lines 5-183 of file: xrst/model/model_variable.xrst {xrst_begin model_variables} {xrst_spell ik } The Model Variables ################### 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 :ref:`prior_table@prior_id` . If the corresponding lower and upper limits are equal, or it is specified to have a constant value in the :ref:`smooth_grid_table-name` , 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 :ref:`db2csv_command@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 :ref:`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. :ref:`bilinear-name` interpolation is used to define the function for all values of age and time. The :ref:`smooth_table@smooth_id` for one of these variables specifies its prior as follows: .. list-table:: :widths: auto * - :ref:`smooth_table@n_age` - number of age points used to represent the function * - :ref:`smooth_table@n_time` - number of time points used to represent the function * - :ref:`smooth_grid_table@age_id` - identifies age value for a variable * - :ref:`smooth_grid_table@time_id` - identifies time value for a variable * - :ref:`smooth_grid_table@const_value` - null or a value that a variable is constrained to * - :ref:`smooth_grid_table@value_prior_id` - value prior for a variable * - :ref:`smooth_grid_table@dage_prior_id` - difference prior for a variable (and next variable) in age direction * - :ref:`smooth_grid_table@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 :ref:`option table` . The :ref:`node_table@parent@Children` corresponding to the parent node. (The children is a set of nodes not a set of variables.) Fixed Effects, theta ******************** We use :math:`\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 :math:`i =` :ref:`smooth_table@smooth_id` , there are three smoothing standard deviation multiplier variables: :math:`\lambda_i^v`, :math:`\lambda_i^a` and :math:`\lambda_i^t`. The corresponding priors as specified by :ref:`smooth_table@mulstd_value_prior_id` , :ref:`smooth_table@mulstd_dage_prior_id` , and :ref:`smooth_table@mulstd_dtime_prior_id` . Parent Rates ============ For each :ref:`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 :ref:`rate_table@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 :ref:`q_k` in the average integrand model. Group Covariate Multipliers =========================== For each :ref:`mulcov_table@mulcov_id` there is a corresponding function (set of variables) specified by the :ref:`mulcov_table@group_smooth_id` . These variables are fixed effects. For more clarification, see the discussion for :ref:`mulcov_table@mulcov_type` . Random Effects, u ***************** we use :math:`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 :ref:`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 :ref:`rate_table@child_smooth_id` ) or it can have a different value for each child (see :ref:`rate_table@child_nslist_id` ). If :ref:`u_ik` is a random effect for a rate and child, the rate for the child is :math:`\exp( u_{i,k} )` times the rate for the parent; see the adjusted rates :ref:`r_ik` in the average integrand model. Subgroup Covariate Multipliers ============================== For each :ref:`mulcov_table@mulcov_id` there is a corresponding smoothing specified by the mulcov table :ref:`mulcov_table@group_smooth_id` . For each :ref:`subgroup_table@subgroup_id` that corresponds to this :ref:`subgroup_table@group_id` in the subgroup table, there is a corresponding function (set of variables) specified by the :ref:`mulcov_table@subgroup_smooth_id` . These variables are random effects. Age and Time Variation ********************** Smoothing Standard Deviation Multiplier ======================================= Each smoothing standard deviation multiplier :math:`\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. {xrst_end model_variables}