1 Introduction
Specification document - a mathematical description of models used by bage.
Note: some features described here have not been implemented yet.
2 Likelihood
2.1 Input data
- outcome variable: events, numbers of people, or some sort of measure on a continuous variable such as income or expenditure
- exposure/size/weights
- disagg by one or more variables. Almost always includes age, sex/gender, and time. May include other variables eg region, ethnicity, education.
- not all combinations of variables present; may be some missing values
2.2 Models
2.2.1 Poisson
Let \(y_i\) be a count of events in cell \(i = 1, \cdots, n\) and let \(w_i\) be the corresponding exposure measure, with the possibility that \(w_i \equiv 1\). The likelihood under the Poisson model is then \[\begin{align} y_i & \sim \text{Poisson}(\gamma_i w_i) \tag{2.1} \\ \gamma_i & \sim \text{Gamma}\left(\xi^{-1}, (\mu_i \xi)^{-1}\right), \tag{2.2} \end{align}\] using the shape-rates parameterisation of the Gamma distribution. Parameter \(\xi\) governs dispersion, with \[\begin{equation} \text{var}(\gamma_i \mid \mu_i, \xi) = \xi \mu_i^2 \end{equation}\] and \[\begin{equation} \text{var}(y_i \mid \mu_i, \xi, w_i) = (1 + \xi \mu_i w_i ) \times \mu_i w_i. \end{equation}\] We allow \(\xi\) to equal 0, in which case the model reduces to \[\begin{equation} y_i \sim \text{Poisson}(\mu_i w_i). \end{equation}\]
For \(\xi > 0\), Equations (2.1) and (2.2) are equivalent to \[\begin{equation} y_i \sim \text{NegBinom}\left(\xi^{-1}, (1 + \mu_i w_i \xi)^{-1}\right) \end{equation}\] (Norton, Christen, and Fox 2018; Simpson 2022). This is the format we use internally for estimation. When values for \(\gamma_i\) are needed, we generate them on the fly, using the fact that \[\begin{equation} \gamma_i \mid y_i, w_i, \mu_i, \xi \sim \text{Gamma}\left(y_i + \xi^{-1}, w_i + (\xi \mu_i)^{-1}\right). \end{equation}\]
2.2.2 Binomial
The likelihood under the binomial model is \[\begin{align} y_i & \sim \text{Binomial}(w_i, \gamma_i) \tag{2.3} \\ \gamma_i & \sim \text{Beta}\left(\xi^{-1} \mu_i, \xi^{-1}(1 - \mu_i)\right). \tag{2.4} \end{align}\] Parameter \(\xi\) again governs dispersion, with \[\begin{equation} \text{var}(\gamma_i \mid \mu_i, \xi) = \frac{\xi}{1 + \xi} \times \mu_i (1 -\mu_i) \end{equation}\] and \[\begin{equation} \text{var}(y_i \mid w_i, \mu_i, \xi) = \frac{\xi w_i + 1}{\xi + 1} \times w_i \mu_i (1 - \mu_i). \end{equation}\]
We allow \(\xi\) to equal 0, in which case the model reduces to \[\begin{equation} y_i \sim \text{Binom}(w_i, \mu_i). \end{equation}\] Equations (2.3) and (2.4) are equivalent to \[\begin{equation} y_i \sim \text{BetaBinom}\left(w_i, \xi^{-1} \mu_i, \xi^{-1} (1 - \mu_i) \right), \end{equation}\] which is what we use internally. Values for \(\gamma_i\) can be generated using \[\begin{equation} \gamma_i \mid y_i, w_i, \mu_i, \xi \sim \text{Beta}\left(y_i + \xi^{-1} \mu_i, w_i - y_i + \xi^{-1}(1-\mu_i) \right). \end{equation}\]
2.2.3 Normal
\[\begin{equation} y_i \sim \text{N}(\mu_i, w_i^{-1}\xi^2) \end{equation}\] where the \(w_i\) are weights.
Response \(y_i\) is standardized to have mean 0 and standard deviation 1. We set \[\begin{equation} y_i = \frac{y_i^{*} - \bar{y}^*}{s^*} \end{equation}\] where the \(y_i^*\) are the values originally supplied by the user, and \(\bar{y}^*\) and \(s^*\) are the mean and standard deviation of the \(y_i^*\).
3 Model for means
Let \(\pmb{\mu} = (\mu_1, \cdots, \mu_n)^{\top}\). Our model for \(\pmb{\mu}\) is \[\begin{equation} \pmb{\mu} = \sum_{m=0}^{M} \pmb{X}^{(m)} \pmb{\beta}^{(m)} + \pmb{Z} \pmb{\zeta} \tag{3.1} \end{equation}\] where
- \(\beta^{(0)}\) is an intercept;
- \(\pmb{\beta}^{(m)}\), \(m=1,\cdots,M\) is a vector with \(J_m\) elements describing a main effect or interaction formed from the dimensions of data \(\pmb{y}\);
- \(\pmb{X}^{(m)}\) is an \(n \times J_m\) matrix of 1s and 0s, the \(i\)th row of which picks out the element of \(\pmb{\beta}^{(m)}\) that is used with cell \(i\);
- \(\pmb{Z}\) is a \(n \times P\) matrix of covariates; and
- \(\pmb{\zeta}\) is a coefficient vector with \(P\) elements.
4 Priors for Intercept, Main Effects, and Interactions
The algorithm for assigning default priors:
- If \(\pmb{\beta}^{(m)}\) has one or two elements, assign \(\pmb{\beta}^{(m)}\) a fixed-normal prior (Section 4.2);
- otherwise, if \(\pmb{\beta}^{(m)}\) is an age or time main effect, assign \(\pmb{\beta}^{(m)}\) a random walk prior (Section 4.3);
- otherwise, assign \(\pmb{\beta}^{(m)}\) a normal prior (Section 4.1)
The intercept term \(\pmb{\beta}^{(0)}\) can only be given a fixed-normal prior (Section 4.2) or a Known prior (Section 4.17).
