Assessing the dimensionality of food-security measures

Abstract

We explore the dimensionality of the U.S. Department of Agriculture’s household food security survey module among households with children. Using a novel methodological approach to measuring food security, we find that there is multidimensionality in the module for households with children that is associated with the overall household, adult, and child dimensions of food security. Additional analyses suggest official estimates of food security among households with children are robust to this multidimensionality. However, we also find that accounting for the multidimensionality of food security among these households provides new insights into the correlates of food security at the household, adult, and child levels of measurement.

Keywords

Adult food insecurity bifactor model child food insecurity household food insecurity Item Response Theory multidimensionality

1. Introduction

Measures of food insecurity – defined as limited or uncertain access to adequate food because of a lack of money or other resources [1] – are increasingly being used as measures of well-being for households with children in empirical analyses. Examples of analyses based on these measures include assessing the extent and severity of food insecurity among households in the United States [2, 3] and internationally [4], understanding the causes and consequences of food insecurity [5, 6, 7], and evaluating the effectiveness of food and nutrition assistance programs [6, 8, 9, 10, 11, 12]. These measures are based on household’s responses to a series of 18 food hardship questions designed to capture the conditions and behaviors of households having difficulty meeting their basic food needs.

In the United States, information on household’s food insecurity is officially collected by the Household Food Security Survey Module (HFSSM) as part of the Current Population Survey Food Security Supplement (CPS-FSS). Since 1995, the U.S. Department of Agriculture (USDA) combines household’s responses to the HFSSM using a Rasch measurement model [13] that is related to Item Response Theory (IRT; van der Linden, [14]) to develop [15, 16] and continually calibrate the U.S. food security scale [17, 18, 19, 20]. Rasch measurement theory requires strict assumptions, in the form of restrictions on the measurement model, for achieving measurement invariance [21]. As pointed out by Wells and Hambleton [22], invariance is a key advantage of IRT models as compared to Classical Test Theory (CTT).

Under the Rasch measurement model, responses to the HFSSM derive from a single underlying latent variable, such as household food insecurity, and are assumed to be independent, conditional on the latent variable. These assumptions imply that each household’s measure of food insecurity can be ranked and compared using simple counts of the affirmed items (i.e., raw score.). USDA uses this property of the model to assign each household a food insecurity status [2, 15, 16]. Each food insecurity status category represents a meaningful range of food hardship and allows for comparisons of the severity of household’s food insecurity. Indicators of the household’s food insecurity status are used to officially assess the well-being of households. For example, 16.5 percent of households with children were food insecure at some time during the previous 12 months in 2016 [2].

Accurately estimating measures of food insecurity relies on the assumptions of the Rasch measurement model being satisfied (i.e., model-data fit). Previous studies have examined household-invariant calibration of the HFSSM [19, 20, 23], and household fit related to item-invariant measurement of households [18]. Early studies of the HFSSM focused on developing a unidimensional scale of household food insecurity that captures the household’s ability to meet its food consumption needs [15, 16], known as the central dimension of food insecurity [1]. More recent studies of the HFSSM have recognized the adult and child questions in the HFSSM may represent dimensions of food insecurity distinct from household food insecurity [23, 24, 25].

The HFSSM consists of ten questions referring to the food hardships of the household and adult members, and an additional eight questions that collect information on the food hardships of child members. As such, food insecurity among households with children, as measured by the HFSSM, may be best represented by more than one dimension of food insecurity. Bias will ensure for these measures if the degree of the empirical dimensionality of food insecurity detected is large enough to suggest a departure from unidimensionality under the Rasch measurement model, resulting in potential misclassification of the household’s food insecurity status based on the raw score. To increase the comparability of indicators of the household’s food insecurity, Nord and Coleman-Jensen [25] developed a new food security status classification system based on separate calibrations of the adult and child HFSSM questions, which are then used to assign household’s a new household-level food insecurity status based on their adult and child food insecurity statuses.

We improve upon prior studies of the measurement of food insecurity among households with children by considering the bifactor measurement model for food insecurity measurement. All our empirical analyses are based on data on households with children from the 2012–2016 CPS-FSS. The bifactor measurement model allows us to conceptualize food insecurity among households with children as multidimensional, depending on the households’ overall food insecurity and food insecurity among adult and child members. While other measurement models can address the multidimensionality of the HFSSM, we chose the bifactor measurement model because of its ability to recover measures of household, adult, and child food insecurity, which are useful for future research on food insecurity and because of the model’s computational advantages. Estimates of the prevalence of household-food insecurity remain the most policy relevant indicators of household’s food hardships and well-being. We exploit the fact that the unidimensional two-parameter (2-PL) measurement model is nested within the bifactor measurement model to formally test for dimensionality in the HFSSM for the first time.

In addition, we use information on the explained common variance (ECV) to construct versions of the HFSSM that are truly unidimensional. The sensitivity of estimates of the prevalence of household-level food insecurity to the dimensionality of the HFSSM is also considered by estimating trends in food insecurity among households with children between 2001 and 2016. Finally, we modify the bifactor measurement model to include latent regression models for household, adult, and child food insecurity to assess the extent to which our understanding of the correlates of food insecurity are affected by the dimensionality of the HFSSM. Examining the correlates of food insecurity this way offers several advantages over the methodology used in previous studies. First, this approach is more efficient since the latent regressions for household, adult, and child food insecurity are jointly estimated, rather than estimates in three separate models. Second, multidimensional modeling of food insecurity allows us to control for the confounding presence of household-level food insecurity in our regression analyses of adult and child food insecurity. Finally, aspects of our modeling approach (described below) allow us to formally test for differences in the correlates of adult and child food insecurity compared to household food insecurity.

2. Data

Data for our empirical analyses of food insecurity are from the 2012–2016 CPS-FSS. Each year the CPS-FSS is administered in December to households as part of the Current Population Survey (CPS). The CPS-FSS asks households questions about their food expenditure, basic food needs, and participation in public- and private-food and nutrition assistance programs.

Our empirical analyses examine responses to the 18 HFSSM questions used to construct the U.S. food security scale [2, 15] as dependent variables. These questions are designed to capture the conditions and behaviors characterizing households having difficulty meeting their food needs. For more information on the HFSSM, see [2, 15, 24]. Note, some of the questions include multiple response categories. USDA recommends researchers recode all the HFSSM questions dichotomously for their empirical analyses of food insecurity. Nord [19] assessed the impact of dichotomizing responses to the HFSSM, and he found that it had no significant effect on the U.S food security scale. We follow USDA’s recommendation for coding responses to the HFSSM here. The first ten HFSSM questions are related to issues of household and adult food insecurity, and the last eight questions relate to issues of food insecurity for children in the household. A complete listing of the questions, and the methods used to convert them into binary indicators, is in Appendix A.

Information on reference person and household characteristics commonly associated with food insecurity in households with children are included in our regression analyses of food insecurity. For a detailed conceptual motivation of these characteristics based on economic theory and review of the literature, see [9]. Our regression analyses include measures of the household’s income and assets. The CPS-FSS asks respondents to report their total household income categorically. We use categorical income to construct continuous measures of household income using the midpoints of the income ranges and adjust for inflation using the Bureau of Labor Statistics’ Consumer Price Index for Urban Consumers (CPI-U). We use household income to measure the household’s short-term economic resources. While the CPS has limited information on household assets, it does include information on homeownership of the household’s current residence. We include a measure of homeownership as a proxy variable for long-term economic resources.

The CPS also includes information on the reference person and household’s demographic, geographic, and economic characteristics. Our regression analyses also include measures of the respondent’s age (and age squared), race, ethnicity, nativity, and marital status. Information on adult (age 18 and older) household member’s employment status and educational attainment are used to construct categorical measures of the highest educational attainment in the household and the employment status of the primary wage earner. The analyses also include measures of the number of adults, children in the household; the presence of an elderly (age 60 or older) member, disabled member; the age of the youngest child; residence in an urban area; and region and year fixed effects.

We consider two samples of households with at least one child (under age 18) in our empirical analyses. Our first sample is used to assess the fit of several measurement models useful for measuring food insecurity. We follow common convention in the empirical literature and exclude households with income above 185 percent of the federal poverty line from our measurement sample [18, 19, 20]. Households in the CPS-FSS are administered the HFSSM if they have income below 185 percent of the federal poverty line, or if their income is above 185 percent of the federal poverty line and they show any signs of food insufficiency (i.e., food stress). In 2012–2016, only a fraction of all households with children and with income above the screening threshold were administered the HFSSM. Omitting households with higher income from our empirical measurement analyses has the benefit of mitigating bias associated with CPS-FSS screening [19]. After excluding households with extreme responses to the HFSSM, our measurement sample includes a total of 11,666 households.

Our second sample is used to explore the empirical associations between household, adult, and child food insecurity and a series of explanatory variables using regression analyses. We include all households with children, surveyed in the 2012–2016 CPS-FSS, who provided valid information to the questions used to construct the explanatory variables in our regression sample. The resulting sample includes a total of 53,964 households. Summary statistics for this sample are available in Appendix B.

3. Methodology

Item response theory models, including the Rasch, one-parameter (1-PL), and 2-PL measurement models, are used to define unidimensional scales [26]. The 1-PL measurement model is conceptually equivalent to the Rasch measurement model, albeit with a different scaling of the model parameters. For the purposes of our measurement analyses, we will consider the 1-PL and 2-PL measurement models when estimating unidimensional models of food insecurity. There are a variety of perspectives on the concept of dimensionality for measurement models in the social, behavioral, health, and economic sciences. In unidimensional models, a single latent variable is hypothesized to account for the correlations or covariances among a set of items. Multidimensional measurement models, such as the bifactor measurement model, hypothesize that more than one latent variable can account for the correlations or covariances among the items [27].

