Binomial data

The response variable y is considered to follow a binomial distribution when it is a count out of a total (such as the percentage survival of individuals, for example). First, we read in the binomial example from pastebin, prepare the data for analysis, and then inspect the dataset as well as the “concentration” or x variable, in this case raw_x.

binom_data <- "https://pastebin.com/raw/zfrUha88" %>%
  read.table(header = TRUE, dec = ",", stringsAsFactors = FALSE) %>%
  dplyr::rename(raw_x = raw.x) %>%
  dplyr::mutate(raw_x = as.numeric(as.character(raw_x)),
                log_x = log(raw_x))

str(binom_data)
range(binom_data$raw_x)
hist(binom_data$raw_x)

In this case for x, lowest concentration is 0.1 and the highest is 400. The data are right skewed and on the continuous scale. This type of distribution for the x data are common for concentration—response experiments, where the x ‘concentration’ data are the concentration of contaminants, or dilutions. The current default in bayesnec is to estimate the appropriate distribution(s) and priors for the family argument, but it is possible to supply family directly. We are going to model this with the x data on a log scale, as this is the scaling clearly used in the experimental design and thus will provide more stable results.

The data are clearly binomial, with the column suc—indicating the number of ‘successes’ in the binomial call, with tot clearly indicating the number of trials.

The main ‘working’ function in bayesnec is the function bnec, which calls the other necessary functions and fits the brms model. We run bnec by supplying data: a data.frame containing the data for the model fitting, here, binom_data; x_var: the name of the column in data which contains the concentration data or ‘x’ data to be used in the NEC model fit, and y_var: the name of the column in data which contains the response or ‘y’ data to be used in the NEC model fit. In our example here, as this is binomial, we must also supply trials_var, which is the name of the column in data which contains the number of trials in the binomial call.

bnec will guess the data types for use, although as mentioned above we could manually specify family as “binomial”. bnec will also generate appropriate priors for the brms model call, although these can also be specified manually (see the Priors vignette for more details).

library(bayesnec)

set.seed(333)
exp_1 <- bnec(data = binom_data, x_var = "log_x",
              y_var = "suc", model = "neclin",
              trials_var = "tot")

The function shows the progress of the brms fit and returns the usual brms output (with a few other elements added to this list). The function plot(exp_1$fit) can be used to plot the chains, so we can assess mixing and look for other potential issues with the model fit. Initially bayesnec will attempt to use starting values generated for that type of model formula and family. It will run the iterations and then test if all chains are valid. If the model does not have valid chains bayesnec with try up to n_tries more times to fit the data using either bayesnec generated or brms default initial values and obtain a successfully fitted model with good mixing. If no model is successfully fit an error will be returned indicating the model could not be fit successfully.

plot(exp_1$fit)

In our example, the chains are well mixed, and the probability density plots look good. We can also run a pairs plot that can help to assess issues with identifiability, and which also looks ok. There are a range of other model diagnostics that can be explored for [brms] model fits, using the $fit syntax. We encourage you to explore the rich material already on github regarding use and validation of [brms] (https://github.com/paul-buerkner/brms) models.

pairs(exp_1$fit)

We can see the summary of our fitted model parameters using:

summary(exp_1)

Note the Rhat values in this example are one, indicating convergence.

The function plot can be used to plot the fitted model. You can also make your own plot from the data included in the returned bayesnecfit object from the call to bnec. Here we use the default plot method from bayesnec, where we add the ‘exp’ function to plot the tick labels on the original scaling of the concentration data.

plot(exp_1, lxform = exp)

Alternatively, we can use the built in brms methods to plot the brms fit directly. For example:

df <- exp_1$fit$data %>%
  dplyr::mutate(prop = y / trials)
plot(brms::conditional_effects(exp_1$fit))[[1]] +
     geom_point(data = df, aes(x = x, y = prop),
                inherit.aes = FALSE)