4.1 Normal
4.1.1 Model
\[\begin{align} \beta_j^{(m)} & \sim \text{N}\left(0, \tau_m^2 \right) \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \end{align}\]
4.1.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{j=1}^{J_m} \text{N}(\beta_j^{(m)} \mid 0, \tau_m^2) \end{equation}\]
4.1.3 Simulation
\[\begin{align} \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \\ \beta_j^{(m)} & \sim \text{N}\left(0, \tau_m^2 \right) \end{align}\]
4.2 Fixed normal
4.2.1 Model
\[\begin{equation} \beta_j^{(m)} \sim \text{N}\left(0, A_{\beta}^{(m)2}\right) \end{equation}\]
4.2.2 Contribution to posterior density
\[\begin{equation} \prod_{j=1}^{J_m} \text{N}(\beta_j^{(m)} \mid 0, A_{\beta}^{(m)2}) \end{equation}\]
4.2.3 Simulation
\[\begin{equation} \beta_j^{(m)} \sim \text{N}\left(0, A_{\beta}^{(m)2}\right) \end{equation}\]
4.3 First-order random walk
4.3.1 Model
\[\begin{align} \beta_{u1}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m \\ \beta_{uv}^{(m)} & \sim \text{N}(\beta_{u,v-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, v = 2, \cdots, V_m \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \end{align}\]
4.3.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \text{N}(\beta_{u1}^{(m)} \mid 0, 1) \prod_{u=1}^{U_m} \prod_{v=2}^{V_m} \text{N}\left(\beta_{uv}^{(m)} \mid \beta_{u,v-1}^{(m)}, \tau_m^2 \right) \end{equation}\]
4.3.3 Simulation
\[\begin{align} \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \\ \beta_{u,1}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, \\ \beta_{u,v}^{(m)} & \sim \text{N}(\beta_{u,v-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, v = 2, \cdots, V_m \end{align}\]
4.3.4 Forecasting
\[\begin{equation} \beta_{u,V_m+h}^{(m)} \sim \text{N}(\beta_{u,V_m+h-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m \end{equation}\]
4.3.5 Code
RW(s = 1, along = NULL)-
sis \(A_{\tau}^{(m)}\). Defaults to 1. -
alongused to identify “along” and “by” dimensions
4.4 Second-order random walk
4.4.1 Model
\[\begin{align} \beta_{u,v}^{(m)} & \sim \text{N}(0, 1), \quad u = 1,\cdots, U_m, \quad v = 1,2 \\ \beta_{u,v}^{(m)} & \sim \text{N}(2 \beta_{u,v-1}^{(m)} - \beta_{u,v-2}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, \quad v = 3, \cdots, V_m \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \end{align}\]
4.4.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \prod_{v=1}^2 \text{N}(\beta_{u,v}^{(m)} \mid 0, 1) \prod_{u=1}^{U_m}\prod_{v=3}^{V_m} \text{N}\left(\beta_{u,v}^{(m)} - 2 \beta_{u,v-1}^{(m)} + \beta_{u,v-2}^{(m)} \mid 0, \tau_m^2 \right) \end{equation}\]
4.4.3 Simulation
\[\begin{align} \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \\ \beta_{u,v}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, \quad v = 1,2 \\ \beta_{u,v}^{(m)} & \sim \text{N}(2 \beta_{u,v-1}^{(m)} - \beta_{u,v-2}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, \quad v = 3,\cdots,V_m \end{align}\]
4.4.4 Forecasting
\[\begin{equation} \beta_{u,V_m+h}^{(m)} \sim \text{N}(2 \beta_{u,V_m+h-1}^{(m)} - \beta_{u,V_m+h-2}^{(m)}, \tau_m^2) \end{equation}\]
4.4.5 Code
RW2(s = 1, sd = 1, along = NULL)-
sis \(A_{\tau}^{(m)}\) -
sdis \(A_{\eta}^{(m)}\) -
alongused to identify “along” and “by” dimensions
4.5 Random walk with seasonal effect
4.5.1 Model
\[\begin{align} \beta_{u,v}^{(m)} & = \alpha_{u,v} + \lambda_{u,s_v}, \quad u = 1, \cdots, U_m, \quad v = 1, \cdots, V_m \\ \alpha_{u,1}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m \\ \alpha_{u,v}^{(m)} & \sim \text{N}(\alpha_{u,v-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, \quad v = 2, \cdots, V_m \\ \lambda_{u,v}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, \quad v = 1, \cdots, S_m \\ \lambda_{u,v}^{(m)} & \sim \text{N}(\lambda_{u,v-S_m}^{(m)}, \omega_m^2), \quad u = 1, \cdots, U_m, \quad v = S_m + 1, \cdots, V_m \tag{4.1} \\ \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \\ \omega_m & \sim \text{N}^+\left(0, A_{\omega}^{(m)2}\right) \end{align}\]
We allow \(A_{\omega}^{(m)2}\) to be set to zero, in which case (4.1) reduces to \[\begin{equation} \lambda_{u,v}^{(m)} = \lambda_{u,v-S_m}^{(m)}, \quad u = 1, \cdots, U_m, \quad v = S_m + 1, \cdots, V_m, \end{equation}\] implying that seasonal effects are constant across years.
4.5.2 Contribution to posterior density
\[\begin{equation} \begin{split} & \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \text{N}(\omega_m \mid 0, A_{\omega}^{(m)2}) \\ \quad \times & \prod_{u=1}^{U_m} \left( \text{N}(\alpha_{u,1}^{(m)} \mid 0, 1 ) \prod_{v=2}^{V_m} \text{N}(\alpha_{u,v}^{(m)} \mid \alpha_{u,v-1}^{(m)}, \tau_m^2 ) \prod_{v=1}^{S_m} \text{N}(\lambda_{u,v}^{(m)} \mid 0, 1) \prod_{v=S_m+1}^{V_m} \text{N}(\lambda_{u,v}^{(m)} \mid \lambda_{u,v-S_m}^{(m)}, \omega_m^2) \right) \end{split} \end{equation}\]
4.6 Second-order randomw walk, with seasonal effect
4.6.1 Model
\[\begin{align} \beta_{u,v}^{(m)} & = \alpha_{u,v} + \lambda_{u,s_v}, \quad u = 1, \cdots, U_m, \quad v = 1, \cdots, V_m \\ \alpha_{u,v}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, \quad \quad v = 1,2, \\ \alpha_{u,v}^{(m)} & \sim \text{N}(2 \alpha_{u,v-1}^{(m)} - \alpha_{u,v-2}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, \quad v = 3, \cdots, V_m \\ \lambda_{u,v}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, \quad v = 1, \cdots, S_m \\ \lambda_{u,v}^{(m)} & \sim \text{N}(\lambda_{u,v-S_m}^{(m)}, \omega_m^2), \quad u = 1, \cdots, U_m, \quad v = S_m + 1, \cdots, V_m \tag{4.1} \\ \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \\ \omega_m & \sim \text{N}^+\left(0, A_{\omega}^{(m)2}\right) \end{align}\]
We allow \(A_{\omega}^{(m)2}\) to be set to zero, in which case (4.1) reduces to \[\begin{equation} \lambda_{u,v}^{(m)} = \lambda_{u,v-S_m}^{(m)}, \quad u = 1, \cdots, U_m, \quad v = S_m + 1, \cdots, V_m, \end{equation}\] implying that seasonal effects are constant across years.
