

Not controlling for differences in dry weight between treatments

new <- expand_grid(sex = c("Male", "Female"),
                   treatment = c("Monogamy", "Polyandry"),
                   Dryweight = NA, line = NA) %>%
  mutate(type = 1:n())

levels <- c("Carbohydrate", "Chitin", "Glycogen", "Lipid", "Protein", "Dryweight")

# Estimate mean dry weight for each of the 4 sex/treatment combinations
evolved_mean_dryweights <- data.frame(
  new[,1:2], 
  fitted(brms_metabolite_SEM, re_formula = NA,
         newdata = new %>% select(-Dryweight), 
         summary = TRUE, resp = "Dryweight")) %>%
  as_tibble()

# Find the mean dry weight for males and females (across treatments)
male_dryweight <- mean(evolved_mean_dryweights$Estimate[1:2])
female_dryweight <- mean(evolved_mean_dryweights$Estimate[3:4])

new_metabolites <- bind_rows(
  expand_grid(sex = c("Male", "Female"),
              treatment = c("Monogamy", "Polyandry"),
              Dryweight = c(male_dryweight, female_dryweight), line = NA) %>%
    filter(sex == "Male" & Dryweight == male_dryweight |
             sex == "Female" & Dryweight == female_dryweight) %>%
    mutate(type = 1:4),
  evolved_mean_dryweights %>% select(sex, treatment, Dryweight = Estimate) %>%
    mutate(line = NA, type = 5:8)
)


# Predict data from the SEM of metabolites...

# Because we use sum contrasts for "line" and line=NA in the new data, 
# this function predicts at the global means across the 4 lines (see ?posterior_epred)
fitted_values <- posterior_epred(
  brms_metabolite_SEM, newdata = new_metabolites, re_formula = NA,
  summary = FALSE) %>% 
  reshape2::melt() %>% rename(draw = Var1, type = Var2, variable = Var3) %>% 
  as_tibble() %>%
  left_join(new_metabolites, by = "type") %>%
  select(draw, variable, value, sex, treatment, Dryweight) %>%
  mutate(variable = factor(variable, levels))


treat_diff_standard_dryweight <- fitted_values %>%
  filter(Dryweight %in% c(male_dryweight, female_dryweight)) %>%
  spread(treatment, value) %>%
  mutate(`Difference in means (Poly - Mono)` = Polyandry - Monogamy) 

treat_diff_actual_dryweight <- fitted_values %>%
  filter(!(Dryweight %in% c(male_dryweight, female_dryweight))) %>% 
  select(-Dryweight) %>%
  spread(treatment, value) %>%
  mutate(`Difference in means (Poly - Mono)` = Polyandry - Monogamy) 

summary_dat1 <- treat_diff_actual_dryweight %>%
  filter(variable != 'Dryweight') %>%
  rename(x = `Difference in means (Poly - Mono)`) %>%
  group_by(variable, sex)  %>%
  summarise(`Difference in means (Poly - Mono)` = median(x),
            `Lower 95% CI` = quantile(x, probs = 0.025), 
            `Upper 95% CI` = quantile(x, probs = 0.975),
            p = 1 - as.numeric(bayestestR::p_direction(x)),
            ` ` = ifelse(p < 0.05, "\\*", ""),
            .groups = "drop")

summary_dat2 <- treat_diff_standard_dryweight %>%
  filter(variable != 'Dryweight') %>%
  rename(x = `Difference in means (Poly - Mono)`) %>%
  group_by(variable, sex)  %>%
  summarise(`Difference in means (Poly - Mono)` = median(x),
            `Lower 95% CI` = quantile(x, probs = 0.025), 
            `Upper 95% CI` = quantile(x, probs = 0.975),
            p = 1 - as.numeric(bayestestR::p_direction(x)),
            ` ` = ifelse(p < 0.05, "\\*", ""),
            .groups = "drop")

sampled_draws <- sample(unique(fitted_values$draw), 100)

ylims <- c(-1.8, 1.8)

