Weakly-informative default priors for a failure-rate hazard model

Supplies a sensible starting prior for the model build_hazard_spec assembles, keyed to each parameter's intercept and unaffected by whatever covariates the dpar_formulas add. For every parameter the chosen functions need, it sets a weakly-informative prior on the intercept (the parameter's value at the reference temperature and start of exposure), a generic Normal(0, 1) on any covariate slopes, and an Exponential(2) on the standard deviation of any random effect. The slope and random-effect priors are added only when the parameter actually has those terms, so brms raises no "no parameter" warning. Override the centre of any parameter through intercept_priors, or the shared slope and random-effect standard-deviation priors through slope_prior and sd_prior; or replace the whole set with your own prior when fitting with fit_hazard().

Usage

build_hazard_prior(
  m_form,
  baseline = "mem",
  dpar_formulas = list(),
  intercept_priors = list(),
  slope_prior = "normal(0, 1)",
  sd_prior = "exponential(2)",
  costressors = list(),
  long
)

Arguments

m_form: A character string denoting a key of M_catalogue (e.g. "exp" or "exp_threshold").
baseline: A character string denoting a key of h0_catalogue: "mem" (default), "weibull", "llogis" or "gompertz".
dpar_formulas: A named list of one-sided formulas, one per parameter, as in build_hazard_spec. A parameter's fixed and random terms decide which slope and standard-deviation priors are added. A formula without an intercept (~ 0 + acclim) gives the parameter one coefficient per level in place of the intercept, and each of them takes the intercept's prior.
intercept_priors: A named list of prior strings, one per parameter, that replace the default intercept (centre) for those parameters, e.g. list(Tc = "normal(26, 4)"). A parameter left out keeps its default; a name that is not a parameter of the chosen functions is ignored with a warning.
slope_prior: A single prior string for every covariate slope (a population-level effect beyond the intercept), applied to each parameter that has one. Defaults to "normal(0, 1)".
sd_prior: A single prior string for every random-effect standard deviation, applied to each parameter that has one. Defaults to "exponential(2)".
costressors: An optional list of costressor objects, as in build_hazard_spec. Each adds an intercept prior for its main effect and, when the temperature interaction is kept, its interaction coefficient. Defaults to none.
long: A data frame, the row-per-interval table from build_hazard_data that the prior is for. The defaults read from it the reference time tau_ref, the reference temperature Tr and the temperatures at which failures were observed. It has no default. fit_hazard() passes its own table when it builds the default prior.

Value

A brmsprior.

Details

The defaults take three things from the data in long, so that they say the same thing whatever the organism, the time unit and the length of the exposures: the reference time tau_ref, the reference temperature Tr, and the median temperature at which failures were observed, T_f. They are weakly-informative and meant to be changed to suit your study:

loga – log amplitude. Stated where the failures are: at T_f, the cumulative hazard reached over tau_ref is centred at exp(-1), about 0.37, with a standard deviation of 2 on the log scale (about 0.007 to 20). With the memoryless and Gompertz baselines a is a rate per unit time, so that centre is -1 - log(tau_ref); with the Weibull and log-logistic it is -1. The prior is then carried from T_f back to the temperature a refers to (see below).
logb ~ normal(-0.7, 1) – log temperature sensitivity
logc – log baseline term of the power function (see below)
Tc ~ normal(T_f, 3) – critical temperature, centred where failures were observed
logshape ~ normal(0, 1) – log Weibull / log-logistic shape
logscale ~ normal(log(tau_ref), 1) – log log-logistic scale
gam ~ normal(0, 3 / tau_ref) – Gompertz rate: the baseline hazard changes by a factor of exp(gamma tau_ref) across the reference time, within about 400-fold either way with 95% probability. It is centred at zero, the memoryless case, so unless the data show a trend the fit stays near a flat baseline.

with normal(0, 1) on any covariate slope and exponential(2) on any random-effect standard deviation. Each co-stressor in costressors adds a normal(0, 1) on its main effect c<name> and a tighter normal(0, 0.5) on its temperature interaction dT<name>, both overridable through intercept_priors.

The constants in these priors (the centre of exp(-1) and the standard deviation of 2 for the amplitude, the centre of -0.7 for log b, the rate of 2 for the group standard deviations, the 0.5 for an interaction) are judgements of what is plausible for a thermal failure assay, not values estimated from data, and the centres that are read from long make the prior depend on the data in a weak way, as a standard deviation of 2 on the log scale is wide. The prior on a covariate slope is per unit of the covariate, so a covariate measured in large units (body mass in grams, say) should be centred and scaled, or the slope prior widened, before normal(0, 1) is taken as weak. With few groups, the prior on a group standard deviation can matter for predictions of new groups (which cross-validation by group scores, see kfold_hazard()); a sensitivity check against exponential(1) is cheap insurance.