Reise [28] has described the rediscovery of the bifactor measurement model for examining data with certain substantive characteristics. Bifactor measurement models provide a perspective on construct-relevant multidimensionality that emerges when measures are constructed with several distributional components or domains. For example, the HFSSM has been successfully modeled by USDA with the unidimensional Rasch measurement model [2, 10, 11, 12, 15, 20], even though there are two distinct components or subsets of items that represent topics on adult and child behaviors related to food insecurity. It is quite common in the development of unidimensional scales to design items in this manner [29], but it does raise questions about whether meaningful subscales reflecting multidimensionality can be constructed.

Holzinger and Swineford [30] first introduced the bifactor method for conducting factor analysis. Factor analyses yield indeterminant solutions with researchers adding additional conditions and constraints to yield meaningful results [31]. One set of constraints is related to a simple structure as reflected in the bifactor measurement model [32]. Specifically, the original bifactor method focused on factor analyses of correlations between items in order to identify a simple structure that meets the following constraints: one general factor or latent variable that underlies all of the items, group factors underlying subsets of items, and the additional condition that each item would be represented only by two factors (the general and one group factor). The principles of simple structure for factor loadings underlying the bifactor method have also been applied to item factor analysis [28, 33].

The general form of the bifactor measurement model can be written as follows:

$\displaystyle y_{j}^{\ast}=\lambda_{j,\textit{Household}}\theta_{\textit{% Household}}+\lambda_{j,\textit{Adult}}\theta_{\textit{Adult}}+\lambda_{j,% \textit{Child}}\theta_{\textit{Child}}+\varepsilon_{j}$ (1)

where $y_{j}^{\ast}$ represents the outcome of a continuous underlying latent response process related to the factor loadings ( $\lambda$ ) on the three latent variables ( $\theta_{\textit{Household}}$ , $\theta_{\textit{Adult}}$ , and $\theta_{\textit{Child}}$ ), with a residual term ( $\varepsilon_{j}$ ). This model is essentially the multiple factor model proposed by Thurstone [32] and adapted by Bock and Aitkin [34] for item-level factor analyses with dichotomous items. For example, the factor loadings for the bifactor pattern with four items and three latent variables ( $\theta_{\textit{Household}}$ , $\theta_{\textit{Adult}}$ , and $\theta_{\textit{Child}}$ ) can be represented in matrix notation as

$\displaystyle\lambda=\left[\begin{array}[]{*{20}c}{\lambda_{1,\textit{% Household}}}&{\lambda_{1,\textit{Adult}}}&0\\ {\lambda_{2,\textit{Household}}}&{\lambda_{2,\textit{Adult}}}&0\\ {\lambda_{3,\textit{Household}}}&0&{\lambda_{3,\textit{Child}}}\\ {\lambda_{4,\textit{Household}}}&0&{\lambda_{4,\textit{Child}}}\\ \end{array}\right].$

All four items have loadings on the first factor (Household), the first two items also load on the second factor (Adult), and the last two items load on the third factor (Child).

The 1-PL and 2-PL measurement models are nested within the bifactor measurement model when specific constraints are imposed on the factor loadings. Since the 1-PL and 2-PL measurement models assume all responses to the items are related to a single underlying latent variable, we must assume the factor loadings for the group factors are set to zero, forcing any correlation between the items to be generated through the general (Household) factor. In addition, the factor loadings on the general factor must be constrained to a constant across the items, suggesting the items are equally informative on the underlying factor. While this may appear to be a strong assumption for food insecurity measurement, Hamilton et al. [16] found that most of the HFSSM items had similar factor loadings. Figure 1 shows the generalized path diagrams that represent the multidimensional bifactor measurement model in Panel A and the unidimensional 1-PL and 2-PL measurement models in Panel B.

Figure 1.

Generalized path diagrams for the bifactor, 1-PL, and 2-PL IRT logistic models for measuring food insecurity. Note. The GHFI latent variable represents general household food insecurity with the Adult latent variable based on the adult items and the Child latent variable based on the child items. The discrimination parameters, represented by the paths, are constrained to be equal under the 1-PL measurement model. This assumption is relaxed under the 2-PL and bifactor measurement models, where the discrimination parameters are free to vary by item.

It is well known that item factor loadings for bifactor models can also be described in terms of dichotomous multidimensional IRT (MIRT) logistic measurement models:

$\displaystyle P({y_{j}|\theta_{\textit{Household}},\theta_{\textit{Adult}},% \theta_{\textit{Child}}})=\frac{\exp({\alpha_{j,\textit{Household}}\theta_{% \textit{Household}}+\alpha_{j,\textit{Adult}}\theta_{\textit{Adult}}+\alpha_{j% ,\textit{Child}}\theta_{\textit{Child}}+c_{j}})}{1+\exp({\alpha_{j,\textit{% Household}}\theta_{\textit{Household}}+\alpha_{j,\textit{Adult}}\theta_{% \textit{Adult}}+\alpha_{j,\textit{Child}}\theta_{\textit{Child}}+c_{j}})}.$ (2)

In Eq. (2), $\alpha_{j,\textit{Household}}$ is the discrimination parameter for item $j$ on the Household (general) factor, $\alpha_{j,\textit{Adult}}$ and $\alpha_{j,\textit{Child}}$ are the discrimination parameters for item $j$ on the Adult and Child group factors, respectively, and $c_{j}$ is an intercept parameter reflecting the item easiness. There is a direct relationship between the parameters in the factor analytic and MIRT models. Equations (1) and (2) are conceptually equivalent, although there may be numerical advantages by estimating the parameters from a factor-analytic versus a MIRT perspective [35, 36]. Various IRT models, including the Rasch measurement model, are nested within the MIRT measurement models [27]. We report the results from both perspectives with statistical analyses performed in Stata 15 Multiprocessor 4-Core [37] on a Windows 10 Professional platform. The results were independently confirmed in Mplus [38] and the MIRT package in R [39] on a Windows 10 Professional platform, but these results are not reported here.

Bifactor measurement models offer several conceptual and numerical advantages over standard item factor analyses. First, there are two perspectives on bifactor measurement models: multidimensional IRT [27] and structural equation modeling (SEM; [40]). The second advantage of bifactor measurement models is that they reflect the structure of many constructs based on various distributional components for items that may reflect construct-relevant variation. These distributional components are sometimes scored as separate subtests. For example, the construct of food insecurity is represented by items that reflect facets of food insecurity related to both adults (10 items) and children (8 items). Finally, bifactor models can be used to create orthogonal latent variables [28, 41]. Using the uncorrelated estimation of $\theta$ for each latent variable offers advantages for these scores in subsequent analyses.

3.1 Incorporating a latent regression model into the bifactor measurement model

In our regression analyses of food insecurity, we estimate direct associations between household, adult, and child food insecurity and a series of explanatory variables by modifying the bifactor measurement model to include a latent regression model. For a general description of this methodology, see Wilson and De Boeck [42]. Prior studies have used a similar methodological approach to modeling food insecurity, but under the assumption the HFSSM is unidimensional [8, 10, 11, 12, 43]. Specifically, a latent regression model may be incorporated in the bifactor measurement model, described above, by re-expressing the latent food insecurity variables $\theta_{\textit{Household}}$ , $\theta_{\textit{Adult}}$ , and $\theta_{\textit{Child}}$ as

$\displaystyle\theta_{j}=X_{j}\beta_{j}+e_{j},\text{ where }j\in\{{\text{% Household, Adult, Child}}\}$ (3)

and $X_{j}$ is a matrix of observable explanatory variables hypothesized to be related to household, adult, and child food insecurity; $\beta_{j}$ is a vector of coefficients relating latent household, adult, and child food insecurity to the explanatory variables; and $e_{j}$ is the new vector of residual terms for household, adult, and child food insecurity after controlling for $X_{j}$ , and it is assumed to be standard normally distributed. For simplicity, we assume that the matrices of explanatory variables for each dimension of food insecurity ( $X_{j}$ ) perfectly overlap; however, this need not be satisfied. Future analyses of food insecurity using this model may wish to consider using different sets of explanatory variables for each dimension of food insecurity. Here, we are interested in reexamining the correlates of household, adult, and child food insecurity, and how they are affected by the multidimensionality of food insecurity implied by the HFSSSM for households with children.

Regression coefficients from our empirical analyses of food insecurity are like those found in prior studies with a few key exceptions. First, coefficients from the latent regression model for the general dimension represent direct associations between the household’s overall latent food insecurity and the explanatory variables. Therefore, direct comparisons, in terms of the direction of the associations, can be made with prior studies of the correlates of household food insecurity. However, our coefficients represent changes in latent food insecurity so comparisons of the magnitude of these associations cannot be made without the appropriate marginal effects [8, 10, 11, 12]. Second, interpreting coefficients for the specific dimensions of adult and child food insecurity should be undertaken with caution since these are discrepancy dimensions. These coefficients describe how associations between the adult and child dimensions of food insecurity and the explanatory variables differ from the general household dimension of food insecurity. Finally, we constrain the variances of the dimensions of food insecurity in all our regression analyses to unity. As a result, the coefficients may be interpreted in terms of the standard deviation of each dimension of food insecurity.

We estimated all our latent regression models of food insecurity using the maximum likelihood method with robust standard errors in Stata 15.1 Multiprocessor 4-Core on an Intel Core I7 CPU desktop running at 3.80 GHz with 128 GB of memory, running on a Windows 10 64-bit Professional operating system [37]. Results from the latent regression model parameters are presented below, and the measurement model parameters are listed in Appendix Table C.1.

4. Results

4.1 Food security measurement – estimates from 1-PL, 2-PL, and bifactor measurement models

We begin our empirical analyses of the measurement of food insecurity among households with children by comparing the fit of several measurement models. First, we examined the matrix of tetrachoric correlations based on the 18 items in the HFSSM. Our results indicate several negative values. Specifically, our examination of the bivariate distributions of the items indicates that three item pairs are not suitable for our measurement analyses of the HFSSM because of local dependencies. As a result, we removed the following items: the frequency follow-up for adult(s) cut or skipped meals, the frequency follow-up for adult(s) did not eat for a whole day, and the frequency follow-up for child(ren) skipped meals. Removing these items from our empirical analyses corrected the issue of negative eigenvalues we encountered in our preliminary analyses. Only the remaining 15 HFSSM items are considered further in our measurement and regression analyses of food insecurity, and we refer to the modified version of the HFSSM as the HFSSM-15.

