Bayesian predictive inference of small area proportions under selection bias
Section 3. Heterogeneous nonignorable selection model

Table of contents

In Section 3.1, we describe the heterogeneous nonignorable selection (HeS) model. We also show how to perform the computations in Section 3.2. We show how to fit this model and how to make inference about the small area proportions. Inference about the small area proportions under the HeS model is obtained using surrogate samples (Nandram, 2007).

3.1 Methodology

We assume that the sample selection probabilities $(π_{i 1}, \dots, π_{i n_{ì}})$ have the different supports over the set $π_{i u}^{*},$ $u = 1, \dots, U_{i},$ for $i = 1, \dots, l .$ That is, $π_{i j},$ $j = 1, \dots, n_{i}$ have a histogram where the midpoints of the categories are the $π_{i u}^{*},$ for $i = 1, \dots, l .$ These $π_{i u}^{*}$ are assumed known and the $π_{i j}$ are assumed to be random quantities. For notational convenience, let $θ_{i} = (θ_{i 10}, \dots, θ_{i U_{i} 0}, θ_{i 11}, \dots, θ_{i U_{i} 1}) = (θ_{i 0}, θ_{i 1})$ (say), $θ = (θ_{1}, \dots, θ_{ℓ}),$ and $π = (π_{11}, \dots, π_{1 n_{1}}, \dots, π_{l 1}, \dots, π_{l n_{l}}) .$ The distribution of the selection probabilities, given the binary response $y_{i j},$ is

$P (π_{i j} = π_{i u}^{*} | θ, y_{i j} = y) = θ_{i u y}, u = 1, \dots, U_{i}, y = 0, 1, j = 1, \dots, n_{i}, i = 1, \dots, l$

and

$y_{i j} | p_{i} \overset{iid}{~} Bernoulli (p_{i}), j = 1, \dots, N_{i}, i = 1, \dots, l .$

Again, it is worth noting that the $θ_{i u y}$ are not selection probabilities.

To accommodate the sample selection scheme, we assume that $θ_{i u 0}$ and $θ_{i u 1},$ for $i = 1, \dots, l,$ are different. Note that we consider the heterogeneity assumption for the sample selection probabilities. We replace the homogeneity assumption with the heterogeneity assumption for the sample selection probabilities of the HoS model, so that the sample selection probabilities have different supports and the distributions of the selection probabilities are different by areas.

Let $δ_{i j},$ $π_{i j}$ and $y_{i j}$ denote the selection indicator, the selection probability and the binary response of the $j^{th}$ unit in the $i^{th}$ small area in the population respectively. Essentially, NBBS postulated that the $(δ_{i j}, π_{i j}, y_{i j})$ within the $i^{th}$ small area are independently distributed with

$\begin{array}{l} P (δ_{i j} = δ, π_{i j} = π_{i u}^{*}, y_{i j} = y | θ_{i}, p_{i}) & = P_{1} (δ_{i j} = δ | π_{i j} = π_{i u}^{*}) P_{2} (π_{i j} = π_{i u}^{*} | y_{i j} = y, θ_{i}) P_{3} (y_{i j} = y | p_{i}) \\ = {(π_{i u}^{*})}^{δ} {(1 - π_{i u}^{*})}^{1 - δ} θ_{i u y} p_{i}^{y} {(1 - p_{i})}^{1 - y}, \\ δ = 0, 1, u = 1, \dots, U_{i}, y = 0, 1, j = 1, \dots, n_{i}, i = 1, \dots, l . \end{array}$

Thus, there is a joint probability mass function for the selection indicator and the response indicator. Therefore, the model that NBBS specified is a nonignorable selection model (i.e., NBBS assumed that the selection mechanism is selection not at random (SNAR)). Since there are no data when $δ = 0$ (i.e., $π$ and $y$ are both unobserved), NBBS used the conditional probability mass function

$P (π_{i j} = π_{i u}^{*}, y_{i j} = y | δ_{i j} = 1, θ_{i}, p_{i}) = \frac{π_{i u}^{*} θ_{i u y} p_{i}^{y} {(1 - p_{i})}^{1 - y}}{\sum_{y =0}^{1} \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u y} p_{i}^{y} {(1 - p_{i})}^{1 - y}}, i = 1, \dots, l;$

see the probability mass function in (4) of NBBS.

