The Rate Table

@(@\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 Rate Table
Discussion
rate_id
rate_name
     pini
     iota
     rho
     chi
     omega
parent_smooth_id
     null
child_smooth_id
     Random Effects Hessian
     null
child_nslist_id
     null
Child Value Priors
     Constant
     Not Constant
Child Difference Priors
Example

Discussion
The rate table is used to set the smoothing for each of the rates. It has the following columns:

rate_id
This column has type integer and is the primary key for this table. Its initial value is zero, and it increments by one for each row.

rate_name
The following values, and only the following values, appear in the rate_name column:

`rate_name`	Description
`pini`	initial prevalence
`iota`	model susceptible incidence
`rho`	model remission
`chi`	model excess mortality
`omega`	model other cause mortality

In addition, these values must appear in the order above. (Note that this column is unique; i.e., each name only appears once.)

pini
The initial prevalence pini is the model for prevalence at the initial age; i.e., the minimum age corresponding in the age table; see p_i,0(t) . If rate_name is pini, the parent_smooth_id and child_smooth_id smoothings must have n_age equal to one. This is because prevalence at the initial age is only a function of time, and does not depend on age. Note that, unlike the other rates, initial prevalence is not a rate in the ordinary differential equation . On the other hand, and like the other rates, initial prevalence is a necessary input to solve the ODE.

iota
The model for susceptible incidence rate iota is the rate at which people who are in the susceptible state S transfer to the with condition state C ; see iota_i(a, t) .

rho
The model for remission rate rho is the rate at which people who are in the with condition state C transfer to the susceptible state S ; see rho_i(a, t) .

chi
The model for excess mortality rate chi is the rate at which people who are in the with condition state C die from the condition; see chi_i(a, t) .

omega
The model for other mortality rate omega is the rate at which people who are in either state, S or C , die from causes other than the condition; see omega_i(a, t) .

parent_smooth_id
This column has type integer and its value is a smooth_id in the smooth_grid table. It specifies the smoothing used as a prior for this rate when fitting the parent (fixed effects); see parent node .

null
If the parent_smooth_id is null, the corresponding parent rate is always zero and no model_variables are allocated for it.

child_smooth_id
This column has type integer and its value is a smooth_id in the smooth_grid table. It specifies a single smoothing used as a prior for the child rate effects for this rate.

Random Effects Hessian
The Hessian of the joint likelihood with respect to the random effects must be positive definite. If the data does not ensure this, the user will have to make the standard deviation for the prior of the random effects small.

null
If child_smooth_id is not null, child_nslist_list must be null. If both are null, there are no random effects . for this rate.

child_nslist_id
This column has type integer and is a nslist_id . If it is not null, it identifies a list of ( node_id , smooth_id ) pairs. The node_id for each of the children must appear in the list. The corresponding smoothing is used for the child rate effects for the rate corresponding to this row of the rate table.

null
If child_nslist_list is not null, child_smooth_id must be null. If both are null, there are no random effects . for this rate.

Child Value Priors
The priors corresponding to child smoothing value_prior_id are called child value priors. It is important to remember that these priors correspond to the random effects, u . The rate in the differential equation, used for a child, is exp(u)*q where q is the corresponding rate for the parent; see r_ik .

Constant
If the lower and upper limit are equal, they must be finite. In this case we refer to a child value prior as constant.

Not Constant
If a child value prior is not constant, it must satisfy the following conditions:

The density_id must not correspond to a nonsmooth density.
The lower limit (upper limit) must be null, which gets interpreted as minus infinity (plus infinity); see bounds .
Note that bound_random is used for the lower and upper limits of during the optimization of the random effects. On the other hand, plus and minus infinity is used for the lower and upper limit when cppad_mixed approximates the integrals with respect to the random effects .

Child Difference Priors
The priors corresponding to child smoothing dage_prior_id and dtime_prior_id are called child difference priors. These priors must satisfy the following conditions:

The density_id must not correspond to a nonsmooth density.
The lower limit (upper limit) must be null, which gets interpreted as minus infinity (plus infinity).

Example
The file rate_table.py contains an example rate table.

Input File: omh/table/rate_table.omh