4.6.2 Contribution to posterior density
\[\begin{equation} \begin{split} & \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \text{N}(\omega_m \mid 0, A_{\omega}^{(m)2}) \\ \quad \times & \prod_{u=1}^{U_m} \left( \text{N}(\alpha_{u,1}^{(m)} \mid 0, 1 ) \prod_{v=2}^{V_m} \text{N}(\alpha_{u,v}^{(m)} \mid 2 \alpha_{u,v-1}^{(m)} - \alpha_{u,v-2}^{(m)}, \tau_m^2 ) \prod_{v=1}^{S_m} \text{N}(\lambda_{u,v}^{(m)} \mid 0, 1) \prod_{v=S_m+1}^{V_m} \text{N}(\lambda_{u,v}^{(m)} \mid \lambda_{u,v-S_m}^{(m)}, \omega_m^2) \right) \end{split} \end{equation}\]
4.7 Autoregressive
4.7.1 Model
\[\begin{equation} \beta_{u,v}^{(m)} \sim \text{N}\left(\phi_1^{(m)} \beta_{u,v-1}^{(m)} + \cdots + \phi_{K_m}^{(m)} \beta_{u,v-{K_m}}^{(m)}, \omega_m^2\right), \quad u = 1, \cdots, U_m, \quad v = K_m + 1, \cdots, V_m. \end{equation}\] Internally, TMB derives values for \(\beta_{u,v}^{(m)}, v = 1, \cdots, K_m\), and for \(\omega_m\), that imply a stationary distribution, and that give every term \(\beta_{u,v}^{(m)}\) the same marginal variance. We denote this marginal variance \(\tau_m^2\), and assign it a prior \[\begin{equation} \tau_m \sim \text{N}^+(0, A_{\tau}^{(m)2}). \end{equation}\] Each of the individual \(\phi_k^{(m)}\) is restricted to the interval \((-1, 1)\), and the \(\phi_k^{(m)}\) are jointly restricted to values that yield stationary models. Let \[\begin{equation} r^{(m)} = \sqrt{\phi_1^{(m)2} + \cdots + \phi_{K_m}^{(m)2}}. \end{equation}\] We assign \(r^{(m)}\) the prior \[\begin{equation} r^{(m)} \sim \text{Beta}(2, 2). \end{equation}\]
4.7.2 Contribution to posterior density
\[\begin{equation} \text{N}^+\left(\tau_m \mid 0, A_{\tau}^{(m)2} \right) \text{Beta}\left( r^{(m)} \mid 2, 2 \right) \prod_{u=1}^{U_m} p\left( \beta_{u,1}^{(m)}, \cdots, \beta_{u,V_m}^{(m)} \mid \phi_1^{(m)}, \cdots, \phi_{K_m}^{(m)}, \tau_m \right) \end{equation}\] where \(p\left( \beta_{u,1}^{(m)}, \cdots, \beta_{u,V_m}^{(m)} \mid \phi_1^{(m)}, \cdots, \phi_{K_m}^{(m)}, \tau_m \right)\) is calculated internally by TMB.
4.7.3 Forecasting
\[\begin{equation} \beta_{u,V_m + h}^{(m)} \sim \text{N}\left(\phi_1^{(m)} \beta_{u,V_m + h - 1}^{(m)} + \cdots + \phi_{K_m}^{(m)} \beta_{u,V_m+h-K_m}^{(m)}, \tau_m^2\right) \end{equation}\]
4.7.4 Code
AR(n = 2, s = 1, along = NULL)-
nis \(K_m\) -
sis \(A_{\tau}^{(m)}\) -
alongis used to indentify the “along” and “by” dimensions
4.8 AR1
Special case or AR, but with extra options for autocorrelation coefficient.
4.8.1 Model
\[\begin{align} \beta_{u,1}^{(m)} & \sim \text{N}(0, \tau_m^2), \quad u = 1, \cdots, U_m \\ \beta_{u,v}^{(m)} & \sim \text{N}(\phi_m \beta_{u,v-1}^{(m)}, (1 - \phi_m^2) \tau_m^2), \quad u = 1, \cdots, U_m, \quad v = 2, \cdots, V_m \\ \phi_m & = a_{0,m} + (a_{1,m} - a_{0,m}) \phi_m^{\prime} \\ \phi_m^{\prime} & \sim \text{Beta}(2, 2) \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right). p\end{align}\] This is adapted from the specification used for AR1 densities in TMB. It implies that the marginal variance of all \(\beta_{u,v}^{(m)}\) is \(\tau_m^2\). We require that \(-1 < a_{0m} < a_{1m} < 1\).
4.8.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \text{Beta}( \phi_m^{\prime} \mid 2, 2) \prod_{u=1}^{U_m} \text{N}\left(\beta_{u,1}^{(m)} \mid 0, \tau_m^2 \right) \prod_{u=1}^{U_m} \prod_{j=2}^{V_m} \text{N}\left(\beta_{u,v}^{(m)} \mid \phi_m \beta_{u,v-1}^{(m)}, (1 - \phi_m^2) \tau_m^2 \right) \end{equation}\]
4.8.3 Forecasting
\[\begin{equation} \beta_{J_m + h}^{(m)} \sim \text{N}\left(\phi_m \beta_{J_m + h - 1}^{(m)}, (1 - \phi_m^2) \tau_m^2\right) \end{equation}\]
4.8.4 Code
AR1(min = 0.8, max = 0.98, s = 1, along = NULL)-
minis \(a_{0m}\) -
maxis \(a_{1m}\) -
sis \(A_{\tau}^{(m)}\). Defaults to 1. -
alongis used to identify “along” and “by” dimensions
The defaults for min and max are based on the defaults for function ets() in R package forecast (Hyndman and Khandakar 2008).
