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avg_integrand

@(@\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 . Model for the Average Integrand
Ordinary Differential Equation
Data or Avgint Table Notation
     i
     a_i
     b_i
     c_i
     d_i
     s_i
     g_i
     Covariate Difference, x_ij
     w_i
     n_i
Rate Functions
     Parent Rate, q_k
     Child Rate Effect, u_ik
     J_k
     Group Rate Covariate Multiplier, alpha_jk
     Subgroup Rate Covariate Multiplier, Delta alpha_jk
     Adjusted Rate, r_ik
     pini, p_i0(t)
     iota_i(a,t)
     rho_i(a,t)
     chi_i(a,t)
     omega_i(a,t)
S_i(a,t)
C_i(a,t)
Differential Equation
Integrand, I_i(a,t)
     Sincidence
     remission
     mtexcess
     mtother
     mtwith
     susceptible
     withC
     prevalence
     Tincidence
     mtspecific
     mtall
     mtstandard
     relrisk
     mulcov
Measurement Value Covariates
     K_i
     Group Measurement Covariate Multiplier, beta_j
     Subgroup Measurement Covariate Multiplier, Delta beta_j
     Measurement Effect
Adjusted Integrand
Weight Integral, wbar_i
Average Integrand, A_i

Ordinary Differential Equation
In the case where the rates do not depend on time, the dismod_ode ordinary differential equation is @[@ \begin{array}{rcl} S'(a) & = & - [ \iota(a) + \omega (a) ] S(a) + \rho(a) C(a) \\ C'(a) & = & + \iota(a) S(a) - [ \rho(a) + \chi(a) + \omega (a) ] C(a) \end{array} @]@ with the initial condition @(@ C(0) = p_0 @)@ and @(@ S(0) = 1 - p_0 @)@. This equation is made more complicated by the fact that the rates vary with time as well as with each data point. The reason for the variation between data points due both to the random effects as well as the rate_value covariates.

Data or Avgint Table Notation

i
We use @(@ i @)@ to denote either, data_id for a row in the data table or, avgint_id for a row in the avgint table.

a_i
We use @(@ a_i @)@ to denote the corresponding age_lower in the data table or avgint table .

b_i
We use @(@ b_i @)@ to denote the corresponding age_upper in the data table or avgint table .

c_i
We use @(@ c_i @)@ to denote the corresponding time_lower in the data table or avgint table .

d_i
We use @(@ d_i @)@ to denote the corresponding time_upper in the data table or avgint table .

s_i
We use @(@ s_i @)@ to denote the corresponding subgroup_id in the data table or avgint table .

g_i
We use @(@ g_i @)@ to denote the group_id that @(@ s_i @)@ is part of.

Covariate Difference, x_ij
We use @(@ x_{i,j} @)@ to denote the covariate difference for the i-th data point and the j-th covariate. Here i denotes a data_id in the data table and j denotes a covariate_id in the covariate table. The difference is the corresponding data table covariate value minus the covariate table reference value .

w_i
We use @(@ w_i (a, t) @)@ for the weighting as a function of age and time that corresponds to the weight_id for this data_id .

n_i
We use @(@ n_i @)@ to denote the corresponding node_id value.

Rate Functions

Parent Rate, q_k
We use @(@ k @)@ to denote a rate_id and @(@ q_k (a, t) @)@ the piecewise linear rate function for the corresponding to the parent rate . This is the model for the rate corresponding to the parent node and the reference value for the covariates; i.e. @(@ x_{i,j} = 0 @)@. The adjusted rate r_ik is defined below.

Child Rate Effect, u_ik
If the node for this data point @(@ n_i @)@ is a child node, or a descendant of a child node, @(@ u_{i,k} (a, t) @)@ is the piecewise linear random effect for the corresponding child and rate . If @(@ n_i @)@ is the parent node, there is no random effect for this data and @(@ u_{i,k} (a, t) = 0 @)@. Otherwise @(@ n_i @)@ is not the parent node or a descendant of the parent node and the corresponding data is not used.

J_k
We use @(@ J_k @)@ to denote the set of covariate_id values in the mulcov table such that the corresponding rate_id is equal to @(@ k @)@, mulcov_type is rate_value, and group_id equal to @(@ g_i @)@. These covariates that affect the k-th rate for this group of measurements.

