IntroductionDataModelResultsConclusionAppendixA Simple Latent Variable Model of Perceived Abusein Long-Term CareConner, T., Fang, Y., Page, C., Prokhorov, A., Xiao, Y.Michigan State UniversityFirst draft: March 31, 2006This draft: May 9, 2006AbstractThe primary difficulty in studying factors of abuse in long-term care is that the levelof abuse is not directly observed. Latent variable models allow to make inference aboutabuse factors using various indicators of the extent of abuse. As such indicators we usereported incidents of perceived abuse obtained in a survey of Michigan families who havea relative in long term care. The survey also provides data on demographic and medicalcharacteristics of each patient. We use this data to construct variables that are commonlybelieved to be determinants of susceptibility to abuse. We then evaluate statistical andpractical significance of such determinants.1 IntroductionThe extent of abuse in long-term care is arguably determined by characteristics of the care-giver and care giving facility (type of long-term care facility, nurses per patient, etc.) as wellas of the person in care (increased frailty, cognitive decline, etc.). This paper considers onlyhealth characteristics of the person in care. Papers that look at the facility-driven factors ofabuse include Korzeniewski et al. (2005), ....The biggest difficulty in modelling abuse is that it is never fully observed. This happensbecause of problems with definition and measurement of something that people tend to conceal,feel uncomfortable reporting, or fail to clearly identify.This paper attempts to solve this difficulty by using a linear latent variable representationin which inference about the relationship between unobserved variables is made on the basesof their observed (reported) indexes.2 DataThe data is from a survey of Michigan families who have a relative in long-term care. Amongother things, the survey asked questions about perceived level of various types of abuse andabout aspects of the person’s health. Appendix contains the relevant questions: Q1 throughQ7 address the extent of abuse while Q8 through Q10 are about health. Answers to Q1-Q7were coded 0, 1.5, 4, 8, or 11 (responses “Don’t Know/Refuse” were coded missing). Wetherefore have seven measures of abuse Y1, . . . , Y7. For answers to Q8 we construct the1Table 1: Descriptives of indicators and risk factors of abuse.Variable Obs Min Max Mean Std. Dev.Y1976 .0 11.0 .151 .887Y2976 .0 11.0 .585 2.038Y3965 .0 11.0 .446 1.677Y4960 .0 11.0 .672 2.178Y5972 .0 11.0 .783 2.327Y6972 .0 11.0 3.549E-02 .484Y7973 .0 11.0 .319 1.400X1887 -2.74 2.55 -.3896 .8773X2990 .00 1.00 .2212 .4153X3875 0 12 8.00 3.34X4984 11 97 75.92 17.60RASCH score (X1). From answers to Q9, we only use the dummy variable for presence (1) orabsence (0) of behavioral problems (X2). In Q10, we calculate the sum of negative answersto get an index (between 0 and 12) that represents the number of restricted activities of dailyliving (X3). We also use the patient’s age (X4).Summary statistics of the indexes are in Table 1.3 ModelWe use a covariance structure model that has become known as the Multiple Indicator MultipleCauses (MIMIC) model. It was pioneered by Karl J¨oreskog in the 70s and since then becamepopular in psychometrics and social s cience s that deal with latent features. See, e.g., J¨oreskog(1970) and J¨oreskog and S¨orbom (1977). The model consists of two parts. The first part,called the measurement model, connects two unobserved factors η and ξ with their indicatorsY1, ..., Y7and X1, X2, X3, respectively. It can be written asY = Λη + ε, (1)X = Γξ + δ, (2)where Y = (Y1, ..., Y7)0, X = (X1, X2, X3)0, Λ = (λ1, ..., λ7)0, Γ = (γ1, γ2, γ3)0, η and ξ areknown as common factors, ε = (ε1, ..., ε7)0and δ = (δ1, δ2, δ3)0as unique factors.The second part, known as the structural equation, represents the relationship between thelatent variables (η and ξ). Interest usually lies in estimation of parameters of this equation.We can write this part of our LISREL model asη = βξ + ν (3)Distinction is often made between the exogenous factor ξ and endogenous factor η, similarto the distinction between the dependent and independent variables in the usual structuralequations.2If the observed variables are deviations from means then the expected values of Y, X, η, ξ, ε, δand ν are all zero. The unique and common factors are uncorrelated like in usual factor analy-sis. The unique factors for Y are uncorrelated with the unique factors for X. Variance matrixof εiis denoted by θ2i, variance matrix of δiis denoted by σ2i, variance of ν is denoted by ψ2.This is a covariance structure model b e cause (1)-(3) impose a structure on the covariancematrix of the observed variables. In fact, after some algebra, variance of (Y0, X0)0can be ex-pressed in terms of the unknown parameters, i.e. in terms of β, λ1, ...λ7, γ1, γ2, γ3, ψ2, θ21, ..., θ27, σ21, σ22, σ23.Denote this covariance matrix by Σ; it is a matrix function of the unknown parameters.The traditional estimation of the model is essentially finding such values of the parametersthat minimize the difference between Σ (the covariance matrix dictated by the model) andthe sample covariance matrix S of the formlog |Σ| + tr(SΣ−1).Minimizing this objective function is equivalent to the maximum likelihood estimation if thedata are multivariate normal. Our data are far from normal. So we use the AsymptoticDistribution Free estimation procedure which minimizes a certain Euclidean distance betweenthe two matrices (see, e.g., Browne, 1984).4 ResultsFigure 1 shows the path diagram for this covariance structure model. The latent abuse variableand all disturbances are in circles, observed variables - in boxes. Arrows indicate the causaldirection. One of the indicators for abuse and one of the determinants of risk are set equal toone to define units of measurement. The figure also contains the parameter estimates.We drop Y6(sexual abuse) and allow for a certain pattern of correlation between distur-bances of the abuse indicators. These are results of a preliminary factor analysis on abusemeasures.The partial correlations between the risk factors are based on the assumption that thebehavioral problem is only related to age through medical conditions or daily living restrictions.While the medical score, the number of restrictions and age are all directly related.Figures 2-3