4.9 Linear
4.9.1 Model
\[\begin{align} \beta_{u,v}^{(m)} & \sim \text{N}(\alpha_u^{(m)} + v \eta_u^{(m)}, \tau_m^2), \quad u = 1,\cdots,U_m, \quad v = 1, \cdots, V_m \\ \alpha_u^{(m)} & \sim \text{N}(0, 1), \quad u = 1,\cdots,U_m \\ \eta_u^{(m)} & \sim \text{N}\left(0, A_{\eta}^{(m)2}\right), \quad u = 1, \cdots, U_m \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \end{align}\]
4.9.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \text{N}(\alpha_u^{(m)} \mid 0, 1) \text{N}(\eta_u^{(m)} \mid 0, A_{\eta}^{(m)2}) \prod_{u=1}^{U_m} \prod_{v=1}^{V_m} \text{N}\left(\beta_{u,v}^{(m)} | \alpha_u^{(m)} + v \eta_u^{(m)}, \tau_m^2 \right) \end{equation}\]
4.9.3 Forecasting
\[\begin{equation} \beta_{u,V_m + h}^{(m)} \sim \text{N}(\alpha_u^{(m)} + (V_m + h) \eta_u^{(m)}, \tau_m^2) \end{equation}\]
4.9.4 Code
Lin(s = 1, sd = 1, along = NULL)-
sis \(A_{\tau}^{(m)}\) -
sdis \(A_{\eta}^{(m)}\) -
alongis used to indentify “along” and “by” dimensions
4.10 Linear-AR
4.10.1 Model
\[\begin{align} \beta_{u,v}^{(m)} & = \alpha_u^{(m)} + \eta_u^{(m)} v + \epsilon_{u,v}^{(m)}, \quad u = 1, \cdots, U_m, \quad v = 1, \cdots, V_m \\ \alpha_u^{(m)} & \sim \text{N}\left(0, 1\right), \quad u = 1, \cdots, U_m \\ \eta_u^{(m)} & \sim \text{N}\left(0, A_{\eta}^{(m)2}\right), \quad u = 1, \cdots, U_m \\ \epsilon_{u,v}^{(m)} & \sim \text{N}\left(\phi_1^{(m)} \epsilon_{u,v-1}^{(m)} + \cdots + \phi_{K_m}^{(m)} \epsilon_{u,v-{K_m}}^{(m)}, \omega_m^2\right), \quad u = 1, \cdots, U_m, \quad v = K_m + 1, \cdots, V_m. \end{align}\]
Internally, TMB derives values for \(\epsilon_{u,v}^{(m)}, v = 1, \cdots, K_m\), and for \(\omega_m\), that provide the \(\epsilon_{u,v}^{(m)}\) with a stationary distribution in which each term has the same marginal variance. We denote this marginal variance \(\tau_m^2\), and assign it a prior \[\begin{equation} \tau_m \sim \text{N}^+(0, A_{\tau}^{(m)2}). \end{equation}\] Each of the individual \(\phi_k^{(m)}\) is restricted to the interval \((-1, 1)\), and the \(\phi_k^{(m)}\) are jointly restricted to values that yield stationary models. Let \[\begin{equation} r^{(m)} = \sqrt{\phi_1^{(m)2} + \cdots + \phi_{K_m}^{(m)2}}. \end{equation}\] We assign \(r^{(m)}\) the prior \[\begin{equation} r^{(m)} \sim \text{Beta}(2, 2). \end{equation}\]
4.10.2 Contribution to posterior density
\[\begin{equation} \text{N}^+\left(\tau_m \mid 0, A_{\tau}^{(m)2} \right) \text{Beta}\left( r_k^{(m)} \mid 2, 2 \right) \prod_{u=1}^{U_m} \text{N}(\alpha_u^{(m)} \mid 0, 1) \text{N}(\eta_u^{(m)} \mid 0, A_{\eta}^{(m)2}) p\left( \epsilon_{u,1}^{(m)}, \cdots, \epsilon_{u,V_m}^{(m)} \mid \phi_1^{(m)}, \cdots, \phi_{K_m}^{(m)}, \tau_m \right) \end{equation}\] where \(p\left( \epsilon_{u,1}^{(m)}, \cdots, \epsilon_{u,V_m}^{(m)} \mid \phi_1^{(m)}, \cdots, \phi_{K_m}^{(m)}, \tau_m \right)\) is calculated internally by TMB.
4.10.3 Forecasting
\[\begin{align} \beta_{u, V_m + h}^{(m)} & = \alpha_u^{(m)} + \eta_u^{(m)} (V_m + h) + \epsilon_{u,V_m+h}^{(m)} \\ \epsilon_{u,V_m+h}^{(m)} & \sim \text{N}\left(\phi_1^{(m)} \epsilon_{u,V_m + h - 1}^{(m)} + \cdots + \phi_{K_m}^{(m)} \epsilon_{u,V_m+h-K_m}^{(m)}, \omega_m^2\right) \end{align}\]
4.10.4 Code
Lin_AR(s = 1, sd = 1, along = NULL)-
sis \(A_{\tau}^{(m)}\) -
sdis \(A_{\eta}^{(m)}\) -
alongis used to indentify “along” and “by” variables
4.11 P-Spline
4.11.1 Model
\[\begin{equation} \pmb{\beta}_u^{(m)} = \pmb{B}^{(m)} \pmb{\alpha}_u^{(m)}, \quad u = 1, \cdots, U_m \end{equation}\] where \(\pmb{\beta}_u^{(m)}\) is the subvector of \(\pmb{\beta}^{(m)}\) composed of elements from the \(u\)th combination of the “by” variables, \(\pmb{B}^{(m)}\) is a \(V_m \times K_m\) matrix of B-splines, and \(\pmb{\alpha}_u^{(m)}\) has a second-order random walk prior (Section 4.4).
\(\pmb{B}^{(m)} = (\pmb{b}_1^{(m)}(\pmb{v}), \cdots, \pmb{b}_{K_m}^{(m)}(\pmb{v}))\), with \(\pmb{v} = (1, \cdots, V_m)^{\top}\). The B-splines are centered, so that \(\pmb{1}^{\top} \pmb{b}_k^{(m)}(\pmb{v}) = 0\), \(k = 1, \cdots, K_m\).
4.11.2 Contribution to posterior density
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \prod_{k=1}^2 \text{N}(\alpha_{u,k}^{(m)} \mid 0, 1) \prod_{u=1}^{U_m}\prod_{k=3}^{K_m} \text{N}\left(\alpha_{u,k}^{(m)} - 2 \alpha_{u,k-1}^{(m)} + \alpha_{u,k-2}^{(m)} \mid 0, \tau_m^2 \right) \end{equation}\]
4.11.4 Code
Sp(n = NULL, s = 1)-
nis \(K_m\). Defaults to \(\max(0.7 J_m, 4)\). -
sis the \(A_{\tau}^{(m)}\) from the second-order random walk prior. Defaults to 1. -
alongis used to identify “along” and “by” variables
4.12 SVD
4.12.1 Model
Age but no sex or gender
Let \(\pmb{\beta}_u\) be the age effect for the \(u\)th combination of the ‘by’ variables. With an SVD prior, \[\begin{equation} \pmb{\beta}_u^{(m)} = \pmb{F}^{(m)} \pmb{\alpha}_u^{(m)} + \pmb{g}^{(m)}, \quad u = 1, \cdots, U_m \end{equation}\] where \(\pmb{F}^{(m)}\) is a \(V_m \times K_m\) matrix, and \(\pmb{g}^{(m)}\) is a vector with \(V_m\) elements, both derived from a singular value decomposition (SVD) of an external dataset of age-specific values for all sexes/genders combined. The construction of \(\pmb{F}^{(m)}\) and \(\pmb{g}^{(m)}\) is described in Appendix 12.2. The centering and scaling used in the construction allow use of the simple prior \[\begin{equation} \alpha_{u,k}^{(m)} \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, k = 1, \cdots, K_m. \end{equation}\]
Joint model of age and sex/gender
In the joint model, vector \(\pmb{\beta}_u\) represents the interaction between age and sex/gender for the \(u\)th combination of the ‘by’ variables. Matrix \(\pmb{F}^{(m)}\) and vector \(\pmb{g}^{(m)}\) are calculated from data that separate sexes/genders. The model is otherwise unchanged.