It is important to note that the removal of these items does not affect item-parameter invariance and the comparability of our findings with prior studies of food insecurity among households with children. Table 1 provides non-centered and centered item locations for the HFSSM-15 from 1-PL, 2-PL, and bifactor measurement models. Item calibrations from the bifactor measurement model are presented with respect to the general dimension of household food insecurity, which is captured by the 1-PL and 2-PL measurement models. For details on how we calculated these item calibrations, see [28]. Alternatively, we could have calculated item calibrations for the specific dimensions as well. As expected, the item calibrations are robust to the multidimensionality of the HFSSM. Since the item calibrations are essentially transformations of the relative proportions of the affirmed HFSSM items, their lack of sensitivity is not surprising [44]. In addition, analyses of the HFSSM-15 yield measures that are comparable to measures obtained from the HFSSM based on the invariant measurement properties of the measurement model (Fig. 2).

In addition, we also conducted preliminary factor analyses of household’s responses to the HFSSM-15. Our results, which are not shown but available upon request, indicate there are up to three meaningful dimensions of food insecurity that relate household’s responses. The first factor is the strongest and appears to represent the household’s overall food insecurity, while the second factor appears to relate household’s responses to the adult items. The third factor is the weakest, but it appears to represent the child items.

Figure 2.

Parameter invariance of item calibrations of bifactor, 1-PL, and 2-PL IRT logistic measurement models compared to the USDA’s official standard item calibrations. Note. Item calibrations are centered with mean of zero and standard deviation of one. See Table 1 for original and re-scaled versions of the item calibrations from the 1-PL, 2-PL, and bifactor IRT logistic models. Official USDA item calibrations were obtained from Bickel et al. [15].

Table 1

Item-severity parameters from 1-PL, 2-PL, and bifactor IRT logistic measurement models of food insecurity

		Uncentered item difficulties			Centered item difficulties
Item number	Items	1-PL model	2-PL model	Bi factor model	1-PL model	2-PL model	Bi factor model
Household and adult HFSSM items
1	Worried food would run out	$-$ 1.361	$-$ 1.826	$-$ 1.884	$-$ 1.776	$-$ 2.126	$-$ 1.795
		(0.021)	(0.069)	(0.073)	(0.015)	(0.053)	(0.033)
2	Food bought would not last	$-$ 0.642	$-$ 0.688	$-$ 0.724	$-$ 1.265	$-$ 1.249	$-$ 1.269
		(0.018)	(0.024)	(0.025)	(0.013)	(0.019)	(0.011)
3	Could not afford to eat balanced meals	$-$ 0.288	$-$ 0.301	$-$ 0.378	$-$ 1.013	$-$ 0.952	$-$ 1.112
		(0.018)	(0.023)	(0.057)	(0.013)	(0.018)	(0.026)
4	Adult(s) cut size or skipped meals	0.557	0.506	1.177	$-$ 0.413	$-$ 0.330	$-$ 0.406
		(0.020)	(0.017)	(0.051)	(0.014)	(0.013)	(0.023)
5	Respondent ate less than felt should have	0.474	0.428	1.166	$-$ 0.471	$-$ 0.390	$-$ 0.411
		(0.020)	(0.016)	(0.060)	(0.014)	(0.012)	(0.027)
7	Respondent hungry but did not eat	1.247	1.063	2.631	0.078	0.100	0.254
		(0.025)	(0.021)	(0.097)	(0.018)	(0.016)	(0.044)
8	Respondent lost weight	1.814	1.615	3.121	0.481	0.525	0.477
		(0.030)	(0.033)	(0.106)	(0.021)	(0.025)	(0.048)
9	Adult(s) did not eat for whole day	2.166	1.875	3.674	0.731	0.725	0.727
		(0.035)	(0.038)	(0.121)	(0.025)	(0.029)	(0.055)
Child HFSSM items
11	Relied on low-cost foods for child(ren)	$-$ 0.192	$-$ 0.203	$-$ 0.239	$-$ 0.945	$-$ 0.876	$-$ 1.049
		(0.018)	(0.025)	(0.030)	(0.013)	(0.020)	(0.014)
12	Could not feed child(ren) balanced meals	0.521	0.597	1.021	$-$ 0.438	$-$ 0.259	$-$ 0.477
		(0.020)	(0.023)	(0.051)	(0.014)	(0.018)	(0.023)
13	Child(ren) not eating enough	1.417	1.402	2.866	0.199	0.361	0.361
		(0.026)	(0.033)	(0.150)	(0.019)	(0.025)	(0.068)
14	Cut size of child(ren)s meals	2.041	1.839	3.270	0.643	0.698	0.544
		(0.034)	(0.040)	(0.104)	(0.024)	(0.031)	(0.047)
15	Child(ren) hungry	2.435	1.940	4.495	0.922	0.775	1.100
		(0.041)	(0.038)	(0.275)	(0.029)	(0.030)	(0.125)
16	Child(ren) skipped meals	2.857	2.303	4.802	1.222	1.055	1.240
		(0.050)	(0.057)	(0.207)	(0.035)	(0.044)	(0.094)
18	Child(ren) not eat for whole day	4.012	3.455	6.033	2.043	1.943	1.798
		(0.106)	(0.206)	(0.746)	(0.075)	(0.159)	(0.339)
	Mean	1.137	0.934	2.069	0.000	0.000	0.000
	Standard deviation	1.407	1.298	2.200	1.000	1.000	1.000
	Number of households	11,666	11,666	11,666	11,666	11,666	11,666

Returning to Table 1, we find that the item difficulty parameters, when ordered by increasing severity of food insecurity, are consistent with prior studies of the measurement of food insecurity [15, 19, 20]. Although, modest differences in the ordering of the items respondent ate less than felt should have, could not feed child(ren) balanced meals, and adult(s) cut or skipped meals based on the 2-PL and bifactor measurement models are observed.

Table 2 presents estimates of item discrimination and factor loading parameters from the 1-PL, 2-PL, and bifactor measurement models. The 1-PL measurement model includes an estimate of the average item-level discrimination (i.e., equal discrimination across all items), while the 2-PL and bifactor measurement models relax this assumption by allowing the item discrimination parameters to vary between items, and in the case of the bifactor measurement model across dimensions. Comparing the item discrimination parameters for the (general) household food insecurity dimension from the bifactor measurement model with the discrimination parameters from the 2-PL measurement model, we note that, with a few exceptions, the discrimination parameters are larger for the 2-PL measurement model. Similar comparisons between the bifactor and 1-PL measurement models indicate the discrimination parameters from the bifactor measurement model are larger than the average discrimination of the 1-PL measurement model. Exceptions to this include the items food bought would not last and relied on low-cost foods for child(ren).

The discrimination parameter for could not afford to eat balanced meals is negative and statistically indistinguishable from zero on the specific dimension of adult food insecurity from the bifactor measurement model. In addition, the bifactor measurement model’s discrimination parameters for the general dimension of household food insecurity are generally larger than those from the specific dimensions of adult and child food insecurity. Therefore, there is likely a single dimension of food insecurity underlying households with children’s responses to the HFSSM-15.

Interpreting the item discrimination parameters from the bifactor measurement model is challenging and potentially misleading because these parameters capture conditional relationships among the household, adult, and child dimensions of food insecurity. Because of this, it is common to appeal to the factor-analytic parameterization of the model since it provides a simpler interpretation of the parameter estimates. The corresponding factor loading parameters were obtained using the following transformation:

$\displaystyle\lambda_{jd}=\frac{\alpha_{jd}}{\sqrt{1.7^{2}+\sum_{d=1}^{D}{% \alpha_{jd}^{2}}}},$

Table 2

Item-discrimination and factor-loading parameterizations of the 1-PL, 2-PL, and bifactor IRT logistic measurement models of food insecurity

Household and adult HFSSM items
	Factor loading parameterization					IRT discrimination parameterization
	Bifactor model			1-PL model	2-PL model	Bifactor model			1-PL model	2-PL model
Item number	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Household food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Household food insecurity
1	0.262	0.702		0.734	0.546	0.671	1.801		1.839	1.108
	(0.042)	(0.030)		(0.003)	(0.018)	(0.102)	(0.132)		(0.019)	(0.052)
2	0.512	0.437		0.734	0.654	1.177	1.003		1.839	1.469
	(0.025)	(0.031)		(0.003)	(0.013)	(0.063)	(0.077)		(0.019)	(0.049)
3	0.827	$-$ 0.080		0.734	0.603	2.524	$-$ 0.245		1.839	1.283
	(0.045)	(0.075)		(0.003)	(0.013)	(0.490)	(0.263)		(0.019)	(0.043)
4	0.666	0.611		0.734	0.870	2.639	2.421		1.839	2.995
	(0.028)	(0.032)		(0.003)	(0.008)	(0.145)	(0.157)		(0.019)	(0.108)
5	0.673	0.638		0.734	0.889	3.067	2.907		1.839	3.297
	(0.028)	(0.029)		(0.003)	(0.007)	(0.185)	(0.184)		(0.019)	(0.124)
7	0.662	0.631		0.734	0.875	2.777	2.648		1.839	3.076
	(0.025)	(0.027)		(0.003)	(0.008)	(0.146)	(0.163)		(0.019)	(0.113)
8	0.594	0.638		0.734	0.813	2.063	2.217		1.839	2.375
	(0.028)	(0.029)		(0.003)	(0.010)	(0.118)	(0.146)		(0.019)	(0.088)
9	0.626	0.608		0.734	0.826	2.178	2.116		1.839	2.490
	(0.024)	(0.026)		(0.003)	(0.010)	(0.118)	(0.135)		(0.019)	(0.097)
Child HFSSM items
11	0.526		0.638	0.734	0.534	1.587		1.925	1.839	1.075
	(0.029)		(0.029)	(0.003)	(0.014)	(0.125)		(0.157)	(0.019)	(0.040)
12	0.764		0.466	0.734	0.678	2.907		1.774	1.839	1.569
	(0.022)		(0.028)	(0.003)	(0.012)	(0.179)		(0.119)	(0.019)	(0.052)
13	0.704		0.577	0.734	0.741	2.890		2.336	1.839	1.875
	(0.046)		(0.060)	(0.003)	(0.012)	(0.227)		(0.310)	(0.019)	(0.068)