Group Rate Covariate Multiplier, alpha_jk
For each rate index @(@ k @)@, and each covariate index @(@ j \in J_k @)@, we use @(@ \alpha_{j,k} (a, t) @)@ to denote the piecewise linear function corresponding to the group covariate multiplier . Note that these are rate_value covariate multipliers specified by the mulcov table.

Subgroup Rate Covariate Multiplier, Delta alpha_jk
For each rate index @(@ k @)@, and each covariate index @(@ j \in J_k @)@, we use @(@ \Delta \alpha_{j,k} (a, t) @)@ to denote the piecewise linear function corresponding to the @(@ s_i @)@ subgroup covariate multiplier .

Adjusted Rate, r_ik
We define the adjusted k-th rate for the i-th data point by @[@ r_{i,k} (a , t) = \exp \left \{ u_{i,k} (a, t) + \sum_{j \in J(k)} x_{i,j} [ \alpha_{j,k} (a, t) + \Delta \alpha_{j,k} (a, t) ] \right \} q_k ( a, t ) @]@ If @(@ n_i @)@ is the parent node, the random effects is zero @(@ u_{i,k} (a, t) = 0 @)@. If this data point also has the reference value for the covariates, @(@ r_{i,k} (a, t) = q_k (a, t) @)@.

pini, p_i0(t)
We use @(@ p_{i,0} (t) @)@ to denote the model value (as apposed to a measurement value) for prevalence at the initial age as a function of time. Note that this function is constant with respect to age @(@ a @)@; see pini . This is denoted by @(@ r_{i,0} (a, t) @)@ above.

iota_i(a,t)
We use iota_i(a, t) and @(@ \iota_i (a,t) @)@ to denote the model value for adjusted susceptible incidence as a function of age and time; see iota . This is denoted by @(@ r_{i,1} ( a, t ) @)@ above.

rho_i(a,t)
We use rho_i(a, t) and @(@ \rho_i (a,t) @)@ to denote the model value for adjusted remission as a function of age and time; see rho . This is denoted by @(@ r_{i,2} ( a, t ) @)@ above.

chi_i(a,t)
We use chi_i(a, t) and @(@ \chi_i (a,t) @)@ to denote the model value for adjusted excess mortality (mortality due to the cause) as a function of age and time. This is denoted by @(@ r_{i,3} ( a, t ) @)@ above.

omega_i(a,t)
We use omega_i(a, t) and @(@ \omega_i (a,t) @)@ to denote the model value for adjusted other cause mortality as a function of age and time; see omega . This is denoted by @(@ r_{i,4} ( a, t ) @)@ above.

S_i(a,t)
We use @(@ S_i (a,t) @)@ to denote the model value for susceptible fraction of the population.

C_i(a,t)
We use @(@ C_i (a,t) @)@ to denote the model value for with condition fraction of the population.

Differential Equation
We drop the subscript @(@ i @)@ in the adjusted rates to simplify notation in the equations below. The with condition and susceptible fractions at age zero are @[@ C (0, t) = p_0 (t) \; , \; S (0, t) = 1 - p_0 (t) @]@ We use @(@ c @)@ to denote cohort; i.e., @(@ t = a + c @)@, Given the rates (initial prevalence is called a rate), the functions @(@ S (a,t) @)@ and @(@ C (a,t) @)@ for @(@ a > 0 @)@ are defined by @[@ \begin{array}{rcl} ( \B{d} / \B{d} a ) S(a, a+c) & = & - [ \iota (a, a+c) + \omega (a, a+c) ] S(a, a+c) + \rho(a, a+c) C(a, a+c) \\ ( \B{d} / \B{d} a ) C(a, a+c) & = & + \iota(a, a+c) S(a, a+c) - [ \rho(a, a+c) + \chi(a, a+c) + \omega(a, a+c) ] C(a, a+c) \end{array} @]@

Integrand, I_i(a,t)
We use @(@ I_i (a, t) @)@ to denote the integrand as a function of age and time. Depending on the value of see integrand_id for data index @(@ i @)@, the function @(@ I_i (a, t) @)@ is defined below. The age and time arguments @(@ (a, t) @)@ and the subscript @(@ i @)@ are dropped to simplify notation. The rates are adjusted rates. (Integrands that do not use @(@ S @)@, @(@ C @)@ or @(@ P @)@, do not require solving the differential equation.)