Independent models for each sex/gender
In the independent model, vector \(\pmb{\beta}_{s,u}\) represents age effects for sex/gender \(s\) and the \(u\)th combination of the ‘by’ variables, and we have \[\begin{equation} \pmb{\beta}_{s,u}^{(m)} = \pmb{F}_s^{(m)} \pmb{\alpha}_{s,u}^{(m)} + \pmb{g}_s^{(m)}, \quad s = 1, \cdots, S; \quad u = 1, \cdots, U_m \end{equation}\] Matrix \(\pmb{F}_s^{(m)}\) and vector \(\pmb{g}_s^{(m)}\) are calculated from data that separate sexes/genders. The prior is \[\begin{equation} \alpha_{s,u,k}^{(m)} \sim \text{N}(0, 1), \quad s = 1, \cdots, S; \quad u = 1, \cdots, U_m; \quad k = 1, \cdots, K_m. \end{equation}\]
4.12.2 Contribution to posterior density
\[\begin{equation} \prod_{u=1}^{U_m}\prod_{k=1}^{K_m} \text{N}\left(\alpha_{uk}^{(m)} \mid 0, 1 \right) \end{equation}\] for the age-only and joint models, and \[\begin{equation} \prod_{s=1}^S \prod_{u=1}^{U_m}\prod_{k=1}^{K_m} \text{N}\left(\alpha_{s,u,k}^{(m)} \mid 0, 1 \right) \end{equation}\] for the independent model
4.12.4 Code
SVD(ssvd, n_comp = NULL, indep = TRUE)where
- ssvd is an object containing \(\pmb{F}\) and \(\pmb{g}\)
- n_comp is the number of components to be used (which defaults to ceiling(n/2), where n is the number of components in ssvd
- indep determines whether and independent or joint model will be used if the term being modelled contains a sex or gender variable.
4.13 SVD_RW
4.13.1 Model
The SVD_RW() prior is identical to the SVD() prior except that the coefficients evolve over time, following independent random walks. For instance, in the combined-sex/gender and joint models,
\[\begin{align} \pmb{\beta}_{u,t}^{(m)} & = \pmb{F}^{(m)} \pmb{\alpha}_{u,t}^{(m)} + \pmb{g}^{(m)}, \quad u = 1, \cdots, U_m; \quad t = 1, \cdots, T \\ \alpha_{u,k,1}^{(m)} & \sim \text{N}(0, 1), \quad u = 1, \cdots, U_m, k = 1, \cdots, K_m \\ \alpha_{u,k,t}^{(m)} & \sim \text{N}(\alpha_{u,k,t-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m, k = 1, \cdots, K_m; t = 2, \cdots, T \\ \tau_m & \sim \text{N}^+\left(0, A_{\tau}^{(m)2}\right) \end{align}\]
4.13.2 Contribution to posterior density
In the combined-sex/gender and joint models,
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \prod_{k=1}^{K_m} \text{N}(\alpha_{u,k,1}^{(m)} \mid 0, 1) \prod_{t=2}^{T} \text{N}\left(\alpha_{u,k,t}^{(m)} \mid \alpha_{u,k,t-1}^{(m)}, \tau_m^2 \right), \end{equation}\]
and in the independent model,
\[\begin{equation} \text{N}(\tau_m \mid 0, A_{\tau}^{(m)2}) \prod_{u=1}^{U_m} \prod_{s=1}^{S} \prod_{k=1}^{K_m} \text{N}(\alpha_{u,s,k,1}^{(m)} \mid 0, 1) \prod_{t=2}^{T} \text{N}\left(\alpha_{u,s,k,t}^{(m)} \mid \alpha_{u,s,k,t-1}^{(m)}, \tau_m^2 \right) \end{equation}\]
4.13.4 Forecasting
\[\begin{align} \alpha_{u,k,T+h}^{(m)} & \sim \text{N}(\alpha_{u,k,T+h-1}^{(m)}, \tau_m^2), \quad u = 1, \cdots, U_m; \quad k = 1, \cdots, K_m \\ \pmb{\beta}_{u,T+h}^{(m)} & = \pmb{F}^{(m)} \pmb{\alpha}_{u,T+h}^{(m)} + \pmb{g}^{(m)}, \quad u = 1, \cdots, U_m \end{align}\]
4.13.5 Code
SVD_RW(ssvd, n_comp = NULL, s = 1, indep = TRUE)where
- ssvd is an object containing \(\pmb{F}\) and \(\pmb{g}\)
- n_comp is \(K_m\) (which defaults to ceiling(n/2), where n is the number of components in ssvd
- s is \(A_{\tau}^{(m)}\)
- indep determines whether and independent or joint model will be used if the term being modelled contains a sex or gender variable.
4.14 SVD_RW2
Same structure as SVD_RW(). TODO - write details.
4.15 SVD_AR
Same structure as SVD_RW(). TODO - write details.
4.16 SVD_AR1
Same structure as SVD_RW(). TODO - write details.
5 Covariates
5.1 Model
The columns of matrix \(\pmb{Z}\) are assumed to be standardised to have mean 0 and standard deviation 1. \(\pmb{Z}\) does not contain a column for an intercept.
We implement two priors for coefficient vector \(\pmb{\zeta}\). The first prior is designed for the case where \(P\), the number of colums of \(\pmb{Z}\), is small, and most \(\zeta_p\) are likely to distinguishable from zero. The second prior is designed for the case where \(P\) is large, and only a few \(\zeta_p\) are likely to be distinguishable from zero.
5.1.1 Standard prior
\[\begin{align} \zeta_p \mid \varphi & \sim \text{N}(0, \varphi^2) \\ \varphi & \sim \text{N}^+(0, 1) \end{align}\]
5.1.2 Shrinkage prior
Regularized horseshoe prior (Piironen and Vehtari 2017)
\[\begin{align} \zeta_p \mid \vartheta_p, \varphi & \sim \text{N}(0, \vartheta_p^2 \varphi^2) \\ \vartheta_p & \sim \text{Cauchy}^+(0, 1) \\ \varphi & \sim \text{Cauchy}^+(0, A_{\varphi}^2) \\ A_{\varphi} & = \frac{p_0}{p_0 + P} \frac{\hat{\sigma}}{\sqrt{n}} \end{align}\] where \(p_0\) is an initial guess at the number of \(\zeta_p\) that are non-zero, and \(\hat{\sigma}\) is obtained as follows:
- Poisson. Using maximum likelihood, fit the GLM \[\begin{align} y_i & \sim \text{Poisson}(w_i \gamma_i) \\ \log \gamma_i & = \sum_{m=0}^{M} \pmb{X}^{(m)} \pmb{\beta}^{(m)}, \end{align}\] and set \[\begin{equation} \hat{\sigma} = \frac{1}{n}\sum_{i=1}^n \frac{1}{w_i \hat{\gamma}_i}. \end{equation}\]
- Binomial. Using maximum likelihood, fit the GLM \[\begin{align} y_i & \sim \text{binomial}(w_i, \gamma_i) \\ \text{logit} \gamma_i & = \sum_{m=0}^{M} \pmb{X}^{(m)} \pmb{\beta}^{(m)}, \end{align}\] and set \[\begin{equation} \hat{\sigma} = \frac{1}{n}\sum_{i=1}^n \frac{1}{w_i \hat{\gamma}_i (1 - \hat{\gamma}_i)}. \end{equation}\]
- Normal. Using maximum likelihood, fit the linear model \[\begin{equation} y_i \sim \text{N}\left(\sum_{m=0}^{M} \pmb{X}^{(m)} \pmb{\beta}^{(m)}, w_i^{-1}\xi^2 \right) \end{equation}\] and set \(\hat{\sigma} = \hat{\xi}\).
