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Department of Statistics and * Department of Economics and Center for Agricultural and Rural Development, Iowa State University, Ames, IA 50011
2To whom correspondence should be addressed. E-mail: jopsomer{at}iastate.edu
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ABSTRACT |
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 TOP ABSTRACT SUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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KEY WORDS: • food insecurity • household hunger • Rasch model
During the last decade, new information has been developed and collected to measure the extent of food insecurity and hunger in the United States. The U.S. Department of Agriculture (USDA) has sponsored data collection to obtain information on food insecurity and hunger in the U.S. population since 1995, including support for annual food security supplements to the Current Population Survey (CPS)3 conducted by the U.S. Census Bureau. Common measurement of the phenomenon of hunger and food insecurity has become possible through efforts of the USDA and others to develop a set of survey questions that can be used to obtain estimates of the prevalence and severity of food insecurity (1
–3
). The data provide the basis for estimates of prevalence and severity of poverty-linked food insecurity and hunger in the United States and are used to help identify those groups in the population at greatest risk.
On the basis of earlier research, the USDA has analyzed data taken from a set of survey questions about food insecurity using an item-response-model approach, to estimate food insecurity experienced at the household level. Hamilton et al. (4
) discuss the approach selected to quantify food security, based on a one-parameter logistic item response model, also referred to as a Rasch model. The results from the original 1995 analysis were made available to the public (5
), and the USDA has also reported findings for subsequent years (6
–8
). The resulting measure, or index, of degree of food insecurity or hunger has shown itself to be remarkably robust across different time periods and across some subpopulation groups (9
–11
). However, further research is needed to understand the robustness of the method across different subgroups and in the context of alternative survey modes and experiences of hunger (12
–14
).
This paper has two main purposes: to evaluate the robustness of the approach currently used for the measurement of food insecurity, and to measure the effect of household-level variables on measured food insecurity. Both purposes will be addressed by fitting the food security data with a class of models that generalizes the Rasch model and comparing the estimates obtained from the different models on several years of CPS data.
Earlier research has shown that the Rasch model is a useful way to assign "food security" measures, or scores, to households participating in the CPS survey modules. The model assumes, however, that the food insecurity questions are interpreted in the same manner by all households interviewed. If this assumption is violated, then the estimated question scores and the food security estimates derived from them are potentially biased. Although it is possible to check this assumption by performing Rasch model fits for subsets of the population and comparing the results, this approach is somewhat cumbersome and the results are problematic to interpret statistically. In contrast, the generalized linear mixed model (GLMM) used here makes it very easy to incorporate household variables as well as interactions between these variables and the question scores. By having these types of variables explicitly in the model, we can answer questions like the following:
Hence, not only will it be possible to test some of the assumptions underlying the Rasch model on the CPS data, but we will also be able to estimate the effect of household variables on food insecurity using a single statistical model. An additional advantage of using a GLMM instead of a Rasch model is that many of the standard statistical software programs already contain procedures for fitting the former models, whereas the latter require specialized software.
The outline of the paper is as follows. First, we address methods of measurement. We review the Rasch and the generalized linear mixed models and show how the latter can be viewed as a direct generalization of the former. We fit the model to data from the CPS Food Security Module. Then, we use the results to validate the food security scale and to examine the effect of demographic and survey design variables on the scale.
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SUBJECTS AND METHODS |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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The Rasch model currently used by the USDA for estimating household food security scores is a type of item response theory (IRT) model developed for the purpose of measuring the ability of respondents based on their answers to a set of questions (15
). The model implies the existence of a continuous "scale" on which the items (questions) can be placed based on their difficulty levels and on which respondents can be placed based on their ability levels. The main objective of item response models is to estimate where respondents will fall on that scale. In the USDA Food Security scale, respondents are asked a set of questions related to their experiences of food insecurity to measure the phenomenon as perceived by their household. Hence, the questions’ "difficulty" is the level of food insecurity they capture, and the scale on which a household’s "ability" is measured is the severity of the household’s food insecurity.
