This catalogue contains every function the failure rate may take: the nine
failure rate functions M(T) of Kingsolver and Umbanhowar (2018, Table 1), each
catalogue key named for its model there. Without a threshold: const (a
constant rate), exp (exponential), zeroexp (exponential minus its value at
the reference temperature Tr, so the rate is zero there), power (a power
of temperature above a baseline rate) and zeropower (the same power with no
baseline, zero at Tr). The threshold variants const_threshold,
exp_threshold, zeroexp_threshold and zeropower_threshold hold the rate at
zero below a fitted critical temperature Tc; above Tc it is constant for
const_threshold and accelerates for the other three. The exponential
functions give a straight thermal death time curve, whereas the power functions
give a curved one.
Details
Tr (T0 in Kingsolver and Umbanhowar's notation) is a single reference
temperature set for the whole model, not the per-trial starting temperature. The
exponential is anchored at Tr and the zero forms vanish there, but because
the failure rate must depend only on the current temperature, Tr cannot be
each trial's own starting temperature. Trials that begin at different
temperatures (acclimation groups, field conditions) enter through their
temperature trajectories, not through Tr.
To keep fitting, simulation and reporting from diverging, all three evaluate the same expression strings, which are held in brms_log_M.
Each entry gives the integer the Stan code dispatches on (stan_id), the
parameters the function needs (dpars), and the steepness of the threshold
switch (k_sw). The rate parameters a, b and c are represented on the log
scale (loga, logb, logc), so the linear predictor is unconstrained and
their strictly positive values are recovered by exponentiation; the critical
temperature Tc is left on the natural scale, since it is not sign-constrained.
k_sw is roughly the inverse of the width of the transition in temperature, so
a larger value is a sharper switch over a narrower band. The switch is smoothed
rather than hard because Stan fits the model with Hamiltonian Monte Carlo,
which explores the posterior by following the gradient of the log-density with
respect to each parameter; the gradient of a hard step with respect to the
critical temperature is zero on either side of it and undefined at the step
itself, so the sampler gets no slope to move that temperature along, whereas a
smooth switch supplies one.
The units of M(T)
The hazard is h0(tau) M(T), and which of the two factors carries the unit of reciprocal time depends on the baseline (see h0_catalogue). With the memoryless and Gompertz baselines h0(tau) has no unit, so M(T) is a rate per unit time, and a is the failure rate at the temperature where M(T) = a. With the Weibull and log-logistic baselines h0(tau) is itself per unit time, so M(T) is a multiplier with no unit; with the Weibull, a is the cumulative hazard reached over the reference time at that temperature. The value of a therefore cannot be compared between models with different baselines, nor between datasets timed in different units when the baseline is memoryless or Gompertz.
Whatever the baseline, survival at a constant temperature is
exp(-M(T) H0(tau)), with H0 the cumulative baseline hazard. derive_tdt_ttl()
derives the critical thermal maximum, the thermal death time curves and the
thermal tolerance landscape from that expression, and those can be compared
between models.
References
Kingsolver, J. G. and Umbanhowar, J. (2018). The analysis and interpretation of critical temperatures. Journal of Experimental Biology 221, jeb167858. doi:10.1242/jeb.167858