The quantities used for Poisson and binomial likelihoods are derived from normal approximations to GLMs (Piironen and Vehtari 2017; Gelman et al. 2014, sec. 16.2).
5.4 Code
set_covariates(formula, data = NULL, n_coef = NULL)-
formulais a one-sided R formula describing the covariates to be used -
dataA data frame. If a value fordatais supplied, thenformulais interpreted in the context of this data frame. If a value fordatais not supplied, thenformulais interpreted in the context of the data frame used for the original call tomod_pois(),mod_binom(), ormod_norm(). -
n_coefis the effective number of non-zero coefficients. If a value is supplied, the shrinkage prior is used; otherwise the standard prior is used.
Examples:
set_covariates(~ mean_income + distance * employment)
set_covariates(~ ., data = cov_data, n_coef = 5)6 Prior for dispersion terms
6.1 Model
Use exponential distribution, parameterised using mean, \[\begin{equation} \xi \sim \text{Exp}(\mu_{\xi}) \end{equation}\]
7 Data models
7.1 Data models for outcome
7.1.1 Random Rounding to Base 3
Random rounding to base 3 (RR3) is a confidentialization method used by some statistical agencies. It is applied to counts data. Each count \(x\) is rounded randomly as follows:
- If \(x \mod 3 = 0\), then \(x\) is left unchanged;
- if \(x \mod 3 = 1\) then \(x\) is changed to \(x-1\) with probability 2/3, and is changed to \(x + 2\) with probability 1/3; and
- if \(x \mod 3 = 2\) then \(x\) is changed to \(x-2\) with probability 1/3, and is changed to \(x + 1\) with probability 2/3.
RR3 data models can be used with Poisson or binomial likelihoods. Let \(y_i\) denote the observed value for the outcome, and \(y_i^*\) the true value. The likelihood with a RR3 data model is then
\[\begin{align} p(y_i | \gamma_i, w_i) & = \sum_{y_i^*} p(y_i | y_i^*) p(y_i^* | \gamma_i, w_i) \\ & = \sum_{k = -2, -1, 0, 1, 2} p(y_i | y_i + k) p(y_i + k | \gamma_i, w_i) \\ & = \tfrac{1}{3} p(y_i - 2 | \gamma_i, w_i) + \tfrac{2}{3} p(y_i - 1 | \gamma_i, w_i) + p(y_i | \gamma_i, w_i) + \tfrac{2}{3} p(y_i + 1 | \gamma_i, w_i) + \tfrac{1}{3} p(y_i + 2 | \gamma_i, w_i) \end{align}\]
8 Estimation
8.1 Filtering and Aggregation
The data that we supply to TMB is a a filtered and aggregated version of the data that the user provides through the data argument.
In the filtering stage, we remove any rows where (i) the offset is 0 or NA, or (ii) the outcome variable is NA.
In the aggregation stage, we identify any rows in the data that duplicated combinations of classification variables. For instance, if the classification variables are age and sex, and we have two rows where age is "20-24" and sex is "Female", then these rows would count as duplicated combinations. We aggregate offset and outcome variables across these duplicates. With Poisson and binomial models, the aggregation formula for outcomes is
\[\begin{equation}
y^{\text{new}} = \sum_{i=1}^D y_i^{\text{old}},
\end{equation}\]
and the aggregation formula for exposure/size is
\[\begin{equation}
w^{\text{new}} = \sum_{i=1}^D w_i^{\text{old}},
\end{equation}\]
where \(D\) is the number of times a particular combination is duplicated. With normal models, the aggregation formula for outcomes is
\[\begin{equation}
y^{\text{new}} = \frac{\sum_{i=1}^D y_i^{\text{old}}}{D},
\end{equation}\]
and the aggregation formuala for weights is
\[\begin{equation}
w^{\text{new}} = \frac{1}{\sum_{i=1}^D \frac{1}{w_i^{\text{old}}}}.
\end{equation}\]
8.2 Inner-Outer Approximation
8.2.1 Step 0: Select ‘inner’ and ‘outer’ variables
Select variables to be used in inner model. By default, these are the age, sex, and time variables in the model. All remaining variables are ‘outer’ variables.
8.2.2 Step 1: Fit inner model
Aggregate the data using the classification formed by the inner variables. (See Section 8.1 on aggregation procedures.) Remove all terms not involving ‘inner’ variables, other than the intercept term, from the model. Set dispersion to 0. Fit the resulting model.
8.2.3 Step 2: Fit outer model
Let \(\hat{\mu}_i^{\text{in}}\) be point estimates for the linear predictor \(\mu_i\) obtained from the inner model.
Poisson model
Aggregate the data using the classification formed by the outer variables. Remove all terms involving the ‘inner’ variables, plus the intercept, from the model. Set dispersion to 0. Set exposure to \(w_i^{\text{out}} = \hat{\mu}^{\text{in}} w_i\). Fit the model.
Binomial model
Fit the original model, but set dispersion to 0, and for all terms from the ‘inner’ model, use Known priors using point estimates from the inner model.
Normal model
Aggregate the data using the classification formed by the outer variables. Remove all terms involving the ‘inner’ variables, plus the intercept, from the model. Set the outcome variable to to $y_i^{} = y_i - ^{}. Fit the model.
8.2.4 Step 4: Concatenate estimates
Concatenate posterior distributions for the inner terms from the inner model to posterior distributions for the outer terms from the outer model.
8.2.5 Step 5: Calculate dispersion
If the original model includes a dispersion term, then estimate dispersion. Let \(\hat{\mu}_i^{\text{comb}}\) be point estimates for the linear predictor obtained from the concatenated estimates.
Poisson model
Use the original disaggregated data, or, if the original data contains more then 10,000 rows, select 10,000 rows at random from the original data. Remove all terms from the original model except for the intercept. Set exposure to \(w_i^{\text{out}} = \hat{\mu}^{\text{comb}} w_i\).
Binomial model
Fit the the original model, but with all terms except the intercept having Known priors, where the values are obtained from point estimates from the concatenated estimates.
Normal model
Use the original disaggregated data, or, if the original data contains more then 10,000 rows, select 10,000 rows at random from the original data. Remove all terms from the original model except for the intercept. Set the outcome to \(y_i^{\text{out}} = y_i - \hat{\mu}^{\text{comb}}\).
