\frac{\partial}{\partial \theta_j} g(\eta(\theta)) = \int f(x) \frac{\partial}{\partial \theta_j} p_{\eta(\theta)}(x) d\mu(x)

By the chain rule, the $\frac{\partial}{\partial \theta_j} g(\eta(\theta)) = \frac{d g(\eta(\theta))}{d \eta(\theta)}\frac{\partial \eta(\theta)}{\partial \theta_j}$.
By Theorem 2.4 in <cit.>, $\frac{d g(\eta(\theta))}{d \eta(\theta)}$ exists and can be
obtained via differentiating under the integral sign:

\frac{d}{d \eta(\theta)} g(\eta(\theta)) = \int f(x) \frac{dp_{\eta(\theta)}(x)}{d \eta(\theta)} d\mu(x).

Using this result and the chain rule again yields

\begin{align}
\frac{\partial}{\partial \theta_j} g(\eta(\theta)) & = \frac{d g(\eta(\theta))}{d \eta(\theta)}\frac{\partial \eta(\theta)}{\partial \theta_j} \\
& = \int f(x) \frac{dp_{\eta(\theta)}(x)}{d \eta(\theta)} \frac{\partial \eta(\theta)}{\partial \theta_j}d\mu(x) \\
& = \int f(x) \frac{\partial}{\partial \theta_j} dp_{\eta(\theta)}(x)
\end{align}

§.§ Lemmas and theorems in service of Fisher Info

Our local identifiability result is based on the following theorem that is referenced, though not proven, in <cit.>.
To our knowledge an explicit proof has not been given, though it follows directly from the proof of the Cramér-Rao lower bound in <cit.>.
This proof provides a slightly different route to showing local identifiability compared to that of <cit.>.
We give a proof below where we use the same notation as used in <cit.> for clarity's sake.

Suppose we have observations $X_n$ and let $\mathbf{x} \in \R^{N}$ be the collection of all observations, with the $n$-th element equal to $X_n$, where $\pi(\mathbf{x}, \boldsymbol{\theta})$, parameterized by $\boldsymbol{\theta} \in \R^d$ with $i^\textrm{th}$ element $\theta_i$ is the joint density of the observations.
Let $f_1(\mathbf{x}), \dots, f_r(\mathbf{x})$ be $r$ statistics for which $\Exp{f_i(\mathbf{x})} = g_i(\boldsymbol{\theta})$.
Further, assume that $\frac{\partial}{\partial \theta_j} \int f_i(\mathbf{x}) \pi(\mathbf{x}, \boldsymbol{\theta}) d\mathbf{x} = \int f_i(\mathbf{x}) \frac{\partial}{\partial \theta_j} \pi(\mathbf{x}, \boldsymbol{\theta}) d\mathbf{x} = \frac{\partial g_i(\boldsymbol{\theta})}{\partial \theta_j}$. Let $\Delta$ be a matrix in $\R^{r \times d}$ with elements $\Delta_{ij} = \frac{\partial g_i(\boldsymbol{\theta})}{\partial \theta_j}$.
Let $\mathcal{I}$ be a matrix in $\R^{d \times d}$ where the $(i,j)$-th element is defined as $\mathcal{I}_{ij} = \Cov{\frac{\partial \log P(\mathbf{x}, \boldsymbol{\theta})}{\partial \theta_i}}{\frac{\partial \log P(\mathbf{x}, \boldsymbol{\theta})}{\partial \theta_j}}$.
Let $\mathbf{V}$ be the matrix in $\R^{r \times r}$ with $(i,j)$ elements $\mathbf{V}_{ij} = \Cov{f_i(\mathbf{x})}{f_j(\mathbf{x})}$.
If $\mathbf{V}$ is positive definite, and $\Delta$ is full-rank, then the Fisher information matrix $\mathcal{I}$
is positive definite.

Let $\mathbf{f}$ be the ordered collection of elements $f_i(\mathbf{x})$, and let
$\nabla \log \pi(\mathbf{x}, \boldsymbol{\theta})$ be the score vector. Then let the random vector $\rho = (\mathbf{f}, \nabla \log \pi(\mathbf{x},\boldsymbol{\theta}))$. The covariance
matrix associated with $\rho$, $\Sigma$, is a block matrix. Under
$\pi(\mathbf{x}, \boldsymbol{\theta})$, $\Cov{f_i}{\frac{\partial \log \pi(\mathbf{x}, \boldsymbol{\theta})}{\partial \theta_j}} = \frac{\partial g_i(\boldsymbol{\theta})}{\partial \theta_j}$, which
is element $(i,j)$ of the matrix $\Delta$. The block covariance matrix for $\rho$ is:

\begin{align*}
\Sigma = \begin{bmatrix}
\mathbf{V} & \Delta \\
\Delta^T & \mathcal{I}
\end{bmatrix}
\end{align*}

Suppose $\mathbf{V}$ is positive definite. We know $\mathcal{I} - \Delta^T \mathbf{V}^{-1} \Delta \succeq 0$,
because $\Sigma$ is a covariance matrix which ensures it is positive semi definite.
Furthermore, $\mathcal{I} \succeq \Delta^T \mathbf{V}^{-1} \Delta$. If $\Delta$ is full-rank then $\mathcal{I} \succ 0$.

We need the following two lemmas to prove that the Fisher information is positive definite.
First, the following lemma is stated in <cit.>:

Let $\mathbf{A}^\dagger$ be the Moore-Penrose inverse of a matrix $\mathbf{A} \in \R^{L \times M}$. Let $\mathbf{B} \in \R^{L \times T}$.
The rank of a block matrix $[\mathbf{A}\,\, \mathbf{B}]$ is the rank of $\mathbf{A}$ plus the rank of $\mathbf{B} - \mathbf{A}\mathbf{A}^\dagger \mathbf{B}$.

Next, we will need this lemma later on:

Suppose $\mathbf{W} \in \R^{I \times K}$ and
$\mathbf{E} \in \R^{I \times J}$. Let $\mathbf{E}_{[:,j]}$ be an $I$-vector of the $j^\textrm{th}$ column of matrix $\mathbf{E}$, let $\mathbf{F}_j = {\rm diag}(\mathbf{E}_{[:,j]})$.

* $I \geq J + K$

* ${\rm rank}\lp\mathbf{E}\rp = J$

* ${\rm rank}\lp\mathbf{W}\rp = K$

* $\forall i \in 1, \dots, I$ $\sum_{j=1}^J E_{ij} > 0$

* $\lambda_j > 0 \forall j \in 1, \dots, J$

* ${\rm rank}\lp \begin{bmatrix}
\mathbf{F}_1\mathbf{W} & \dots & \mathbf{F}_J\mathbf{W} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]} & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}
\end{bmatrix}\rp > J + K$

Then the matrix

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]} & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}
\end{bmatrix}

is rank $J + K$.

Given conds:rank-lemma-dimconds:rank-lemma-pos-lambda above , the matrix
$\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W}$ is rank $K$. Using lemma
<ref> we proceed sequentially, showing first that

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1]}
\end{bmatrix}\rp = K + 1

By <ref>,

\begin{align*}
\text{rank} \lp
\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1]}
\end{bmatrix}
\rp = &
\text{rank} \lp
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W}
\rp \\
& + \text{rank} \lp\mathbf{E}_{[:,1]} -
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} \lp\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W}\rp^\dagger\mathbf{E}_{[:,1]}
\rp
\end{align*}

We can show that
$\text{rank} \lp\mathbf{E}_{[:,1]} - \lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} \lp\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W}\rp^\dagger\mathbf{E}_{[:,1]}\rp = 1$.
We show by contradiction that given conditions on the coefficient matrix, no
solution to the equation in $a_k$

\begin{align}\label{eqn:base-case-sys}
\mathbf{E}_{[:,1]} = \sum_{k=1}^K a_k \lp \sum_{j=1}^J \lambda_j \mathbf{E}_{[:,j]} \odot \mathbf{W}_k\rp
\end{align}

can be found.
This follows from examining the matrix of coefficients of the equation, shown
here in block form:

\begin{align}\label{eqn:base-case-coef-mat}
\begin{bmatrix}
\mathbf{E}_{[:,1]} & \lambda_1 \mathbf{F}_1 \mathbf{W} & \dots & \lambda_J \mathbf{F}_J \mathbf{W}
\end{bmatrix}
\end{align}

Given <ref>,  $\lambda_j > 0$ for all $j$, for a fixed set of parameters $\lambda_j$ the matrix in <ref> has the same column space as

\begin{align}\label{eqn:base-case-col-space-coef-mat}
\begin{bmatrix}
\mathbf{E}_{[:,1]} & \mathbf{F}_1 \mathbf{W} & \dots & \mathbf{F}_J \mathbf{W}
\end{bmatrix}
\end{align}

Given <ref>, matrix <ref> has rank greater than $K$, so the system of equations in <ref> will not have a solution in $a_k, k \in [1, \dots, K]$. Thus the rank of matrix

\begin{align}
\mathbf{E}_{[:,1]} - \lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} \lp\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W}\rp^\dagger\mathbf{E}_{[:,1]}
\end{align}

is $1$ so the rank of .

