Thermal tolerance derived from a fitted failure-rate model

Derives, from a model fitted with fit_hazard(), the failure-rate parameters on their natural scale, the static critical thermal maximum sCTmax, the thermal death time curves and the thermal tolerance landscape, each with its posterior median and equal-tailed credible interval (the interval between the (1 - prob) / 2 and 1 - (1 - prob) / 2 quantiles of the draws), at the covariate values in each row of newdata. The model's settings (failure-rate and baseline functions, reference temperature, softening steepness, co-stressors) are read from the fit. Random effects are set to zero, so the results are those of a typical group.

Usage

derive_tdt_ttl(
  fit,
  newdata = NULL,
  p = c(0.1, 0.5, 0.9),
  t_ref,
  Ta = NULL,
  tau = NULL,
  levels = NULL,
  prob = 0.95,
  n_draws = 1000
)

Arguments

fit: A brmsfit, a model fitted with fit_hazard().
newdata: A data frame of covariate values, one row per setting at which to derive. It must hold every covariate in the model's parameter formulas. May be NULL for a model without covariates.
p: A numeric vector of the fractions of individuals failing at which sCTmax and the thermal death time curves are derived, each strictly between 0 and 1.
t_ref: A single positive number, the reference exposure time for sCTmax, in the time unit of the data the model was fitted to (for example 60 for one hour when times are in minutes). It has no default, because the fit does not record that unit.
Ta: An optional numeric vector of the constant temperatures for the thermal death time curves and the tolerance landscape. NULL (the default) takes 50 evenly spaced across the temperatures the data cover.
tau: An optional numeric vector of the elapsed exposure times for the tolerance landscape, in the time unit of the data, none negative. NULL (the default) takes 50 spaced evenly on a log scale across the times the data cover. A time of zero is allowed, and gives a survival of 1, so that a survival curve can start there.
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 strictly between 0 and 1, the coverage of the credible intervals.
n_draws: An optional single number, the number of posterior draws to use, evenly spaced through the posterior. Defaults to 1000, which keeps the derivation fast while the limits of a 95% interval still rest on about 25 draws in each tail; NULL uses them all.

Value

A list of four tibbles, each starting with the columns of newdata and ending with the posterior summary median, lower, upper, prob, n_draws and n_unreached:

params: one row per parameter, named in quantity;
sctmax: one row per value of p, with p and t_ref;
tdt_curves: one row per value of p and temperature, with p and Ta; the summary is of the time until that fraction has failed;
ttl_surface: one row per temperature and elapsed time, with Ta and tau; the summary is of survival.

Details

sCTmax is the constant temperature at which a fraction p of individuals fail within the reference exposure time t_ref, the static critical thermal maximum of Jorgensen et al. (2021), who report it for one hour. Under a constant temperature, survival is exp(-M(T) H0(t_ref)), so for each posterior draw sCTmax solves M(T) = -log(1 - p) / H0(t_ref). It is not the critical thermal maximum a ramping experiment records (their dynamic CTmax).

The parameters reported are a, and where the failure-rate function has them b, c and Tc; the baseline's shape (Weibull, log-logistic), its scale (log-logistic; the Weibull's is fixed at the reference time, see h0_catalogue) or its rate constant gamma (Gompertz); and each co-stressor's coefficients. With a co-stressor that interacts with temperature, b_at_levels is the thermal sensitivity b + dT z at the chosen levels. The thermal sensitivity z, the temperature change that shortens the time to failure tenfold, is reported only where the thermal death time curve is a straight line in log time against temperature, so that one number describes it: the exponential failure rate with the memoryless or Weibull baseline, where z = k log(10) / b (k = 1 for the memoryless baseline, the Weibull shape otherwise). Elsewhere the curve bends and no single z stands for it.

The thermal death time curve for a fraction p gives, at each constant temperature in Ta, the time until that fraction has failed: t_p = H0^-1(-log(1 - p) / M(T)). The thermal tolerance landscape gives survival exp(-M(T) H0(tau)) over the grid of temperatures Ta and elapsed exposure times tau. "Thermal death time" is the literature's name for the curve, whatever the failure endpoint.

Draws that never reach the quantity

A posterior draw for which a quantity is never reached (an sCTmax the rate cannot attain within 1000 degrees of the reference temperature, or at all; a time to failure that never comes, under a Gompertz baseline whose cumulative hazard levels off) counts as infinite rather than being dropped, so the summaries describe the whole posterior. n_unreached counts these draws, and a message lists the quantities that have any. With coverage prob, let u = (1 - prob) / 2 (0.025 for a 95% interval):

fewer than a share u of the draws unreached: both bounds are finite; report the summary as it stands.
between u and half: the outer bound is infinite; report it as unbounded (for example "above 40 degC", "not reached"), and consider whether the question lies beyond the temperatures or exposure times the data cover.
half or more: the median is infinite; the quantity is not estimable there. Ask at a lower p or a shorter t_ref.

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.

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))

tol <- derive_tdt_ttl(fit, newdata = data.frame(mass = c(8, 12)), p = 0.5,
                      t_ref = 60)
tol$sctmax
} # }