We have the data $(π_{i j}, y_{i j}), j = 1, \dots, n_{i}, i = 1, \dots, l .$ Since the sampling units are independent, the likelihood function is given by

$\prod_{i =1}^{l} \prod_{j =1}^{n_{i}} P (π_{i j} = π_{i u}^{*}, Y_{i j} = y_{i j} | δ_{i j} = 1, θ_{i}, p) = \prod_{i =1}^{l} \prod_{j =1}^{n_{i}} \frac{π_{i u}^{*} θ_{i u y_{i j}} p_{i}^{y_{i j}} {(1 - p_{i})}^{1 - y_{i j}}}{\sum_{y_{i j} = 0}^{1} \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u y_{i j}} p_{i}^{y_{i j}} {(1 - p_{i})}^{1 - y_{i j}}},$

where $π_{i u}^{*}, u = 1, \dots, U_{i}, i = 1, \dots, l$ are known. The likelihood function can be rewritten as

$\begin{array}{l} P (y, π | θ, p) & = \frac{\prod_{i =1}^{l} \prod_{j =1}^{n_{i}} π_{i u}^{*} θ_{i u y_{i j}} \prod_{i =1}^{l} \prod_{j =1}^{n_{i}} p_{i}^{y_{i j}} {(1 - p_{i})}^{1 - y_{i j}}}{\prod_{i =1}^{l} \prod_{j =1}^{n_{i}} \sum_{y_{i j} = 0}^{1} \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u y_{i j}} p_{i}^{y_{i j}} {(1 - p_{i})}^{1 - y_{i j}}} \\ = \frac{\prod_{i =1}^{l} \prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 0})}^{g_{i u 0}} \prod_{i =1}^{l} \prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 1})}^{g_{i u 1}} \prod_{i =1}^{l} p_{i}^{s_{i}} {(1 - p_{i})}^{n_{i} - s_{i}}}{\prod_{i =1}^{l} {[p_{i} \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 1} + (1 - p_{i}) \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 0}]}^{n_{i}}}, \end{array}$

where $s_{i} = \sum_{j =1}^{n_{i}} y_{i j},$ $g_{i u 0}$ is the cell count for category $u$ at $y = 0$ and $g_{i u 1}$ is the cell count for category $u$ at $y = 1$ under the area $i .$ Note that $\sum_{u =1}^{U_{i}} g_{i u 0} = n_{i} - s_{i},$ $\sum_{u =1}^{U_{i}} g_{i u 1} = s_{i}$ and $\sum_{u =1}^{U_{i}} (g_{i u 0} + g_{i u 1}) = n_{i} .$ This likelihood includes the selection bias.

Let $a_{i y} = \sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u y}, y = 0, 1, i = 1, \dots, l .$ Differences between $a_{i 0}$ and $a_{i 1}$ for some $i$ indicate that there is selection bias. The likelihood function can be expressed as

$P (y, π | θ, p) = \prod_{i =1}^{l} \prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 0})}^{g_{i u 0}} \prod_{i =1}^{l} \prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 1})}^{g_{i u 1}} \prod_{i =1}^{l} \frac{p_{i}^{s_{i}} {(1 - p_{i})}^{n_{i} - s_{i}}}{{[a_{i 1} p_{i} + a_{i 0} (1 - p_{i})]}^{n_{i}}} .$

We make a one-to-one transformation from $p_{i}$ to $q_{i}$ via

$q_{i} = \frac{a_{i 1} p_{i}}{a_{i 1} p_{i} + a_{i 0} (1 - p_{i})} .$

Note that if $θ_{i u y}$ does not depend on $y,$ $q_{i}$ and $p_{i}$ are the same. In this case the likelihood would be the same as the ignorable case. Let $q = (q_{1}, \dots, q_{ℓ}) .$ Then the likelihood function can be expressed as