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1]}
\end{bmatrix}

is $K + 1$.

Now for the induction step:
Suppose that

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]}
\end{bmatrix}

is rank $M + K$
and suppose we want to determine the rank of

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]} & \mathbf{E}_{[:,M+1]}
\end{bmatrix}.

By lemma <ref>, the rank of the above matrix is

\begin{align}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]}
\end{bmatrix}\label{eqn:ind-step-induction-hypo} \\
& +
\text{rank}\lp\mathbf{E}_{[:,M+1]} - \begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]}
\end{bmatrix}\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]}
\end{bmatrix}^\dagger\mathbf{E}_{[:,M+1]} \rp \label{eqn:ind-step-to-prove}
\end{align}

By the induction hypothesis, we know that <ref> equals $K + M$ and we
look to see whether <ref> equals 1 by determining whether the system of equations below:

\begin{align}\label{eqn:ind-step-sys}
\mathbf{E}_{[:,M+1]} = \sum_{k=1}^K a_k \lp \sum_{j=1}^J \lambda_j \mathbf{E}_j \odot \mathbf{W}_k\rp
+ \sum_{m=1}^M d_m \mathbf{E}_{[:,m]}
\end{align}

has a solution in the variables $a_k, k \in [1,\dots,K]$, $d_m, m \in [1,\dots,M], M < J$.
Then for a fixed $\{\lambda_j, j \in [1, \dots, J]\}$ the coefficient matrix for the system of equations in <ref> has the same column space as

\begin{align}\label{eqn:ind-step-coef-mat}
\begin{bmatrix}
\mathbf{E}_{[:,M+1]}  & \mathbf{F}_1 \mathbf{W} & \dots & \mathbf{F}_J \mathbf{W} & \mathbf{E}_{[:,1:M]}
\end{bmatrix}
\end{align}

By <ref>, the rank of matrix (<ref>) is greater than $M + K$, which precludes $\mathbf{E}_{[:,M+1]}$ from lying in the column space of

\begin{bmatrix}\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]}\end{bmatrix}
. Thus the line (<ref>) is equal to $1$ and summing with line (<ref>) shows that the rank of

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1:M]} & \mathbf{E}_{[:,M+1]}
\end{bmatrix}

is $M + 1 + K$.
Therefore by induction,

\begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{W} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]} & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}
\end{bmatrix}

is rank $J + K$.

§ FULL MODEL LOCAL IDENTIFIABILITY PROOF

In order to draw on the results of <ref>, we must ensure $\frac{\partial}{\partial \theta_j} \int f_i \pi(\mathbf{x}, \theta) d\nu = \int f_i \frac{\partial}{\partial \theta_j} \pi(\mathbf{x}, \theta) d\nu$ holds for model (<ref>).
This condition does indeed hold by lemma <ref> because our observational density is an exponential family density.
Now we look for moment estimators, $\mathbf{f} = (f_1, \dots, f_r)$ with full-rank $\Delta$ and positive definite covariance matrices.
As shown in section <ref>, conditionally on unknown parameters, known covariates $\mathbf{z}_i$ and population counts $\mathbf{e}_i$, $X_{ij}$ is independent of $M_i$ for all $i$ and $j$.
Let $\mathbf{x}_j = (X_{1j}, X_{2j},\dots, X_{Ij})$,
and set

$$\mathbf{f} = (\mathbf{x}_{1},\mathbf{x}_{2}, \dots, \mathbf{x}_{J}, M_1, M_2, \dots, M_I).$$

From the independence of the elements of the vector $(X_{i1},X_{i2}, \dots,X_{iJ},,  M_i)^T$, and the conditional independence between observations $i$, $\text{Cov}(\mathbf{f})$ is diagonal, and is positive definite because cond:full-model-pcond:full-model-E are assumed to hold.
The final condition for <ref> to hold, which establishes a lower bound on the positive definiteness of the Fisher information matrix is to ensure that $\Delta$ has full column-rank.
To that end, let us calculate $\Delta$, beginning with $\Exp{\mathbf{f}}$:
The expected value of $\mathbf{f}$ is the vector:

\begin{align*}
\begin{bmatrix}
E_{11} p_{11} e^{\mathbf{z}_1^T \boldsymbol{\beta}}\lambda_1 \\
E_{21} p_{21} e^{\mathbf{z}_2^T \boldsymbol{\beta}}\lambda_1 \\
\vdots \\
E_{I1} p_{I1} e^{\mathbf{z}_I^T \boldsymbol{\beta}}\lambda_1 \\
E_{12} p_{12} e^{\mathbf{z}_1^T \boldsymbol{\beta}}\lambda_2 \\
E_{22} p_{22} e^{\mathbf{z}_2^T \boldsymbol{\beta}}\lambda_2 \\
\vdots \\
E_{I2} p_{I2} e^{\mathbf{z}_I^T \boldsymbol{\beta}}\lambda_2 \\
\vdots \\
E_{1J} p_{1J} e^{\mathbf{z}_1^T \boldsymbol{\beta}}\lambda_J \\
E_{2J} p_{2J} e^{\mathbf{z}_2^T \boldsymbol{\beta}}\lambda_J \\
\vdots \\
E_{IJ} p_{IJ} e^{\mathbf{z}_I^T \boldsymbol{\beta}}\lambda_J \\
\sum_j  E_{1j} (1 - p_{1j}) e^{\mathbf{z}_1^T \boldsymbol{\beta}}\lambda_j \\
\vdots \\
\sum_j  E_{Ij} (1 - p_{Ij}) e^{\mathbf{z}_I^T \boldsymbol{\beta}}\lambda_j
\end{bmatrix}
\end{align*}

For efficiency of notation,
let $c_{ij} = p_{ij} \lambda_j$ $p^{\prime}_{ij} = p_{ij}(1 - p_{ij})$, and $q_{ij} = (1 - p_{ij})$
Let $\mathbf{H}$ by the $(J + 1)I \times 2K$ matrix of partial derivatives of $\Exp{\mathbf{f}}$ with respect to the vector $(\boldsymbol{\beta}, \boldsymbol{\gamma})$:

\begin{align*}
\mathbf{H} =
\begin{bmatrix}
\mathbf{z}_1^T E_{11} c_{11}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{11} p^{\prime}_{11}\lambda_1e^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{21} c_{21}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{21} p^{\prime}_{21}\lambda_1e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{I1} c_{I1}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{I1} p^{\prime}_{I1}\lambda_1e^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\mathbf{z}_1^T E_{12} c_{12}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{12} p^{\prime}_{12}\lambda_2e^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{22} c_{22}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{21} p^{\prime}_{22}\lambda_2e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{I2} c_{I2}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{I2} p^{\prime}_{I2}\lambda_2e^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_1^T E_{1J} c_{1J}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{1J} p^{\prime}_{1J}\lambda_Je^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{2J} c_{2J}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{2J} p^{\prime}_{2J}\lambda_J e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{IJ} c_{IJ}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{IJ} p^{\prime}_{IJ}\lambda_Je^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\mathbf{z}_1^T e^{\mathbf{z}_1^T \boldsymbol{\beta}}\sum_{j}E_{1j} q_{1j} \lambda_j & -\mathbf{z}_1^T e^{\mathbf{z}_1^T \boldsymbol{\beta}}\sum_{j} E_{1j} p^\prime_{1j} \lambda_j \\
\vdots & \vdots \\
\mathbf{z}_I^T e^{\mathbf{z}_I^T \boldsymbol{\beta}}\sum_{j}E_{Ij} q_{Ij} \lambda_j & -\mathbf{z}_I^T e^{\mathbf{z}_I^T \boldsymbol{\beta}}\sum_{j} E_{Ij} p^\prime_{Ij} \lambda_j
\end{bmatrix}
\end{align*}