As in the case of the USDA Food Security scale, we consider items to have only two answer categories ("yes/no" or "true/false"). Suppose a set of such dichotomous questions was administered to a sample of households (or, more precisely, household representatives or respondents). Each household responds to each question according to its latent food security: the more severe the food insecurity of the household, the larger the probability of giving a positive response to any given question. At the same time, each question has an implied food insecurity level, with "harder" questions more likely to be answered negatively than "easier" questions, regardless of the household’s food insecurity level.
Specifically, suppose that a sample of n households was administered a set of m dichotomous items, with each household receiving the whole set of m items. Based on their responses, the goal is to estimate each household’s severity (or, in IRT terminology, its ability) as well as each item’s implied food security (or its inherent difficulty). To formalize, let 
i be the ith household’s ability parameter for i = 1, ..., n and let 
j be the jth item’s difficulty parameter for j = 1, ..., m. If Iij is an indicator random variable that gives the dichotomous answer of household i to item j, then its distribution is
 | (1) |
The indicator variables Iij are assumed to be independent of each other, conditional on the parameters (i, i = 1, ..., n and j, j = 1, ..., m). An easy way to interpret the model (eq. 1) is to note that when i = j, household i has a 50% chance of answering question j affirmatively. When i > j, the household is >50% likely to answer affirmatively, and conversely, when i < j, the household is <50% likely to answer affirmatively.
The Rasch model provides a convenient framework in which to simultaneously estimate the abilities and the item difficulty levels, based on a set of questions administered to a group of households or individuals. The model makes it possible to estimate parameters even in the presence of item nonresponse, or if different but partially overlapping sets of questions are presented to respondents. In the USDA Food Security scale, for instance, households with children are asked 18 dichotomous questions, whereas households without children are asked only 10 questions [see Hamilton et al. (5
)]. Also, it is relatively easy to generalize to more complicated settings in which the items have different discriminating power; the respondents are assumed to randomly guess the answers to some or all of the questions; and so on (15
).
Even though the Rasch model (eq. 1) leads to an exponential family model, it cannot be fitted directly by maximum likelihood methods because of overparameterization. Hence, the estimated values are not unique. To solve this problem and get unique estimates, constraints are added, for instance, 
j=1m j = 0. Several methods are then available to fit this kind of Rasch model, including unconditional maximum likelihood as used in the USDA approach (4
).
No such simple adjustments are available, however, for incorporating household-level covariates such as the number of children or the gender of the household head into the model. Therefore, the Rasch model (eq. 1) will be replaced by a GLMM. This model reduces the number of parameters by assuming that the household parameters i, i = 1, ..., n follow a parametric distribution. Specifically, we assume that the severity parameter for the ith household can be written as
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where the x1i, ..., xpi are covariates of interest for household i, the ß0, ..., ßp are unknown parameters determining the effect of the covariates on households in the population, and 
i is a household-specified deviation from this population trend. The i, i = 1, ..., n follow a parametrically specified distribution. As is common practice in GLMM fitting, we will assume here that they are identically and independently normally distributed with mean 0 and unknown variance 
2. Although normally distributed household food insecurity levels might at first seem like an overly restrictive assumption, especially because food insecurity is documented to be a highly skewed phenomenon (8
), it should be noted that this distribution is only for the remainder after all the fixed household effects are taken into account, not for the food insecurity level itself.
The model we therefore are considering in this paper is
 | (2) |
with i 
N (0, 2), where "logit" is the commonly used notation for the logistic transformation [i.e., logit (p) = log (p/1 - p) (16
, p. 108)]. The parameters ß0, ..., ßp, j(j = 1, ..., m) and 2 are to be estimated from the data. Conditional on the i and the parameters, the response indicators Iij are again independent of each other. The GLMM shares most of the good features of the Rasch model, including the ability to handle nonresponse, guessing and so forth mentioned above. The model (eq. 2) will be fitted using restricted pseudo-likelihood (REPL) maximization, which is implemented in the GLIMMIX routines available for the Statistical Analysis System (SAS) (17
). The GLIMMIX program contains an additional parameter for deviation between the observed dispersion of the observations and that predicted by the exponential family model. We will not discuss that parameter here.
