Are We Underestimating Food Insecurity? Partial Identification with a Bayesian 4-Parameter IRT Model

Abstract

This paper addresses measurement error in food security in the USA. In particular, it uses a Bayesian 4-parameter IRT model to look at the likelihood of over- or under-reporting of the conditions that comprise the food security module (FSM), the data collection administered in many US surveys to assess and monitor food insecurity. While this model’s parameters are only partially identified, we learn about the likely values of these parameters by using a Bayesian framework. My results suggest significant under-reporting of more severe food security items, particularly those in the child module. I find no evidence of over-reporting of food hardships. I show that, under conservative assumptions, this model predicts food insecurity prevalence between 1 and 3 percentage points higher than current estimates, or roughly 4 to 15 percent of prevalence, for the years 2007–2015. Results suggest much larger increases—on the order of 50 percent of prevalence—for very low food security among households that were screened into the food security module.

Appendices

Appendix A: Items in Food Security Module

The items in the FSM are listed below, along with the variable names used in tables to denote them.

Appendix B: Estimation Details

1.1 B.I Statement of the Problem

As mentioned in the text, the conditional distribution of Y_ij given W_ij shows the indirect relationship between $\theta _{i}$ and Y_ij:

$$ P(Y_{ij}|W_{ij}) = \left\{\begin{array}{ll} \delta_{j}^{Y_{ij}}(1-\delta_{j})^{1-Y_{ij}} \text{if} W_{ij}=0 \\ \tau_{j}^{1-Y_{ij}}(1-\tau_{j})^{Y_{ij}} \text{if} W_{ij}=1 \end{array} \right. $$

In the 4PL, W_ij assigns some probability to a misreport of food hardship, captured in the parameters δ and τ. The probability of an over-report of food hardship is $P(Y_{ij}=1|W_{ij}=0) = \delta _{j}$, while the probability of an under report is $P(Y_{ij}=0|W_{ij}=1) = \tau _{j}$.

The full model implied by this discussion is as follows:

$$ \begin{array}{@{}rcl@{}} Y_{ij}|W_{ij},\delta_{j}, \tau_{j} &\sim & \left\{\begin{array}{ll} \text{Bernoulli}(\delta_{j}), W_{ij}=0 \\ \text{Bernoulli}(1-\tau_{j}), W_{ij} = 1 \end{array} \right. \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} W_{ij} &=& \mathbf{I}(Z_{ij}>0) \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} Z_{ij} &\sim & N(\nu_{ij},1), \nu_{ij} = (\theta_{i}-\beta_{j}) \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} \theta_{i} & \sim & N(\mu_{\theta},\sigma_{\theta}^{2}) \end{array} $$

(14)

$$ \begin{array}{@{}rcl@{}} (\beta_{j}) & \sim & N(\mathbf{\mu}_{\beta},\mathbf{\Sigma}_{\beta}) \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} \pi(\delta_{j}, \tau_{j}) & \propto & \delta_{j}^{a_{\delta}-1}(1-\delta_{j})^{b_{\delta}-1} \tau_{j}^{a_{\tau}-1}(1-\tau_{j})^{b_{\tau}-1}\mathbf{I}((\delta_{j},\tau_{j}) \in {\Omega}) , \end{array} $$

(16)

where I is an indicator function and Ω is the support of τ_j and δ_j. The only difference between this model and that in Culpepper (2016) is that we constrain α_j = 1∀j. (See Eqs. 25 and 27, below.) As mentioned above, the variable W_ij is a discrete latent variable that relies on the value of the continuous latent variable Z_ij, which is partially identified by the most severe question affirmed and the number of questions affirmed. Equations 12 and 13 describe the relationship between W_ij and ν_ij as a probit model. We employ the normalizing assumption that 𝜃_i has a standard normal distribution.

The posterior distribution that we are interested in is as follows:

$$ p(\mathbf{Z},\mathbf{W},\mathbf{\theta},\mathbf{\gamma},\mathbf{\delta},\mathbf{\tau}) \propto p(\mathbf{Y}|\mathbf{W},\mathbf{\delta},\mathbf{\tau})p(\mathbf{Z},\mathbf{W}|\mathbf{\theta},\mathbf{\beta})p(\theta)p(\beta)p(\delta,\tau) $$

(17)

In a way similar to Y, we write W = (W₁...W_J) and $\mathbf {W_{j}}^{\prime } = (W_{1j} ... W_{Nj}$. The conditional probability of observing Y given W is

$$ \begin{array}{@{}rcl@{}} p(\mathbf{Y}|\mathbf{W},\delta,\tau) & = & \prod\limits_{i=1}^N \prod\limits_{j=1}^{J}\left[(1-\tau_{j})^{W_{ij}} \delta_{j}^{(1-W_{ij})}\right]^{Y_{ij}}\left[\tau_{j}^{W_{ij}} (1-\delta_{j})^{(1-W_{ij})} \right]^{(1-Y_{ij})} \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} & = & \prod\limits_{j=1}^{J}\tau_{j}^{\hat{n}_{10j}} (1-\tau_{j})^{\hat{n}_{11j}} \delta_{j}^{\hat{n}_{01j}} (1-\delta_{j})^{\hat{n}_{00j}}, \end{array} $$

(19)

where $\hat {n}_{00j}, \hat {n}_{01j}, \hat {n}_{10j}, \text {and} \hat {n}_{11j},$ are counts produced by the cross tabulation of W_ij and Y_ij. For example, $\hat {n}_{00j}$ is the count of respondents for whom the latent variable W and the observed response Y are both zero – true negatives.