Sincidence
The incidence rate relative to susceptible population is @(@ I = \iota @)@.

remission
The remission rate is @(@ I = \rho @)@.

mtexcess
The excess mortality rate is @(@ I = \chi @)@.

mtother
The other cause mortality rate is @(@ I = \omega @)@.

mtwith
The with condition mortality rate is @(@ I = \omega + \chi @)@.

susceptible
The susceptible fraction of the population is @(@ I = S @)@.

withC
The with condition fraction of the population is @(@ I = C @)@.

prevalence
The prevalence of the condition is @(@ I = P = C / ( S + C ) @)@.

Tincidence
The incidence rate relative to the total population is @(@ I = \iota ( 1 - P ) @)@.

mtspecific
The cause specific mortality rate is @(@ I = \chi P @)@.

mtall
The all cause mortality rate is @(@ I = \omega + \chi P @)@.

mtstandard
The standardized mortality ratio is @(@ I = ( \omega + \chi ) / ( \omega + \chi P ) @)@.

relrisk
The relative risk is @(@ I = ( \omega + \chi ) / \omega @)@.

mulcov
If the integrand_name is mulcov_mulcov_id , @(@ I @)@ is the covariate multiplier corresponding to mulcov_id . In this case there are no covariate that affect the measurement.

Measurement Value Covariates

K_i
We use @(@ K_i @)@ to the set of covariate_id values in the mulcov table such that the corresponding integrand_id corresponds to @(@ I_i (a, t) @)@, mulcov_type is meas_value, and group_id equal to @(@ g_i @)@. These are the covariates that affect the i-th measurement value.

Group Measurement Covariate Multiplier, beta_j
For each covariate index @(@ j \in K_i @)@, we use @(@ \beta_j (a, t) @)@ to denote the piecewise linear function corresponding to the group covariate multiplier . Note that these are meas_value covariate multipliers specified by the mulcov table.

Subgroup Measurement Covariate Multiplier, Delta beta_j
For each covariate index @(@ j \in K_i @)@, we use @(@ \Delta \beta_j (a, t) @)@ to denote the piecewise linear function corresponding to the @(@ s_i @)@ subgroup covariate multiplier .

Measurement Effect
The effect for the i-th measurement value, as a function of the fixed effects @(@ \theta @)@, is @[@ E_i ( a, t ) = \sum_{j \in K_i} x_{i,j} [ \beta_j (a, t) + \Delta \beta_j (a, t) ] @]@

Adjusted Integrand
The adjusted integrand is the following function of age and time: @[@ I_i (a, t) \; \exp \left[ E_i (a, t) \right] @]@ Note that if @(@ I_i (a, t) @)@ is a covariate multiplier mulcov , the adjusted integrand is equal to @(@ I_i(a,t) @)@; i.e., there is no adjustment.

Weight Integral, wbar_i
We use @(@ \bar{w}_i @)@ to denote the weight integral defined by @[@ \bar{w}_i = \frac{1}{b_i - a_i} \frac{1}{d_i - c_i} \int_{a(i)}^{b(i)} \int_{c(i)}^{d(i)} w_i (a,t) \; \B{d} t \; \B{d} a @]@

Average Integrand, A_i
We use @(@ u @)@ and @(@ \theta @)@ to denote the vector of random effects and fixed effects respectively. The model for the i-th measurement, not counting integrand effects or measurement noise, is @[@ A_i ( u , \theta ) = \frac{1}{b_i - a_i} \frac{1}{d_i - c_i} \left[ \int_{a(i)}^{b(i)} \int_{c(i)}^{d(i)} \frac{ w_i (a,t) }{ \bar{w}_i } I_i (a, t) \; \exp \left[ E_i (a, t) \right] \; \B{d} t \; \B{d} a \right] \; @]@ Note that this is actually a weighted average of the integrand function @(@ I_i (a, t) @)@ times the total measurement covariate effect @(@ E_i (a, t) @)@ Also note that in the case where @(@ a(i) = b(i) @)@, @(@ c(i) = d(i) @)@, or both, the average is defined as the limiting value.

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Input File: omh/model/avg_integrand.omh