Several sets of covariates are of interest. Demographic variables make it possible to study household characteristics potentially affecting food security. Variables related to the interviewing process itself can be included in the model to determine whether "mode effects" are changing the outcome of the survey estimates. In addition, interactions between these variables and the item difficulties will be studied, given that these can point to differences in interpretation of the questions across population groups, a violation of the Rasch model assumptions.
Data.
The data for analysis come from the CPS Food Security Module for the years 1995, 1997 and 1999. A set of 18 food security questions of the 1995 CPS were used in developing the Food Security scale (see Table 1
). These questions correspond to the food insecurity items with parameters j mentioned previously and are coded as dichotomous variables. For a complete description of the questions and the coding of the answers, see Hamilton et al. (5
). The questions NHES40-NHES50 and NHES56-NHES58 were asked only of households with children, whereas the remaining questions were asked of all households. To avoid confounding with the intercept term ß0, question NHES58 was treated as the comparison level and left out of the regression model. This is common practice in regression with categorical variables.
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). The age of the household reference person is treated as a continuous variable. It was included as a linear (AGE) and a quadratic term (AGESQ) to capture some nonlinear effects. The variable PHONEINT is a dummy variable denoting whether the food security interview was conducted by phone (PHONEINT = 1) or in person (PHONEINT = 0). Some of the CPS interviews were conducted in Spanish if the respondent did not have sufficient knowledge of English. The variable SPKSPNSH is a dummy variable, with a value of 1 if Spanish was the only language spoken by all members of the household age 15 and over, and with a value of 0 otherwise. The variable MALE codes the gender of the household reference person, with a value of 1 if male and with 0 otherwise. Ethnicity is also recoded into two groups, with MINORITY = 0 for non-Hispanic white and MINORITY = 1 for all other groups. The variable CHILD is an indicator variable, taking the value of 1 if there are any children present in the household, and taking 0 otherwise. Here, the term "children" refers to anybody who is under 18 y old. UNEMPLOY is an indicator variable for the employment status of the household reference person at the time of the interview; it takes a value of 1 if he or she is unemployed, and takes 0 otherwise. Metropolitan status of the location of the household is included in the model as the indicator variable METRO, equal to 1 if the household is in a Metropolitan Statistical Area.
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Marital status of the household reference person is captured by the indicator SINGLE. In addition to "never married," SINGLE = 1 also includes situations such as married but spouse absent, divorced, separated or widowed. For education, LOWEDU = 0 is the code for household reference persons with high school diploma or above, and LOWEDU = 1 is the code for those without a high school diploma.
These data are available for each of the Food Security Module surveys since 1995. We discuss the model fits for the survey years 1995, 1997 and 1999 in the results section. The numbers of households with complete and valid data available for all the variables (with the exception of the child-specific food security questions for households without children and additional questionnaire skip patterns in 1997 and 1999), and who answered affirmatively to at least one of the food insecurity questions are 16,185 for 1995; 3817 for 1997; and 5475 for 1999. When the Rasch model is used, respondents who did not answer any of the questions affirmatively need to be removed, because their food insecurity level cannot be estimated. We did the same thing in the GLMM analysis, so that our results remain comparable with the Rasch model fits used by the USDA (5
–7
). This also reduced the data set to a size that could be more easily accommodated by the GLIMMIX algorithm. Because of the much larger sample size available for 1995, we will focus primarily on the data from that survey year. It should be noted that, because the models are fitted only on households who displayed at least measurable levels of food insecurity as discussed above, the interpretation of the household estimates is, strictly speaking, valid only for that subset of the overall population.
For 1997 and 1999, the sample sizes are smaller because more stringent screening criteria were used in those years. In addition to a set of initial screening questions, households who responded negatively to subsequent sets of questions were also screened out before completing the full set of Food Security items (3
). Specifically, respondents who answered negatively to a set of "easier" food security items were not asked the remaining "harder" items. As recommended by the USDA, we imputed the answer no (or zero) for these "harder" skipped questions for all the applicable households. In 1997 two of the eight rotation groups that completed the Food Security questionnaire were given some experimental questions as part of the survey. Hence, as recommended by the USDA, households in these rotation groups were removed from the analysis.
