Non-parametric delay distribution family

A delay distribution with no parametric form, for use with the marginal and meta models. Pass it as the family argument of epidist() in the same way as brms::lognormal().

The delay sits on a grid of $K$ bins with boundaries $b_0 < b_1 < \dots < b_K$. All the probability of bin $k$ is placed at its right edge $b_k$, so the delay distribution function is a step function. This is the discrete hazard distribution of primarycensored::pdiscretehazard(), and the likelihood is the primarycensored one for that distribution.

The distribution is written in terms of the discrete time hazard of each bin, the probability that the delay ends in bin $k$ given that it did not end before it. The hazard of the last bin is 1, so all the delays end by $b_K$. The logit hazards of the other bins are a linear predictor over the bins, $$\mathrm{logit}(h_k) = \mu + \sum_q B_{kq} \theta_q,$$ where the basis $B$ comes from formula evaluated on the bins.

Usage

nonparametric(formula = NULL, boundaries = NULL)

Arguments

formula: A one sided formula for the logit hazards over the bins, see Details. The default, NULL, is a spline over the delay.
boundaries: A numeric vector of at least four strictly increasing bin boundaries, $b_0$ to $b_K$. The default, NULL, is set by epidist() from the data, with a bin for every whole delay from 0 up to the longest delay in the data, and at least four bins, that is seq(-1, max(3, max_delay)). The last boundary must be at least as long as the longest observed delay.

Value

A brmsfamily object for use as the family argument of epidist().

The hazard formula

formula is a one sided formula in two variables of the bins: delay, the right edge $b_k$ of each bin, and bin, a factor with one level per bin. It takes the terms of a brms formula that make sense on the bins:

parametric terms such as delay, I(delay^2) or bin,
smooths such as s(delay), s(delay, bs = "ps") or t2(delay), built with mgcv::smoothCon() as brms builds them,
a random intercept per bin, (1 | bin), which is the smooth s(bin, bs = "re").

~ 1 gives the same hazard in every bin, a geometric delay. The default, NULL, is ~ s(delay, k = min(10, K - 1)), a thin plate regression spline over the delay, which smooths the hazard over neighbouring bins. With fewer than three free bins it is ~ (1 | bin).

The intercept of the formula is dropped, because $\mu$ is the intercept. Each column of the basis is centred over the bins, so $\mu$ is the mean logit hazard, and scaled, so that each coefficient is on the scale of the logit hazard.

Parameters

Each coefficient $\theta_q$ is a distributional parameter, so it appears in the output of epidist() and takes a brms formula and prior in the same way as mu:

hb is the coefficient of the $i$th unpenalised column, such as a parametric term or the linear part of a spline.
h<j>sd is the standard deviation of the $j$th penalised term, such as the wiggly part of a spline or the random intercept per bin.
hz is the $i$th standardised coefficient of the penalised terms, so the coefficient is h<j>sd * hz for the term $j$ the column belongs to. This is the non-centred form brms uses for smooths and random effects.

The coefficients element of the np element of the family built by epidist_family() names the term of each coefficient.

Covariates

A covariate in the mu formula shifts the logit hazard of every bin by the same amount, which is a proportional odds model for the hazard. A covariate in the formula of a coefficient changes the shape of the hazard over the bins, a non-proportional effect. For example with formula = ~ delay, bf(mu ~ age_group, h1b ~ age_group) gives each age group its own intercept and slope of the logit hazard over the delay, as delay * age_group would.

Priors

The default priors are normal(logit(1 / K), 1.5) on the intercept of mu, centred on the hazard of the first bin when every bin is equally likely, normal(0, 2) on each hb, normal(0, 1) on the log of each h<j>sd, as primarycensored uses for the spread of the logit hazards, and std_normal() on each hz. Set others with the prior argument of epidist().

Examples

nonparametric(boundaries = -1:10)
#> 
#> Family: nonparametric 
#> Link function: identity 
#> 
nonparametric(~ (1 | bin), boundaries = -1:10)
#> 
#> Family: nonparametric 
#> Link function: identity 
#> 
nonparametric(~ s(delay, bs = "ps", k = 6), boundaries = c(-1:5, 7, 10))
#> 
#> Family: nonparametric 
#> Link function: identity 
#>