---------------------------------------------------
lines 6-165 of file: example/absolute_covariates.py
---------------------------------------------------

{xrst_begin_parent absolute_covariates}
{xrst_spell
  dage
  dtime
}

Example Using absolute_covariates Option in all_node_database
#############################################################
For this example everything is constant w.r.t. age and time.

Nodes
*****
The following is a diagram of the node tree for this example.
The :ref:`glossary@root_node` is n0,
the :ref:`glossary@fit_goal_set`
and :ref:`fit_goal_table-name` are {n3, n4, n2},
and the leaf nodes are {n3, n4, n5, n6}::

                     n0
             /-----/\-----\
          n1             (n2)
         /  \            /  \
     (n3)  (n4)       n5    n6

fit_goal_set
============
{xrst_literal
    # BEGIN_FIT_GOAL
    # END_FIT_GOAL
}

Rates
*****
The only non-zero dismod_at rates for this example are
:ref:`glossary@iota`.and :ref:`glossary@omega`.

Splitting Covariate
===================
This cascade is set up to split by sex reference value; see
:ref:`split_reference_table-name`
{xrst_literal
    # BEGIN split_reference_table
    # END split_reference_table
}

Covariate
*********
There are three covariates for this example, sex, vaccine, and income.
Income is the only :ref:`glossary@Relative Covariate`.
{xrst_literal
    # BEGIN avg_income
    # END avg_income
}
{xrst_literal
    # BEGIN split_reference_list
    # END split_reference_list
}

absolute_covariates
===================
The only absolute covariate in this example is vaccine
(0 for no vaccine, 1 for yes vaccine).
{xrst_literal
    # BEGIN_1 absolute_covariates
    # END_1 absolute_covariates
}

alpha
=====
We use *alpha*\ ``[income]``
and  *alpha*\ ``[vaccine]``
for the :ref:`glossary@rate_value` covariate multipliers
that multiply the income and vaccine covariates.
The true value for *alpha* (used which simulating the data) is
{xrst_literal
    # BEGIN alpha_true
    # END alpha_true
}

Random Effects
**************
There are no random effect for this example.

Simulated Data
**************

rate_true(rate, a, t, n, c)
===========================
For *rate* equal to iota or omega,
this is the true value for *rate*
in node *n* at age *a*, time *t*, and covariate values *c=[sex,income]*.
The covariate values are a list in the
same order as the covariate table.
The values *a*, *t*, *n*, *sex*
are not used by this function for this example.
{xrst_literal
    # BEGIN rate_true
    # END rate_true
}

y_i
===
The only simulated integrand for this example is :ref:`glossary@Sincidence`
which is a direct measurement of iota.
This data is simulated without any noise; i.e.,
the i-th measurement is simulated as
*y_i = rate_true('iota', None, None, None, [None, I_i])*
where *I_i* is the income for the i-th measurement.
The data is modeled as having noise even though there is no simulated noise.

n_i
===
Data is only simulated for the leaf nodes; i.e.,
each *n_i* is in the set { n3, n4, n5, n6 }.
Since the data does not have any nose, the data residuals are a measure
of how good the fit is for the nodes in the fit_goal_set.

Parent Rate Smoothing
*********************
This is the iota smoothing used for the fit_node.
There are no :ref:`glossary@dage` or :ref:`glossary@dtime`
priors because there is only one age and one time point in the smoothing grid.

Value Prior
===========
The following is the value prior used for the root_node
{xrst_literal
    # BEGIN parent_value_prior
    # END parent_value_prior
}
The mean and standard deviation are only used for the root_node.
The :ref:`create_shift_db-name`
routine replaces them for other nodes.

Alpha Smoothing
***************
This is the smoothing used for *alpha* which multiplies the income covariate.
There is only one age and one time point in this smoothing
so it does not have dage or dtime priors.

Value Prior
===========
The following is the value prior used for this smoothing:
{xrst_literal
    # BEGIN alpha_value_prior
    # END alpha_value_prior
}
The mean and standard deviation are only used for the root_node.
The create_shift_db
routine replaces them for other nodes.

Checking The Fit
****************
The results of the fit are checked by check_cascade_node
using the :ref:`check_cascade_node@avgint_table`
that was created by the root_node_db routine.
The node_id for each row is replaced by the node_id for the
fit being checked.

{xrst_end absolute_covariates}