p1 <- treat_diff_actual_dryweight %>%
  filter(variable != 'Dryweight') %>% 
  ggplot(aes(x = sex, y = `Difference in means (Poly - Mono)`,fill = sex)) +
  geom_hline(yintercept = 0, linetype = 2) +
  stat_halfeye() + 
  # geom_line(data = treat_diff_actual_dryweight %>%
  #             filter(draw %in% sampled_draws) %>%
  #             filter(variable != 'Dryweight'), 
  #           alpha = 0.8, size = 0.12, colour = "black", aes(group = draw)) +
  geom_point(data = summary_dat1, pch = 21, colour = "black", size = 3.1) + 
  scale_fill_brewer(palette = 'Pastel1', direction = 1, name = "") +
  scale_colour_brewer(palette = 'Pastel1', direction = 1, name = "") +
  facet_wrap( ~ variable,  nrow = 1) +
  theme_bw() +
  theme(legend.position = 'none',
        strip.background = element_blank(),
        text = element_text(family = nice_font),
        panel.grid.major.x = element_blank()) +
  coord_cartesian(ylim = ylims) + 
  ylab("Difference in means between\nselection treatments (E - M)") + xlab("Sex") +
  #ggsave("figures/metabolite_plot_nolines.pdf", height=2.5, width=6) +
  NULL
p1

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
ffb09dd	MartinGarlovsky	2021-02-08
034431b	lukeholman	2020-12-18
b6cf554	lukeholman	2020-12-11
a88f037	lukeholman	2020-12-11
bb96acf	lukeholman	2020-12-11
e5c580f	lukeholman	2020-12-11
7fca240	lukeholman	2020-12-10
f7c88a2	lukeholman	2020-12-10
43cc270	lukeholman	2020-12-09

Figure XX: Posterior estimates of the treatment effect size for both sexes, for each of the five metabolites. A positive value means that the mean metabolite concentration is higher in the E treatment than the M treatment, while a negative effects denotes M > E. A strongly supported treatment effect is implied by the majority of the posterior lying to one side of zero. The error bars summarise the 66% and 95% quantiles of the posterior. This plot was created used posterior predictions of the means that were not adjusted for differences in dry weight between treatments.

Controlling for differences in dry weight between treatments

treat_diff_standard_dryweight %>%
  filter(variable != 'Dryweight') %>% 
  ggplot(aes(x = sex, y = `Difference in means (Poly - Mono)`, fill = sex)) +
  geom_hline(yintercept = 0, linetype = 2) +
  stat_halfeye() + 
  # geom_line(data = treat_diff_standard_dryweight %>%
  #             filter(draw %in% sampled_draws) %>%
  #             filter(variable != 'Dryweight'), 
  #           alpha = 0.8, size = 0.12, colour = "black", aes(group = draw)) +
  geom_point(data = summary_dat2, pch = 21, colour = "black", size = 3.1) + 
  scale_fill_brewer(palette = 'Pastel1', direction = 1, name = "") +
  scale_colour_brewer(palette = 'Pastel1', direction = 1, name = "") +
  facet_wrap( ~ variable,  nrow = 1) +
  theme_bw() +
  theme(legend.position = 'none',
        strip.background = element_blank(),
        text = element_text(family = nice_font),
        panel.grid.major.x = element_blank()) +
  coord_cartesian(ylim = ylims) + 
  ylab("Difference in means between\nselection treatments (E - M)") + xlab("Sex") + 
  #ggsave('figures/metabolite_plotCONTROLLED_nolines.pdf', height=2.5, width=6) +
  NULL

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
ffb09dd	MartinGarlovsky	2021-02-08
034431b	lukeholman	2020-12-18
b6cf554	lukeholman	2020-12-11
a88f037	lukeholman	2020-12-11
bb96acf	lukeholman	2020-12-11
e5c580f	lukeholman	2020-12-11
7fca240	lukeholman	2020-12-10
f7c88a2	lukeholman	2020-12-10

Figure XX: Posterior estimates of the treatment effect size for both sexes, for each of the five metabolites. A positive value means that the mean metabolite concentration is higher in the P treatment than the M treatment, while a negative effects denotes M > E. A strongly supported treatment effect is implied by the majority of the posterior lying to one side of zero. The error bars summarise the 66% and 95% quantiles of the posterior. This plot was created used posterior predictions of the means that were adjusted for differences in dry weight between treatments.