Let $\mathbf{T}$ be the $(J + 1)I \times 2J$ matrix of partial derivatives with respect to $(\lambda_1, \lambda_2, \dots, \lambda_{J}, \eta_1,\eta_2, \dots, \eta_{J})$,

\begin{align*}
\mathbf{T} =   \begin{bmatrix}
E_{11} p_{11}e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{11} p^\prime_{11}\lambda_1e^{\mathbf{z}_1^T \boldsymbol{\beta}} & 0 & \dots & 0\\
E_{21} p_{21}e^{\mathbf{z}_2^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{21} p^\prime_{21}\lambda_1e^{\mathbf{z}_2^T \boldsymbol{\beta}} & 0 & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
E_{I1} p_{I1}e^{\mathbf{z}_I^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{I1} p^\prime_{I1}\lambda_1e^{\mathbf{z}_I^T \boldsymbol{\beta}} & 0 & \dots & 0\\
0 &  E_{12} p_{12}e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & \dots & 0 & 0 &  E_{12} p^\prime_{12} \lambda_2e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & \dots & 0\\
0 &  E_{22} p_{22}e^{\mathbf{z}_2^T \boldsymbol{\beta}}   & \dots & 0 & 0 &  E_{22} p^\prime_{22} \lambda_2e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
0 &  E_{I2} p_{I2}e^{\mathbf{z}_I^T \boldsymbol{\beta}}  & \dots & 0 & 0 &  E_{I2} p^\prime_{I2} \lambda_2e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \dots & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots  & \ddots & \vdots\\
0 & 0 & \dots &  E_{1J} p_{1J}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{1J} p^\prime_{1J}\lambda_Je^{\mathbf{z}_1^T \boldsymbol{\beta}}\\
0 & 0 & \dots &  E_{2J} p_{2J}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{2J} p^\prime_{2J}\lambda_Je^{\mathbf{z}_2^T \boldsymbol{\beta}}\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
0 & 0 & \dots &  E_{IJ} p_{IJ}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{IJ} p^\prime_{IJ}\lambda_Je^{\mathbf{z}_I^T \boldsymbol{\beta}}\\
E_{11} q_{11}e^{\mathbf{z}_1^T \boldsymbol{\beta}} &  E_{12} q_{12}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \dots &  E_{1J} q_{1J}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & -E_{11} p^\prime_{11}\lambda_1e^{\mathbf{z}_1^T \boldsymbol{\beta}} & - E_{12} p^\prime_{12}\lambda_2e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \dots
& - E_{1J} p^\prime_{1J} \lambda_Je^{\mathbf{z}_1^T \boldsymbol{\beta}}\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots  & \ddots & \vdots\\
E_{I1} q_{I1} e^{\mathbf{z}_I^T \boldsymbol{\beta}}&  E_{I2} q_{I2}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \dots &  E_{IJ} q_{IJ}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & -E_{I1} p^\prime_{I1}\lambda_1 e^{\mathbf{z}_I^T \boldsymbol{\beta}}& - E_{I2} p^\prime_{I2}\lambda_2 e^{\mathbf{z}_I^T \boldsymbol{\beta}}& \dots
& - E_{IJ} p^\prime_{IJ} \lambda_J e^{\mathbf{z}_I^T \boldsymbol{\beta}}\\
\end{bmatrix}
\end{align*}

\begin{align*}
\Delta & = \begin{bmatrix} \mathbf{T} & \mathbf{H} \end{bmatrix}
\end{align*}

Let the matrix $\mathbf{R}_{i,j}(m)$ be the elementary row-addition matrix.
When a matrix $\mathbf{A} \in \R^{M\times N}$ is left-multiplied by $\mathbf{R}_{ij}(m)$, $\tilde{\mathbf{A}} = \mathbf{E}_{ij}(m) \mathbf{A}$, all rows of $\tilde{\mathbf{A}}$ equal that of $\mathbf{A}$ excepting $\tilde{\mathbf{A}}$'s $i$-th row, which is $\tilde{\mathbf{A}}_{[i,:]} = \mathbf{A}_{[i,:]} + m\mathbf{A}_{[j,:]}$.
Let $\tilde{\mathbf{H}}$ and $\tilde{\mathbf{T}}$ be the result of left-multiplying $\mathbf{H}$ and $\mathbf{T}$ by the same product of elementary row-addition matrices, namely:

\begin{align*}
\prod_{i=1}^I\prod_{j=1}^J \mathbf{R}_{JI + j,(j - 1)I + i}(1)
\end{align*}

Then let $\tilde{\Delta}$ be the matrix $\Delta$ after applying the product of elementary row-addition matrices:

\begin{align*}
\tilde{\Delta} & = \prod_{i=1}^I\prod_{j=1}^J \mathbf{R}_{JI + j,(j - 1)I + i}(1) \begin{bmatrix} \mathbf{T} & \mathbf{H} \end{bmatrix} \\
& = \begin{bmatrix} \prod_{i=1}^I\prod_{j=1}^J \mathbf{R}_{JI + j,(j - 1)I + i}(1) \mathbf{T} & \prod_{i=1}^I\prod_{j=1}^J \mathbf{R}_{JI + j,(j - 1)I + i}(1) \mathbf{H} \end{bmatrix}  \\
& = \begin{bmatrix} \tilde{\mathbf{T}} & \tilde{\mathbf{H}} \end{bmatrix}
\end{align*}

\begin{align*}
\tilde{\mathbf{H}} =
\begin{bmatrix}
\mathbf{z}_1^T E_{11} c_{11}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{11} p^{\prime}_{11}\lambda_1e^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{21} c_{21}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{21} p^{\prime}_{21}\lambda_1e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{I1} c_{I1}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{I1} p^{\prime}_{I1}\lambda_1e^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\mathbf{z}_1^T E_{12} c_{12}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{12} p^{\prime}_{12}\lambda_2e^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{22} c_{22}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{21} p^{\prime}_{22}\lambda_2e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{I2} c_{I2}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{I2} p^{\prime}_{I2}\lambda_2e^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_1^T E_{1J} c_{1J}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \mathbf{z}_1^T E_{1J} p^{\prime}_{1J}\lambda_Je^{\mathbf{z}_1^T \boldsymbol{\beta}} \\
\mathbf{z}_2^T E_{2J} c_{2J}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \mathbf{z}_2^T E_{2J} p^{\prime}_{2J}\lambda_J e^{\mathbf{z}_2^T \boldsymbol{\beta}} \\
\vdots & \vdots \\
\mathbf{z}_I^T E_{IJ} c_{IJ}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \mathbf{z}_I^T E_{IJ} p^{\prime}_{IJ}\lambda_Je^{\mathbf{z}_I^T \boldsymbol{\beta}} \\
\mathbf{z}_1^T e^{\mathbf{z}_1^T \boldsymbol{\beta}}\sum_{j}E_{1j} \lambda_j & 0_{1 \times K} \\
\vdots & \vdots \\
\mathbf{z}_I^T e^{\mathbf{z}_I^T \boldsymbol{\beta}}\sum_{j}E_{Ij} \lambda_j & 0_{1 \times K}
\end{bmatrix},
\end{align*}

\begin{align*}
\tilde{\mathbf{T}} =   \begin{bmatrix}
E_{11} p_{11}e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{11} p^\prime_{11}\lambda_1e^{\mathbf{z}_1^T \boldsymbol{\beta}} & 0 & \dots & 0\\
E_{21} p_{21}e^{\mathbf{z}_2^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{21} p^\prime_{21}\lambda_1e^{\mathbf{z}_2^T \boldsymbol{\beta}} & 0 & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
E_{I1} p_{I1}e^{\mathbf{z}_I^T \boldsymbol{\beta}}  & 0 & \dots  & 0 &  E_{I1} p^\prime_{I1}\lambda_1e^{\mathbf{z}_I^T \boldsymbol{\beta}} & 0 & \dots & 0\\
0 &  E_{12} p_{12}e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & \dots & 0 & 0 &  E_{12} p^\prime_{12} \lambda_2e^{\mathbf{z}_1^T \boldsymbol{\beta}}  & \dots & 0\\
0 &  E_{22} p_{22}e^{\mathbf{z}_2^T \boldsymbol{\beta}}   & \dots & 0 & 0 &  E_{22} p^\prime_{22} \lambda_2e^{\mathbf{z}_2^T \boldsymbol{\beta}} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
0 &  E_{I2} p_{I2}e^{\mathbf{z}_I^T \boldsymbol{\beta}}  & \dots & 0 & 0 &  E_{I2} p^\prime_{I2} \lambda_2e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \dots & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots  & \ddots & \vdots\\
0 & 0 & \dots &  E_{1J} p_{1J}e^{\mathbf{z}_1^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{1J} p^\prime_{1J}\lambda_Je^{\mathbf{z}_1^T \boldsymbol{\beta}}\\
0 & 0 & \dots &  E_{2J} p_{2J}e^{\mathbf{z}_2^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{2J} p^\prime_{2J}\lambda_Je^{\mathbf{z}_2^T \boldsymbol{\beta}}\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots  & \vdots & \vdots\\
0 & 0 & \dots &  E_{IJ} p_{IJ}e^{\mathbf{z}_I^T \boldsymbol{\beta}} & 0 & 0 & \dots &  E_{IJ} p^\prime_{IJ}\lambda_Je^{\mathbf{z}_I^T \boldsymbol{\beta}}\\
E_{11} e^{\mathbf{z}_1^T \boldsymbol{\beta}} &  E_{12} e^{\mathbf{z}_1^T \boldsymbol{\beta}} & \dots &  E_{1J} e^{\mathbf{z}_1^T \boldsymbol{\beta}} & 0 & 0 & \dots
& 0\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots  & \ddots & \vdots\\
E_{I1} e^{\mathbf{z}_I^T \boldsymbol{\beta}}&  E_{I2} e^{\mathbf{z}_I^T \boldsymbol{\beta}} & \dots &  E_{IJ} e^{\mathbf{z}_I^T \boldsymbol{\beta}} & 0& 0 & \dots
& 0\\
\end{bmatrix}.
\end{align*}