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RESULTS |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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The mixed logistic model (eq. 2) was fitted to the 1995 data with the item questions and demographic and survey mode variables entering the model linearly, for a total of 31 degrees of freedom (Table 3
). Note that, relative to the model in eq. 2, the signs for the item question parameters were changed to be positive, so that both household and question parameters were estimated and interpreted in the same manner: the higher the estimated parameter value, the higher the probability of observing a positive (yes) answer. The estimate of 2, the variance of the random household effect i was 
2= 8.49.
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). The correlation between that scale and the GLMM parameter estimates was -0.9994, further indicating the high level of agreement between both models. Hence, it appears that fitting the model with household covariates and a household random effect results in virtually identical item scores as those found with fixed household effects and no covariates.
The results for the survey mode variables were mixed. The variable SPKSPNSH was not statistically significant for any reasonable significance level. However, PHONEINT was significant, although less so than most other variables, and the size of the effect was small compared to that of the other significant variables. Overall, this indicates a weak mode effect for phone compared to in-person interviewing, with a slight decrease in the reported food security for phone interviewing. For logistic models, parameter effects are often expressed through their effect on the odds ratio, which is the ratio of the probability of answering a question positively over the probability of answering negatively (16
, p. 110). For instance, a value ß = -0.16 implies that this variable has the effect of multiplying the odds ratio by 0.85 = e(-0.16). In other words, respondents who were interviewed in Spanish were 15% less likely to answer yes to any question than they were to answer no, compared to those who were not interviewed in Spanish. An effect of that magnitude is generally considered weak. In contrast, evidence from the National Health and Nutrition Examination Survey shows being Mexican American was associated with greater food insufficiency (18
).
Among the demographic variables, all parameters except CHILD, the presence of children, were significant for any reasonable significance level, and MALE was almost significant at the 95% level (Table 3)
. Note that positive parameter estimates mean that these variables increase the likelihood of answering yes to any of the questions, implying an increase in food insecurity. The effects of most variables were as expected and consistent with earlier findings [(3
,5
,8
,19
), e.g., and especially the multivariate results of Alaimo et al. (18
)]: minority households (MINORITY) and those with unemployed (UNEMPLOY), single (SINGLE) or lower educated (LOWEDU) household reference persons all had positive estimated coefficients, indicating higher food insecurity. Households in metropolitan areas (METRO) also had slightly higher food insecurity than those living in nonmetropolitan areas. The remaining two demographic variables, age and income relative to poverty level, require additional explanation.
The combined effect of the linear and quadratic age terms (AGE, AGESQ) implies that the reported food insecurity increases until age 35 and then steadily decreases as the household reference person becomes older. Although somewhat counterintuitive, this is consistent with other analyses (20
,21
), which found that households with heads over 60 y old reported less food insecurity than did younger households, and with USDA reports on household food security (6
–8
), which consistently found lower rates of food insecurity among the elderly.
The parameter estimates for the three income categories (POVCATx) in Table 3
imply that food insecurity decreases with increasing income, up to 185% of the poverty level, but then increases sharply to a level above even that of POVCAT1 (which is set to 0 by default). Although at first surprising, this finding is in fact an artifact of the screening procedures used in the 1995 CPS survey. Households with income below 185% of the poverty level were automatically taken through the food security questionnaire in 1995, whereas those with income above that line were included only if they answered several food insecurity "screening" questions affirmatively. Hence, because of this difference in screening procedures, households in the POVCAT4 category were more likely to be food insecure than were those in the other income groups, and this effect is reflected in the parameter estimates in Table 3
. Clearly, this selection bias might have an effect on other aspects of the fitted model as well. However, as discussed below, the data sets for 1997 and 1999 did not exhibit this selection bias, and the remaining aspects of the models fitted on those data are in overall agreement with the findings for 1995. Hence, it appears that the effect of the selection bias is mostly limited to the POVCATx parameter estimates.
The model was refitted after omitting the variables SPKSPNSH, MALE and CHILD. None of the remaining parameters changed significantly, and their results are omitted here. We will keep these variables in the model, because some of the further models’ extensions, as well as the models for other years, did find some of these variables to be significant.
