A variant of disbayes in which data from different areas can be related in a hierarchical model and, optionally, the effect of gender can be treated as additive with the effect of area. This is much more computationally intensive than the basic model in disbayes. Time trends are not supported in this function.

disbayes_hier(
data,
group,
gender = NULL,
inc_num = NULL,
inc_denom = NULL,
inc_prob = NULL,
inc_lower = NULL,
inc_upper = NULL,
prev_num = NULL,
prev_denom = NULL,
prev_prob = NULL,
prev_lower = NULL,
prev_upper = NULL,
mort_num = NULL,
mort_denom = NULL,
mort_prob = NULL,
mort_lower = NULL,
mort_upper = NULL,
rem_num = NULL,
rem_denom = NULL,
rem_prob = NULL,
rem_lower = NULL,
rem_upper = NULL,
age = "age",
cf_init = 0.01,
eqage = 30,
eqagehi = NULL,
cf_model = "default",
inc_model = "smooth",
rem_model = "const",
prev_zero = FALSE,
sprior = c(1, 1, 1),
hp_fixed = NULL,
nfold_int_guess = 5,
nfold_int_upper = 100,
nfold_slope_guess = 5,
nfold_slope_upper = 100,
mean_int_prior = c(0, 10),
mean_slope_prior = c(5, 5),
gender_int_priorsd = 0.82,
gender_slope_priorsd = 0.82,
inc_prior = c(1.1, 0.1),
rem_prior = c(1.1, 1),
method = "opt",
draws = 1000,
iter = 10000,
stan_control = NULL,
...
)

## Arguments

data

Data frame containing some of the variables below. The variables below are provided as character strings naming columns in this data frame. For each disease measure available, one of the following three combinations of variables must be specified:

(1) numerator and denominator (2) estimate and denominator (3) estimate with lower and upper credible limits.

Mortality must be supplied, and at least one of incidence and prevalence. If remission is assumed to be possible, then remission data should also be supplied (see below).

Estimates refer to the probability of having some event within a year, rather than rates. Rates per year $r$ can be converted to probabilities $p$ as $p = 1 - exp(-r)$, assuming the rate is constant within the year.

For estimates based on registry data assumed to cover the whole population, then the denominator will be the population size.

group

Variable in the data representing the area (or other grouping factor).

gender

If NULL (the default) then the data are one homogenous gender, and there should be one row per year of age. Otherwise, set gender to a character string naming the variable in the data representing gender (or other categorical grouping factor). Gender will then treated as a fixed additive effect, so the linear effect of gender on log case fatality is the same in each area. The data should have one row per year of age and gender.

inc_num

Numerator for the incidence data, assumed to represent the observed number of new cases within a year among a population of size inc_denom.

inc_denom

Denominator for the incidence data.

The function ci2num can be used to convert a published estimate and interval for a proportion to an implicit numerator and denominator.

Note that to include extra uncertainty beyond that implied by a published interval, the numerator and denominator could be multiplied by a constant, for example, multiplying both the numerator and denominator by 0.5 would give the data source half its original weight.

inc_prob

Estimate of the incidence probability

inc_lower

Lower credible limit for the incidence estimate

inc_upper

Upper credible limit for the incidence estimate

prev_num

Numerator for the estimate of prevalence, i.e. number of people currently with a disease.

prev_denom

Denominator for the estimate of prevalence (e.g. the size of the survey used to obtain the prevalence estimate)

prev_prob

Estimate of the prevalence probability

prev_lower

Lower credible limit for the prevalence estimate

prev_upper

Upper credible limit for the prevalence estimate

mort_num

Numerator for the estimate of the mortality probability, i.e number of deaths attributed to the disease under study within a year

mort_denom

Denominator for the estimate of the mortality probability (e.g. the population size, if the estimates were obtained from a comprehensive register)

mort_prob

Estimate of the mortality probability

mort_lower

Lower credible limit for the mortality estimate

mort_upper

Upper credible limit for the mortality estimate

rem_num

Numerator for the estimate of the remission probability, i.e number of people observed to recover from the disease within a year.