The joint prior for the latent variables Z and W conditioned on the 2PM parameters γ can be thought of as the density for Z truncated according to the value of W:

$$ p(\mathbf{Z,W}|\theta,\gamma) = \prod\limits_{i=1}^{N}\prod\limits_{j=1}^{J}\phi(Z_{ij};\nu_{ij},1)[\mathbf{I}(Z_{ij}<0)\mathbf{I}(W_{ij}=0)+\mathbf{I}(Z_{ij}\geq 0)\mathbf{I}(W_{ij}=1)] $$

(20)

Once Z is integrated out, the prior distribution for Z given W is

$$ p(\mathbf{Z}|\mathbf{W},\mathbf{\theta},\mathbf{\beta}) = \frac{p(\mathbf{Z,W}|\theta,\beta)}{p(\mathbf{W}|\mathbf{\theta,\beta})}= \prod\limits_{i=1}^{N}\prod\limits_{j=1}^{J}\phi(Z_{ij};\nu_{ij},1)\left[\frac{\mathbf{I}(Z_{ij}<0)}{1-{\Phi}(\nu_{ij})}\right]^{(1-W_{ij})} \left[\frac{\mathbf{I}(Z_{ij} \geq 0)}{\Phi(\nu_{ij})}\right]^{W_{ij}}, $$

(21)

which is the full conditional prior of Z, given W.

1.2 B.II Sampling Distributions

The full conditional distributions of the parameters are as follows.

$$ \begin{array}{@{}rcl@{}} W_{ij}|Y_{ij}, \nu_{ij}, \delta_{j}, \tau_{j} &\sim & \left\{\begin{array}{ll} \text{Bernoulli}\left( \frac{\tau_{j}{\Phi}(\nu_{ij})}{1-\delta_{j}-(1-\tau_{j}-\delta_{j}){\Phi}(\nu{ij})}\right), Y_{ij}=0 \\ \text{Bernoulli}\left( \frac{(1-\tau_{j}){\Phi}(\nu_{ij})}{\delta_{j}+(1-\tau_{j}-\delta_{j}){\Phi}(\nu{ij})}\right),Y_{ij}=1 \end{array} \right. \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} Z_{ij}|W_{ij}, \nu_{ij} &\sim & \left\{\begin{array}{ll} N(\nu_{ij},1)\mathbf{I}(Z_{ij}\leq 0), W_{ij} = 0 \\ N(\nu_{ij},1)\mathbf{I}(Z_{ij} > 0), W_{ij} = 1 \end{array} \right. \end{array} $$

(23)

$$ \begin{array}{@{}rcl@{}} \theta_{i}|\mathbf{Z}_{i},\boldsymbol{\varepsilon} &\sim& N \left( \frac{(\mathbf{Z}_{i}+\beta)+\mu_{\theta}\sigma_{\theta}^{-2}}{\sigma_{\theta}^{-2}},\frac{1}{\sigma_{\theta}^{-2}} \right) \end{array} $$

(24)

$$ \begin{array}{@{}rcl@{}} (\beta_{j})|\mathbf{Z}^{\prime}_{j} & \sim & N(\boldsymbol{\mu}^{*}_{\beta},\boldsymbol{\Sigma}_{\beta}^{*}) \end{array} $$

(25)

$$ \begin{array}{@{}rcl@{}} \mathbf{Z}_{i} & = & (Z_{1j}...Z_{Nj}) \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \mathbf{X}_{\theta} & = & (\mathbf{-1_{N}}) \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\Sigma}_{\varepsilon}^{*} &=& \left( \mathbf{X}_{\theta}^{\prime}\mathbf{X}_{\theta} + \boldsymbol{\Sigma}_{\varepsilon}^{-1}\right)^{-1} \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\mu}_{\varepsilon}^{*}&=& \boldsymbol{\Sigma}_{\varepsilon}^{*}\left( \mathbf{X}_{\theta}^{\prime}\mathbf{Z}_{j} + \boldsymbol{\Sigma}_{\varepsilon}^{-1}\boldsymbol{\mu}_{\varepsilon} \right) \end{array} $$

(29)

$$ \begin{array}{@{}rcl@{}} p(\delta_{j}|\mathbf{Y}_{j}, \mathbf{W}_{j}) & \propto & \delta_{j}^{\hat{a}_{\delta}-1} (1-\delta_{j}^{\hat{b}_{\delta}-1}) \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} p(\tau_{j}|\mathbf{Y}_{j}, \mathbf{W}_{j}) & \propto & \tau_{j}^{\hat{a}_{\tau}-1} (1-\tau_{j}^{\hat{b}_{\tau}-1}) \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} \hat{a}_{\delta} & = & \hat{n}_{01j}+ a_{\delta}, \hat{b}_{\delta} =\hat{n}_{00j}+b_{\delta} \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} \hat{a}_{\tau} & = & \hat{n}_{10j}+ a_{\tau}, \hat{b}_{\tau} =\hat{n}_{11j}+b_{\tau} \end{array} $$

(33)