We can represent $\tilde{\mathbf{T}}$ and $\tilde{\mathbf{H}}$ as block matrices.
We let $\mathbf{E}$ be the $I \times J$ matrix with $(i,j)^\text{th}$ elements $E_{ij}$,
and we similarly define the matrix $\mathbf{C}$ to be in $\R^{I \times J}$ with
its $i,j$ element equal to $c_{ij}$. Furthermore, let $\mathbf{p}^\prime_{ij} = p^\prime_{ij}$.
Let the matrix $\boldsymbol{\omega}$ be the diagonal matrix in $\R^{I \times I}$ with $i, j$ elements $e^{\mathbf{z}_i^T \boldsymbol{\beta}} \mathbbm{1}(i = j)$. Let $\mathbf{E}_{[:,j]} \odot \mathbf{C}_{[:,j]}$ be the element-wise multiplication between the two matrices $\mathbf{E}_{[:,j]}$ and $ \mathbf{C}_{[:,j]}$.
Let $\mathbf{Z} \in \R^{I \times K}$ with rows $\mathbf{Z}_{[i,:]} = \mathbf{z}_i^T$.
Let $\mathbf{1}$ be the $I$-dimensional vector with each element equal to 1 and let

\begin{align*}
\mathbf{D}_j & = \text{diag}(\mathbf{E}_{[:,j]} \odot \mathbf{C}_{[:,j]}) \\
\mathbf{D}^\prime_j & = \text{diag}(\mathbf{E}_{[:,j]} \odot \mathbf{p}^\prime_{[:,j]}) \\
\mathbf{F}_j & = \text{diag}(\mathbf{E}_{[:,j]}).
\end{align*}

Let $\boldsymbol{\Omega}$ be the block matrix:

\begin{align}
\boldsymbol{\Omega} =
\begin{bmatrix}
\boldsymbol{\omega} & 0_{I\times I}& \dots &0_{I\times I}&0_{I\times I}&0_{I\times I}\\
0_{I\times I} & \boldsymbol{\omega} & \dots &0_{I\times I}&0_{I\times I}&0_{I\times I}\\
\vdots & \vdots & \ddots & \vdots& \vdots& \vdots\\
0_{I\times I} & 0_{I\times I} & \dots & \boldsymbol{\omega} & 0_{I\times I}& 0_{I\times I}\\
0_{I\times I} & 0_{I\times I} & \dots  & 0_{I\times I}& \boldsymbol{\omega}& 0_{I\times I} \\
0_{I\times I} & 0_{I\times I} & \dots  & 0_{I\times I}& 0_{I\times I} & \boldsymbol{\omega}
\end{bmatrix}
\end{align}

Then we can write $\tilde{\mathbf{H}}$ as

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-a}
\boldsymbol{\Omega}
\begin{bmatrix}
\mathbf{D}_1 \mathbf{Z} & \mathbf{D}_1^\prime \mathbf{Z}\lambda_1  \\
\mathbf{D}_2 \mathbf{Z} & \mathbf{D}_2^\prime \mathbf{Z}\lambda_2  \\
\vdots & \vdots\\
\mathbf{D}_{J-1} \mathbf{Z} & \mathbf{D}_{J-1}^\prime \mathbf{Z}\lambda_{J-1}  \\
\mathbf{D}_J \mathbf{Z} & \mathbf{D}_J^\prime \mathbf{Z} \lambda_{J} \\
\lp \sum_{j} \lambda_j \mathbf{F}_j \rp \mathbf{Z} & 0_{I \times K}  \\
\end{bmatrix}
\end{align}

and $\tilde{\mathbf{T}}$ as

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-b}
\boldsymbol{\Omega}
\begin{bmatrix}
\mathbf{D}_1 \mathbf{1} & 0_{I \times 1} &
\dots & 0_{I \times 1} & 0_{I \times 1} & \lambda_{1}\mathbf{D}_1^\prime\mathbf{1}  & 0_{I \times 1}& \dots & 0_{I \times 1} & 0_{I \times 1} \\
0_{I \times 1} & \mathbf{D}_2 \mathbf{1} &
\dots & 0_{I \times 1} & 0_{I \times 1} & 0_{I \times 1} & \lambda_{2}\mathbf{D}^\prime_2\mathbf{1}  & \dots & 0_{I \times 1} & 0_{I \times 1}    \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots  & \vdots & \vdots \\
0_{I \times 1} &  0_{I \times 1} &
\dots & \mathbf{D}_{J-1}\mathbf{1}  & 0_{I \times 1} & 0_{I \times 1} & 0_{I \times 1}& \dots &  \lambda_{J-1}\mathbf{D}^\prime_{J-1}\mathbf{1} & 0_{I \times 1}\\
0_{I \times 1} &  0_{I \times 1} &
\dots & 0_{I \times 1} & \mathbf{D}_{J}\mathbf{1}  & 0_{I \times 1} & 0_{I \times 1}& \dots &  0_{I \times 1}&  \lambda_{J}\mathbf{D}^\prime_{J} \mathbf{1}\\
\mathbf{E}_{[:,1]} & \mathbf{E}_{[:,2]}  & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}  &
0_{I \times 1} & 0_{I \times 1} & \dots & 0_{I \times 1} & 0_{I \times 1}
\end{bmatrix}
\end{align}

Rearranging the columns of $\tilde{\Delta}$ to form $\tilde{\Delta}^\prime$ does not change the rank of the matrix, and clarifies the conditions needed for the matrix to be full-rank:

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-comb}
\tilde{\Delta}^\prime = \boldsymbol{\Omega}
\begin{bmatrix}
\mathbf{D}_1 \mathbf{Z} & \mathbf{D}_1\mathbf{1}  & 0_{I \times 1} &  \dots & 0_{I \times 1} & 0_{I \times 1} & \lambda_1\mathbf{D}_1^\prime \mathbf{Z} & 0_{I \times 1}& \lambda_1\mathbf{D}_1^\prime \mathbf{1} & 0_{I \times 1}& \dots & 0_{I \times 1} \\
\mathbf{D}_2 \mathbf{Z} & 0_{I \times 1} & \mathbf{D}_2 \mathbf{1}& \dots & 0_{I \times 1} & 0_{I \times 1} & \lambda_2\mathbf{D}_2^\prime \mathbf{Z} & 0_{I \times 1} & 0_{I \times 1} & \lambda_2\mathbf{D}^\prime_2\mathbf{1}  & \dots & 0_{I \times 1}\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots  & \vdots\\
\mathbf{D}_{J-1} \mathbf{Z}  & 0_{I \times 1} & 0_{I \times 1} &  \dots &  \mathbf{D}_{J-1}\mathbf{1}  & 0_{I \times 1} & \lambda_{J-1}\mathbf{D}^\prime_{J-1} \mathbf{Z} & 0_{I \times 1} & 0_{I \times 1} & 0_{I \times 1} & \dots & \lambda_{J-1}\mathbf{D}_{J-1}^\prime\mathbf{1}\\
\mathbf{D}_J \mathbf{Z} & 0_{I \times 1} & 0_{I \times 1} & \dots & 0_{I \times 1} &\mathbf{D}_J \mathbf{1} &
\lambda_{J}\mathbf{D}^\prime_J \mathbf{Z}  & \lambda_{J}\mathbf{D}_{J}^\prime\mathbf{1} & 0_{I \times 1} & 0_{I \times 1} & \dots & 0_{I \times 1}\\
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{Z} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]}  & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}  & 0_{I \times K}  &  0_{I \times 1} & 0_{I \times 1}  & 0_{I \times 1}& \dots & 0_{I \times 1}
\end{bmatrix}
\end{align}

is $\in \R^{\lp I(J + 1)\rp \times \lp 2K + 2J\rp}$. The first matrix $\boldsymbol{\Omega}$
is a $I(J + 1) \times I(J + 1)$ block diagonal matrix of diagonal matrices and by condition <ref> is rank $I(J + 1)$.
Then the product in equation (<ref>) is rank $2K + 2J$ if the second matrix above
is rank $2K + 2J$ by Sylvester's rank inequality.
The second matrix above is rank $2K + 2J$ if the following three sub-blocks are full column rank:

