This catalogue contains the four baseline hazard functions h0(tau), the factor
in elapsed exposure time that multiplies the failure rate M(T). mem is the
memoryless case, in which the hazard does not depend on how long the individual
has already been exposed; weibull, llogis and gompertz allow
it to rise or fall with elapsed exposure.
Details
mem is the baseline canonical to Kingsolver and Umbanhowar (2018), and is
the default throughout. Their framework has no baseline-hazard-in-time
component at all: the only hazard is M(T) evaluated along the temperature
trajectory T(t). The other three are extensions of their framework.
gompertz, h0(tau) = exp(gamma * tau), is the only one of the four baselines
whose hazard increases exponentially with elapsed exposure time, which is also
the shape an exponential M(T) exhibits as temperature increases through a ramp.
Because the two share that shape, a single ramp rate cannot separate them, as
it only sees the combined acceleration, not how much comes from the increasing
temperature and how much from time under exposure. Two or more ramp rates can
separate them, because the temperature part steepens with the ramp rate while the
elapsed time part does not. The same confound occurs between weibull, whose
hazard is a power of elapsed time, and a power M(T) (zeropower or
zeropower_threshold), since a power of temperature along a ramp is also a power of
elapsed time. So the two are confounded under a single ramp and, as for
gompertz, are separated by two or more ramp rates. llogis, whose hazard
rises then falls, matches none of the failure-rate functions of temperature,
which all rise monotonically, so it is not confounded with any of them.
A Gompertz baseline hazard is a different object from the Gompertz distribution of failure times that Kingsolver and Umbanhowar's Eqn 6 derives. The distribution is an outcome of the model: an exponential M(T) traversed along a linear ramp, under a memoryless baseline, produces Gompertz-distributed failure times. The baseline hazard presented here is an assumption about how risk depends on elapsed exposure time, independent of temperature.
Each entry gives the integer the Stan code dispatches on (stan_id) and the
parameters of each baseline function (dpars), as in M_catalogue. mem has
no parameters. weibull has a shape parameter and llogis shape and scale
parameters, on the log scale (logshape, logscale). gompertz has the rate
constant gamma (gam) parameter in reciprocal time, left on the natural scale and
unconstrained in sign (a positive value is a hazard rising with elapsed exposure
time and a negative one a hazard falling). The amplitude of the Gompertz hazard is
not included, because it is the same quantity as loga in M(T), and mem omits a
parameter for the same reason.
The scale of the weibull baseline is not estimated. It is fixed at the reference
time tau_ref set in build_hazard_data, so that
h0(tau) = (k / tau_ref) (tau / tau_ref)^(k - 1) and the cumulative baseline hazard
at tau_ref is 1. A Weibull scale s and the amplitude a of M(T) enter the hazard
only through a / s^k, so the data cannot tell them apart and only one of them can
be estimated. With the scale fixed, a is the cumulative hazard reached over the
reference time at the temperature where M(T) = a (the reference temperature Tr,
for the functions anchored there). The scale of llogis also enters where its
hazard turns from rising to falling, so it is estimated; the data inform it only
weakly when few failures come after that turn, and its posterior then leans on
its prior.
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