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Logistic plus linear (LPL) Model

This model proposes a latent true coverage curve, which is subject to observation error. A hierarchy accounts for the effects of categorical features.

Terminology and notation

  • \(C_{sgm}\): reported, estimated coverage (proportion of people vaccinated) in season \(s\), geography (i.e., state) \(g\), and month \(m\) (estimate in the data)
  • \(N_{sgm}\): number of people surveyed (sample_size in the data)
  • \(v_{sg}(t)\): latent true coverage at time \(t\) (measured in years)

Model equations

\[ \begin{align*} X_{sgm} &= \mathrm{round}(C_{sgm} \cdot N_{sgm}) \\ v_{sg}(t) &= \frac{A_{sg}}{1 + \exp\{- K \cdot (t - \tau)\}} + B_{sg} t \\ A_{sg} &= \beta^{(A)} + \beta_s^{(AS)} + \beta_g^{(AG)} + \beta_{sg}^{(ASG)} \\ B_{sg} &= \beta^{(B)} + \beta_s^{(BS)} + \beta_g^{(BG)} + \beta_{sg}^{(BSG)} \\ \\ X_{sgm} &\sim \mathrm{BetaBinom}\big( N_{sgm}, v_{sg}(t_m) \cdot D, [1-v_{sg}(t_m)] \cdot D \big) \\ K &\sim \text{Gamma}(\text{shape} = 25.0, \text{rate} = 1.0) \\ \tau &\sim \text{Beta}(100.0, 225.0) \\ D &\sim \text{Gamma}(\text{shape} = 350.0, \text{rate} = 1.0) \\ \beta^{(A)} &\sim \text{Beta}(100.0, 180.0) \\ \beta_s^{(AS)} &\sim \mathcal{N}\left(0, \sigma^{(AS)}\right) \\ \beta_g^{(AG)} &\sim \mathcal{N}\left(0, \sigma^{(AG)}\right) \\ \beta_{sg}^{(ASG)} &\sim \mathcal{N}\left(0, \sigma^{(ASG)}\right) \\ \beta^{(B)} &\sim \text{Gamma}(\text{shape} = 1.0, \text{rate} = 10.0) \\ \beta_s^{(BS)} &\sim \mathcal{N}\left(0, \sigma^{(BS)}\right) \\ \beta_g^{(BG)} &\sim \mathcal{N}\left(0, \sigma^{(BG)}\right) \\ \beta_{sg}^{(BSG)} &\sim \mathcal{N}\left(0, \sigma^{(BSG)}\right) \\ \sigma^{(\bullet)} &\sim \text{Exp}(40.0) \\ \end{align*} \]

Notes:

  • The latent coverage \(v_{sg}(t)\) is assumed to be a sum of a logistic curve and a line with intercept at \(t=0\)
  • The shape parameter \(K\) and midpoint \(\tau\) of the logistic curve are assumed to be common to all groups
  • The height \(A_{sg}\) of the logistic curve is a grand mean \(\beta^{(A)}\) plus effects for the season, state, and season-state interaction. The slopes \(M_g\) follow a similar pattern.
\[ \begin{align*} \mathbb{E}[X_{sgk}] &= v_{sg}(t_k) \cdot N_{sgk} \\ \mathrm{Var}[X_{sgk}] &= v_{sg}(t_k) \cdot [1-v_{sg}(t_k)] \cdot \frac{N_{sgk} (N_{sgk} + D)}{D+1} \end{align*} \]