$P (y, π | θ, q) = \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 0})}^{g_{i u 0}}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 0})}^{n_{i} - s_{i}}} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 1})}^{g_{i u 1}}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 1})}^{s_{i}}} \prod_{i =1}^{l} q_{i}^{s_{i}} {(1 - q_{i})}^{n_{i} - s_{i}} .$

We assume that $q,$ $θ_{i 0},$ $θ_{i 1}$ are independent, and we take

$q_{i} | μ, τ_{1} \overset{iid}{~} Beta (μ τ_{1}, (1 - μ) τ_{1}), i = 1, \dots, l,$

$θ_{i 0} | τ_{0} \overset{ind}{~} Dirichlet (θ_{i 0}^{( 0 )} τ_{0}) and θ_{i 1} | τ_{0} \overset{ind}{~} Dirichlet (θ_{i 1}^{( 0 )} τ_{0}),$

where $θ_{i 0}^{( 0 )}$ and $θ_{i 1}^{( 0 )}$ are to be specified. Recall that $x | μ, τ ~ Dirichlet (μ τ)$ has the density $f (x | μ, τ) = \frac{\prod_{i =1}^{k} x_{i}^{μ_{i} τ - 1}}{D (μ τ)},$ $0 < x_{i} < 1,$ $\sum_{i =1}^{k} x_{i} = 1,$ where $D (μ τ) = \prod_{i =1}^{k} Γ (μ_{i} τ) / Γ ( τ ),$ $0 < μ_{i} < 1,$ $\sum_{i =1}^{k} μ_{i} = 1,$ and $τ > 0.$

Finally, a priori we assume

$p (μ, τ_{0}, τ_{1}) = \frac{1}{{(1 + τ_{0})}^{2}} \frac{1}{{(1 + τ_{1})}^{2}}, 0 < μ < 1, τ_{0}, τ_{1} \geq 0.$

Of course one can use a half Cauchy density but these are very similar. This latter prior is used to avoid the difficulties associated with improper priors of the form $p ( τ ) \propto 1 / τ$ (e.g., see Gelman, 2006).

Hence, the joint prior density of $q,$ $θ_{10}, \dots, θ_{l 0},$ $θ_{11}, \dots, θ_{l 1},$ $μ,$ $τ_{0},$ $τ_{1}$ is

$\begin{array}{l} π (q, θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1}) & \propto \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 0}^{θ_{i u 0}^{( 0 )} τ_{0} - 1}}{D (θ_{i 0}^{( 0 )} τ_{0})} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 1}^{θ_{i u 1}^{( 0 )} τ_{0} - 1}}{D (θ_{i 1}^{( 0 )} τ_{0})} \\ \times \frac{\prod_{i =1}^{l} q_{i}^{μ τ_{1} - 1} {(1 - q_{i})}^{(1 - μ) τ_{1} - 1}}{{[B (μ τ_{1}, (1 - μ) τ_{1})]}^{l}} \frac{1}{{(1 + τ_{0})}^{2}} \frac{1}{{(1 + τ_{1})}^{2}} . \end{array}$

Using Bayes’ theorem, the joint posterior density of $q,$ $θ_{10}, \dots, θ_{l 0},$ $θ_{11}, \dots, θ_{l 1},$ $μ,$ $τ_{0},$ $τ_{1},$ given the data $y,$ is

$\begin{array}{l} π (q, θ_{10}, \dots, θ_{ℓ 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1} | y) & \propto \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 0})}^{g_{i u 0}}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 0})}^{n_{i} - s_{i}}} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} {(π_{i u}^{*} θ_{i u 1})}^{g_{i u 1}}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 1})}^{s_{i}}} \\ \times \prod_{i =1}^{l} q_{i}^{s_{i}} {(1 - q_{i})}^{n_{i} - s_{i}} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 0}^{θ_{i u 0}^{( 0 )} τ_{0} - 1}}{D (θ_{i 0}^{( 0 )} τ_{0})} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 1}^{θ_{i u 1}^{( 0 )} τ_{0} - 1}}{D (θ_{i 1}^{( 0 )} τ_{0})} \\ \times \frac{\prod_{i =1}^{l} q_{i}^{μ τ_{1} - 1} {(1 - q_{i})}^{(1 - μ) τ_{1} - 1}}{{[B (μ τ_{1}, (1 - μ) τ_{1})]}^{l}} \frac{1}{{(1 + τ_{0})}^{2}} \frac{1}{{(1 + τ_{1})}^{2}} . \end{array}$