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-block-1}
\mathbf{L}_1  =   \begin{bmatrix}
\lambda_1\mathbf{D}_1^\prime\mathbf{1}  & 0_{I \times 1}& \dots & 0_{I \times 1} \\
0_{I \times 1} & \lambda_2\mathbf{D}^\prime_2\mathbf{1}  & \dots & 0_{I \times 1}    \\
\vdots & \vdots & \ddots  & \vdots \\
0_{I \times 1} & 0_{I \times 1} & \dots &  \lambda_{J-1} \mathbf{D}^\prime_{J-1} \mathbf{1}
\end{bmatrix}
\end{align}

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-block-2}
\mathbf{L}_2  =    \begin{bmatrix}
\lp \sum_{j} \lambda_j\mathbf{F}_j \rp \mathbf{Z} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]} & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}
\end{bmatrix}
\end{align}

\begin{align} \label{eq:gradient-modeled-missingness-cat-effects-block-3}
\mathbf{L}_3  =    \begin{bmatrix}
\lambda_{J}\mathbf{D}^\prime_J \mathbf{Z} & \lambda_{J}\mathbf{D}^\prime_J \mathbf{1}
\end{bmatrix}
\end{align}

Sufficient conditions for $\mathbf{L}_1$ to be full column rank is

* $I \geq (J-1)S$

* $\text{diag}(\mathbf{E}_{[:,j]})\mathbf{1}$ is full column rank for all $j$

* $\lambda_j > 0 \forall j \leq J$

* $\eta_{j} \in (-\infty, \infty) \forall j \leq J$

* $\beta_k, \gamma_k \in (-\infty, \infty) \forall k \leq J$

Sufficient conditions for $\mathbf{L}_2$ to be full column rank is

* $I \geq K + J$

* $\forall i \in 1, \dots, I$ $\sum_{j=1}^J E_{ij} > 0$

* $\text{rank}\lp\begin{bmatrix} \mathbf{Z} & \mathbf{E}\end{bmatrix}\rp = J + K$

* ${\rm rank}\lp \begin{bmatrix}
\mathbf{F}_1\mathbf{Z} & \dots & \mathbf{F}_J\mathbf{Z} & \mathbf{E}_{[:,1]}  & \mathbf{E}_{[:,2]} & \dots & \mathbf{E}_{[:,J-1]}  & \mathbf{E}_{[:,J]}
\end{bmatrix}\rp > J + K$

Sufficient conditions for $\mathbf{L}_3$ to be full column rank is

* $I \geq K + S$

* $\mathbf{F}_J \begin{bmatrix} \mathbf{Z} & \mathbf{1}\end{bmatrix}$
is full column rank

We recognize matrix $\mathbf{L}_2$ as the same matrix as in <ref>
and that cond:full-model-rank-Econds:full-model-rank-big-mat are a superset of
the conds:rank-lemma-dimconds:rank-lemma-rank-big-mat in <ref>. Then by <ref>, matrix $\mathbf{L}_2$ is rank $J + K$.
cond:full-model-rank-Econds:full-model-rank-big-mat ensure that $\mathbf{L}_1$ and $\mathbf{L}_3$ are full rank as well, so $\text{rank}(\tilde{\Delta})$ is full rank.

Given that $\text{rank}(\tilde{\Delta}^\prime) = \text{rank}(\tilde{\Delta}) = \text{rank}(\Delta) = 2J + 2K$, the column dimension of $\Delta$, $\text{Cov}(\mathbf{f})$ is positive definite, and that the observational density, $\pi_\theta(\mathbf{f})$ is Poisson so $\frac{\partial}{\partial \theta_j} \int f_i \pi_\theta(\mathbf{x}) d\nu = \int f_i \frac{\partial}{\partial \theta_j} \pi_\theta(\mathbf{x}) d\nu$, we can apply lemma <ref>, which bounds the positive definiteness below by 0 for the Fisher information matrix $\mathcal{I}$.
In other words, the Fisher Information matrix is positive definite.
By Theorem 1 in <cit.>, model <ref> is locally identifiable for any $(\mathbf{p}, \boldsymbol{\lambda}, \boldsymbol{\beta}, \boldsymbol{\gamma}) \in ((0, 1)^J \times (0,\infty)^J \times \R^K \times \R^K)$, where $((a,b)^n)$ is the $n$-fold Cartesian product of the set $(a,b)$.

§ FURTHER SIMULATION STUDY RESULTS

§.§ Mean squared error plots

Log-mean squared error across simulated datasets for the standardized incidence ratio, or $\text{SIR}_j$ for Blacks, Hispanic/Latinos, Others, and Whites plotted against the proportion of cases observed with race data. The blue color corresponds to the joint model in equation (<ref>), while the red color corresponds to a the model defined in equation (<ref>), or a  complete-case analysis. Smaller magnitude is better.

Log-mean squared error across simulated datasets for the relative risk ratio, or $\mathbbm{I}_j / \mathbbm{I}_J$ for Blacks, Hispanic/Latinos, Others relative to Whites plotted against the proportion of cases observed with race data. The blue color corresponds to the joint model in equation (<ref>), while the red color corresponds to a the model defined in equation (<ref>), or a  complete-case analysis. Smaller magnitude is better.

Log-mean squared error across simulated datasets for the population relative risk ratio, or $\exp((\boldsymbol{\alpha}_\lambda)_j - (\boldsymbol{\alpha}_\lambda)_J)$ for Blacks, Hispanic/Latinos, Others relative to Whites plotted against the proportion of cases observed with race data. The blue color corresponds to the joint model in equation (<ref>), while the red color corresponds to a the model defined in equation (<ref>), or a  complete-case analysis. Smaller magnitude is better.

§.§ 80% posterior interval coverage

Table shows 80% posterior credible interval coverages and lengths for estimands of interest from the simulation study. Coverage proportion is calculated across 200 simulated datasets for each model/simulation scenario. Column headers for percentages (e.g. 20%) indicate the missing-data simulation scenario which corresponds to the statistic calculated in the table column; the simulation scenario corresponds to the proportion of cases observed with completely observed race covariates.

4c80% interval coverage     4c80% mean interval length

Parameter     Model      20%     60%     80%     90%     20%     60%     80%     90%

$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Blacks}$     Complete Case     0.70     0.70     0.68     0.60     0.51     0.47     0.47     0.44

$ \quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Joint     0.78     0.81     0.81     0.78     0.62     0.55     0.55     0.53

MI-Ad-hoc    1.00     0.88     0.77     0.66     0.41     0.40     0.43     0.42

MI-Gibbs     0.75     0.70     0.67     0.61     0.52     0.46     0.46     0.44
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Hispanics/Latinos}$     Complete Case     0.77     0.80     0.83     0.77     0.63     0.56     0.56     0.53

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Joint     0.56     0.77     0.84     0.78     0.68     0.61     0.57     0.56

MI-Ad-hoc    0.99     0.91     0.89     0.82     0.42     0.44     0.49     0.49

MI-Gibbs     0.07     0.68     0.82     0.78     1.13     0.65     0.58     0.55
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Others}$     Complete Case     0.42     0.33     0.27     0.28     0.44     0.38     0.37     0.36

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Joint     0.37     0.73     0.77     0.82     0.63     0.67     0.60     0.58

MI-Ad-hoc    1.00     0.62     0.41     0.30     0.40     0.36     0.36     0.35

MI-Gibbs     0.70     0.48     0.32     0.29     0.71     0.42     0.39     0.37
$\mathbbm{SI}_\text{Blacks}$     Complete Case     0.06     0.04     0.01     0.00     0.03     0.02     0.01     0.01

