discretised_logit_hazard.stan File Reference

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Functions
real	loglogistic_lcdf (real y, real alpha, real beta)
vector	lcdf_discretised (real mu, real sigma, int u, int dist)
vector	normalise_lcdf_as_uniform_double_censored (vector lcdf, int u, int max_strat)
vector	lcdf_to_uniform_double_censored_log_prob (vector lcdf, int u)
vector	lprob_to_uniform_double_censored_log_hazard (vector lprob, vector lcdf, int u)
	Compute discretised logit hazard.
vector	log_hazard_to_logit_hazard (vector lhaz)
vector	discretised_logit_hazard (real mu, real sigma, int dmax, int dist, int max_strat, int ref_as_p)

Function Documentation

discretised_logit_hazard()

vector discretised_logit_hazard	(	real	mu,
		real	sigma,
		int	dmax,
		int	dist,
		int	max_strat,
		int	ref_as_p
	)

Calculate logit hazard or log probability for discretised delay distribution

Computes logit hazards or log probabilities for a specified discretised parametric distribution up to a maximum possible delay. Employs various normalisation strategies and assumes that delays are double-censored and that the interval width is approximately uniformly distributed.

Parameters

mu	Location parameter of the parametric distribution.
sigma	Scale parameter of the parametric distribution.
dmax	Maximum possible delay. The parametric distribution will be truncated at dmax.
dist	Distribution type indicator (e.g., exponential, lognormal, ...).
max_strat	Strategy for normalising LCDF values (e.g., handling probability mass beyond the maximum observed value).
ref_as_p	Flag indicating whether to return log probabilities directly (1) or to convert to logit hazards (0).

Returns

A vector of logit hazards or log probabilities of the discretised distribution, representing discrete delays from 0 to (dmax-1).

Note

This function integrates several steps:

1. Generates LCDF values using lcdf_discretised.
2. Normalises these LCDF values with normalise_lcdf_as_uniform_double_censored.
3. Converts LCDF to log probabilities using lcdf_to_uniform_double_censored_log_prob.
4. If ref_as_p is 0, further processes these probabilities into logit hazards using lprob_to_uniform_double_censored_log_hazard and log_hazard_to_logit_hazard.

Dependencies:

• lcdf_discretised
• normalise_lcdf_as_uniform_double_censored
• lcdf_to_uniform_double_censored_log_prob
• lprob_to_uniform_double_censored_log_hazard
• log_hazard_to_logit_hazard

Definition at line 286 of file discretised_logit_hazard.stan.

lcdf_discretised()

vector lcdf_discretised	(	real	mu,
		real	sigma,
		int	u,
		int	dist
	)

Compute the discretised LCDF for various distributions

Computes the LCDF at integer points from 1 to u for a selected parametric distribution. Supports exponential, lognormal, gamma, and log-logistic distributions.

Parameters

mu	Location parameter for the distribution.
sigma	Scale parameter for the distribution.
u	Upper bound of the discretised LCDF.
dist	Integer flag for distribution type (1: exponential, 2: lognormal, 3: gamma, 4: log-logistic).

Returns

Vector of LCDF values for the specified distribution.

Note

Uses loglogistic_lcdf for the log-logistic distribution.

Definition at line 40 of file discretised_logit_hazard.stan.

lcdf_to_uniform_double_censored_log_prob()

vector lcdf_to_uniform_double_censored_log_prob	(	vector	lcdf,
		int	u
	)

Compute LPMF from discretised LCDF under double censoring

Converts LCDF values to log-scale probability mass function (LPMF) assuming double censoring and a uniform interval approximation. Suitable for discretised LCDFs evaluated at integers 1 to u.

Parameters

lcdf	Vector of LCDF values.
u	Upper bound of the discretised LCDF / LPMF.

Returns

Vector of log-scale PMF values.

Note

While the elements of the LCDF vector correspond to delays 1 to u, the elements of the PMF correspond to delays 0:(u-1). Hence, log(p_0) = lpmf[1].