This model fit doesn’t look great. You can see that the error bounds around the fit are far too narrow for this data, suggesting over dispersion of this model (meaning that the data are more variable than this model fit predicts). An estimate of dispersion is provided by bayesnec, and this can be extracted using exp_1$dispersion. Values >1 indicate overdispersion and values <1 indicate underdispersion. In this case the overdispersion value is much bigger than 1, suggesting extreme overdispersion (meaning our model doesn’t properly capture the true variability represented in this data). We would need to consider alternative ways of modelling this data using a different distribution, such as the Beta-Binomial.

exp_1$dispersion

Beta-Binomial

The Beta-Binomial model can be useful for overdispersed Binomial data.

set.seed(333)
exp_1b <- bnec(data = binom_data, x_var = "log_x",
              y_var = "suc", model = "neclin",
              family = beta_binomial2,
              trials_var = "tot")

Fitting this data with the betabinomial2 yields a much more realistic fit in terms of the confidence bounds and the spread in the data. Note that a dispersion estimate is not provided here, as overdispersion is only relevant for Poisson and Binomial data.

plot(exp_1b, lxform = exp)
exp_1b$dispersion
#[1] NA NA NA

Now we have a good fit to these data, we can interpret the results. The estimated NEC value can be obtained directly from the fitted model object, using exp_1b$nec, or we can plot the posterior distribution of the NEC values.

exp_1b$nec
#Estimate     Q2.5    Q97.5 
#3.436315 2.674583 4.00693

hist(exp_1b$nec_posterior)

ECx estimates can also be obtained from the NEC model fit, using the function ecx. Note these may differ from a typical 4-parameter non-linear model, as the NEC model is a broken stick non-linear regression and will often fall more sharply than a smooth 4-parameter non-linear curve. See both the Model details and Comparing posterior predictions vignettes for more information.

ecx(exp_1b)
#  ec_10_Q50  ec_10_Q2.5 ec_10_Q97.5 
#   3.799644    3.276595    4.223064 

A summary method has also been developed for bayesnecfit objects that gives an overall summary of the model statistics, which also include the estimate for NEC, as the nec_intercept population level effect in the model.

summary(exp_1b)
# Object of class bayesnecfit containing the following non-linear model: neclin
# 
#  Family: beta_binomial2 
#   Links: mu = logit; phi = identity 
# Formula: y | trials(trials) ~ top - slope * (x - nec) * step(x - nec) 
#          top ~ 1
#          slope ~ 1
#          nec ~ 1
#    Data: structure(list(x = c(-2.30258509299405, -2.3025850 (Number of observations: 48) 
# Samples: 4 chains, each with iter = 10000; warmup = 9000; thin = 1;
#          total post-warmup samples = 4000
# 
# Population-Level Effects: 
#                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
# top_Intercept       1.83      0.23     1.38     2.34 1.00     1667     1751
# slope_Intercept     1.52      0.24     1.06     2.04 1.00     1947     2129
# nec_Intercept       3.41      0.30     2.67     3.94 1.00     1625     1618
# 
# Family Specific Parameters: 
#     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
# phi     5.74      1.30     3.55     8.75 1.00     2276     2206
# 
# Samples were drawn 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).

Beta data

Sometimes the response variable is distributed between 0 and 1 but is not a straight-forward Binomial. A common example in coral ecology is maximum quantum yield (the proportion of light used for photosynthesis when all reaction centres are open) which is a measure of photosynthetic efficiency calculated from PAM data. Here we have a proportion value that is not based on trials and successes. In this case there are no theoretical ‘trials’ and the data must be modelled using a beta distribution.

prop_data <- "https://pastebin.com/raw/123jq46d" %>%
  read.table(header = TRUE, dec = ",", stringsAsFactors = FALSE) %>%
  dplyr::rename(raw_x = raw.x) %>%
  dplyr::mutate(raw_x = log(as.numeric(as.character(raw_x)) + 1),
                resp = as.numeric(as.character(resp)))

set.seed(333)
exp_2 <- bnec(data = prop_data, x_var = "raw_x",
              y_var = "resp", model = "neclin")

plot(exp_2$fit)

plot(exp_2)