Joint     0.67     0.77     0.80     0.78     0.09     0.04     0.03     0.02

MI-Ad-hoc    0.09     0.20     0.06     0.00     0.01     0.01     0.01     0.01

MI-Gibbs     0.03     0.01     0.00     0.00     0.02     0.01     0.01     0.01
$\mathbbm{SI}_\text{Hispanics/Latinos}$     Complete Case     0.34     0.24     0.12     0.01     0.13     0.07     0.06     0.06

Joint     0.46     0.74     0.81     0.70     0.47     0.23     0.14     0.10

MI-Ad-hoc    0.12     0.23     0.27     0.06     0.07     0.06     0.06     0.06

MI-Gibbs     0.00     0.05     0.01     0.01     0.16     0.08     0.06     0.06
$\mathbbm{SI}_\text{Others}$     Complete Case     0.03     0.00     0.00     0.00     0.11     0.06     0.05     0.05

Joint     0.27     0.70     0.76     0.77     0.49     0.36     0.25     0.20

MI-Ad-hoc    0.17     0.14     0.01     0.00     0.07     0.06     0.05     0.05

MI-Gibbs     0.20     0.01     0.00     0.00     0.15     0.06     0.05     0.05
$\mathbbm{SI}_\text{Whites}$     Complete Case     0.04     0.01     0.00     0.00     0.02     0.01     0.01     0.01

Joint     0.50     0.80     0.82     0.84     0.06     0.03     0.02     0.01

MI-Ad-hoc    0.14     0.17     0.00     0.00     0.01     0.01     0.01     0.01

MI-Gibbs     0.21     0.01     0.00     0.00     0.02     0.01     0.01     0.01
$\mathbbm{I}_\text{Blacks}/\mathbbm{I}_\text{Whites}$     Complete Case     0.06     0.01     0.00     0.00     0.04     0.02     0.02     0.02

Joint     0.73     0.79     0.81     0.81     0.12     0.05     0.04     0.03

MI-Ad-hoc    0.10     0.20     0.02     0.00     0.02     0.02     0.02     0.02

MI-Gibbs     0.12     0.01     0.00     0.00     0.03     0.02     0.02     0.02
$\mathbbm{I}_\text{Hispanics/Latinos}/$     Complete Case     0.17     0.18     0.57     0.83     0.07     0.04     0.04     0.03

$\mathbbm{I}_\text{Whites}$       Joint     0.43     0.73     0.81     0.73     0.28     0.14     0.09     0.06

MI-Ad-hoc    0.12     0.21     0.32     0.41     0.04     0.04     0.04     0.03

MI-Gibbs     0.00     0.14     0.24     0.17     0.10     0.05     0.04     0.04
$\mathbbm{I}_\text{Others}/\mathbbm{I}_\text{Whites}$     Complete Case     0.00     0.00     0.00     0.00     0.06     0.03     0.03     0.03

Joint     0.26     0.72     0.74     0.76     0.32     0.24     0.17     0.13

MI-Ad-hoc    0.14     0.11     0.01     0.00     0.05     0.04     0.03     0.03

MI-Gibbs     0.21     0.00     0.00     0.00     0.10     0.04     0.03     0.03

Table shows 50% posterior credible interval coverages and lengths for estimands of interest from the simulation study. Coverage proportion is calculated across 200 simulated datasets for each model/simulation scenario. Column headers for percentages (e.g. 20%) indicate the missing-data simulation scenario which corresponds to the statistic calculated in the table column; the simulation scenario corresponds to the proportion of cases observed with completely observed race covariates.

4c50% interval coverage     4c50% mean interval length

Parameter     Model      20%     60%     80%     90%     20%     60%     80%     90%

$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Blacks}$     Complete Case     0.39     0.35     0.41     0.32     0.27     0.24     0.24     0.23

$ \quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Joint     0.52     0.53     0.49     0.47     0.32     0.28     0.29     0.27

MI-Ad-hoc    0.96     0.56     0.47     0.30     0.21     0.21     0.22     0.22

MI-Gibbs     0.48     0.38     0.38     0.29     0.27     0.24     0.24     0.23
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Hispanics/Latinos}$     Complete Case     0.48     0.47     0.47     0.44     0.33     0.29     0.29     0.28

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Joint     0.27     0.48     0.47     0.43     0.35     0.31     0.30     0.29

MI-Ad-hoc    0.96     0.62     0.53     0.47     0.22     0.23     0.25     0.25

MI-Gibbs     0.01     0.39     0.46     0.45     0.58     0.33     0.30     0.28
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Others}$     Complete Case     0.20     0.10     0.12     0.09     0.23     0.20     0.19     0.19

$ \quad \quad -(\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Joint     0.17     0.41     0.47     0.51     0.32     0.35     0.31     0.30

MI-Ad-hoc    0.91     0.29     0.15     0.10     0.21     0.18     0.19     0.18

MI-Gibbs     0.38     0.23     0.12     0.10     0.36     0.22     0.20     0.19
$\mathbbm{SI}_\text{Blacks}$     Complete Case     0.04     0.02     0.00     0.00     0.01     0.01     0.01     0.01

Joint     0.39     0.49     0.48     0.49     0.04     0.02     0.01     0.01

MI-Ad-hoc    0.04     0.09     0.03     0.00     0.01     0.01     0.01     0.01

MI-Gibbs     0.03     0.01     0.00     0.00     0.01     0.01     0.01     0.01
$\mathbbm{SI}_\text{Hispanics/Latinos}$     Complete Case     0.17     0.12     0.05     0.00     0.07     0.04     0.03     0.03

Joint     0.27     0.47     0.51     0.41     0.24     0.12     0.07     0.05

MI-Ad-hoc    0.07     0.15     0.18     0.01     0.04     0.03     0.03     0.03

MI-Gibbs     0.00     0.01     0.01     0.00     0.08     0.04     0.03     0.03
$\mathbbm{SI}_\text{Others}$     Complete Case     0.01     0.00     0.00     0.00     0.06     0.03     0.03     0.02

Joint     0.12     0.45     0.48     0.41     0.25     0.18     0.13     0.10

MI-Ad-hoc    0.08     0.07     0.01     0.00     0.04     0.03     0.03     0.03

MI-Gibbs     0.09     0.00     0.00     0.00     0.08     0.03     0.03     0.02
$\mathbbm{SI}_\text{Whites}$     Complete Case     0.03     0.01     0.00     0.00     0.01     0.01     0.00     0.00

Joint     0.28     0.51     0.54     0.51     0.03     0.02     0.01     0.01

MI-Ad-hoc    0.07     0.08     0.00     0.00     0.01     0.00     0.00     0.00

MI-Gibbs     0.08     0.00     0.00     0.00     0.01     0.01     0.00     0.00
$\mathbbm{I}_\text{Blacks}/\mathbbm{I}_\text{Whites}$     Complete Case     0.03     0.01     0.00     0.00     0.02     0.01     0.01     0.01

Joint     0.48     0.54     0.52     0.48     0.06     0.03     0.02     0.01

MI-Ad-hoc    0.04     0.10     0.01     0.00     0.01     0.01     0.01     0.01

MI-Gibbs     0.07     0.01     0.00     0.00     0.02     0.01     0.01     0.01
$\mathbbm{I}_\text{Hispanics/Latinos}/$     Complete Case     0.09     0.08     0.33     0.53     0.03     0.02     0.02     0.02

$\mathbbm{I}_\text{Whites}$      Joint     0.24     0.46     0.51     0.45     0.14     0.07     0.05     0.03

MI-Ad-hoc    0.09     0.12     0.17     0.24     0.02     0.02     0.02     0.02

MI-Gibbs     0.00     0.09     0.07     0.05     0.05     0.02     0.02     0.02
$\mathbbm{I}_\text{Others}/\mathbbm{I}_\text{Whites}$     Complete Case     0.00     0.00     0.00     0.00     0.03     0.02     0.02     0.01

Joint     0.12     0.43     0.47     0.41     0.16     0.12     0.09     0.07

MI-Ad-hoc    0.09     0.06     0.00     0.00     0.02     0.02     0.02     0.02

MI-Gibbs     0.09     0.00     0.00     0.00     0.05     0.02     0.02     0.02

§ FURTHER WAYNE COUNTY APPLIED DATA ANALYSIS RESULTS

§.§ Age-Race/Ethnicity posterior predictive checks

Posterior predictive checks for cumulative incidence by age group by race for Blacks and Whites.

§.§ Rootogram

Posterior predictive rootogram for missing case counts.