Table 2, continued
	Factor loading parameterization					IRT discrimination parameterization
	Bifactor model			1-PL model	2-PL model	Bifactor model			1-PL model	2-PL model
Item number	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Household food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Household food insecurity
14	0.734		0.444	0.734	0.797	2.433		1.471	1.839	2.245
	(0.056)		(0.089)	(0.003)	(0.012)	(0.219)		(0.294)	(0.019)	(0.096)
15	0.845		0.315	0.734	0.885	3.321		1.238	1.839	3.229
	(0.065)		(0.141)	(0.003)	(0.011)	(0.473)		(0.495)	(0.019)	(0.182)
16	0.794		0.379	0.734	0.864	2.837		1.353	1.839	2.917
	(0.109)		(0.216)	(0.003)	(0.014)	(0.480)		(0.748)	(0.019)	(0.181)
18	0.658		0.531	0.734	0.802	2.093		1.691	1.839	2.285
	(0.213)		(0.304)	(0.003)	(0.032)	(0.573)		(1.118)	(0.019)	(0.258)
Num. of HHs	11,666			11,666	11,666	11,666			11,666	11,666

Note: Models estimated with variances constrained to unity using weighted data on low-income households with children from the 2012–2016 CPS-FSS. Standard errors for the factor-loading and discrimination parameters are in parenthesis. We used the methodology described in Reise [28] to transform the discrimination parameters into their corresponding factor-loading parameters and calculated their standard errors using the delta method. Table 1 provides a detailed description of the food security items and their correspondence with the item numbers in this table. HH $=$ households.

where $j$ is an item index, $d$ is a dimension index, $\lambda_{jd}$ is the factor loading parameter for item $j$ on dimension $d$ , and $\alpha_{jd}$ is the discrimination parameter for item $j$ on dimension $d$ . Returning to Table 2, we can see how important it is to use the factor loadings to compare the unidimensional 1-PL and 2-PL measurement models with the multidimensional bifactor measurement model. When comparing the factor loading parameters for the general dimension of household food insecurity, we find that only three items have factor loadings based on the bifactor measurement model that are larger in magnitude than the 1-PL and 2-PL measurement models. These items are could not afford balanced meals, could not feed child(ren) balanced meals, and child(ren) skipped meals.

Since the 1-PL and 2-PL measurement models are nested within each other and the bifactor measurement model the likelihood ratio test can be used to assess the overall fit of the bifactor measurement model relative to the 1-PL and 2-PL measurement models. Table 3 provides a summary of this, and other fit statistics, for all the measurement models. Based on the likelihood ratio test, the 1-PL measurement model is rejected in favor of the 2-PL measurement model, and the 2-PL measurement model is rejected in favor of the bifactor measurement model. All these tests are statistically significant at the one percent level or better. Similar conclusions about the model comparisons are reached when using the Akaike information criterion (AIC) and Bayesian information criterion (BIC).

Table 3

Model-fit statistics from the 1-PL, 2-PL, and bifactor IRT logistic measurement models of food insecurity

Model statistics	1-PL model	2-PL model	Bifactor model
Free parameters	16	30	45
Log-likelihood	$-$ 61,466.43	$-$ 60,484.24	$-$ 57,865.02
Likelihood ratio test of 1-PL vs. 2-PL model		1,964.40
		[0.000]
Likelihood ratio test of 2-PL vs. bifactor model			5,238.43
			[0.000]
AIC	122,964.9	121,028.5	115,820.0
BIC	123,082.7	121,249.4	116,151.4
Number of households	11,666	11,666	11,666

Note: Models estimated with variances constrained to unity using weighted data on low-income households with children from the 2012–2016 CPS-FSS. $P$ -values are in brackets. AIC $=$ Akaike Information Criterion, and BIC $=$ Bayesian Information Criterion.

4.2 Assessing the dimensionality of the HFSSM based on the bifactor measurement model

Up to this point, we have assessed the fit of the 1-PL, 2-PL, and bifactor measurement models to the HFSSM-15, and we have found evidence that supports the unidimensionality of the HFSSM for households with children. An additional benefit of using the bifactor measurement model to measure food insecurity in households with children is that the relative strength of the general dimension of household food insecurity relative to the specific dimensions of adult and child food insecurity may be examined. Comparing the dimensions of food insecurity provides additional insights into the behaviors and conditions related to food insecurity among households with children and can be used to select sets of items for which multidimensionality of the HFSSM is weak or negligible. When data conform to the bifactor measurement model assumptions, an index of the degree of unidimensionality may be constructed [28]. Specifically, the explained common variance (ECV) may be calculated for each dimension as follows:

$\displaystyle\textit{ECV}=\frac{\sum{\lambda_{\textit{Household}}^{2}}}{\sum{% \lambda_{\textit{Household}}^{2}}+\sum{\lambda_{\textit{Adult}}^{2}+\sum{% \lambda_{\textit{Child}}^{2}}}},$

where the $\lambda$ ’s are the factor loadings for the items on the household, adult, and child dimensions of food insecurity. The value of the ECV index of 0.610 for the general dimension of food insecurity listed at the bottom of Table 4 indicates this dimension captures roughly two-thirds of the common variance in responses to the HFSSM-15. Higher values of the ECV indicate a “stronger” general dimension relative to the specific dimensions in a bifactor measurement model, reinforcing our confidence that a unidimensional latent variable may represent the underlying construct being measured by the HFSSM-15. Unfortunately, no formal guidelines exist for ECV values. The prior literature suggests that ECV index values ranging from 0.20 [45] to 0.85 or higher are required for a set of items to be considered sufficiently unidimensional [46, 47, 48].

Table 4
Factor-loading parameters from the bifactor IRT logistic measurement model of household, adult, and child food insecurity

	Factor loading parameters				Marginal slope parameters
Items	Household food insecurity	Adult food insecurity	Child food insecurity	I-ECV	Household food insecurity	Adult food insecurity	Child food insecurity
Household and adult HFSSM items
Worried food would	0.262	0.702		0.122	0.171	0.460
run out	(0.042)	(0.030)			(0.027)	(0.022)
Food bought would	0.512	0.437		0.579	0.328	0.280
not last	(0.025)	(0.031)			(0.016)	(0.020)
Could not afford to	0.827	$-$ 0.080		0.991	0.557	$-$ 0.054
eat balanced meals	(0.045)	(0.075)			(0.042)	(0.052)
Adult(s) cut size or	0.666	0.611		0.543	0.462	0.424
skipped meals	(0.028)	(0.032)			(0.020)	(0.022)
Respondent ate less	0.673	0.638		0.527	0.473	0.448
than felt should have	(0.028)	(0.029)			(0.020)	(0.021)
Respondent hungry	0.662	0.631		0.524	0.462	0.440
but did not eat	(0.025)	(0.027)			(0.018)	(0.019)
Respondent lost	0.594	0.638		0.464	0.407	0.437
weight	(0.028)	(0.029)			(0.019)	(0.020)
Adult(s) did not eat	0.626	0.608		0.514	0.429	0.417
for whole day	(0.024)	(0.026)			(0.017)	(0.019)
Child HFSSM items
Relied on low-cost	0.526		0.638	0.405	0.354		0.429
foods for child(ren)	(0.029)		(0.029)		(0.020)		(0.021)
Could not feed child	0.764		0.466	0.729	0.528		0.323
(ren) balanced meals	(0.022)		(0.028)		(0.016)		(0.019)
Child(ren) not eating	0.704		0.577	0.599	0.491		0.402
enough	(0.046)		(0.060)		(0.032)		(0.043)
Cut size of child(ren)s	0.734		0.444	0.732	0.500		0.303
meals	(0.056)		(0.089)		(0.038)		(0.060)
Child(ren) hungry	0.845		0.315	0.878	0.586		0.218
	(0.065)		(0.141)		(0.048)		(0.097)
Child(ren) skipped	0.794		0.379	0.815	0.546		0.260
meals	(0.109)		(0.216)		(0.077)		(0.148)
Child(ren) not eat for	0.658		0.531	0.605	0.446		0.360
whole day	(0.213)		(0.304)		(0.140)		(0.210)
ECV	0.610	0.239	0.151

Note: Models estimated with variances constrained to unity using weighted data on low-income households with children from the 2012–2016 CPS-FSS. Standard errors are in parenthesis. We used the methodology described in Reise [28] to transform the discrimination parameters into their corresponding factor-loading parameters Marginal slopes were calculated using the factor-loading parameters based on the methodology described in Stucky and Edelen [47]. Standard errors for the factor-loading and marginal-slope parameters were calculated using the delta method. ECV $=$ Explained common variance, and I-ECV $=$ Item-level explained common variance.

The values of ECV indices for the specific dimensions of adult and child food insecurity in Table 4, 0.239 and 0.151, respectively, also indicate that the HFSSM-15 adult and child items represent unique constructs as indicated by their low factor loadings on the general household food insecurity dimension. Therefore, we find additional evidence supporting the analyses of adult and child food insecurity as unique constructs beyond household food insecurity. The larger ECV index values for the adult items suggest there is stronger support for a unique adult food-insecurity construct compared to a child food-insecurity construct.

4.3 Trends in household food insecurity under different versions of the HFSSM

Next, we assess the implications of the dimensionality of the HFSSM on measures of food insecurity for households with children used in the empirical literature. Specifically, we use a binary measure of food insecurity to capture the food insecurity of households with children. This measure is constructed using the methodology developed by USDA to categorize households into meaningful ranges of food insecurity (i.e., food insecurity status categories) based on the measures of food insecurity from the Rasch measurement model. Households who affirm at least three of the HFSSM items are classified as food insecure. For more details on the methodology used to assign each household a food-insecurity status, see [2].