Processes output of normalise_lcdf_as_uniform_double_censored.

Definition at line 138 of file discretised_logit_hazard.stan.

log_hazard_to_logit_hazard()

vector log_hazard_to_logit_hazard

(

vector

lhaz

)

Convert log hazards to logit hazards efficiently

Transforms log hazards to logit hazards without converting to the natural scale. Used in the final step of converting discretised probability distributions to logit hazards.

Parameters

lhaz	Vector of log hazards.

Returns

Vector of logit hazards.

Note

Final transformation step in discretised_logit_hazard.

Definition at line 233 of file discretised_logit_hazard.stan.

loglogistic_lcdf()

real loglogistic_lcdf	(	real	y,
		real	alpha,
		real	beta
	)

Compute the log cumulative distribution function (LCDF) for a log-logistic distribution

Calculates the LCDF of a log-logistic distribution for a given value using specified scale and shape parameters.

Parameters

y	The value at which to compute the LCDF.
alpha	Scale parameter of the log-logistic distribution (alpha > 0).
beta	Shape parameter of the log-logistic distribution (beta > 0).

Returns

The log-logistic LCDF of y.

Definition at line 16 of file discretised_logit_hazard.stan.

lprob_to_uniform_double_censored_log_hazard()

vector lprob_to_uniform_double_censored_log_hazard	(	vector	lprob,
		vector	lcdf,
		int	u
	)

Compute discretised logit hazard.

Compute logit hazards from log probabilities.

Derives logit hazards from log probabilities of a delay LPMF, assuming double censoring and a uniform interval approximation for discretised delays.

Parameters

lprob	Vector of log probabilities from the LPMF.
lcdf	Vector of corresponding LCDF values for the LMPF. This could also be computed from lprob, but by re-using the precomputed LCDF from elsewhere we reduce overall computation.
u	Upper bound of discretised LCDF.

Returns

Vector of logit hazards.

Note

While the elements of the LCDF vector correspond to delays 1 to u, the elements of the PMF correspond to delays 0:(u-1). Hence, log(h_0) = lhaz[1].

Processes output of lcdf_to_uniform_double_censored_log_prob and `normalise_lcdf_as_uniform_double_censored

\[ \begin{align*} \text{Let } & h_d = \text{ hazard at delay } d, \\ & p_d = \text{ probability at delay } d, \\ & F_d = \text{ cumulative distribution function (cdf) evaluated at } d, \\ & 1 - F_d = \text{ complementary cumulative distribution function (ccdf) evaluated at } d. \\ \\ % Hazard definition h_d &= \frac{p_d}{1 - \sum_{i=0}^{d-1} p_i} \\ &= \frac{F_{d+1} - F_{d-1}}{1 - \sum_{i=0}^{d-1} (F_{i+1} - F_{i-1})} \\ &= \frac{F_{d+1} - F_{d-1}}{1 - (F_d + F_{d-1})} \\ &= \frac{F_{d+1} - F_{d-1}}{(1 - F_d) - F_{d-1}}. \\ \\ % Log transformation \log(h_d) &= \log(F_{d+1} - F_{d-1}) - \log((1 - F_d) - F_{d-1}). \end{align*} \]

Definition at line 195 of file discretised_logit_hazard.stan.

normalise_lcdf_as_uniform_double_censored()

vector normalise_lcdf_as_uniform_double_censored	(	vector	lcdf,
		int	u,
		int	max_strat
	)

Normalise a discretised LCDF under double censoring

Adjusts LCDF values for the probability mass beyond the upper bound of the discretised LCDF, assuming double censoring and a uniform interval approximation. Different strategies are applied depending on max_strat.

Parameters

lcdf	Vector of LCDF values to be adjusted.
u	Upper bound of the discretised LCDF.
max_strat	Strategy for handling probability mass beyond upper bound.

Returns

Vector of adjusted LCDF values.

Note

Used on the output of lcdf_discretised.

Definition at line 84 of file discretised_logit_hazard.stan.