§.§ Tables for posterior summaries for estimands of interest

This table presents posterior summary statistics for the Wayne-County estimands of interest. Post. mean stands for Posterior Mean, and MCSE stands for Monte Carlo Standard Error, which is the standard error in the posterior estimator, which can be estimated assuming that the MCMC central limit theorem holds. See <cit.> and <cit.> for more details

Post.          10% Post.          90% Post.

Estimand     Model     Mean     MCSE    quant.     MCSE     quant.     MCSE

Joint     3.93     1.28e-02     3.16     1.40e-02     4.76     2.06e-02

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Complete Case model     3.56     1.02e-02     2.89     1.09e-02     4.29     1.79e-02

Ad-Hoc MI     2.96     1.80e-03     2.43     1.60e-03     3.52     2.15e-03

Gibbs MI     3.49     1.75e-03     2.78     1.84e-03     4.24     2.59e-03
Joint     2.10     6.34e-03     1.65     6.39e-03     2.59     8.22e-03

$ \quad \quad- (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$      Complete Case model     1.93     4.35e-03     1.58     4.66e-03     2.29     8.52e-03

Ad-Hoc MI     1.67     1.83e-03     1.35     1.72e-03     2.01     1.55e-03

Gibbs MI     2.09     3.22e-03     1.65     2.62e-03     2.56     3.87e-03
Joint     8.88     4.78e-02     6.09     5.51e-02     11.81     6.13e-02

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     5.21     1.30e-02     4.26     1.55e-02     6.22     1.91e-02

Ad-Hoc MI     4.08     2.75e-03     3.32     2.52e-03     4.90     2.79e-03

Gibbs MI     5.27     5.03e-03     4.18     4.14e-03     6.42     4.88e-03
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Asians/Pacific Islanders}$
Joint     1.35     4.79e-03     1.01     5.14e-03     1.73     7.37e-03

$ \quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     1.29     3.30e-03     1.00     3.59e-03     1.61     6.06e-03

Ad-Hoc MI     1.21     1.98e-03     0.96     1.63e-03     1.48     2.43e-03

Gibbs MI     1.66     4.44e-03     1.22     3.10e-03     2.15     6.17e-03
Joint     1.59     6.62e-04     1.55     1.20e-03     1.64     7.33e-04

Complete Case model     1.58     8.02e-05     1.57     1.44e-04     1.60     1.53e-04

Ad-Hoc MI     1.56     1.47e-04     1.55     1.63e-04     1.57     1.37e-04

Gibbs MI     1.60     1.65e-04     1.58     1.72e-04     1.61     1.65e-04
Joint     1.16     1.59e-03     1.02     1.55e-03     1.30     2.20e-03

Complete Case model     1.17     3.29e-04     1.12     6.04e-04     1.23     6.36e-04

Ad-Hoc MI     1.15     7.02e-04     1.10     6.59e-04     1.20     7.41e-04

Gibbs MI     1.23     9.89e-04     1.17     1.05e-03     1.28     1.01e-03
Joint     4.64     1.81e-02     3.50     2.24e-02     5.64     1.38e-02

Complete Case model     3.06     8.03e-04     2.93     1.33e-03     3.19     1.77e-03

Ad-Hoc MI     2.68     1.21e-03     2.57     1.03e-03     2.80     1.38e-03

Gibbs MI     3.15     2.40e-03     3.01     NA     3.28     2.69e-03
$\mathbbm{SI}_\text{Asians/Pacific Islanders}$
Joint     0.61     1.09e-03     0.53     4.95e-04     0.71     2.52e-03

Complete Case model     0.65     3.27e-04     0.60     7.57e-04     0.71     7.64e-04

Ad-Hoc MI     0.66     8.58e-04     0.61     8.63e-04     0.72     9.36e-04

Gibbs MI     0.72     1.14e-03     0.66     NA     0.78     1.34e-03
Joint     0.46     4.25e-04     0.44     3.98e-04     0.49     4.65e-04

Complete Case model     0.52     5.20e-05     0.52     7.63e-05     0.53     1.08e-04

Ad-Hoc MI     0.55     1.00e-04     0.54     9.79e-05     0.56     1.02e-04

Gibbs MI     0.50     9.26e-05     0.49     1.11e-04     0.51     8.00e-05
Joint     3.01     2.86e-03     2.84     2.98e-03     3.19     3.31e-03

Complete Case model     2.63     3.77e-04     2.57     5.53e-04     2.70     7.52e-04

Ad-Hoc MI     2.49     6.53e-04     2.44     6.89e-04     2.55     6.51e-04

Gibbs MI     2.79     7.12e-04     2.73     6.31e-04     2.85     8.37e-04
Joint     1.69     3.26e-03     1.47     3.26e-03     1.92     4.36e-03

$\mathbbm{I}_\text{Whites}$       Complete Case model     1.50     4.66e-04     1.42     9.17e-04     1.58     1.13e-03

Ad-Hoc MI     1.42     9.31e-04     1.35     9.26e-04     1.49     9.86e-04

Gibbs MI     1.66     1.43e-03     1.57     1.44e-03     1.74     1.50e-03
Joint     6.55     2.96e-02     4.74     3.24e-02     8.12     2.06e-02

Complete Case model     3.78     1.11e-03     3.60     2.05e-03     3.96     2.31e-03

Ad-Hoc MI     3.20     1.67e-03     3.04     1.40e-03     3.35     1.90e-03

Gibbs MI     4.10     3.36e-03     3.90     NA     4.30     4.10e-03
$\mathbbm{I}_\text{Asians/Pacific Islanders}/$
Joint     1.08     2.59e-03     0.92     1.60e-03     1.27     5.65e-03

$\mathbbm{I}_\text{Whites}$       Complete Case model     1.01     5.26e-04     0.92     1.04e-03     1.10     1.02e-03

Ad-Hoc MI     0.99     1.32e-03     0.91     NA     1.07     1.39e-03

Gibbs MI     1.18     1.89e-03     1.08     NA     1.28     2.25e-03

The table shows sampling efficiency for population estimands of interest presented in table <ref>. ESS stands for effective sample size; Bulk ESS and Tail ESS are measures of the equivalent number of independent samples generated from a MCMC procedure. See <cit.> for more detail. Bulk and Tail ESS efficiency are the Bulk and Tail ESS figures divided by the total number of MCMC samples, which is $16,000$. As noted in <cit.> it is possible for MCMC samplers to generate Tail and Bulk ESS values that are larger than the total number of samples.

Estimand     Model     $\hat{R}$     Bulk ESS     Tail ESS     Bulk ESS eff.     Tail ESS eff.

Joint     1.00     2465     4753     0.15     0.30

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     1.00     3037     5645     0.19     0.35

Ad-Hoc MI     1.01     57036     223009     0.07     0.28

Gibbs MI     1.00     114009     259368     0.14     0.32
Joint     1.00     3506     6994     0.22     0.44

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     1.00     4440     6910     0.28     0.43

Ad-Hoc MI     1.01     19965     91952     0.02     0.11

Gibbs MI     1.02     12435     68983     0.02     0.09
Joint     1.00     2098     4315     0.13     0.27

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     1.00     3742     6528     0.23     0.41

Ad-Hoc MI     1.01     51519     256045     0.06     0.32

Gibbs MI     1.01     30320     220330     0.04     0.28
$\exp\big((\boldsymbol{\alpha}_\lambda)_\text{Asians/Pacific Islanders}$
Joint     1.00     3850     7799     0.24     0.49

$\quad \quad - (\boldsymbol{\alpha}_\lambda)_\text{Whites}\big)$       Complete Case model     1.00     5511     8283     0.34     0.52

Ad-Hoc MI     1.02     11078     45632     0.01     0.06

Gibbs MI     1.03     7298     30332     0.01     0.04
Joint     1.00     2531     4877     0.16     0.30

Complete Case model     1.00     16204     14672     1.01     0.92

Ad-Hoc MI     1.05     4322     15046     0.01     0.02

Gibbs MI     1.06     3490     13738     0.00     0.02
Joint     1.00     4523     9442     0.28     0.59

Complete Case model     1.00     16596     12657     1.04     0.79

Ad-Hoc MI     1.07     3392     14038     0.00     0.02

Gibbs MI     1.12     1990          0.00     NA
Joint     1.00     2034     4928     0.13     0.31

Complete Case model     1.00     16338     11881     1.02     0.74

Ad-Hoc MI     1.04     5584     18672     0.01     0.02

Gibbs MI     1.12     1938          0.00     NA
$\mathbbm{SI}_\text{Asians/Pacific Islanders}$
Joint     1.00     6412     6513     0.40     0.41

