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@(@\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\W}[1]{ \; #1 \; }@)@This is dismod_at-20221105 documentation: Here is a link to its current documentation .
The Model Variables

Introduction
Prior for a Variable
     Standard Deviation Multipliers
     Functions of Age and Time
Children
Fixed Effects, theta
     Smoothing Standard Deviation Multipliers, lambda
     Parent Rates
     Group Covariate Multipliers
Random Effects, u
     Child Rate Effects
     Subgroup Covariate Multipliers
Age and Time Variation
     Smoothing Standard Deviation Multiplier
     Initial Prevalence
     Other Cases

Introduction
Model variables are scalar values, not functions, that are inputs to the model, and are the possibly unknown. They are often referred to as model parameters in the statistical literature. Each variable has a statistical prior on its value; i.e., a corresponding prior_id . If the corresponding lower and upper limits are equal, or it is specified to have a constant value in the smooth_grid_table , the variable is known. Variables that are used to define a function of age and time have priors on their forward difference in age and time. Each row of the variable.csv file corresponds to a single variable.

Prior for a Variable
There are two types of variables:

Standard Deviation Multipliers
The first type are the standard deviation multipliers for a smoothing. The prior for these variables are specified directly; see lambda below.

Functions of Age and Time
The second type of variables represents a functions of age and time by specifying its value at one (age, time) point in a smoothing. Bilinear interpolation is used to define the function for all values of age and time. The smooth_id for one of these variables specifies its prior as follows:
n_age number of age points used to represent the function
n_time number of time points used to represent the function
age_id identifies age value for a variable
time_id identifies time value for a variable
const_value null or a value that a variable is constrained to
value_prior_id value prior for a variable
dage_prior_id difference prior for a variable (and next variable) in age direction
dtime_prior_id difference prior for a variable (and next variable) in time direction
The number of variables (number of grid points) corresponding to a smoothing is n_age*n_time . The age and time difference priors specify the smoothing in a mathematical sense.

Children
The parent node is specified in the option table . The children corresponding to the parent node. (The children is a set of nodes not a set of variables.)

Fixed Effects, theta
We use @(@ \theta @)@ to denote the vector of fixed effects; i.e., all of the variables except for the random effects. Below is a list of the types fixed effects:

Smoothing Standard Deviation Multipliers, lambda
These variables do not represent a function of age and time. For each @(@ i = @)@ smooth_id , there are three smoothing standard deviation multiplier variables: @(@ \lambda_i^v @)@, @(@ \lambda_i^a @)@ and @(@ \lambda_i^t @)@. The corresponding priors as specified by mulstd_value_prior_id , mulstd_dage_prior_id , and mulstd_dtime_prior_id .

Parent Rates
For each rate (pini, iota, rho, chi, and omega) there is a function (set of variables) corresponding to the parent value for the rate. The smoothing for each of these functions is specified by the corresponding parent_smooth_id . The smoothing determines the number of variables in the set as well as the corresponding age and time values; see the unadjusted rates q_k in the average integrand model.

Group Covariate Multipliers
For each mulcov_id there is a corresponding function (set of variables) specified by the group_smooth_id . These variables are fixed effects. For more clarification, see the discussion for mulcov_type .

Random Effects, u
we use @(@ u @)@ to denote the vector of random effects; i.e., all of the variables except for the fixed effects. There are two types of random effects:

Child Rate Effects
For each rate (pini, iota, rho, chi, and omega) there is a function (set of variables) corresponding to the child random effects for the rate. The smoothing can be the same for all the children (see child_smooth_id ) or it can have a different value for each child (see child_nslist_id ). If u_ik is a random effect for a rate and child, the rate for the child is @(@ \exp( u_{i,k} ) @)@ times the rate for the parent; see the adjusted rates r_ik in the average integrand model.

Subgroup Covariate Multipliers
For each mulcov_id there is a corresponding smoothing specified by the mulcov table group_smooth_id . For each subgroup_id that corresponds to this group_id in the subgroup table, there is a corresponding function (set of variables) specified by the subgroup_smooth_id . These variables are random effects.

Age and Time Variation

Smoothing Standard Deviation Multiplier
Each smoothing standard deviation multiplier @(@ \lambda @)@ is scalar fixed effect (not a function) and has a prior specified above.

Initial Prevalence
Initial prevalence is a function of time but must be constant with respect to age. Hence a smoothing corresponding to initial prevalence must have only one age point. This holds for the parent initial prevalence function and each child initial prevalence effect function.

Other Cases
All the other variable are members of a set that represents a function of age and time using a smoothing with an arbitrary number of age and time points.
Input File: omh/model/model_variable.omh