So far, our measurement analyses have provided evidence supporting USDA’s assumption that a single underlying dimension of food insecurity relates responses to the HFSSM. Yet, there is some evidence that a subset of the HFSSM items may be best represented by multiple dimensions of food insecurity. To explore this further, we use the item-level ECVs (I-ECV) because they allow us to identify which of the HFSSM items are best represented by a single dimension. The I-ECVs are calculated as the ratio of the item-level variance accounted for by the general dimension of food insecurity relative to the total item-level variance [47].

Estimates of the I-ECVs are listed in Table 4 and may be used to select a subset of items from the HFSSM to form a unidimensional version of the HFSSM since they represent the extent to which an item is representative of the general dimension of food insecurity alone. The I-ECV indices range from a low of 0.122 for worried food would run out to a high of 0.991 for could not afford to eat balanced meals. Higher I-ECV values indicate items that would be strong candidates for a unidimensional version of the HFSSM.

We explore possible recommendations for a set of items that best represent a unidimensional HFSSM based on our estimates of the I-ECVs calculated using the bifactor measurement model parameters. There are a range of possible cutoffs for including items in a unidimensional scale based on the I-ECVs. Since there is no specific recommendation for selection criteria based on the I-ECVs, we consider several cutoffs between 0.85 and 0.50.

Table 5
Summary of item-ECVs from the HFSSM-15 by various thresholds useful for constructing unidimensional measures of food insecurity

	Item-ECV thresholds for a unidimensional measure of food insecurity based on the HFSSM
Items	$>$ 0.85	$>$ 0.80	$>$ 0.70	$>$ 0.60	$>$ 0.50
Worried food would run out	No	No	No	No	No
Food bought would not last	No	No	No	No	Yes
Could not afford to eat balanced meals	Yes	Yes	Yes	Yes	Yes
Relied on low-cost foods for child(ren)	No	No	No	No	No
Could not feed child(ren) balanced meals	No	No	Yes	Yes	Yes
Respondent ate less than should have	No	No	No	No	Yes
Adult(s) cut size or skipped meals	No	No	No	No	Yes
Respondent hungry but did not eat	No	No	No	No	Yes
Child(ren) not eating enough	No	No	No	No	Yes
Respondent lost weight	No	No	No	No	No
Cut size of child(ren)s meals	No	No	Yes	Yes	Yes
Adult(s) not eat for whole day	No	No	No	No	Yes
Child(ren) hungry	Yes	Yes	Yes	Yes	Yes
Child(ren) skipped meals	No	Yes	Yes	Yes	Yes
Child(ren) not eat for whole day	No	No	No	Yes	Yes

Note. All items are listed in increasing severity order and item-ECVs were calculated using the bifactor measurement model factor loadings listed in Table 4.

Table 5 summarizes the HFSSM items that would be included in a unidimensional scale of food insecurity based on different I-ECV selection criteria. All the items are ordered in increasing severity of food insecurity based on the item difficulty parameters so that the items are in a self-scoring format. This allows researchers to make a better judgement of how each scale would function compared to the official U.S. food security scale.

The number of items included in a unidimensional HFSSM depends on the I-ECV selection criteria. As we relax the criteria for item inclusion, the number of items in the unidimensional HFSSM increases. Under the most stringent criteria (I-ECV $>$ 0.85) only the items could not afford balanced meals and child(ren) hungry are included, while 12 out of the 15 items considered are included under the most flexible criteria (I-ECV $>$ 0.50). The items could not afford balanced meals and child(ren) hungry are included in the HFSSM regardless of what selection criteria are used.

Now that we have recommendations for constructing unidimensional versions of the HFSSM, we use them to estimate the prevalence of food insecurity among households with children in 2001–2016 using the CPS-FSS. First, we estimate the prevalence of food insecurity using the official version of the HFSSM. We use these estimates as a baseline for comparisons with USDA’s official estimates [2], allowing us to determine how dimensionality in the HFSSM affects estimates of the prevalence of food insecurity. Second, we use the HFSSM-15 to estimate the prevalence of food insecurity using the set of items considered in our empirical analyses. Comparisons between estimates of the prevalence of food insecurity based on the HFSSM and the HFSSM-15 allow us to assess the comparability of our findings with prior studies. Finally, we estimate the prevalence of food insecurity using alternative versions of the HFSSM constructed using the I-ECV selection criteria described in Table 5. The thresholds used to assign each household’s binary food insecurity status are selected so that they capture the same measured level of food insecurity based on the household’s measure of food insecurity.

Figure 3 describes estimates of the prevalence of food insecurity for households with children in 2001–2016 based on different versions of the HFSSM. Estimates of the prevalence of food insecurity based on the HFSSM and HFSSM-15 are essentially the same. Therefore, our findings are directly comparable with prior studies of food insecurity among households with children. In addition, the various forms of the unidimensional HFSSM produce similar estimates of the prevalence of food insecurity between 2001 and 2011. After 2011, estimates of the prevalence of food insecurity under the more stringent criteria used to construct a unidimensional HFSSM (I-ECV $>$ 0.60 or higher) produce slightly higher estimates of food insecurity.

Figure 3.

Trends in food insecurity for households with children based on different versions of the HFSSM, 2001–2016. Note. Means calculated using weighted data on households with children from the 2001–2016 CPS-FSS.

4.4 Revisiting the correlates of household, adult, and child food insecurity using behavioral 2-PL and bifactor models

Finally, we revisit the associations between household, adult, and child food insecurity and a series of explanatory variables known to be related to food insecurity among households with children. Unlike prior studies, we assume food insecurity among these households depends on their overall household food insecurity, and food insecurity among adult and child members. The explanatory variables are incorporated into the bifactor measurement model using a latent regression modeling approach [12, 42]. Under this approach, we can assess whether the dimensionality of the HFSSM affects our understanding of the correlates of food insecurity. In addition, our modeling approach also allows us to directly test whether the correlates of adult and child food insecurity are statistically significantly different from household food insecurity.

Table 6 lists coefficient estimates and standard errors from latent regression models of household, adult, and child food insecurity based on the behavioral 2-PL and bifactor models. Behavioral 1-PL models were also estimated, but they are not reported here because they were like the 2-PL model estimates. A complete list of estimates from these models is available in Appendix C. Columns 2–4 contain estimates of associations between household, adult, and child food insecurity and the explanatory variables from separate behavioral 2-PL models, and columns 5–7 describe associations between household, adult, and child food insecurity and the explanatory variables from a single behavioral bifactor model that jointly estimates the latent regression models for household, adult, and child food insecurity. The 2-PL model estimates are particularly useful for making comparisons with prior studies of food insecurity among households with children.

Table 6
Correlates of household, adult, and child food insecurity based on latent regression model parameters from the behavioral 2-PL and bifactor and IRT logistic models

Variable	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity
	2-PL model			Bifactor model
Female resp.	0.111 ${}^{***}$	0.108 ${}^{***}$	0.143 ${}^{***}$	0.101 ${}^{***}$	$-$ 0.073 ${}^{*}$	0.096 ${}^{***}$
	(0.016)	(0.016)	(0.020)	(0.016)	(0.039)	(0.032)
Age of resp.	0.024 ${}^{***}$	0.032 ${}^{***}$	0.030 ${}^{***}$	0.021 ${}^{***}$	0.009	0.015 ${}^{*}$
	(0.002)	(0.002)	(0.003)	(0.002)	(0.010)	(0.008)
Age of resp. squared	0.00 ${}^{***}$	0.000 ${}^{***}$	0.000 ${}^{***}$	0.000 ${}^{***}$	0.000	0.000
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Black, non-Hispanic resp.	0.052 ${}^{**}$	0.033	0.063 ${}^{**}$	0.053 ${}^{**}$	$-$ 0.381 ${}^{***}$	0.091 ${}^{**}$
	(0.023)	(0.024)	(0.027)	(0.026)	(0.050)	(0.045)
Other-race, non-Hispanic	0.001	0.011	0.014	$-$ 0.002	$-$ 0.067	0.006
resp.	(0.034)	(0.035)	(0.041)	(0.034)	(0.075)	(0.066)
Hispanic resp.	0.014	$-$ 0.007	0.068 ${}^{***}$	0.002	$-$ 0.452 ${}^{***}$	0.161 ${}^{***}$
	(0.023)	(0.024)	(0.028)	(0.027)	(0.055)	(0.046)
Married, spouse present	$-$ 0.234 ${}^{***}$	$-$ 0.236 ${}^{***}$	$-$ 0.223 ${}^{***}$	$-$ 0.226 ${}^{***}$	$-$ 0.078 ${}^{*}$	$-$ 0.039
	(0.018)	(0.019)	(0.022)	(0.019)	(0.043)	(0.034)
Highest educated adult in	0.030	0.038 ${}^{**}$	0.030	0.021	0.240 ${}^{***}$	0.009
HH, some college	(0.018)	(0.018)	(0.021)	(0.019)	(0.041)	(0.035)
Highest educated adult in	$-$ 0.380 ${}^{***}$	$-$ 0.377 ${}^{***}$	$-$ 0.312 ${}^{***}$	$-$ 0.382 ${}^{***}$	0.377 ${}^{***}$	0.015
HH, college grad. or	(0.022)	(0.023)	(0.027)	(0.025)	(0.051)	(0.048)
beyond
Number of adults in HH	0.076 ${}^{***}$	0.087 ${}^{***}$	0.021 ${}^{*}$	0.082 ${}^{***}$	$-$ 0.028	$-$ 0.078 ${}^{***}$
	(0.010)	(0.010)	(0.012)	(0.010)	(0.021)	(0.019)
Number of children in	0.115 ${}^{***}$	0.097 ${}^{***}$	0.169 ${}^{***}$	0.098 ${}^{***}$	$-$ 0.045 ${}^{***}$	0.176 ${}^{***}$
HH	(0.008)	(0.008)	(0.009)	(0.008)	(0.017)	(0.014)
Immigrant	$-$ 0.021	$-$ 0.061 ${}^{***}$	0.045 ${}^{*}$	$-$ 0.040	$-$ 0.396 ${}^{***}$	0.199 ${}^{***}$
	(0.023)	(0.023)	(0.027)	(0.026)	(0.060)	(0.049)
Age of youngest in HH	0.006 ${}^{***}$	0.002	0.018 ${}^{***}$	0.003	$-$ 0.007	0.030 ${}^{***}$
	(0.002)	(0.002)	(0.002)	(0.002)	(0.004)	(0.003)
Any elderly members in	$-$ 0.078 ${}^{**}$	$-$ 0.056	$-$ 0.082 ${}^{*}$	$-$ 0.075 ${}^{*}$	$-$ 0.162 ${}^{*}$	$-$ 0.057
HH	(0.039)	(0.039)	(0.048)	(0.040)	(0.093)	(0.080)
Any disabled members	0.358 ${}^{***}$	0.370 ${}^{***}$	0.299 ${}^{***}$	0.368 ${}^{***}$	0.005	$-$ 0.044
in HH	(0.026)	(0.027)	(0.030)	(0.027)	(0.055)	(0.047)
Urban residence	$-$ 0.022	$-$ 0.011	$-$ 0.019	$-$ 0.018	0.070	$-$ 0.034
	(0.020)	(0.020)	(0.023)	(0.020)	(0.044)	(0.037)
Head employed	$-$ 0.047 ${}^{***}$	$-$ 0.051 ${}^{***}$	$-$ 0.020	$-$ 0.058 ${}^{***}$	0.008	0.068 ${}^{**}$
	(0.018)	(0.018)	(0.022)	(0.018)	(0.041)	(0.034)
LN total HH income,	$-$ 0.577 ${}^{***}$	$-$ 0.561 ${}^{***}$	$-$ 0.498 ${}^{***}$	$-$ 0.567 ${}^{***}$	0.029	$-$ 0.073 ${}^{***}$
$2016	(0.012)	(0.013)	(0.015)	(0.013)	(0.031)	(0.025)
Home owned by HH	$-$ 0.354 ${}^{***}$	$-$ 0.362 ${}^{***}$	$-$ 0.296 ${}^{***}$	$-$ 0.358 ${}^{***}$	0.066 ${}^{*}$	0.032
member	(0.017)	(0.018)	(0.021)	(0.018)	(0.040)	(0.032)
Residence in Midwest	0.031	0.036	0.078 ${}^{***}$	0.017	0.021	0.099 ${}^{**}$
	(0.025)	(0.026)	(0.030)	(0.025)	(0.059)	(0.051)
Residence in South	0.036	0.044 ${}^{*}$	0.061 ${}^{**}$	0.030	$-$ 0.076	0.047
	(0.023)	(0.023)	(0.028)	(0.024)	(0.055)	(0.047)