The prior on a when the reference is far from the failures

For the rates anchored at the reference temperature (exp, zeroexp, zeropower, power), a is the rate at Tr, or its scale there, and Tr may lie a distance d = T_f - Tr from where failures were observed. The log rate at T_f is then loga plus a term in b and d (b d for exp), so loga is the log rate at T_f less that term, and it inherits the uncertainty in b across the distance:

its centre is the centre at T_f less the term at the prior median of b;
its variance is 2^2 plus the square of (the change in the term per unit of b, times the prior standard deviation of b).

For exp this is normal(centre - 0.5 d, sqrt(4 + (1.07 d)^2)): the usual prior when the reference is at the failures (d = 0), and a much wider one 14 degrees below them. A prior of standard deviation 2 on loga at a distant reference would in effect be a tight prior on b, and biases it. The threshold functions are anchored at Tc, not Tr, and take the centre at T_f unchanged. The calculation uses the default prior on logb; if you change that prior and keep a distant reference, set loga to match.

The baseline term c of the power function

In M(T) = c + a (T - Tr)^b, c is the rate that remains at the reference temperature: failure that happens whatever the warming. Its default is centred so that about 1% of individuals would fail over tau_ref for reasons unrelated to temperature, normal(log(0.01) - log(tau_ref), 2) with the memoryless or Gompertz baseline and normal(log(0.01), 2) otherwise, which covers background failure from about 0.02% to 40%. The 1% is a judgement, not a measured value. Where controls were held at the reference temperature, set it from them: if a fraction q of controls failed over a time t, c is about -log(1 - q) / t, so for instance intercept_priors = list(logc = "normal(log(-log(1 - 0.02) / 120), 1)") for 1 in 50 failing over 120 minutes.

Examples

# A ramp from 20 to 35 degrees over 60 minutes, with two failures.
path   <- ramp_path("A", T_start = 20, rate = 0.25, t_max = 60)
checks <- data.frame(path_id = "A", time = seq(0, 60, by = 10))
ind    <- data.frame(id = 1:3, path_id = "A", event = c(1, 1, 0),
                     t_upper = c(30, 40, 60), acclim = c(10, 15, 15))
long   <- build_hazard_data(ind, path, "interval", covariates = "acclim",
                            checks = checks)
#> ℹ The intervals at risk were divided further, into 277 rows from 13, so that
#>   temperature changes by at most 0.1 degrees within each.
#> ℹ The hazard is taken as constant within an interval, which biases the fit when
#>   temperature changes much within it. Failures at a known time are also given a
#>   short closing interval, so their hazard is that of the moment of failure.
#> ℹ Set `grid_dT` to change the step, or to `Inf` to keep the intervals as given.
#> Warning: ! For 2 individuals, temperature changes by up to 2.5 degrees within the
#>   interval in which failure was found.
#> ℹ That interval cannot be divided, since where in it the individual failed is
#>   not known, and the hazard is taken at its midpoint, so the estimates may be
#>   somewhat biased.

# Default weakly-informative priors for an exponential-threshold + Weibull model.
build_hazard_prior("exp_threshold", baseline = "weibull", long = long)
#>            prior class      coef group resp dpar    nlpar   lb   ub tag source
#>    normal(-1, 2)     b Intercept                     loga <NA> <NA>       user
#>  normal(-0.7, 1)     b Intercept                     logb <NA> <NA>       user
#>  normal(27.5, 3)     b Intercept                       Tc <NA> <NA>       user
#>     normal(0, 1)     b Intercept                 logshape <NA> <NA>       user

# Override a couple of priors (a cooler species, a tighter amplitude) and narrow
# the covariate-slope prior; every other parameter keeps its default.
build_hazard_prior("exp_threshold", baseline = "weibull",
  dpar_formulas = list(loga = ~ 1 + acclim),
  intercept_priors = list(Tc = "normal(26, 4)", loga = "normal(-6, 1.5)"),
  slope_prior = "normal(0, 0.5)", long = long)
#>            prior class      coef group resp dpar    nlpar   lb   ub tag source
#>  normal(-6, 1.5)     b Intercept                     loga <NA> <NA>       user
#>   normal(0, 0.5)     b                               loga <NA> <NA>       user
#>  normal(-0.7, 1)     b Intercept                     logb <NA> <NA>       user
#>    normal(26, 4)     b Intercept                       Tc <NA> <NA>       user
#>     normal(0, 1)     b Intercept                 logshape <NA> <NA>       user