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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

  • feature: A categorical feature of the data (e.g., season, geography) that partially determines coverage
  • level: A value taken on by a feature (e.g., New Jersey, or 2018/2019)
  • group: A unique combination of features (e.g., New Jersey in 2018/2019)
  • \(t\): time since the start of the season, measured in \(\text{year}^{-1}\)
  • \(n_{gt}\): number of people in group \(g\), surveyed at time \(t\). Drawn from the sample_size column of the NIS data; called N_tot in the codebase.
  • \(x_{gt}\): number of people in group \(g\), surveyed at time \(t\), who are vaccinated. Approximated as \(\mathrm{round}(\hat{v}_{gt}, n_{gt})\), where \(\hat{v}_{gt}\) is the estimate column.
  • \(v_g(t)\): latent true coverage among group \(g\) at time \(t\)
  • \(z_{gj}\): integer index indicating the level of the \(j\)-th feature for group \(g\).

For example, let the features be season and geography, in that order. Let group 5 be associated with the fourth season and the third geography. Then \(z_{51} = 4\) and \(z_{52} = 3\).

Model overview

For each group \(g\), the latent coverage \(v_g(t)\) 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 \(A_g\) of the logistic curve is a grand mean \(\mu_A\) plus effects \(\delta_{A,j,z_{gj}}\) for each feature \(j\). For example, the \(A_g\) for Alaska in 2018/2019 will be the grand mean \(\mu_A\), plus the Alaska effect, plus the 2018/2019 effect. In fact, the model uses a third interaction term season-geography, so there is also an Alaska-in-2018/2019 term.

The slopes \(M_g\) follow a similar pattern.

The observations \(x_{gt}\) are beta binomial-distributed around the mean \(v_g(t) \cdot n_{gt}\), with variance modified by an extra parameter \(D\).

Model equations

\[ \begin{align*} x_{gt} &\sim \mathrm{BetaBinom}\big(v_g(t) \cdot D, [1-v_g(t)] \cdot D, n_{gt}\big) \\ v_g(t) &= \frac{A_g}{1 + \exp\{- K \cdot (t - \tau)\}} + M_g t \\ A_g &= \mu_A + \sum_j \delta_{Aj z_{gj}} \\ M_g &= \mu_M + \sum_j \delta_{Mj z_{gj}} \\ \mu_A &\sim \text{Beta}(100.0, 180.0) \\ \mu_M &\sim \text{Gamma}(\text{shape} = 1.0, \text{rate} = 10.0) \\ \delta_{Ajk} &\sim \mathcal{N}(0, \sigma_{Aj}) \\ \delta_{Mjk} &\sim \mathcal{N}(0, \sigma_{Mj}) \\ \sigma_{Aj} &\sim \text{Exp}(40.0) \\ \sigma_{Mj} &\sim \text{Exp}(40.0) \\ K &\sim \text{Gamma}(\text{shape} = 25.0, \text{rate} = 1.0) \\ \tau &\sim \text{Beta}(100.0, 225.0) \\ D &\sim \text{Gamma}(\text{shape} = 350.0, \text{rate} = 1.0) \\ \end{align*} \]

Note that:

\[ \begin{align*} \mathbb{E}[x_{gt}] &= v_g(t) \cdot n_{gt} \\ \mathrm{Var}[x_{gt}] &= v_g(t) \cdot [1-v_g(t)] \cdot \frac{n_{gt} (n_{gt} + D)}{D+1} \end{align*} \]