Next, the model was fitted with interaction terms between the household variables (excluding AGESQ) and each of the questions (i.e., items), resulting in a model with 236 parameters for estimation. This model is of interest for studying the effect of household-specific variables on the probability of answering individual questions rather than on the underlying food security level as in the linear model. This extended model is found to fit the data better, as measured by both Akaike Information Criterion (AIC) and Bayes Information Criterion (BIC) (22
), frequently used goodness-of-fit criteria in generalized regression and mixed models.
Many but not all of the individual interactions between the household variables and the questions were significant, and the relationships between them were complex. Because of the large number of interactions, the individual parameter estimates are not reported here. The direction of the interactions that were statistically significant at the 95% confidence level are displayed in Table 4
, as well as the number of interactions that were significant for each of the household variables and the food insecurity items. A large number of significant interactions for a household variable might indicate that households with that characteristic tend to respond to many individual food insecurity questions differently than do other households. Equivalently, food insecurity questions with a large number of interactions indicate that these are interpreted differently by households with different characteristics.
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Demographic characteristics with the largest numbers of interactions were the metropolitan status (METRO) and the minority status (MINORITY). It certainly seems plausible that households living in metropolitan areas or those belonging to minority groups might interpret some of the questions on the Food Security questionnaires differently (13
,14
). Therefore, further study on such interpretation differences among these population groups also is recommended.
Finally, a number of food security questions displayed much higher numbers of significant interactions than did others. For example, many of the measures associated with more severe food insecurity showed statistically significant interactions: "Adult cut size or skipped meals" (NHES24 and NHES25), "adult did not eat for a whole day" (NHES28 and NHES29), "adult eats less than should" (NHES32) and "adult hungry but didn’t eat" (NHES35). These results suggest differences in response or reporting of adults’ behaviors with respect to the food insecurity measure.
To assess the practical importance of these interactions, the food security question severity levels were recalculated for specific demographic subgroups. For instance, if we consider the minority respondents only, then the question severity as applied to that demographic group (ignoring the interactions with the other variables) is the sum of the question effect and the interaction effect between minority and that question. In this manner, a new food insecurity scale can be obtained for that demographic group and compared with the no-interaction scale or the original Rasch scale (at least up to a linear transformation of the scale).
Such demographic group-specific scales were calculated for minority, unemployed, metropolitan and single subgroups, as well as for all of the two-way intersections between those subgroups. In all of those cases, the correlation between the subgroup-specific scale and the original Rasch scale remained above 96%. This analysis was repeated using all the interaction effect estimates for each subgroup or using only the ones that were found to be statistically significant. The correlations remained equally high in both cases. Hence, it appears that, although the interactions were indeed statistically significant (because of the large sample size), they were not large enough to indicate significant departures from the overall food security for subpopulations.
1997 and 1999 Food Security results.
The no-interaction analysis was performed on the 1997 and 1999 data (Table 5
). The variables SKSPNSH, MINORITY and LOWEDU were not significant at the 95% confidence level in both years, and MALE and METRO were not significant in one of the two years. There was a high degree of agreement between both sets of parameter estimates, with all significant coefficients for household characteristics having the same signs and similar sizes. There was also a high level of agreement with the 1995 analysis. The most noticeable difference was that the higher income category (POVCAT4) no longer represented the most food insecure households. Instead, households in POVCAT4 were now the least food insecure, which was more in line with expectations of the relationship between food insecurity and income. Another difference between these and the 1995 results was that the presence of children (CHILD) had a significant negative parameter estimate in 1997 and 1999. Note, however, that the variable CHILD was potentially confounded with the presence/absence of the additional children-related questions in the data, so that its parameter estimates should be interpreted with some care.
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We also fitted the models with interactions, and, unlike in 1995, the AIC and BIC criteria indicated that the model without interactions provided a better fit to the data relative to the number of parameters used. This agrees with the finding in the previous section that the interactions did not appear to be of major practical importance.