Complete Case model     1.00     17496     11891     1.09     0.74

Ad-Hoc MI     1.10     2439          0.00     NA

Gibbs MI     1.15     1673          0.00     NA
Joint     1.00     1722     5572     0.11     0.35

Complete Case model     1.00     16277     13527     1.02     0.85

Ad-Hoc MI     1.05     4083     17557     0.01     0.02

Gibbs MI     1.05     4488          0.01     NA
Joint     1.00     2197     5606     0.14     0.35

Complete Case model     1.00     16184     13675     1.01     0.85

Ad-Hoc MI     1.05     4225     16200     0.01     0.02

Gibbs MI     1.05     4673          0.01     NA
Joint     1.00     2928     6147     0.18     0.38

$\mathbbm{I}_\text{Whites}$       Complete Case model     1.00     16631     12370     1.04     0.77

Ad-Hoc MI     1.06     3557     14586     0.00     0.02

Gibbs MI     1.11     2144          0.00     NA
Joint     1.00     1933     4788     0.12     0.30

Complete Case model     1.00     16306     11978     1.02     0.75

Ad-Hoc MI     1.04     5156          0.01     NA

Gibbs MI     1.11     2114          0.00     NA
$\mathbbm{I}_\text{Asians/Pacific Islanders}/$
Joint     1.00     3827     6402     0.24     0.40

$\mathbbm{I}_\text{Whites}$       Complete Case model     1.00     17423     11926     1.09     0.75

Ad-Hoc MI     1.10     2447          0.00     NA

Gibbs MI     1.14     1731          0.00     NA

The table shows the posterior means, 80% credible interval endpoints and the Monte Carlo standard errors of these estimates. $\text{CC}$ stands for the complete-case model while $\text{J}$ stands for the joint model.

Post.          10% Post.          90% Post.

Estimand      Mean     MCSE     quant.     MCSE      quant.     MCSE

$\mathbb{I}^{\text{CC}}_\text{Blacks} / \mathbb{I}^\text{J}_\text{Blacks}$
0.81     3.50e-04     0.79     3.55e-04     0.84     5.80e-04
$\mathbb{I}^{\text{CC}}_\text{Hispanics/Latinos} / \mathbb{I}^\text{J}_\text{Hispanics/Latinos}$
0.83     1.14e-03     0.73     1.14e-03     0.94     1.70e-03

$\mathbb{I}^{\text{CC}}_\text{Others} / \mathbb{I}^\text{J}_\text{Others}$
0.55     2.41e-03     0.44     1.36e-03     0.71     4.26e-03
$\mathbb{I}^{\text{CC}}_\text{Asians/Pacific Islanders} / \mathbb{I}^\text{J}_\text{Asians/Pacific Islanders}$
0.88     1.42e-03     0.73     2.50e-03     1.03     1.71e-03
$\mathbb{I}^{\text{CC}}_\text{Whites} / \mathbb{I}^\text{J}_\text{Whites}$
0.93     8.51e-04     0.88     7.85e-04     0.98     7.79e-04

$\Prob{\text{Race observed}}_\text{Blacks}$
0.85     7.55e-04     0.81     7.25e-04     0.91     1.35e-03
$\Prob{\text{Race observed}}_\text{Hispanics/Latinos}$
0.87     8.63e-04     0.78     1.43e-03     0.95     6.59e-04
$\Prob{\text{Race observed}}_\text{Others}$
0.58     2.93e-03     0.45     1.20e-03     0.77     5.31e-03
$\Prob{\text{Race observed}}_\text{Asians/Pacific Islanders}$
0.90     8.25e-04     0.81     2.36e-03     0.97     3.80e-04
$\Prob{\text{Race observed}}_\text{Whites}$
0.94     7.09e-04     0.89     8.70e-04     0.98     4.33e-04

§ STAN CODE FOR NEGATIVE BINOMIAL LIKELIHOOD

The following Stan code computes the likelihood related to the following generative model using an efficient dynamic programming algorithm :

\begin{align*}
Y_{ij} & \sim \text{Binomial}(E_{ij}, \theta_{ij}) \\
X_{ij} | Y_{ij} & \sim \text{Binomial}(Y_{ij}, p_{ij})
\end{align*}

after marginalizing over all combinations of $Y_{ij}$ such that $\sum_{j} Y_{ij} = T$ where $T$ is the total identified cases of disease in stratum $i$, a known quantity.

The code was derived from <cit.>.

real binomial_2_lpmf(int y_obs, int y_miss,
real p, real theta, int E)
return binomial_lpmf(y_obs | y_miss, p)
+ binomial_lpmf(y_miss | E, theta);

real miss_lpmf(int[] y, int n_miss,
vector p, vector theta,
int[] E)
int N = rows(theta);
real alpha[N + 1, n_miss + 1];

// alpha[n + 1, tot + 1] = log p of tot missing cases
// distributed among first n categories
alpha[1, 1:(n_miss + 1)] = rep_array(0, n_miss+1);
for (n in 1:N)
// tot = 0
alpha[n + 1, 1] = alpha[n, 1]
+ binomial_2_lpmf(y[n]|y[n],p[n],theta[n], E[n]);

// 0 < tot < n

for (tot in 1:n_miss)
if (n > 1)
vector[tot + 1] vec;
for (i in 1:(tot + 1))
vec[i] = alpha[n,i]
+ binomial_2_lpmf(y[n] |
y[n] + tot - (i - 1),
p[n],theta[n], E[n]);

alpha[n + 1, tot + 1] = log_sum_exp(vec);
alpha[n + 1,tot + 1]
= binomial_2_lpmf(y[n]| y[n]
+ tot,p[n],theta[n], E[n]);

return alpha[N + 1, n_miss + 1];

We would like to thank Mitzi Morris, Andrew Gelman, and Bob Carpenter for their feedback on an earlier draft of the paper.

JZ & RT were supported by award #6 U01 IP00113801-01 from the U.S. Centers for Disease Control and Prevention, and award #812255 from the Simons Foundation.
This research was supported in part through computational resources and services provided by Advanced Research Computing (ARC), a division of Information and Technology Services (ITS) at the University of Michigan, Ann Arbor.

Alternative generative model:

\begin{align*}
Y_{i0} & \sim \text{Poisson}(\lambda_0) \\
Y_{ij} | Y_{i0} & \sim \text{Poisson}(E_{ij} \lambda_j + \beta_j Y_{i0}) \\
X_{ij} | Y_{ij} & \sim \text{Binomial}(Y_{ij}, p_j)
\end{align*}

Leads to the observed data model:

\begin{align*}
Y_{i0} & \sim \text{Poisson}(\lambda_0) \\
X_{ij} | Y_{i0} & \sim  \text{Poisson}(p_j (E_{ij} \lambda_j + \beta_j Y_{i0})) \\
M_{i}| Y_{i0} & \sim  \text{Poisson}(\sum_j (1-p_j) (E_{ij} \lambda_j + \sum_j (1-p_j) \beta_j Y_{i0}))
\end{align*}

It turns out that the joint marginal distribution for $\{X_{ij}, M_i\}$ has an analytic, finite series representation:

If we further assume that $R_n$ are conditionally independent, Bernoulli distributed random variables:

R_{n} | p_{S_n,C_n},  \sim \text{Bernoulli}(p_{S_n,C_n}),

where $p_{S_n, C_n}$ is equal to $p_{ij}$ if $S_n = i$ and $C_n = j$.

Then let the random variable $X_{ij}$ be the cases for whom we have observed $C_n$:

X_{ij} | \mathcal{U} = \sum_{\{n \,|\, S_n = i, C_n = j, U_n = 1\}} R_n

By the fact that $R_{n}$ are conditionally independent Bernoulli distributed random variables and are identically distributed within a strata defined as $(S_n = i, C_n = j)$, the following distributional equivalence holds:

X_{ij} | \mathcal{U} \overset{d}{=} X_{ij} | Y_{ij},

and, furthermore,

X_{ij} | Y_{ij}, p_{ij} \sim \text{Binomial}(Y_{ij}, p_{ij})

Let $W_{ij}$ be the cases for which we do not observe $C_n$:

W_{ij} | \mathcal{U} = \sum_{\{n \,|\, S_n = i, C_n = j, U_n = 1\}} (1 - R_n).

X_{ij} | \mathcal{U}, p_{ij} \sim \text{Binomial}(Y_{ij}, (1 - p_{ij})).

We can observe only $X_{ij}$ and $M_i = \sum_j W_{ij}$; while the observations stratified by $(i,j)$, $W_{ij}$, are not observed. In the appendix section need ref we derive the likelihood corresponding to the observed data likelihood and generative model.