Table

Not controlling for differences in dry weight between treatments

summary_dat1 %>%
  kable(digits = 3) %>%
  kable_styling(full_width = FALSE)

variable	sex	Difference in means (Poly - Mono)	Lower 95% CI	Upper 95% CI	p
Carbohydrate	Female	-0.198	-0.759	0.370	0.237
Carbohydrate	Male	-0.650	-1.320	0.034	0.030	*
Chitin	Female	-0.374	-0.884	0.143	0.077
Chitin	Male	-0.518	-1.087	0.065	0.041	*
Glycogen	Female	0.391	-0.139	0.931	0.076
Glycogen	Male	0.672	0.035	1.285	0.021	*
Lipid	Female	0.674	0.184	1.150	0.006	*
Lipid	Male	0.432	-0.139	0.979	0.064
Protein	Female	-0.321	-0.899	0.268	0.138
Protein	Male	0.101	-0.575	0.773	0.386

Controlling for differences in dry weight between treatments

summary_dat2 %>%
  kable(digits = 3) %>%
  kable_styling(full_width = FALSE)

variable	sex	Difference in means (Poly - Mono)	Lower 95% CI	Upper 95% CI	p
Carbohydrate	Female	-0.245	-0.838	0.353	0.205
Carbohydrate	Male	-0.663	-1.326	0.015	0.027	*
Chitin	Female	-0.124	-0.658	0.446	0.326
Chitin	Male	-0.435	-1.002	0.147	0.074
Glycogen	Female	0.214	-0.359	0.803	0.233
Glycogen	Male	0.612	-0.023	1.225	0.029	*
Lipid	Female	0.390	-0.152	0.918	0.075
Lipid	Male	0.343	-0.217	0.888	0.113
Protein	Female	-0.210	-0.826	0.402	0.251
Protein	Male	0.132	-0.539	0.803	0.343

Posterior difference in treatment effect size between sexes

This section essentially examines the treatment \(\times\) sex interaction term, by calculating the difference in the effect size of the E/M treatment between sexes, for each of the five metabolites. We find no strong evidence for a treatment \(\times\) sex interaction, i.e. the treatment effects did not differ detectably between sexes.

Figure

Not controlling for differences in dry weight between treatments

treatsex_interaction_data1 <- treat_diff_actual_dryweight %>%
  select(draw, variable, sex, d =  `Difference in means (Poly - Mono)`) %>%
  arrange(draw, variable, sex) %>%
  group_by(draw, variable) %>%
  summarise(`Difference in effect size between sexes (male - female)` = d[2] - d[1],
            .groups = "drop") # males - females


p2 <- treatsex_interaction_data1 %>%
  filter(variable != 'Dryweight') %>% 
  ggplot(aes(x = `Difference in effect size between sexes (male - female)`, y = 1, fill = stat(x < 0))) +
  geom_vline(xintercept = 0, linetype = 2) +
  stat_halfeye() +
  facet_wrap( ~ variable) +
  scale_fill_brewer(palette = 'Pastel2', direction = 1, name = "") +
  theme_bw() +
  theme(legend.position = 'none',
        text = element_text(family = nice_font),
        strip.background = element_blank()) +
  ylab("Posterior density") +
  #ggsave("figures/metabolite_interaction_plot.pdf", height=4, width=6) +
  NULL

p2

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
034431b	lukeholman	2020-12-18
e5c580f	lukeholman	2020-12-11
f7c88a2	lukeholman	2020-12-10
43cc270	lukeholman	2020-12-09

Figure XX: Posterior estimates of the difference in the treatment effect size (i.e. mean of E minus mean of M) between males and females, for each of the five metabolites. A positive value means that the effect size is more positive in males, and negative means it is more positive in females. A strongly supported sex difference in effect size would be implied by the majority of the posterior lying to one side of zero. The error bars summarise the 66% and 95% quantiles of the posterior. This plot was created used posterior predictions of the means that were not adjusted for differences in dry weight between treatments.

Controlling for differences in dry weight between treatments

treatsex_interaction_data2 <- treat_diff_standard_dryweight %>%
  select(draw, variable, sex, d =  `Difference in means (Poly - Mono)`) %>%
  arrange(draw, variable, sex) %>%
  group_by(draw, variable) %>%
  summarise(`Difference in effect size between sexes (male - female)` = d[2] - d[1],
            .groups = "drop") # males - females

treatsex_interaction_data2 %>%
  filter(variable != 'Dryweight') %>% 
  ggplot(aes(x = `Difference in effect size between sexes (male - female)`, y = 1, fill = stat(x < 0))) +
  geom_vline(xintercept = 0, linetype = 2) +
  stat_halfeye() +
  facet_wrap( ~ variable) +
  scale_fill_brewer(palette = 'Pastel2', direction = 1, name = "") +
  theme_bw() +
  theme(legend.position = 'none',
        text = element_text(family = nice_font),
        strip.background = element_blank()) +
  ylab("Posterior density")

