CHAPTER 3: REVIEW OF LATENT VARIABLE ANALYSIS AND SEM TECHNIQUES
3.4 The MIMIC model
A popular SEM which contains observed covariates is the Multiple-Indicators, Multiple-Causes (MIMIC) model. The MIMIC model of Jöreskog & Goldberger (1975) considers the relationships among observable endogenous “indicator” variables, exogenous “cause” variables, and latent constructs. This approach allows for the identification and estimation of latent variable indices and the impact of various factors on these indices (Richards & Jeffrey, 2000).
The MIMIC model is a variation of SEM that has gained popularity as a research framework due to its flexibility in a wide range of research contexts (Thompson & Green, 2006, as cited by Finch
& French, 2011). The notable advantages of SEM, including MIMIC, over observed variable
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modelling are: 1) the ability to consider latent variables that cannot be estimated by any single measure; and 2) the ability to consider error due to measurement or omission, rather than assuming that measurements are made free of error (Finch & French, 2011).
Figure 3.1 provides an illustration of a basic MIMIC model in which a single latent variable (η) is determined by several indicator variables, response items (Xq), and observed “cause” variables, regressors (Yp).
Figure 3.1: A one-factor MIMIC model Source: Adapted from Muthén (1989)
The observable cause variables are generally regarded as some of the most important determinants of the latent variable (Dell’Anno, 2007). The relationship between the cause and indicator
Regressors Response items
X 1 Y 1
X q Y p
η
Latent construct
Structural Equations Measurement Equations
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variables and the latent dependent variable is captured through the specification of two separate equations. In the structural equation, the latent dependent variable is linearly determined by a set of observable exogenous causes (Xi), while in the measurement equation, the latent dependent variable determines, linearly, a set of observable endogenous indicators (Yi). These two sets of equations are simultaneously solved, often using maximum likelihood (ML) estimation, to determine the effects of the various cause and indicator variables on the latent dependent variable (Jorëskog & Goldberger, 1975). It is important to note that there are no special rules of identification associated with the MIMIC model and estimation of model parameters proceeds in the same way as in general structural equation modelling (Kaplan, 2000).
Figure 3.2 shows a path diagram of a multiple-factor MIMIC model. In this specification, there are four latent variables (n, X1, X2, X3), or factors, to be identified, whereby the latent response variable (n) is determined by three latent constructs (X1, X2, X3), which represent “cause”
variables. Each of these latent “cause” variables are then identified by three “indicator” variables (V1, V2, …, V9). For example: the latent “cause” variable X1 would be identified by V1, V2, and V3.
The MIMIC model has been a popular choice of model framework for latent variable analysis, which has been adopted extensively in a number of different disciplines, from behavioural psychology to marketing and economics (Macias & Cazzavillan, 2010). Proitsi et al. (2011) recently considered an application of the MIMIC model to assess the behavioural and psychological symptoms in dementia. The authors credited the MIMIC model for its ability to efficiently capture the complexity of inter-relationships between symptoms, factors and clinical variables considered in the study. Shehzad (2006) adopted a MIMIC model framework in his study on the determinants of child health in Pakistan. Results indicated that the use of MIMIC models allowed for a more comprehensive understanding of the determinants of child health compared to studies relying on single health measures. Furthermore, the unobservable nature of child health was successfully overcome using latent variable analysis.
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Figure 3.2: Path diagram for a multiple-factor MIMIC model Source: own illustration adapted from Kaplan (2000)
The MIMIC model has received several applications in the analysis of the informal (underground) economy. Breusch (2005) adopted the MIMIC model in an attempt to quantify the underground economy of various countries. The attractiveness of the MIMIC model in this instance comes in the form of being able to represent the size of the underground (informal) economy as a latent, unobservable variable, which cannot be directly quantified but has a number of causes and effects which are observable. Dell’ Anno (2007), in a similar study, analysed the “shadow” economy in Portugal using a MIMC model approach. The authors noted that the MIMIC model could be considered a useful methodology when taking other econometric alternatives into account.
More recently, Macias & Cazzavillan (2010) investigated the Mexican informal economy using a MIMIC model approach. Dell’ Anno (2007) and Macias & Cazzavillan (2010) highlighted two
n
X1
1 V1 2
V2 3
V3 4
X2
5 V4 6
V5 7
V6 8
X3
9 V7 10
V8 11
V9 12
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potential limitations associated with the MIMIC, the first of which pertains to the difficulties that arise when undertaking time series analysis. Since the MIMIC model only provides a group of estimated coefficients that can be used to create an index, benchmarking or calibration techniques must be used to convert the resulting index into values that can be used to construct a time series (Dell’ Anno, 2007). Secondly, assigning a specific meaning to the latent dependent variable is subjective since the actual meaning of the estimated latent variable may be conceptually different.
There have been a limited number of applications of the MIMIC model to farm level data. As mentioned in the opening subsection of this chapter, Ivaldi et al. (1994) and Ivaldi et al. (1995) implemented covariance structure analysis, with similar equation structure to the MIMIC method, to estimate stochastic production functions of French grain and fruit producers, respectively. As mentioned earlier in the chapter, Esposti & Pierani (2000) adopted a MIMIC model approach to the measurement of technical change and determine sources of output growth in Italian agriculture.
Richards & Jeffrey (2000) adopted a MIMIC model framework to estimate the efficiency and economic performance of a sample of Canadian dairy farmers, treating economic performance as a latent variable for which several imperfect indicators exist. Measures of technical, allocative and economic efficiency, estimated using a stochastic cost function framework, were then incorporated into the MIMC model as indicators of economic performance. Furthermore, the authors constructed latent quality indices to determine the effect of the quality of the breeding, feeding and labour programmes on latent economic performance.
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CHAPTER 4: MILK PRODUCTION IN SOUTH AFRICA