Dynamic critical thermal maximum under a ramp, from a fitted model

Derives, from a model fitted with fit_hazard(), the dynamic critical thermal maximum dCTmax of Jorgensen et al. (2021): the temperature at which a fraction p of individuals has failed while warmed from T_start at a constant rate. It is the critical thermal maximum a ramping trial would record under that ramp, with its posterior median and credible interval, at the covariate values in each row of newdata. Random effects are set to zero, so the results are those of a typical group.

Usage

derive_dctmax(
  fit,
  rate,
  T_start,
  newdata = NULL,
  p = 0.5,
  levels = NULL,
  prob = 0.95,
  n_draws = 1000,
  T_max = NULL
)

Arguments

fit

A brmsfit, a model fitted with fit_hazard().

rate

A numeric vector of ramp rates, in degrees per unit of time of the data the model was fitted to (e.g. degrees C per minute), all positive.

T_start

A numeric vector of starting temperatures, one per rate or a single one for all.

newdata

A data frame with one row per set of covariate values to report at, holding every covariate in the parameters' formulas. NULL (the default) suits a model with no covariates.

p

A numeric vector of fractions of individuals failing, each strictly between 0 and 1. Defaults to 0.5.

levels

A named numeric vector giving the level of each co-stressor, on its measurement scale, e.g. c(DO = 6). Required when the model has co-stressors.

prob

A single number between 0 and 1, the coverage of the credible intervals. Defaults to 0.95.

n_draws

A single number, the number of posterior draws to use, evenly spaced through the posterior. Defaults to 1000.

T_max

A single number, the temperature up to which the ramp is followed. NULL (the default) takes 15 degrees above the warmest temperature the model was fitted to.

Value

A tibble with a row for each row of newdata, ramp and p: the columns of newdata, rate, T_start, p, and the posterior median, lower and upper of dCTmax, with prob, n_draws and n_unreached.

Details

Unlike the static critical thermal maximum from derive_tdt_ttl(), dCTmax depends on the ramp. Individuals accrue hazard throughout the ramp, so a slower one gives them more time to fail before a given temperature is reached, and its dCTmax is lower. Giving several rates shows how much. The ramp need not be one that was run: the model was fitted to failure times, from a static or a ramping assay, and is applied here to any ramp.

Under the ramp T(t) = T_start + rate t, the cumulative hazard is the integral of M(T(t)) h0(t) from the start of the ramp, and dCTmax solves, for each posterior draw, cumulative hazard = -log(1 - p). With a baseline other than the memoryless one, the hazard also depends on the time elapsed since the ramp began.

The ramp is followed up to T_max. A posterior draw whose fraction failed has not reached p by then counts as infinite, as in derive_tdt_ttl(), and is counted in n_unreached. A message is given when a dCTmax lies above the temperatures the model was fitted to, where the data do not constrain the failure-rate function.

References

Jorgensen, L. B., Malte, H., Orsted, M., Klahn, N. A. and Overgaard, J. (2021). A unifying model to estimate thermal tolerance limits in ectotherms across static, dynamic and fluctuating exposures to thermal stress. Scientific Reports 11, 12840.

See also

derive_tdt_ttl() for the static critical thermal maximum, and average_dctmax() to average dCTmax over several models.

Examples

if (FALSE) { # \dontrun{
# a model fitted to a small ramping trial: 20 individuals warmed from 20 to 35
# degrees over 60 minutes and checked every 10, in two trials (see fit_hazard())
path   <- data.frame(path_id = "A", time = c(0, 60), temp = c(20, 35))
checks <- data.frame(path_id = "A", time = seq(0, 60, by = 10))
ind    <- data.frame(id = 1:20, path_id = "A", event = 1, mass = runif(20, 8, 12),
                     trial = rep(c("a", "b"), each = 10),
                     t_upper = sample(seq(10, 60, 10), 20, replace = TRUE))
long   <- build_hazard_data(ind, path, "interval", covariates = c("mass", "trial"),
                            checks = checks)
fit    <- fit_hazard(long, "exp", dpar_formulas = list(loga = ~ 1 + mass))

# the CTmax expected under three ramps from 14 degrees
derive_dctmax(fit, rate = c(0.1, 0.2, 0.3), T_start = 14,
              newdata = data.frame(mass = 15))
} # }