Table 6, continued
	2-PL model			Bifactor model
Variable	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity
Residence in West	0.019	0.023	0.043	0.011	$-$ 0.033	0.059
	(0.025)	(0.025)	(0.030)	(0.025)	(0.060)	(0.050)
2012	0.142 ${}^{***}$	0.143 ${}^{***}$	0.159 ${}^{***}$	0.138 ${}^{***}$	$-$ 0.027	0.049
	(0.023)	(0.023)	(0.028)	(0.023)	(0.052)	(0.045)
2013	0.113 ${}^{***}$	0.119 ${}^{***}$	0.133 ${}^{***}$	0.113 ${}^{***}$	$-$ 0.026	0.029
	(0.023)	(0.024)	(0.028)	(0.024)	(0.053)	(0.046)
2014	0.110 ${}^{***}$	0.117 ${}^{***}$	0.101 ${}^{***}$	0.113 ${}^{***}$	$-$ 0.053	$-$ 0.011
	(0.023)	(0.024)	(0.028)	(0.024)	(0.054)	(0.047)
2015	0.031	0.036	0.038	0.033	$-$ 0.014	$-$ 0.005
	(0.024)	(0.025)	(0.030)	(0.025)	(0.057)	(0.048)
Number of households	53,964	53,964	53,964	53,964

Note: Models estimated with variances constrained to unity using weighted data on low-income households with children from the 2012–2016 CPS-FSS. Standard errors are in parenthesis. Results exclude all estimated measurement model parameters for conciseness. See Table C.1 in the appendix for a complete listing of the estimated parameters from the behavioral bifactor and 2-PL IRT logistic models. ${}^{*}$ Significant at the 0.10 level. ${}^{**}$ Significant at the 0.05 level. ${}^{***}$ Significant at the 0.01 level.

Like prior studies, we find that several respondent and household characteristics are associated with household, adult, and child food insecurity. Specifically, our results from the behavioral 2-PL models (columns 2–4) suggest households with a female, older, or black non-Hispanic respondent; more adults or more children; older children; or disabled members are more likely to be food insecure. Conversely, households with a married respondent, elderly members, an employed primary wage earner, higher total household income, or own their home are less likely to experience household food insecurity.

Many of the respondent and household characteristics that are associated with household food insecurity are also associated with adult and child food insecurity and similarly signed. In several cases, the associations are between household, adult, and child food insecurity and the explanatory variables are stronger for some dimensions of food insecurity and weaker for others based on the behavioral 2-PL models. For example, the association between child food insecurity and the number of children is statistically significantly larger for child food insecurity than it is for household and adult food insecurity. In addition, having a disabled household member is more strongly associated with adult and child food insecurity than household food insecurity.

Columns 4–7 of Table 6 list coefficient estimates and standard errors from the latent regression models for the bifactor model. This model assumes food insecurity in households with children is multidimensional, depending on a general overall household dimension of food insecurity, and specific dimensions capturing adult and child food insecurity. This model also has the added benefit of being more efficient because the latent regression models for household, adult, and child food insecurity are estimated jointly, rather than in three separate regression models. One major difference is evident after controlling for the multidimensionality of food insecurity among households with children. Specifically, age of the youngest in the household is no longer statistically significantly associated with the household dimension of food insecurity. The lack of statistical significance for this coefficient may suggest that the effect of the explanatory variable on food insecurity is operating through its relationships with the specific dimension of child food insecurity rather than the general dimension of household food insecurity. Therefore, the behavioral bifactor model considered here could be used to disentangle complex relationships between food insecurity and other factors not feasible using traditional regression methods.

The estimated associations between the household dimension of food insecurity and the explanatory variables are generally similar for the behavioral bifactor model compared to the 2-PL model. We also find that the estimated associations between the specific dimensions of adult and child food insecurity and the explanatory variables differ when compared to the 2-PL model results. For example, under the behavioral bifactor model households with children with a black non-Hispanic respondent are less likely to experience adult food insecurity. This result is obtained by summing the black non-Hispanic coefficients for the general household and adult specific dimensions of food insecurity. Conversely, under the 2-PL model for adult food insecurity the black non-Hispanic coefficient is statistically indistinguishable from zero. In addition, the association between adult food insecurity and highest educational attainment is a bachelor’s degree or higher is effectively zero under the bifactor model but negative and highly statistically significant under the 2-PL model.

5. Discussion and conclusions

The growing use of food insecurity measures as indicators of well-being for households with children in empirical research reinforces the need for a careful examination of the dimensionality of the HFSSM. To date, empirical research based on measures of food insecurity assumes the responses to the HFSSM are consistent with the Rasch measurement model and are derived from a single underlying construct that is assumed to represent the household’s food insecurity. Each year, these measures of food insecurity are used by USDA to assess the well-being of households with children and inform program administrators and policy makers as to the effectiveness of the food and nutrition assistance programs. Prior research pointed out the potential multidimensionality of the HFSSM for households with children, but they did not formally assess it [24, 25]. Rather, these studies focus on how the multidimensionally of the HFSSM affects comparisons of food insecurity between households with and without children. We extend the literature on the measurement and secondary analyses of food insecurity by providing the first formal assessment of the dimensionality of the HFSSM for households with children.

Our findings suggest there is moderate evidence that the HFSSM is represented by multiple dimensions of food insecurity for households with children. Several components of our measurement analyses support this. First, our preliminary factor analyses indicate responses to the HFSSM may be represented by up to three meaningful dimensions of food insecurity. Second, we continued to explore the dimensionality of the HFSSM by estimating unidimensional and multidimensional measurement models. Based on our findings, the bifactor measurement model provides a better fit to the HFSSM than traditional unidimensional measurement models. The improved fit of the bifactor relative to the 2-PL measurement model is not surprising since the bifactor measurement model specifically models the covariances among the adult and child items in the HFSSM while the 2-PL measurement model assumes these covariances are zero. Conceptually, the 2-PL measurement model can be viewed as a more restrictive model that is nested within the bifactor measurement model. We exploit this fact and formally test for dimensionality in the HFSSM using a likelihood ratio test to compare the 2-PL and bifactor measurement models, rejecting the 2-PL model in favor of the bifactor model at the one percent level or better.

While our findings do suggest the HFSSM is multidimensional for households with children, it is important to note that our results suggest there is also strong evidence for a primary dimension of food insecurity that captures the household’s overall food insecurity. To explore this further, we calculated the explained common variance (ECV) for each dimension of food insecurity measured by the bifactor measurement model. Higher values of the ECV index indicate a more prominent general dimension, reinforcing the use of unidimensional measurement models. Our estimate of the ECV for the general dimension of household food insecurity (0.610) indicates that the general dimension explains nearly two-thirds of the variation in household’s responses to the HFSSM. While this ECV index value falls below the more conservative threshold of 0.85 recommended by prior research [46, 47, 48], it does provide strong evidence of the usefulness of continuing to measure the food insecurity of households with children using unidimensional measurement models, such as the Rasch measurement model, since it is above the more flexible threshold of 0.20 proposed by Reckase [45]. In addition, the ECV index values for the adult and child specific dimensions reinforce the importance of viewing these as distinct constructs representing adult and child food insecurity.