9 Deriving outputs
Running TMB yields a set of means \(\pmb{m}\), and a precision matrix \(\pmb{Q}^{-1}\), which together define the approximate joint posterior distribution of
- \(\pmb{\beta}^{(m)}\) for terms with independent normal, fixed normal, multivariate normal, random walk, second-order random walk, AR1, Linear, and Linear-AR1 priors,
- \(\pmb{\alpha}\) for terms with Spline and SVD priors,
- hyper-parameters for \(\pmb{\beta}^{(m)}\) and \(\pmb{\alpha}^{(m)}\) typically transformed to another scale, such as a log scale,
- dispersion term \(\xi\), and
- seasonal effects \(\pmb{\lambda}\), together with associated hyper-parameters \(\tau_{\lambda}\) (on a log scale).
We use \(\tilde{\pmb{\theta}}\) to denote a vector containing all these quantities.
We perform a Cholesky decomposition of \(\pmb{Q}^{-1}\), to obtain \(\pmb{R}\) such that \[\begin{equation} \pmb{R}^{\top} \pmb{R} = \pmb{Q}^{-1} \end{equation}\] We store \(\pmb{R}\) as part of the model object.
We draw generate values for \(\tilde{\pmb{\theta}}\) by generating a vector of standard normal variates \(\pmb{z}\), back-solving the equation \[\begin{equation} \pmb{R} \pmb{v} = \pmb{z} \end{equation}\] and setting \[\begin{equation} \tilde{\pmb{\theta}} = \pmb{v} + \pmb{m}. \end{equation}\]
Next we convert any transformed hyper-parameters back to the original units, and insert values for \(\pmb{\beta}^{(m)}\) for terms that have Known priors. We denote the resulting vector \(\pmb{\theta}\).
Finally we draw from the distribution of \(\pmb{\gamma} \mid \pmb{y}, \pmb{\theta}\) using the methods described in Section 2.
9.1 Standardization
9.1.1 The need for standardization
NOTE - We are ignoring covariates for the moment. We can probably just subtract the covariates term from \(\pmb{\mu}\) and then proceed as before.
Although the sum \(\pmb{\mu} = \sum_{m=0}^{M} \pmb{X}^{(m)} \pmb{\beta}^{(m)}\) is well identified by the data, the individual \(\pmb{\beta}^{(m)}\) are not. For instance, adding \(\Delta\) to \(\pmb{\beta}^{(0)}\) and subtracting it from every term in \(\pmb{\beta}^{(m)}\) for some \(m > 0\) leaves the likelihood unchanged. The use of proper priors for the \(\pmb{\beta}^{(m)}\) mean that posterior distributions for the \(\pmb{\beta}^{(m)}\), and for related terms such as trend, cyclical, and seasonal effects, are proper. However, they can be diffuse and hard to interpret.
To help with interpretation of the \(\pmb{\beta}^{(m)}\) and related terms, we standardize. We in fact appy two forms of standardization, each designed for a different purpose.
9.1.2 Standardization of estimates
9.1.2.1 ANOVA-style standardization
The first type of standardization is applied only to the \(\pmb{\beta}^{(m)}\). The aim is to partition the total variation in \(\mu_i\) into variation associated with each main effect or interaction, in the style, for instance, of Section 15.6 of Gelman et al. (2014). Let \(\tilde{\pmb{\beta}}^{(m)}\) be the standardized version of \(\pmb{\beta}^{(m)}\). We obtain the \(\tilde{\pmb{\beta}}^{(m)}\) through the following algorithm:
Set \[\begin{equation} \pmb{r}^{(0)} = \pmb{\mu} \end{equation}\]
For \(m = 0, \cdots, M\):
- Set \(\tilde{\pmb{\beta}}^{(m)} = (\pmb{X}^{(m)\top} \pmb{X}^{(m)})^{-1} \pmb{X}^{(m)\top} \pmb{r}^{(m)}\)
- Set \(\pmb{r}^{(m+1)} = \pmb{r}^{(m)} - \pmb{X}^{(m)} \tilde{\pmb{\beta}}^{(m)}\)
9.1.2.2 Term-level standardization
Term-level standardization aims to clarify the behaviour of the terms making up the model, including trend, cyclical, and seasonal effects. The intercept term is left untouched. All other \(\pmb{\beta}^{(m)}\) are centered across the “along” dimension, and each of the “by” dimensions.
9.1.3 Standardization of forecasts
9.1.3.1 ANOVA-style standardization
The forecasted values for the time-varying \(\pmb{\beta}^{(m)}\) are shifted up or down so that they line up with the estimated values. The standardization algorithm is then applied to these shifted values.
9.1.3.2 Term-level standardization
The forecasted values for the time-varying \(\pmb{\beta}^{(m)}\), and for SVD coefficients, and trend, cyclical, and seasonal components, are shifted up or down so that they line up with the estimated values. All terms are then centered along all dimensions other than time.
10 Simulation
To generate one set of simulated values, we start with values for exposure, trials, or weights, \(\pmb{w}\), and possibly covariates \(\pmb{Z}\), then go through the following steps:
- Draw values for any parameters in the priors for the \(\pmb{\beta}^{(m)}\), \(m = 1, \cdots, M\).
- Conditional on the values drawn in Step 1, draw values the \(\pmb{\beta}^{(m)}\), \(m = 0, \cdots, M\).
- If the model contains seasonal effects, draw the standard deviation \(\kappa_m\), and then the effects \(\pmb{\lambda}^{(m)}\).
- If the model contains covariates, draw \(\varphi\) and \(\vartheta_p\) where necessary, draw coefficient vector \(\pmb{\zeta}\).
- Use values from steps 2–4 to form the linear predictor \(\sum_{m=0}^{M} \pmb{X}^{(m)} (\pmb{\beta}^{(m)} + \pmb{\lambda}^{(m)}) + \pmb{Z} \pmb{\zeta}\).
- Back-transform the linear predictor, to obtain vector of cell-specific parameters \(\pmb{\mu}\).
- If the model contains a dispersion parameter \(\xi\), draw values from the prior for \(\xi\).
- In Poisson and binomial models, use \(\pmb{\mu}\) and, if present, \(\xi\) to draw \(\pmb{\gamma}\).
- In Poisson and binomial models, use \(\pmb{\gamma}\) and \(\pmb{w}\) to draw \(\pmb{y}\); in normal models, use \(\pmb{\mu}\), \(\xi\), and \(\pmb{w}\) to draw \(\pmb{y}\).