Poisson data

Where data are a count (of, for example, individuals or cells) y is Poisson. Such data are distributed from 0 to Inf and are integers. First we read in the count data example from pastebin, and then plot the “concentration” or x data, Again, this is raw_x, and distributed as in our binomial example above.

count_data <- "https://pastebin.com/raw/ENgNSgf7" %>%
  read.table(header = TRUE, dec = ",", stringsAsFactors = FALSE) %>%
  dplyr::rename(raw_x = raw.x) %>%
  dplyr::mutate(raw_x = as.numeric(as.character(raw_x)),
                sqrt_x = sqrt(raw_x))

str(count_data)
range(count_data$raw_x)

par(mfrow = c(1, 2))
hist(count_data$raw_x, xlab = "'x' variable", main = "")
hist(count_data$count, xlab = "Total counts", main = "")

First, we supply bnec with data (count_data), and specify x_var and y_var. As we have integers of 0 and greater, the family is “poisson”. The default behaviour to guess the variable types works for this example.

set.seed(333)
exp_3 <- bnec(data = count_data, x_var = "sqrt_x",
              y_var = "count", model = "neclin")

We first plot the model chains and parameter estimates to check the fit.

plot(exp_3$fit)

plot(exp_3)
exp_3$dispersion
#Estimate     Q2.5    Q97.5 
#2.915520 1.927265 4.526432 

The chains look ok; however, our plot of the fit is not very convincing. The error bars are very narrow, and this does not seem to be a particularly good model for these data. Note that the dispersion estimate is much greater than one, indicating serious overdispersion. In this case we can try a negative binomial family.

Negative binomial

When count data are overdispersed and cannot be modelled using the poisson family, the negative binomial family is generally used. We can do this by calling family = "negbinomial".

set.seed(333)
exp_3b <- bnec(data = count_data, x_var = "sqrt_x",
              y_var = "count", family = "negbinomial", model = "neclin")

The resultant plot seems to indicate the negative binomial family works better in terms of dispersion (more sensible wider confidence bands) however it still does not seem to be a good model for these data. See the Model details and Multi model usage vignettes for options to fit other models to these data.

plot(exp_3b)

plot(exp_3b$fit)

Measure data

Where data are a measured variable y is family gamma. Good examples of gamma distributed data include measures of body size, such as length, weight, or area. Such data are distributed from 0+ to Inf and are continuous. Here we use the nec_data supplied with bayesnec with y on the exponential scale to ensure the right data range for a gamma as an example.

data(nec_data)
measure_data <- nec_data %>%
  dplyr::mutate(measure = exp(y))

set.seed(333)
exp_4 <- bnec(data = measure_data, x_var = "x",
              y_var = "measure", model = "neclin")

plot(exp_4$fit)

plot(exp_4)
summary(exp_4)
#Object of class bayesnecfit containing the following non-linear model: neclin
#
# Family: gamma 
#  Links: mu = log; shape = identity 
#Formula: y ~ top - slope * (x - nec) * step(x - nec) 
#         top ~ 1
#         slope ~ 1
#         nec ~ 1
#   Data: structure(list(x = c(1.01874617094183, 0.815747457 (Number of observations: 100) 
#Samples: 4 chains, each with iter = 10000; warmup = 9000; thin = 1;
#         total post-warmup samples = 4000
#
#Population-Level Effects: 
#                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#top_Intercept       0.89      0.01     0.88     0.91 1.00     2625     2290
#slope_Intercept     0.55      0.03     0.50     0.60 1.00     1970     2145
#nec_Intercept       1.31      0.04     1.23     1.38 1.00     1648     2038
#
#Family Specific Parameters: 
#      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#shape   248.66     36.61   185.55   328.96 1.00     2690     2229
#
#Samples were drawn 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).

In this case our model has converged well, although it is perhaps not the best model to fit. See the Model details and Multi model usage vignettes for options to fit other models to these data.