Remission data should be supplied if remission is permitted in the model, either as a numerator and denominator or as an estimate and lower credible interval. Conversely, if no remission data are supplied, then remission is assumed to be impossible. These "data" may represent a prior judgement rather than observation - lower denominators or wider credible limits represent weaker prior information.

rem_denom

Denominator for the estimate of the remission probability

rem_prob

Estimate of the remission probability

rem_lower

Lower credible limit for the remission estimate

rem_upper

Upper credible limit for the remission estimate

age

Variable in the data indicating the year of age. This must start at age zero, but can end at any age.

cf_init

Initial guess at a typical case fatality value, for an average age.

eqage

Case fatalities (and incidence and remission rates) are assumed to be equal for all ages below this age, inclusive, when using the smoothed model.

eqagehi

Case fatalities (and incidence and remission rates) are assumed to be equal for all ages above this age, inclusive, when using the smoothed model.

cf_model

The following alternative models for case fatality are supported:

"default" (the default). Random intercepts and slopes, and no further restriction.

"interceptonly". Random intercepts, but common slopes.

"increasing". Case fatality is assumed to be an increasing function of age (note it is constant below "eqage" in all models) with a common slope for all groups.

"common" Case fatality is an unconstrained function of age which is common to all areas, i.e. it has the same parameter values in every area. This and "increasing_common" are used in situations where you want to compare a model with area-specific rates with a single model for the data aggregated over areas. Modelling the area-disaggregated data using a common function for all areas is equivalent to a model for the aggregated data, and can be statistically compared (using cross-validation) with a model with area-specific rates.

"increasing_common" Case fatality is an increasing function of age which is common to all areas.

"const" Case fatality is assumed to be constant with age, for all ages, but different in each area.

"const_common" Case fatality is a constant over all ages and areas.

In all models, case fatality is a smooth function of age.

inc_model

Model for how incidence varies with age.

"smooth" (the default). Incidence is modelled as a smooth spline function of age, independently for each area (and gender).

"indep" Incidence rates for each year of age, area (and gender) are estimated independently.

rem_model

Model for how remission varies with age. Currently supported models are "const" for a constant remission rate over all ages, "const" for a smooth spline, or "indep" for a different remission rates estimated independently for each age with no smoothing.

prev_zero

If TRUE, attempt to estimate prevalence at age zero from the data, as part of the Bayesian model, even if the observed prevalence is zero. Otherwise (the default) this is assumed to be zero if the count is zero, and estimated otherwise.

sprior

Rates of the exponential prior distributions used to penalise the coefficients of the spline model. The default of 1 should adapt appropriately to the data, but Higher values give stronger smoothing, or lower values give weaker smoothing, if required.

This can be a named vector with names "inc","cf","rem" in any order, giving the prior smoothness rates for incidence, case fatality and remission. If any of these are not smoothed they can be excluded, e.g. sprior = c(cf=10, inc=1).

This can also be an unnamed vector of three elements, where the first refers to the spline model for incidence, the second for case fatality, the third for remission. If one of the rates (e.g. remission) is not being modelled with a spline, any number can be supplied here and it is just ignored.

hp_fixed

A list with one named element for each hyperparameter to be fixed. The value should be either

• a number (to fix the hyperparameter at this number)

• TRUE (to fix the hyperparameter at the posterior mode from a training run where it is not fixed)

If the element is either NULL, FALSE, or omitted from the list, then the hyperparameter is given a prior and estimated as part of the Bayesian model.

The hyperparameters that can be fixed are

• scf Smoothness parameter for the spline relating case fatality to age.

• sinc Smoothness parameter for the spline relating incidence to age.

• scfmale Smoothness parameter for the spline defining how the gender effect relates to age. Only for models with additive gender and area effects.

• sd_int Standard deviation of random intercepts for case fatality.

• sd_slope Standard deviation of random slopes for case fatality.