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DISCUSSION |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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The predicted effects for the demographic variables were consistent with earlier findings and in line with prior expectations: those household representatives who speak Spanish, who are from minority groups, who are unemployed, single or who have lower education were more likely to be food insecure. Males were less likely to be food insecure. The income effects for 1995 reflected different screening criteria for different income groups. The effects of income estimated in 1997 and 1999 did not suffer from this screening effect and showed consistent effects of income: those with the least income were most likely to be food insecure. Age had a nonlinear effect on food insecurity, with older respondents, other factors held constant, being less likely to report food insecurity or hunger.
The analysis performed here also makes it possible to test some of the assumptions underlying the Rasch model. In particular, the Rasch model assumes that a unique measure of food security is appropriate for all households. The presence of numerous interactions between the individual questions and both survey mode and demographic variables indicates that further testing of this assumption might be of interest. In particular, it appears that minority respondents and those living in metropolitan areas respond to many of the questions somewhat differently than do other households. Although the overall magnitude of these differences did not appear to be sufficient to invalidate the food security scale currently used by the USDA, additional study of possible differential question interpretation by subpopulations is needed.
The GLMM approach makes it possible to use off-the-shelf software such as the GLIMMIX routine in SAS to fit models to the food security data, whereas the Rasch model requires specialized software. A possible drawback of the GLMM is that, unlike the Rasch model, it can calculate expected household food insecurity levels based on household characteristics, but does not directly yield a hunger score for particular households. In this sense, both approaches are complementary and the choice between them will depend on whether one is primarily interested in quantifying the effect of household characteristics on food insecurity, for which the GLMM is most appropriate, or in predicting the food insecurity level for specific households, where the Rasch model is more useful. In particular, the GLMM is not appropriate for estimating proportions of the U.S. population corresponding to specified food insecurity levels.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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3 Abbreviations used: AIC, Akaike Information Criterion; BIC, Bayes Information Criterion; CPS, Current Population Survey; GLMM, generalized linear mixed model; IRT, item response theory; REPL, restricted pseudo likelihood.
Manuscript received 10 September 2002. Initial review completed 21 October 2002. Revision accepted 21 November 2002.
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LITERATURE CITED |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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1. U.S. Department of Agriculture, Food and Consumer Service, and U.S. Department of Health and Human Services, National Center for Health Statistics, Centers for Disease Control and Prevention (USDA-HHS) (1994) Conference on Food Security Measurement and Research, Papers and Proceedings, January 21–22, Washington, DC 1994.
2. Frongillo, E. A., Jr, Rauschenbach, B. S., Olson, C. M., Kendall, A. & Colmenares, A. G. (1997) Questionnaire-based measures are valid for the identification of rural households with hunger and food insecurity. J. Nutr. 127:699-705.
3. Bickel, G., Carlson, S. & Nord, M. (1999) Household Food Security in the United States 1995–1998 (Advance Report) 1999 U.S. Department of Agriculture, Food and Nutrition Service Alexandria, VA. July. http://www.ers.usda.gov/briefing/FoodSecurity/lit/foodsecurity95-98.htm (accessed August 2002).
4. Hamilton, W., Cook, J., Thompson, W., Buron, L., Frongillo, E., Olson, C. & Wehler, C. (1997a) Household Food Security in the United States in 1995: Technical Report of the Food Security Measurement Project. Technical Report, September 1997a U.S. Department of Agriculture, Food and Consumer Service Washington, DC.
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13. Derrickson, J. P., Fisher, A. G. & Anderson, J.E.L. (2000) The core Food Security Module scale measure is valid and reliable when used with Asians and Pacific Islanders. J. Nutr. 130:2666-2674.
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18. Alaimo, K., Briefel, R. R., Frongillo, E. A., Jr & Olson, C. M. (1998) Food insufficiency exists in the United States: results from the third National Health and Nutrition Examination Survey (NHANES III). Am. J. Public Health 88:419-426.
19. Olson, C. M., Rauschenbach, B. S., Frongillo, E. A., Jr & Kendall, A. (1996) Factors Contributing to Household Food Insecurity in a Rural Upstate New York County. Institute for Research on Poverty Discussion Paper No. 1107-96 1996 University of Wisconsin Madison, WI.
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22. Nishii, R. (1984) Asymptotic properties of criteria for selection of variables in multiple regression. Ann. Stat. 12:758-765.
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