Because we did find evidence of multidimensionality in the HFSSM for households with children we conducted additional analyses to determine whether the methodology used by USDA to produce estimates of the prevalence of food insecurity is robust to the multidimensionality of the HFSSM. Using item-level ECVs, we identified subsets of the HFSSM items that are represented by a single underlying dimension of food insecurity based on a series of index thresholds. We find that estimates of the prevalence of food insecurity are robust to the multidimensionality of the HFSSM. Our estimates of the prevalence of food insecurity were similar, regardless of which thresholds were used to identify the subset of items to include the food security scale.

Finally, we revisited the correlates of household, adult, and child food insecurity using our preferred measurement model, the bifactor measurement model. Latent regression model techniques were used to model direct associations between the dimensions of food insecurity and our explanatory variables, which are then jointly estimated. Our findings based on this model are consistent with previous studies. Yet, we do find evidence that our approach to modeling food insecurity among households with children may open new opportunities for disentangling complex mechanisms that interrelate household, adult, and child food insecurity. In some instances, our findings suggest the correlates of food insecurity may be more complex than prior research suggests.

Footnotes

Acknowledgments

This study was supported by a cooperative agreement from the Economic Research Service of the U.S. Department of Agriculture under Cooperative Agreement no. 58-5000-6-0020R. The study was also supported in part by the intramural research program of the U.S. Department of Agriculture, Economic Research Service. The findings and conclusions in this publication are those of the authors and should not be construed to represent any official USDA or U.S. government determination or policy.

Appendix A: CPS-FSS household food security survey module

Questions asked of all households:

“We worried whether our food would run out before we got money to buy more.” Was that often, sometimes, or never true for you in the last 12 months?

“The food that we bought just didn’t last and we didn’t have money to get more.” Was that often, sometimes, or never true for you in the last 12 months?

“We couldn’t afford to eat balanced meals.” Was that often, sometimes, or never true for you in the last 12 months?

In the last 12 months, did you or other adults in the household ever cut the size of your meals or skip meals because there wasn’t enough money for food? (Yes/No)

(If yes to question 4) How often did this happen – almost every month, some months but not every month, or in only 1 or 2 months?

In the last 12 months, did you ever eat less than you felt you should because there wasn’t enough money for food? (Yes/No)

In the last 12 months, were you ever hungry, but didn’t eat, because there wasn’t enough money for food? (Yes/No)

In the last 12 months, did you lose weight because there wasn’t enough money for food? (Yes/No)

In the last 12 months did you or other adults in your household ever not eat for a whole day because there wasn’t enough money for food? (Yes/No)

10.

(If yes to question 9) How often did this happen – almost every month, some months but not every month, or in only 1 or 2 months?

Questions asked only of households with children under 18 years of age:

11.

“We relied on only a few kinds of low-cost food to feed our children because we were running out of money to buy food.” Was that often, sometimes, or never true for you in the last 12 months?

12.

“We couldn’t feed our children a balanced meal, because we couldn’t afford that.” Was that often, sometimes, or never true for you in the last 12 months?

13.

“The children were not eating enough because we just couldn’t afford enough food.” Was that often, sometimes, or never true for you in the last 12 months?

14.

In the last 12 months, did you ever cut the size of any of the children’s meals because there wasn’t enough money for food? (Yes/No)

15.

In the last 12 months, were the children ever hungry but you just couldn’t afford more food? (Yes/No)

16.

In the last 12 months, did any of the children ever skip a meal because there wasn’t enough money for food? (Yes/No)

17.

(If yes to question 16) How often did this happen – almost every month, some months but not every month, or in only 1 or 2 months?

18.

In the last 12 months did any of the children ever not eat for a whole day because there wasn’t enough money for food? (Yes/No)

Note: “Affirmative” responses indicated in bold [15].

Appendix B. Characteristics of analysis households

Table B.1.

Means of the analysis variables for households with children with incomes below 185 percent of the federal poverty line

Variable	All households
Female respondent	0.539 (0.498)
Age of respondent	40.826 (10.657)
White, non-Hispanic respondent (ref.)	0.649 (0.477)
Black, non-Hispanic respondent	0.106 (0.308)
Other-race, non-Hispanic respondent	0.080 (0.271)
Hispanic respondent	0.166 (0.372)
Married, spouse present	0.667 (0.471)
Highest educated adult in HH, HS diploma or less (ref.)	0.244 (0.430)
Highest educated adult in HH, some college	0.311 (0.463)
Highest educated adult in HH, college graduate or beyond	0.194 (0.395)
Number of adults in HH	2.123 (0.817)
Number of children in HH	1.890 (0.997)
Immigrant	0.184 (0.387)
Age of youngest in HH	7.257 (5.251)
Any elderly members in HH	0.055 (0.228)
Any disabled members in HH	0.074 (0.261)
Urban residence	0.794 (0.405)
Head employed	0.762 (0.426)
LN total household income, $2015	1.643 (0.949)
Home owned by HH member	0.647 (0.478)
Residence in Northeast (ref.)	0.165 (0.371)
Residence in Midwest	0.220 (0.414)
Residence in South	0.343 (0.475)
Residence in West	0.271 (0.445)
2012	0.218 (0.413)
2013	0.200 (0.400)
2014	0.207 (0.405)
2015	0.185 (0.389)
2016 (ref.)	0.190 (0.392)
Number of households	53,964

Note: Means and standard deviations (in parentheses) estimated using weighted household data from the 2012–2016 CPS-FSS.

Appendix C. Parameter estimates from the behavioral IRT logistic models

Table C.1.

Parameter estimates and standard errors from behavioral 1-PL, 2-PL, and bifactor IRT logistic models

	2-PL model			Bifactor model
Variable	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity
Latent regression model parameters
Female resp.	0.111 ${}^{***}$	0.108 ${}^{***}$	0.143 ${}^{***}$	0.101 ${}^{***}$	$-$ 0.073 ${}^{*}$	0.096 ${}^{***}$
	(0.016)	(0.016)	(0.020)	(0.016)	(0.039)	(0.032)
Age of resp.	0.024 ${}^{***}$	0.032 ${}^{***}$	0.030 ${}^{***}$	0.021 ${}^{***}$	0.009	0.015 ${}^{*}$
	(0.002)	(0.002)	(0.003)	(0.002)	(0.010)	(0.008)
Age of resp. squared	0.00 ${}^{***}$	0.000 ${}^{***}$	0.000 ${}^{***}$	0.000 ${}^{***}$	0.000	0.000
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Black, non-Hispanic resp.	0.052 ${}^{**}$	0.033	0.063 ${}^{**}$	0.053 ${}^{**}$	$-$ 0.381 ${}^{***}$	0.091 ${}^{**}$
	(0.023)	(0.024)	(0.027)	(0.026)	(0.050)	(0.045)
Other-race, non-Hispanic resp.	0.001	0.011	0.014	$-$ 0.002	$-$ 0.067	0.006
	(0.034)	(0.035)	(0.041)	(0.034)	(0.075)	(0.066)
Hispanic resp.	0.014	$-$ 0.007	0.068 ${}^{***}$	0.002	$-$ 0.452 ${}^{***}$	0.161 ${}^{***}$
	(0.023)	(0.024)	(0.028)	(0.027)	(0.055)	(0.046)
Married, spouse present	$-$ 0.234 ${}^{***}$	$-$ 0.236 ${}^{***}$	$-$ 0.223 ${}^{***}$	$-$ 0.226 ${}^{***}$	$-$ 0.078 ${}^{*}$	$-$ 0.039
	(0.018)	(0.019)	(0.022)	(0.019)	(0.043)	(0.034)
Highest educated adult in HH,	0.030	0.038 ${}^{**}$	0.030	0.021	0.240 ${}^{***}$	0.009
some college	(0.018)	(0.018)	(0.021)	(0.019)	(0.041)	(0.035)
Highest educated adult in HH,	$-$ 0.380 ${}^{***}$	$-$ 0.377 ${}^{***}$	$-$ 0.312 ${}^{***}$	$-$ 0.382 ${}^{***}$	0.377 ${}^{***}$	0.015
college graduate or beyond	(0.022)	(0.023)	(0.027)	(0.025)	(0.051)	(0.048)
Number of adults in HH	0.076 ${}^{***}$	0.087 ${}^{***}$	0.021 ${}^{*}$	0.082 ${}^{***}$	$-$ 0.028	$-$ 0.078 ${}^{***}$
	(0.010)	(0.010)	(0.012)	(0.010)	(0.021)	(0.019)
Number of children in HH	0.115 ${}^{***}$	0.097 ${}^{***}$	0.169 ${}^{***}$	0.098 ${}^{***}$	$-$ 0.045 ${}^{***}$	0.176 ${}^{***}$
	(0.008)	(0.008)	(0.009)	(0.008)	(0.017)	(0.014)
Immigrant	$-$ 0.021	$-$ 0.061 ${}^{***}$	0.045 ${}^{*}$	$-$ 0.040	$-$ 0.396 ${}^{***}$	0.199 ${}^{***}$
	(0.023)	(0.023)	(0.027)	(0.026)	(0.060)	(0.049)
Age of youngest in HH	0.006 ${}^{***}$	0.002	0.018 ${}^{***}$	0.003	$-$ 0.007	0.030 ${}^{***}$
	(0.002)	(0.002)	(0.002)	(0.002)	(0.004)	(0.003)
Any elderly members in HH	$-$ 0.078 ${}^{**}$	$-$ 0.056	$-$ 0.082 ${}^{*}$	$-$ 0.075 ${}^{*}$	$-$ 0.162 ${}^{*}$	$-$ 0.057
	(0.039)	(0.039)	(0.048)	(0.040)	(0.093)	(0.080)
Any disabled members in HH	0.358 ${}^{***}$	0.370 ${}^{***}$	0.299 ${}^{***}$	0.368 ${}^{***}$	0.005	$-$ 0.044
	(0.026)	(0.027)	(0.030)	(0.027)	(0.055)	(0.047)
Urban residence	$-$ 0.022	$-$ 0.011	$-$ 0.019	$-$ 0.018	0.070	$-$ 0.034
	(0.020)	(0.020)	(0.023)	(0.020)	(0.044)	(0.037)
Head employed	$-$ 0.047 ${}^{***}$	$-$ 0.051 ${}^{***}$	$-$ 0.020	$-$ 0.058 ${}^{***}$	0.008	0.068 ${}^{**}$
	(0.018)	(0.018)	(0.022)	(0.018)	(0.041)	(0.034)
LN total HH income, $2016	$-$ 0.577 ${}^{***}$	$-$ 0.561 ${}^{***}$	$-$ 0.498 ${}^{***}$	$-$ 0.567 ${}^{***}$	0.029	$-$ 0.073 ${}^{***}$
	(0.012)	(0.013)	(0.015)	(0.013)	(0.031)	(0.025)
Home owned by HH member	$-$ 0.354 ${}^{***}$	$-$ 0.362 ${}^{***}$	$-$ 0.296 ${}^{***}$	$-$ 0.358 ${}^{***}$	0.066 ${}^{*}$	0.032
	(0.017)	(0.018)	(0.021)	(0.018)	(0.040)	(0.032)
Residence in Midwest	0.031	0.036	0.078 ${}^{***}$	0.017	0.021	0.099 ${}^{**}$
	(0.025)	(0.026)	(0.030)	(0.025)	(0.059)	(0.051)
Residence in South	0.036	0.044 ${}^{*}$	0.061 ${}^{**}$	0.030	$-$ 0.076	0.047
	(0.023)	(0.023)	(0.028)	(0.024)	(0.055)	(0.047)
Residence in West	0.019	0.023	0.043	0.011	$-$ 0.033	0.059
	(0.025)	(0.025)	(0.030)	(0.025)	(0.060)	(0.050)
2012	0.142 ${}^{***}$	0.143 ${}^{***}$	0.159 ${}^{***}$	0.138 ${}^{***}$	$-$ 0.027	0.049
	(0.023)	(0.023)	(0.028)	(0.023)	(0.052)	(0.045)
2013	0.113 ${}^{***}$	0.119 ${}^{***}$	0.133 ${}^{***}$	0.113 ${}^{***}$	$-$ 0.026	0.029
	(0.023)	(0.024)	(0.028)	(0.024)	(0.053)	(0.046)
2014	0.110 ${}^{***}$	0.117 ${}^{***}$	0.101 ${}^{***}$	0.113 ${}^{***}$	$-$ 0.053	$-$ 0.011
	(0.023)	(0.024)	(0.028)	(0.024)	(0.054)	(0.047)