To improve the computations, we use the more optimal Rao-Blackwellization to get the posterior density of $q$ (hence $p) .$ Given the data, the joint posterior density can be expressed as

$π (q, θ, μ, τ_{0}, τ_{1} | y) = π (q | θ, μ, τ_{0}, τ_{1}, y) π (θ, μ, τ_{0}, τ_{1} | y).$

3.2 Computations

By integrating out $q$ from the joint posterior of $q,$ $θ_{10}, \dots, θ_{l 0},$ $θ_{11}, \dots, θ_{l 1},$ $μ,$ $τ_{0},$ $τ_{1}$ given $y,$ we get the marginal joint posterior density of $θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1}$ given $y,$

$\begin{array}{l} π (θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1} | y) & \propto \frac{\prod_{i =1}^{l} B (s_{i} + μ τ_{1}, n_{i} - s_{i} + (1 - μ) τ_{1})}{{[B (μ τ_{1}, (1 - μ) τ_{1})]}^{l}} \\ \times \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 0}^{g_{i u 0} + θ_{i u 0}^{( 0 )} τ_{0} - 1}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 0})}^{n_{i} - s_{i}}} \prod_{i =1}^{l} \frac{\prod_{u =1}^{U_{i}} θ_{i u 1}^{g_{i u 1} + θ_{i u 1}^{( 0 )} τ_{0} - 1}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 1})}^{s_{i}}} \\ \times \frac{1}{\prod_{i =1}^{l} D (θ_{i 0}^{( 0 )} τ_{0})} \frac{1}{\prod_{i =1}^{l} D (θ_{i 1}^{( 0 )} τ_{0})} \frac{1}{{(1 + τ_{0})}^{2}} \frac{1}{{(1 + τ_{1})}^{2}} . \end{array}$

The conditional posterior density of $q$ is given by

$π (q | θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1}, y) \propto \prod_{i =1}^{l} q_{i}^{s_{i} + μ τ_{1} - 1} {(1 - q_{i})}^{n_{i} - s_{i} + (1 - μ) τ_{1} - 1},$

which we can sample directly.

We obtain a sample of $q$ in the following manner: We draw each element of $(θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1})$ from the conditional posterior density of $θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1} | y$ using the Metropolis-Hastings algorithm and a grid method, and then we draw $q$ from the conditional posterior density of $q | θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1}, y .$

We have monitored the convergence of the Metropolis-Hastings sampler using trace plots, autocorrelation plots and Geweke test of stationarity, which showed satisfactory performance.

The conditional posterior densities needed to execute the Metropolis-Hastings sampler are

$π (θ_{i 0} | θ_{j 0}, j \neq i, θ_{i 1}, i = 1, \dots, l, μ, τ_{0}, τ_{1}, y) \propto \frac{\prod_{u = 1}^{U_{i}} θ_{i u 0}^{g_{i u 0} + θ_{i u 0}^{( 0 )} τ_{0} - 1}}{{(\sum_{u = 1}^{U_{i}} π_{i u}^{*} θ_{i u 0})}^{n_{i} - s_{i}}}, i = 1, \dots, l,$

$π (θ_{i 1} | θ_{j 1}, j \neq i, θ_{i 0}, i = 1, \dots, l, μ, τ_{0}, τ_{1}, y) \propto \frac{\prod_{u = 1}^{U_{i}} θ_{i u 1}^{g_{i u 1} + θ_{i u 1}^{( 0 )} τ_{0} - 1}}{{(\sum_{u = 1}^{U_{i}} π_{i u}^{*} θ_{i u 1})}^{s_{i}}}, i = 1, \dots, l,$

