The framework described above can be estimated through multivariate analysis models. Equation (21.1) is a hierarchical model in which some variables are dependent on the one side and independent of the other. Unobservable (i.e. latent) variables also have to be dealt with. Figure 21.2 shows the path diagram of the model concerned.

In the causal models literature (Spirtes et al. 2000), circles represent latent variables and boxes represent observed variables. Most of the hierarchical or multi-level models

6 Recalling the discussion of the different meanings of resilience in the previous section, it can be argued that high stability and low adaptability create a status closer to the definition of engineering resilience, while ecological resilience is closer to a status characterized by high adaptability and low stability.

7 For example, IFA includes not only household income (which is observable), but also a series of estimated variables related to food consumption and expenditure and to the household's perception of food access and dietary diversity, which are context- and data-specific.

studied in the literature deal with measured variables, so the regression properties are extended. One of the innovative contributions of this research is the estimation of latent variable models in complex survey data.

Considering the complexity of the model concerned, two alternative estimation strategies could be adopted for the estimation of household resilience: structural equation modelling and multi-stage modelling.

Structural equation models (SEMs) are the most appropriate tool for dealing with the kind of model illustrated in Figure 21.2. Structural equation modelling combines factor analysis with regression. It is assumed that the set of measured variables is an imperfect measure of the underlying latent variable of interest. Structural equation modelling uses a factor analysis-type model to measure the latent variables via observed variables, while simultaneously using a regression-type model to identify relationships among the latent variables (Bollen, 1989). Generally, the estimation methods developed for SEMs are limited to normally distributed observed variables, but in most cases (including this one), many variables are nominal or ordinal. The literature has proposed ways to broaden the SEMs to include nominal/ordinal variables, but there are difficulties regarding computational aspects (Muthen, 1984). It is also possible to use generalized latent variable models (Bartholomew and Knott, 1999; Skrondal and Rabe-Hesketh, 2004) to model different response types. A major concern in using SEMs for measuring resilience is that the algorithms of SEM procedures are usually totally data-driven, but this chapter seeks to include some prior knowledge on deterministic relations among measured variables. Bayesian procedures could be used for their flexibility, but the number of variables to be used in this case make parameter identification problems likely to emerge, requiring careful consideration to incorporate proper prior information. In the last decade, the use of Markov chain Monte Carlo (MCMC) simulation methods (Arminger and Muthen, 1998; Lopes and West, 2003; Mezzetti and Billari, 2005; Rowe, 2003) has alleviated the computational concerns.

The other approach explored is a multi-stage strategy for estimating the latent variables separately, based on the relevant observed variables. This involves the use of various sets of observed variables (represented as squares in Figure 21.2) to estimate the underlying latent variables (circles in Figure 21.2). In other words, the circles represent the common pattern in the measured variables. The methods used for generating these latent variables depend on the scales of the observed variables. Traditional multivariate methods are based on continuous variables, but most of the variables in household-level surveys are qualitative (nominal, ordinal or interval), so it is necessary to use different techniques for non-continuous types of variables. The main multivariate techniques relevant for this analysis are SEMs or LISREL methods,8 factorial analysis, principal components analysis, correspondence analysis, multidimensional scaling, and optimal scaling, all of which are usually combined with deterministic decision matrices that are based on prior knowledge of variables.9

In the household surveys considered, most of the variables are categorical or ordinal, and measured at different scales, which makes the situation even more complex and

8 This should not be confused with the holistic SEMs proposed in the first approach. A single latent variable may be estimated through simpler cases of the general structural equations for the latent variable model: y = By + Tf + Z, where y is a vector of latent endogenous variables and f is a vector of exogenous variables.

9 This chapter does not explain these methods, which are described in all multivariate analysis manuals. The following sections provide the information necessary for an understanding of the procedures.

requires a mechanism to identify the optimal scaling. The variables require the use of optimal scaling methods, and scaling categorical variables is an important issue. Usually, these variables are coded with natural numbers (1,2, ..., n), but they may be coded as a,b,c, etc. In many cases, the distance between 1 and 2 may not be equal to the distance between 2 and 3, so categorical variables cannot be treated as quantitative variables. In this chapter, the principal components analysis by alternating least squares (PRINCALS) algorithm (De Leeuw and Van Rijckevorsel, 1980) is used to estimate optimal scaling and principal components simultaneously. The estimation minimizes the following objective function (loss function):

a(X, Y) = m-1J2SSQ(X - GjYj), j where SSQ is the sum of squares, m is the number of variables, X is the matrix of object scores, Gj is the indicator matrix for variable j, and Yj (scaling) is the matrix of category quantifications for j .

There are two main reasons for adopting the second (multi-stage) estimation strategy: the available variables are not all normally distributed, so may require the use of different multivariate techniques; and measuring the different components separately makes the model more flexible, allowing the inclusion of prior information and thus reducing the parameter identification problem.

After estimation of the different components and resilience, the CART methodology is adopted to test the validity of the adopted model and identify the contributions of the original variables to the resilience index.

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