Table C.1., continued
	2-PL model			Bifactor model
Variable	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity
2015	0.031	0.036	0.038	0.033	$-$ 0.014	$-$ 0.005
	(0.024)	(0.025)	(0.030)	(0.025)	(0.057)	(0.048)
Item-discrimination parameters
$\alpha_{1}$	4.674 ${}^{***}$	5.336 ${}^{***}$		5.580 ${}^{***}$	$-$ 0.538 ${}^{***}$
	(0.117)	(0.165)		(0.167)	(0.186)
$\alpha_{2}$	4.006 ${}^{***}$	4.137 ${}^{***}$		5.158 ${}^{***}$	$-$ 0.735 ${}^{***}$
	(0.076)	(0.083)		(0.170)	(0.190)
$\alpha_{3}$	2.956 ${}^{***}$	2.649 ${}^{***}$		2.918 ${}^{***}$	$-$ 0.054
	(0.050)	(0.044)		(0.051)	(0.039)
$\alpha_{4}$	4.093 ${}^{***}$	4.898 ${}^{***}$		5.340 ${}^{***}$	1.797 ${}^{***}$
	(0.097)	(0.137)		(0.175)	(0.157)
$\alpha_{5}$	4.597 ${}^{***}$	5.741 ${}^{***}$		6.067 ${}^{***}$	1.799 ${}^{***}$
	(0.115)	(0.182)		(0.213)	(0.174)
$\alpha_{7}$	4.070 ${}^{***}$	4.951 ${}^{***}$		4.856 ${}^{***}$	1.651 ${}^{***}$
	(0.111)	(0.158)		(0.162)	(0.135)
$\alpha_{8}$	3.202 ${}^{***}$	3.757 ${}^{***}$		3.718 ${}^{***}$	1.568 ${}^{***}$
	(0.088)	(0.116)		(0.128)	(0.128)
$\alpha_{9}$	3.298 ${}^{***}$	3.759 ${}^{***}$		3.692 ${}^{***}$	1.275 ${}^{***}$
	(0.102)	(0.131)		(0.134)	(0.132)
$\alpha_{11}$	2.713 ${}^{***}$		4.272 ${}^{***}$	3.496 ${}^{***}$		1.804 ${}^{***}$
	(0.044)		(0.156)	(0.097)		(0.089)
$\alpha_{12}$	3.032 ${}^{***}$		4.273 ${}^{***}$	4.036 ${}^{***}$		2.171 ${}^{***}$
	(0.058)		(0.144)	(0.138)		(0.109)
$\alpha_{13}$	3.179 ${}^{***}$		4.621 ${}^{***}$	4.288 ${}^{***}$		2.548 ${}^{***}$
	(0.081)		(0.184)	(0.185)		(0.160)
$\alpha_{14}$	3.271 ${}^{***}$		3.514 ${}^{***}$	3.299 ${}^{***}$		1.710 ${}^{***}$
	(0.107)		(0.123)	(0.118)		(0.104)
$\alpha_{15}$	4.571 ${}^{***}$		3.990 ${}^{***}$	4.217 ${}^{***}$		1.645 ${}^{***}$
	(0.216)		(0.190)	(0.206)		(0.146)
$\alpha_{16}$	3.980 ${}^{***}$		3.880 ${}^{***}$	3.656 ${}^{***}$		1.880 ${}^{***}$
	(0.207)		(0.211)	(0.201)		(0.197)
$\alpha_{18}$	3.819 ${}^{***}$		3.776 ${}^{***}$	3.454 ${}^{***}$		2.082 ${}^{***}$
	(0.349)		(0.395)	(0.364)		(0.475)
Item-intercept parameters
c ${}_{1}$	2.038 ${}^{***}$	3.110 ${}^{***}$		1.763 ${}^{***}$
	(0.105)	(0.162)		(0.169)
c ${}_{2}$	2.833 ${}^{***}$	3.520 ${}^{***}$		3.003 ${}^{***}$
	(0.089)	(0.102)		(0.190)
c ${}_{3}$	2.603 ${}^{***}$	2.780 ${}^{***}$		2.224 ${}^{***}$
	(0.068)	(0.062)		(0.080)
c ${}_{4}$	5.214 ${}^{***}$	6.779 ${}^{***}$		6.076 ${}^{***}$
	(0.130)	(0.185)		(0.452)
c ${}_{5}$	5.622 ${}^{***}$	7.658 ${}^{***}$		6.574 ${}^{***}$
	(0.149)	(0.237)		(0.457)
c ${}_{7}$	6.799 ${}^{***}$	8.770 ${}^{***}$		7.595 ${}^{***}$
	(0.171)	(0.258)		(0.417)
c ${}_{8}$	6.697 ${}^{***}$	8.192 ${}^{***}$		7.502 ${}^{***}$
	(0.147)	(0.213)		(0.398)
c ${}_{9}$	7.611 ${}^{***}$	9.093 ${}^{***}$		8.209 ${}^{***}$
	(0.184)	(0.266)		(0.357)
c ${}_{11}$	2.683 ${}^{***}$		5.879 ${}^{***}$	4.880 ${}^{***}$
	(0.063)		(0.289)	(0.354)
c ${}_{12}$	4.071 ${}^{***}$		7.448 ${}^{***}$	7.326 ${}^{***}$
	(0.084)		(0.252)	(0.480)
c ${}_{13}$	5.965 ${}^{***}$		10.541 ${}^{***}$	10.732 ${}^{***}$
	(0.131)		(0.406)	(0.643)

Table C.1., continued
	2-PL model			Bifactor model
Variable	Household food insecurity	Adult food insecurity	Child food insecurity	Household food insecurity	Adult food insecurity	Child food insecurity
c ${}_{14}$	7.300 ${}^{***}$		9.658 ${}^{***}$	9.459 ${}^{***}$
	(0.190)		(0.304)	(0.414)
c ${}_{15}$	10.526 ${}^{***}$		11.720 ${}^{***}$	11.764 ${}^{***}$
	(0.422)		(0.494)	(0.573)
c ${}_{16}$	10.249 ${}^{***}$		12.410 ${}^{***}$	12.191 ${}^{***}$
	(0.426)		(0.577)	(0.689)
c ${}_{18}$	12.222 ${}^{***}$		14.661 ${}^{***}$	14.803 ${}^{***}$
	(0.867)		(1.284)	(1.582)
Number of households	53,964	53,964	53,964	53,964

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Assessing the dimensionality of food-security measures

Abstract

Keywords

1. Introduction

2. Data

3. Methodology

4.1 Food security measurement – estimates from 1-PL, 2-PL, and bifactor measurement models

Table 4 Factor-loading parameters from the bifactor IRT logistic measurement model of household, adult, and child food insecurity

Table 5 Summary of item-ECVs from the HFSSM-15 by various thresholds useful for constructing unidimensional measures of food insecurity

Table 6 Correlates of household, adult, and child food insecurity based on latent regression model parameters from the behavioral 2-PL and bifactor and IRT logistic models

Footnotes

Acknowledgments

Appendix A: CPS-FSS household food security survey module

Appendix B. Characteristics of analysis households

Appendix C. Parameter estimates from the behavioral IRT logistic models

References

Table 4
Factor-loading parameters from the bifactor IRT logistic measurement model of household, adult, and child food insecurity

Table 5
Summary of item-ECVs from the HFSSM-15 by various thresholds useful for constructing unidimensional measures of food insecurity

Table 6
Correlates of household, adult, and child food insecurity based on latent regression model parameters from the behavioral 2-PL and bifactor and IRT logistic models