$π (μ | θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, τ_{0}, τ_{1}, y) \propto \frac{\prod_{i =1}^{l} B (s_{i} + μ τ_{1}, n_{i} - s_{i} + (1 - μ) τ_{1})}{{[B (μ τ_{1}, (1 - μ) τ_{1})]}^{l}},$

and transforming $τ_{0}$ and $τ_{1}$ to respectively $ρ_{0} = 1 / (1 + τ_{0})$ and $ρ_{1} = 1 / (1 + τ_{1}),$

$π (ρ_{0} | θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{1}, y) \propto {[\prod_{i = 1}^{l} \frac{\prod_{u = 1}^{U_{i}} θ_{i u 0}^{θ_{i u 0}^{( 0 )} τ_{0} - 1}}{D (θ_{i 0}^{( 0 )} τ_{0})} \prod_{i =1}^{l} \frac{\prod_{u = 1}^{U_{i}} θ_{i u 1}^{θ_{i u 1}^{( 0 )} τ_{0} - 1}}{D (θ_{i 1}^{( 0 )} τ_{0})}]}_{τ_{0} = (1 - ρ_{0}) / ρ_{0}},$

$π (ρ_{1} | θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, ρ_{0}, y) \propto {[\frac{\prod_{i =1}^{l} B (s_{i} + μ τ_{1}, n_{i} - s_{i} + (1 - μ) τ_{1})}{{[B (μ τ_{1}, (1 - μ) τ_{1})]}^{l}}]}_{τ_{1} = (1 - ρ_{1}) / ρ_{1}} .$

In above formula, the $θ_{i 0},$ $θ_{i 1}$ and $μ$ are conditionally independent.

We use Metropolis steps to sample $θ_{i 0}, i = 1, \dots, l$ and $θ_{i 1}, i = 1, \dots, l$ from conditional distributions $π (θ_{i 0} | θ_{j 0}, j \neq i, θ_{i 1}, i = 1, \dots, l, μ, τ_{0}, τ_{1}, y)$ and $π (θ_{i 1} | θ_{j 1}, j \neq i, θ_{i 0}, i = 1, \dots, l, μ, τ_{0}, τ_{1}, y),$ respectively. Let $τ_{i 0}^{*} = n_{i} - s_{i} + τ_{0},$ $θ_{i u 0}^{*} = (g_{i u 0} + θ_{i u 0}^{( 0 )} τ_{0}) / τ_{i 0}^{*},$ $τ_{i 1}^{*} = s_{i} + τ_{0},$ and $θ_{i u 1}^{*} = (g_{i u 1} + θ_{i u 1}^{( 0 )} τ_{0}) / τ_{i 1}^{*} .$ For notational convenience, let $θ_{i 0}^{*} = (θ_{i 10}^{*}, \dots, θ_{i U_{i} 0}^{*})$ and $θ_{i 1}^{*} = (θ_{i 11}^{*}, \dots, θ_{i U_{i} 1}^{*}) .$ We choose overdispersed Dirichlet distributions as proposal densities for $θ_{i 0}$ and $θ_{i 1}, i = 1, \dots, l .$ In fact, the proposal densities are $Dirichlet (θ_{i 0}^{*} {\tilde{τ}}_{i 0})$ and $Dirichlet (θ_{i 1}^{*} {\tilde{τ}}_{i 1}),$ where ${\tilde{τ}}_{i 0} = τ_{i 0}^{*} / κ_{i 0}$ and ${\tilde{τ}}_{i 1} = τ_{i 1}^{*} / κ_{i 1},$ for all $i = 1, \dots, l .$ Note that as the $κ_{i 0}$ and $κ_{i 1}$ increase, the dispersion tends to increase in the Dirichlet distribution.