Version	Author	Date
aa3e11e	MartinGarlovsky	2021-10-05
21567bb	MartinGarlovsky	2021-03-12
034431b	lukeholman	2020-12-18
e5c580f	lukeholman	2020-12-11
f7c88a2	lukeholman	2020-12-10

Figure XX: Posterior estimates of the difference in the treatment effect size (i.e. mean of E minus mean of M) between males and females, for each of the five metabolites. A positive value means that the effect size is more positive in males, and negative means it is more positive in females. A strongly supported sex difference in effect size would be implied by the majority of the posterior lying to one side of zero. The error bars summarise the 66% and 95% quantiles of the posterior. This plot was created used posterior predictions of the means that were adjusted for differences in dry weight between treatments.

Table

Not controlling for differences in dry weight between treatments

treatsex_interaction_data1 %>%
  filter(variable != 'Dryweight') %>%
  rename(x = `Difference in effect size between sexes (male - female)`) %>%
  group_by(variable)  %>%
  summarise(`Difference in effect size between sexes (male - female)` = median(x),
            `Lower 95% CI` = quantile(x, probs = 0.025), 
            `Upper 95% CI` = quantile(x, probs = 0.975),
            p = 1 - as.numeric(bayestestR::p_direction(x)),
            ` ` = ifelse(p < 0.05, "\\*", ""),
            .groups = "drop") %>%
  kable(digits=3) %>%
  kable_styling(full_width = FALSE)

variable	Difference in effect size between sexes (male - female)	Lower 95% CI	Upper 95% CI	p
Carbohydrate	-0.451	-1.103	0.209	0.092
Chitin	-0.139	-0.743	0.479	0.318
Glycogen	0.279	-0.381	0.900	0.202
Lipid	-0.244	-0.831	0.326	0.207
Protein	0.422	-0.255	1.098	0.112

Controlling for differences in dry weight between treatments

treatsex_interaction_data2 %>%
  filter(variable != 'Dryweight') %>%
  rename(x = `Difference in effect size between sexes (male - female)`) %>%
  group_by(variable)  %>%
  summarise(`Difference in effect size between sexes (male - female)` = median(x),
            `Lower 95% CI` = quantile(x, probs = 0.025), 
            `Upper 95% CI` = quantile(x, probs = 0.975),
            p = 1 - as.numeric(bayestestR::p_direction(x)),
            ` ` = ifelse(p < 0.05, "\\*", ""),
            .groups = "drop") %>%
  kable(digits=3) %>%
  kable_styling(full_width = FALSE)

variable	Difference in effect size between sexes (male - female)	Lower 95% CI	Upper 95% CI	p
Carbohydrate	-0.420	-1.101	0.253	0.117
Chitin	-0.308	-0.962	0.328	0.161
Glycogen	0.399	-0.292	1.059	0.126
Lipid	-0.053	-0.681	0.547	0.431
Protein	0.344	-0.366	1.045	0.166

sessionInfo()

R version 4.0.3 (2020-10-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] showtext_0.9-4   showtextdb_3.0   sysfonts_0.8.5   knitrhooks_0.0.4 knitr_1.33       kableExtra_1.3.4 DT_0.18          tidybayes_3.0.1  brms_2.16.3     
[10] Rcpp_1.0.7       ggridges_0.5.3   gridExtra_2.3    GGally_2.1.2     forcats_0.5.1    stringr_1.4.0    dplyr_1.0.7      purrr_0.3.4      readr_2.0.1     
[19] tidyr_1.1.4      tibble_3.1.5     ggplot2_3.3.5    tidyverse_1.3.1  workflowr_1.6.2 