For example, to fix the case fatality smoothness to 1.2, fix the incidence smoothness to its posterior mode, and estimate all the other hyperparameters, specify hp_fixed = list(scf=1.2, sinc=TRUE).

nfold_int_guess

Prior guess at the ratio of case fatality between a high risk (97.5% quantile) and low risk (2.5% quantile) area.

nfold_int_upper

Prior upper 95% credible limit for the ratio in average case fatality between a high risk (97.5% quantile) and low risk (2.5% quantile) area.

nfold_slope_guess, nfold_slope_upper

This argument and the next argument define the prior distribution for the variance in the random linear effects of age on log case fatality. They define a prior guess and upper 95% credible limit for the ratio of case fatality slopes between a high trend (97.5% quantile) and low risk (2.5% quantile) area. (Note that the model is not exactly linear, since departures from linearity are defined through a spline model. See the Jackson et al. paper for details.).

mean_int_prior

Vector of two elements giving the prior mean and standard deviation respectively for the mean random intercept for log case fatality.

mean_slope_prior

Vector of two elements giving the prior mean and standard deviation respectively for the mean random slope for log case fatality.

gender_int_priorsd

Prior standard deviation for the additive effect of gender on log case fatality

gender_slope_priorsd

Prior standard deviation for the additive effect of gender on the linear age slope of log case fatality

inc_prior

Vector of two elements giving the Gamma shape and rate parameters of the prior for the incidence rate. Only used if inc_model="indep", for each age-specific rate.

rem_prior

Vector of two elements giving the Gamma shape and rate parameters of the prior for the remission rate, used in both rem_model="const" and rem_model="indep".

method

String indicating the inference method, defaulting to "opt".

If method="mcmc" then a sample from the posterior is drawn using Markov Chain Monte Carlo sampling, via rstan's rstan::sampling() function. This is the most accurate but the slowest method.

If method="opt", then instead of an MCMC sample from the posterior, disbayes returns the posterior mode calculated using optimisation, via rstan's rstan::optimizing() function. A sample from a normal approximation to the (real-line-transformed) posterior distribution is drawn in order to obtain credible intervals.

If the optimisation fails to converge (non-zero return code), try increasing the number of iterations from the default 1000, e.g. disbayes(..., iter=10000, ...), or changing the algorithm to disbayes(..., algorithm="Newton", ...).

If there is an error message that mentions chol, then the computed Hessian matrix is not positive definite at the reported optimum, hence credible intervals cannot be computed. This can occur either because of numerical error in computation of the Hessian, or because the reported optimum is wrong. If you are willing to believe the optimum and live without credible intervals, then set draws=0 to skip computation of the Hessian. To examine the problematic Hessian, set hessian=TRUE,draws=0, then look at the $fit$hessian component of the disbayes return object. If it can be inverted, do sqrt(diag(solve())) on the Hessian, and check for NaNs, indicating the problematic parameters. Otherwise, diagonal entries of the Hessian matrix that are very small may indicate parameters that are poorly identified from the data, leading to computational problems.

If method="vb", then variational Bayes methods are used, via rstan's rstan::vb() function. This is labelled as "experimental" by rstan. It might give a better approximation to the posterior than method="opt", but has not been investigated much for disbayes models.

draws

Number of draws from the normal approximation to the posterior when using method="opt".

iter

Number of iterations for MCMC sampling, or maximum number of iterations for optimization.

stan_control

(method="mcmc" only). List of options supplied as the control argument to rstan::sampling() in rstan for the main model fit.

...

Further arguments passed to rstan::sampling() to control MCMC sampling, or rstan::optimizing() to control optimisation, in Stan.

## Value

A list including the following components

call: Function call that was used.

fit: An object containing posterior samples from the fitted model, in the stanfit format returned by the stan

function in the rstan package.

method: Optimisation method that was chosen.

nage: Number of years of age in the data

narea: Number of areas (or other grouping variable that defines the hierarchical model).

ng: Number of genders (or other categorical variable whose effect is treated as additive with the area effect).

groups: Names of the areas (or other grouping variable), taken from the factor levels in the original data.

genders: Names of the genders (or other categorical variable), taken from the factor levels in the original data.

dat: A list containing the input data in the form of numerators and denominators.

stan_data: Full list of data supplied to Stan

stan_inits: Full list of parameter initial values supplied to Stan

trend: Whether a time trend was modelled

hp_fixed Values of any hyperparameters that are fixed during the main model fit.

## References

Jackson C, Zapata-Diomedi B, Woodcock J. "Bayesian multistate modelling of incomplete chronic disease burden data" https://arxiv.org/abs/2111.14100