Assuming that the Metropolis-Hastings sampler is at $θ_{i 0}^{( r )},$ then the probability of accepting $θ_{i 0}^{(r + 1)}$ is

$U_{r, r + 1}^{( 0 )} = min {\frac{ψ (θ_{i 0}^{(r + 1)}, y)}{ψ (θ_{i 0}^{( r )}, y)}, 1},$

where $ψ (θ_{i 0}, y) = \frac{\prod_{u =1}^{U_{i}} θ_{i u 0}^{θ_{i u 0}^{*} τ_{i 0}^{*} (1 - 1 / κ_{i 0})}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 0})}^{n_{i} - s_{i}}},$ for all $i = 1, \dots, l .$ Also, if we assume that the Metropolis-Hastings sampler is at $θ_{i 1}^{( r )},$ then the probability of accepting $θ_{i 1}^{( r + 1 )}$ is

$U_{r, r + 1}^{( 1 )} = min {\frac{ψ (θ_{i 1}^{(r + 1)}, y)}{ψ (θ_{i 1}^{( r )}, y)}, 1},$

where $ψ (θ_{i 1}, y) = \frac{\prod_{u =1}^{U_{i}} θ_{i u 1}^{θ_{i u 1}^{*} τ_{i 1}^{*} (1 - 1 / κ_{i 1})}}{{(\sum_{u =1}^{U_{i}} π_{i u}^{*} θ_{i u 1})}^{s_{i}}},$ for all $i = 1, \dots, l .$

The Metropolis step is obtained as follows: Assume the Markov chain is at $θ_{i 0}^{( r)},$ a random vector $θ_{i 0}^{( r + 1 )}$ is drawn from the proposal density with properly chosen $κ_{i 0},$ and $U_{r, r + 1}^{( 0 )}$ is computed. A random uniform deviate $U$ in $[0, 1]$ is drawn, and if $U \leq U_{r, r + 1}^{( 0 )},$ then random vector $θ_{i 0}^{( r + 1 )}$ is accepted, otherwise the chain stays at $θ_{i 0}^{( r )} .$ This algorithm is applied for all $i = 1, \dots, l .$ The Metropolis step is utilized in a manner similar to that for $θ_{i 1}$ for all $i = 1, \dots, l .$

We generate $μ,$ $ρ_{0}$ and $ρ_{1},$ using the grid method; see CNK. Once we get the sample from the posterior $π (q, θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1} | y),$ by retransforming from $q_{i}$ to $p_{i},$

$p_{i} = \frac{a_{i 0} q_{i}}{a_{i 0} q + a_{i 1} (1 - q_{i})},$

we can draw a sample from the posterior distribution of $π (p, θ_{10}, \dots, θ_{l 0}, θ_{11}, \dots, θ_{l 1}, μ, τ_{0}, τ_{1} | y) .$

Once $p$ is estimated, we draw the entire finite population values, $y_{i 1}, \dots, y_{i N_{i}},$ independently from $Bernoulli (p_{i}),$ $i = 1, \dots, l .$ This is surrogate sampling (e.g., Nandram, 2007). So we have corrected the observed biased sample and replaced it by a surrogate sample for $p$ that we obtained from the heterogeneous nonignorable selection model. We can obtain a sample of $P_{i}$ by drawing $\sum_{i =1}^{N_{i}} y_{i j}$ from $Binomial (N_{i}, p_{i})$ and by dividing the result by $N_{i}$ for all $i = 1, \dots, l .$

The selection mechanism is similar to the missing data mechanism. So it is possible to incorporate missing data into our framework, or independently (i.e., on its own) we can assume that the missing data are “missing not at random” and a nonignorable nonresponse model can be used to adjust a population model, see Nandram and Choi (2010).

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Bayesian predictive inference of small area proportions under selection bias
Section 3. Heterogeneous nonignorable selection model

3.1 Methodology

3.2 Computations

Bayesian predictive inference of small area proportions under selection bias Section 3. Heterogeneous nonignorable selection model

3.1 Methodology

3.2 Computations

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Bayesian predictive inference of small area proportions under selection bias
Section 3. Heterogeneous nonignorable selection model