11 Replicate data
11.1 Model
11.1.1 Poisson likelihood
11.1.1.1 Condition on \(\pmb{\gamma}\)
\[\begin{equation} y_i^{\text{rep}} \sim \text{Poisson}(\gamma_i w_i) \end{equation}\]
11.1.1.2 Condition on \((\pmb{\mu}, \xi)\)
\[\begin{align} y_i^{\text{rep}} & \sim \text{Poisson}(\gamma_i^{\text{rep}} w_i) \\ \gamma_i^{\text{rep}} & \sim \text{Gamma}(\xi^{-1}, (\xi \mu_i)^{-1}) \end{align}\] which is equivalent to \[\begin{equation} y_i^{\text{rep}} \sim \text{NegBinom}\left(\xi^{-1}, (1 + \mu_i w_i \xi)^{-1}\right) \end{equation}\]
11.1.2 Binomial likelihood
11.2 Code
replicate_data(x, condition_on = c("fitted", "expected"), n = 20)12 Appendices
12.1 Definitions
| Quantity | Definition |
|---|---|
| \(i\) | Index for cell, \(i = 1, \cdots, n\). |
| \(y_i\) | Value for outcome variable. |
| \(w_i\) | Exposure, number of trials, or weight. |
| \(\gamma_i\) | Super-population rate, probability, or mean. |
| \(\mu_i\) | Cell-specific mean. |
| \(\xi\) | Dispersion parameter. |
| \(g()\) | Log, logit, or identity function. |
| \(m\) | Index for intercept, main effect, or interaction. \(m = 0, \cdots, M\). |
| \(j\) | Index for element of a main effect or interaction. |
| \(u\) | Index for combination of ‘by’ variables for an interaction. \(u = 1, \cdots U_m\). \(U_m V_m = J_m\) |
| \(v\) | Index for the ‘along’ dimension of an interaction. \(v = 1, \cdots V_m\). \(U_m V_m = J_m\) |
| \(\beta^{(0)}\) | Intercept. |
| \(\pmb{\beta}^{(m)}\) | Main effect or interaction. \(m = 1, \cdots, M\). |
| \(\beta_j^{(m)}\) | \(j\)th element of \(\pmb{\beta}^{(m)}\). \(j = 1, \cdots, J_m\). |
| \(\pmb{X}^{(m)}\) | Matrix mapping \(\pmb{\beta}^{(m)}\) to \(\pmb{y}\). |
| \(\pmb{Z}\) | Matrix of covariates. |
| \(\pmb{\zeta}\) | Parameter vector for covariates \(\pmb{Z}^{(m)}\). |
| \(A_0\) | Scale parameter in prior for intercept \(\beta^{(0)}\). |
| \(\tau_m\) | Standard deviation parameter for main effect or interaction. |
| \(A_{\tau}^{(m)}\) | Scale parameter in prior for \(\tau_m\). |
| \(\pmb{\alpha}^{(m)}\) | Parameter vector for P-spline and SVD priors. |
| \(\alpha_k^{(m)}\) | \(k\)th element of \(\pmb{\alpha}^{(m)}\). \(k = 1, \cdots, K_m\). |
| \(\pmb{V}^{(m)}\) | Covariance matrix for multivariate normal prior. |
| \(h_j^{(m)}\) | Linear covariate |
| \(\eta^{(m)}\) | Parameter specific to main effect or interaction \(\pmb{\beta}^{(m)}\). |
| \(\eta_u^{(m)}\) | Parameter specific to \(u\)th combination of ‘by’ variables in interaction \(\pmb{\beta}^{(m)}\). |
| \(A_{\eta}^{(m)}\) | Standard deviation in normal prior for \(\eta_m\). |
| \(\omega_m\) | Standard deviation of parameter \(\eta_c\) in multivariate priors. |
| \(\phi_m\) | Correlation coefficient in AR1 densities. |
| \(a_{0m}\), \(a_{1m}\) | Minimum and maximum values for \(\phi_m\). |
| \(\pmb{B}^{(m)}\) | B-spline matrix in P-spline prior. |
| \(\pmb{b}_k^{(m)}\) | B-spline. \(k = 1, \cdots, K_m\). |
| \(\pmb{F}^{(m)}\) | Matrix in SVD prior. |
| \(\pmb{g}^{(m)}\) | Offset in SVD prior. |
| \(\pmb{\beta}_{\text{trend}}\) | Trend effect. |
| \(\pmb{\beta}_{\text{cyc}}\) | Cyclical effect. |
| \(\pmb{\beta}_{\text{seas}}\) | Seasonal effect. |
| \(\varphi\) | Global shrinkage parameter in shrinkage prior. |
| \(A_{\varphi}\) | Scale term in prior for \(\varphi\). |
| \(\vartheta_p\) | Local shrinkage parameter in shrinkage prior. |
| \(p_0\) | Expected number of non-zero coefficients in \(\pmb{\zeta}\). |
| \(\hat{\sigma}\) | Empirical scale estimate in prior for \(\varphi\). |
| \(\pi\) | Vector of hyper-parameters |
12.2 SVD prior for age
Let \(\pmb{A}\) be a matrix of age-specific estimates from an international database, transformed to take values in the range \((-\infty, \infty)\). Each column of \(\pmb{A}\) represents one set of age-specific estimates, such as log mortality rates in Japan in 2010, or logit labour participation rates in Germany in 1980.
Let \(\pmb{U}\), \(\pmb{D}\), \(\pmb{V}\) be the matrices from a singular value decomposition of \(\pmb{A}\), where we have retained the first \(K\) components. Then \[\begin{equation} \pmb{A} \approx \pmb{U} \pmb{D} \pmb{V}. \tag{12.1} \end{equation}\]
Let \(m_k\) and \(s_k\) be the mean and sample standard deviation of the elements of the \(k\)th row of \(\pmb{V}\), with \(\pmb{m} = (m_1, \cdots, m_k)^{\top}\) and \(\pmb{s} = (s_1, \cdots, s_k)^{\top}\). Then \[\begin{equation} \tilde{\pmb{V}} = (\text{diag}(\pmb{s}))^{-1} (\pmb{V} - \pmb{m} \pmb{1}^{\top}) \end{equation}\] is a standardized version of \(\pmb{V}\).
We can rewrite (12.1) as \[\begin{align} \pmb{A} & \approx \pmb{U} \pmb{D} (\text{diag}(\pmb{s}) \tilde{\pmb{V}} + \pmb{m} \pmb{1}^{\top}) \\ & = \pmb{F} \tilde{\pmb{V}} + \pmb{g} \pmb{1}^{\top}, \tag{12.2} \end{align}\] where \(\pmb{F} = \pmb{U} \pmb{D} \text{diag}(\pmb{s})\) and \(\pmb{g} = \pmb{U} \pmb{D} \pmb{m}\).
Let \(\tilde{\pmb{v}}_l\) be a randomly-selected column from \(\tilde{\pmb{V}}\). From the construction of \(\tilde{\pmb{V}}\), and the orthogonality of the rows of \(\pmb{V}\), we have \(\text{E}[\tilde{\pmb{v}}_l] = \pmb{0}\) and \(\text{var}[\tilde{\pmb{v}}_l] = \pmb{I}\). This implies that if \(\pmb{z}\) is a vector of standard normal variables, then \[\begin{equation} \pmb{F} \pmb{z} + \pmb{g} \end{equation}\] should look approximately like a randomly-selected column from the original data matrix \(\pmb{A}\).