loaded via a namespace (and not attached):
  [1] readxl_1.3.1         backports_1.2.1      systemfonts_1.0.2    plyr_1.8.6           igraph_1.2.6         svUnit_1.0.6         splines_4.0.3       
  [8] crosstalk_1.1.1      TH.data_1.0-10       rstantools_2.1.1     inline_0.3.19        digest_0.6.28        htmltools_0.5.1.1    rsconnect_0.8.24    
 [15] fansi_0.5.0          magrittr_2.0.1       checkmate_2.0.0      tzdb_0.1.2           modelr_0.1.8         RcppParallel_5.1.4   matrixStats_0.60.0  
 [22] vroom_1.5.4          svglite_2.0.0        xts_0.12.1           sandwich_3.0-1       prettyunits_1.1.1    colorspace_2.0-2     rvest_1.0.1         
 [29] ggdist_3.0.0         haven_2.4.3          xfun_0.25            callr_3.7.0          crayon_1.4.2         jsonlite_1.7.2       lme4_1.1-27.1       
 [36] survival_3.2-12      zoo_1.8-9            glue_1.5.0           gtable_0.3.0         emmeans_1.7.0        webshot_0.5.2        distributional_0.2.2
 [43] pkgbuild_1.2.0       rstan_2.21.3         abind_1.4-5          scales_1.1.1         mvtnorm_1.1-2        DBI_1.1.1            miniUI_0.1.1.1      
 [50] viridisLite_0.4.0    xtable_1.8-4         bit_4.0.4            stats4_4.0.3         StanHeaders_2.21.0-7 datawizard_0.2.0     htmlwidgets_1.5.3   
 [57] httr_1.4.2           DiagrammeR_1.0.6.1   threejs_0.3.3        arrayhelpers_1.1-0   RColorBrewer_1.1-2   posterior_1.0.1      ellipsis_0.3.2      
 [64] pkgconfig_2.0.3      reshape_0.8.8        loo_2.4.1            farver_2.1.0         sass_0.4.0           dbplyr_2.1.1         utf8_1.2.2          
 [71] labeling_0.4.2       tidyselect_1.1.1     rlang_0.4.12         reshape2_1.4.4       later_1.3.0          visNetwork_2.0.9     munsell_0.5.0       
 [78] cellranger_1.1.0     tools_4.0.3          cli_3.1.0            generics_0.1.1       broom_0.7.9          evaluate_0.14        fastmap_1.1.0       
 [85] yaml_2.2.1           bit64_4.0.5          processx_3.5.2       fs_1.5.0             nlme_3.1-152         projpred_2.0.2       whisker_0.4         
 [92] mime_0.11            xml2_1.3.2           compiler_4.0.3       bayesplot_1.8.1      shinythemes_1.2.0    rstudioapi_0.13      curl_4.3.2          
 [99] gamm4_0.2-6          reprex_2.0.1         bslib_0.2.5.1        stringi_1.7.5        highr_0.9            ps_1.6.0             Brobdingnag_1.2-6   
[106] lattice_0.20-44      Matrix_1.3-4         nloptr_1.2.2.2       markdown_1.1         shinyjs_2.0.0        tensorA_0.36.2       vctrs_0.3.8         
[113] pillar_1.6.4         lifecycle_1.0.1      jquerylib_0.1.4      bridgesampling_1.1-2 estimability_1.3     insight_0.14.3       httpuv_1.6.2        
[120] R6_2.5.1             promises_1.2.0.1     codetools_0.2-18     boot_1.3-28          colourpicker_1.1.0   MASS_7.3-54          gtools_3.9.2        
[127] assertthat_0.2.1     rprojroot_2.0.2      withr_2.4.2          shinystan_2.5.0      multcomp_1.4-17      bayestestR_0.10.5    mgcv_1.8-36         
[134] parallel_4.0.3       hms_1.1.0            grid_4.0.3           minqa_1.2.4          coda_0.19-4          rmarkdown_2.10       git2r_0.28.0        
[141] shiny_1.6.0          lubridate_1.7.10     base64enc_0.1-3      dygraphs_1.1.1.6

Metabolite composition

Load packages

Load metabolite composition data

Inspect the raw data

Raw numbers

Simple plots

Plot of correlations between variables

Mean dry weight

Directed acyclic graph (DAG)

Fit brms structural equation model

Define Priors

Define the SEM’s sub-models

Running the model

Posterior predictive check of model fit

Table of model parameter estimates

Formatted table

Complete output from summary.brmsfit()

Posterior effect size of treatment on metabolite abundance, for each sex

Figure

Not controlling for differences in dry weight between treatments

Controlling for differences in dry weight between treatments

Table

Not controlling for differences in dry weight between treatments

Controlling for differences in dry weight between treatments

Posterior difference in treatment effect size between sexes

Figure

Not controlling for differences in dry weight between treatments

Controlling for differences in dry weight between treatments

Table

Not controlling for differences in dry weight between treatments

Controlling for differences in dry weight between treatments

Fit `brms` structural equation model

Complete output from `summary.brmsfit()`