Food Insecurity and Hunger in the United States An Assessment of the Measure (2006) / Chapter Skim
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5 Item Response Theory and Food Insecurity
5 Item Response Theory and Food Insecurity
Pages 71-98
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From page 71... ...
The first section provides a brief history of latent variable models, of which IRT models are a special case. The next section discusses latent variable models in general and IRT models in particular.
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From page 72... ...
Bartholomew (1987) gives a unified discussion of the three related types of latent variable models -- factor analysis, latent class analysis, and item response theory.
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From page 73... ...
, as opposed to the continuous manifest variables of factor analysis or the unordered nominal manifest variables of latent class analysis. In addition to the manifest variables, all latent variable models also assume the existence of a latent variable, the value of which varies from respondent to respondent but that is not directly observable for any respondent.
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From page 74... ...
In the application to food insecurity measurement, the latent variable represents the degree of food insecurity experienced by a given household that in turn influences the likelihood of endorsing or affirming responses to questions about lack of food due to economic constraints. There is a close connection between IRT models and latent class models.
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From page 75... ...
and the product rule in equation (3) are important for understanding the structure of latent variable models.
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Measurement Models One way to understand the role of conditional independence in latent variable models is in terms of measurement models. In this usage, the
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From page 77... ...
is the basic defining assumption of all latent variable models. It says that the joint distribution of the manifest variables simplifies to independence once one conditions on the latent variable, f.
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From page 78... ...
Suppose X is the manifest variable that codes the response to a given food insecurity question as 1 = affirm and 0 = not affirm. For the Rasch model, the item response function is determined by the conditional probability of affirming the binary food insecurity question given f, P{X = 1 | f}, and is given by the formula P{X = 1 | f} = exp{f b} /[1 + exp{f b}]
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From page 79... ...
Threshold Models for Item Response Functions Both the Rasch model and the 2PL model are examples of threshold models that are used in other applications in which observations are made with some degree of measurement error. An example in which such models are often used is the field of signal detection, in which an observer is trying to identify a signal in the midst of a noisy background (Peterson, Birdsall, and Fox, 1954; Birdsall, 1955)
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From page 80... ...
A later section briefly considers more general item response functions that are directly applicable to the case of polytomous ordered responses to the food insecurity questions. The Latent Distribution In order to be able to specify the joint distribution of the manifest variables, X1, X2, .
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From page 81... ...
include both the item parameters from the item response functions and possibly other parameters from the latent distribution. It is this joint distribution for the manifest variables that allows these parameters to be estimated and for the latent variable model to be tested against data.
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USDA should evaluate the amount and consequences of DIF in the FSS. The Latent Posterior Distribution It is evident that what can be deduced about the latent variable for a respondent depends on the values of the manifest variables for that respondent and the features assumed for the latent variable model.
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From page 83... ...
The method of moments estimates are simple and direct but appear to be feasible only for Normal Ogive models that are assumed to have a Gaussian latent distribution. Conditional maximum likelihood estimation eliminates the need for assumptions about the latent distribution but is satisfactory only for the Rasch model and closely related models in which the strength of the connection between the latent variable and the manifest variable is assumed the same for all of the manifest variables.
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From page 84... ...
This monotonicity in the connection between f and the probabilities of endorsing any of the food insecurity questions is a natural assumption, and it is critical to the use of such models for food insecurity measurement. In the use of IRT models by USDA, the item response functions, P{Xj = xj | f}, are specified by the Rasch
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From page 85... ...
Even when the restrictions placed on the latent variable model are sufficient to resolve the identifiability problem just described, there is still a slight problem of identifiability that is well known to users of continuous latent variables. The location and scale parameters of the latent distribution are confounded with the difficulty and discrimination parameters of the measurement model.
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From page 86... ...
For households with children similar comments apply, so this case is not considered further here. The responses to the 10 "adult" food insecurity questions are dichotomized and then the number of affirming responses to these 10 questions is used to classify the households into food insecurity levels.
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From page 87... ...
One of these is the location of the cut points that define the food insecurity categories. There are several ways to use IRT models to do this.
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From page 88... ...
Thus, somewhat against intuition, disagreement among judges about household classifications based on manifest data is a potential indicator of where to locate cut points along a latent scale. Classifying households based on the manifest data: Once a latent variable model is estimated and the latent posterior distributions are available, these can be used to determine cut points along the latent scale.
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From page 89... ...
depends only on the number of affirming responses of the household rather than on which questions are affirmed. This simplifies the classification rule to make it more like the one used by USDA, but it requires that the Rasch model accurately represents the distribution of the observed responses to the dichotomized HFSSM questions.
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From page 90... ...
This can be studied to some extent by trying out different assumptions and seeing what effect they have. As the number of manifest variables increases, the effects of different assumptions about the latent distribution grow less, but in the case of food insecurity the number of manifest variables is too small for this to be assumed.
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From page 91... ...
These estimated values of the latent variable are a side benefit of the unconditional maximum likelihood method of estimating the item parameters. However, when the effect of measurement error is large, as it is in the case of food insecurity measurement, the distribution of the estimated values of f across the sampled households does not form an unbiased estimate of the latent distribution.
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From page 92... ...
Methods of detecting differential item functioning may be used to investigate it with the data from different years being treated as the multiple groups. BETTER MATCH BETWEEN THE MEASUREMENT MODEL AND THE DATA COLLECTED The current approach to IRT modeling used by USDA is to create dichotomous/binary questions out of the several types of questions on the HFSSM, and then to use the Rasch model, which is designed for dichotomous questions.
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From page 93... ...
The panel considers them here because the measurement model is easy to understand, and it is sufficiently general for the currently collected food insecurity data. Different but related IRT models for ordered responses are given in Masters (1982)
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From page 94... ...
To specify a measurement model for this question, continue to assume that there is a latent variable, f, that underlies a respondent's answer to food insecurity questions. The higher f is, the more likely the household is to give a response of "often" for this question and the less likely a response of "never." Because there are three possible responses, there will be two thresholds, b1 and b2, on the f-scale (rather than the single threshold of the dichotomous case)
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From page 95... ...
If it is assumed that V has the normal distribution with mean 0 and variance 1, the result is the Normal Ogive graded response model, one of the earliest IRT models for ordered polytomous responses (Samejima, 1969)
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The independence of the measurement errors implies the conditional independence of the responses to questions given the latent variable f, so that equation (5) holds.
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These conclusions lead to the following recommendations to improve the categorization of households into food insecurity levels: Recommendation 5-1: USDA should consider more flexible alter natives to the dichotomous Rasch model, the latent variable model that underlies the current food insecurity classification scheme. The alternatives should reflect the types of data collected in the Food Security Supplement.
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From page 98... ...
· Studying the stability of the measurement system over time, pos sibly using the methods of differential item functioning. Recommendation 5-3: To implement the underlying latent variable model that results from the recommended research, USDA should develop a new classification system that reflects the measurement error inherent in latent variable models.
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