Normas para el desarrollo de bienes inmuebles en cualquier territorio

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 ψ11

ψ21 ψ22

ψ₃₁ ψ₃₂ ψ₃₃

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

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



(5.7)

Within the SEM framework, a latent growth model can be easily represented by a graphic. Those representations of models have become widely used within the SEM research field. An example is given here for a LGM in Figure 5.3:

Figure 5.3: Latent curve model

The coefficients in the oval boxes represent the latent variables (in this case latent factors); the boxes are the observed variables used to define the latent factors (e.g. drug use measured at five time points). The linear and quadratic development is captured by the factor loadings (λt and λ²_t in the equation, and the arrows in the graphic) that are fixed to represent a quadratic trajectory as specified above. Error terms for the observed outcomes and variance for the latent factor are estimated as well (small arrows pointing to the boxes in the figure), and, if needed, equality restrictions can also be applied on them (Bollen & Curran, 2006; T. Duncan et al., 2006).

5.1.2 Model-fit indices

The choice of the best LGM model follows the same criterias used for structural equation models (Bollen, 1989). The goodness of fit is calculated as the difference between the sample matrix S and the estimated matrixP(θ) : (S − P(θ)). For this purpose different estimation methods are at disposal and the preference for a specific one depends largely on the distribution and characteristics of the observed variables (Bollen, 1989). Commonly

used methods to define model fit in LGM are based on Maximum Likelihood estimation procedure (Eliason, 1993).

There are generally two situations in which model-fit indices are necessary. In the first case the researcher has already formulated a specific model and wants to test it for good-ness of fit. In this situation model-fit indices are needed to determine whether the given model is acceptable or not. For this case, the most common are: the chi-square statistic, the RMSEA, and the CFI (see Bollen, 1989; Bollen & Curran, 2006). In the second situ-ation, the researcher has formulated different model variations, whereas these variations are based on paramater restrictions. One needs statistics which can allow a compari-son among similar nested or non-nested models. Here widely accepted methods are the likelihood-ratio difference test, the AIC, and the BIC. The latter two statistics can be directly compared among models without any further calculation.

For what concern the comparison of models, the more commonly used tests are those belonging to the information-based test family. Among these, there are the AIC (Akaike, 1973) and the BIC (Raftery, 1993). The AIC is calculated as:

AIC = −2ln(L) + 2p (5.8)

The AIC considers the loglikelihood value and applies a penalty where p represents the number of estimated parameters. Here, parsimonious models with few parameters are preferred. The BIC uses a similar approach, with the main difference consisting in taking into account also the sample size n:

BIC = −2ln(L) + pln(n) (5.9)

In both cases, when comparing nested models, the model with the smaller value on these statistics should be preferred to the models with larger values.

The chi-square test of model fit is based on the null hypothesis that the estimated and ob-served mean and covariance matrices are equal. This situation means that the estimated model perfectly reproduces the observed data and thus the reality. Thus, a significant test statistic suggests that the specified model does not perfectly match the mean and covariance structure of the observed data. In fact, the size of a chi-square value can be tested; being chi-square distributed all values can be tested for significance using a normal chi-square table and knowing the degree of freedom (Bollen & Curran, 2006).

Another largely used statistic test is the RMSEA (Root Mean Square Error of Approx-imation) (Steiger, 1990). The value of RMSEA shows how close the estimated and the observed covariance matrix are. It has a minimum of zero and no maximum, where zero represents perfect model fit. There are accepted guidelines for assessing the result of a RMSEA test: Browne and Cudeck (1993) suggest that values smaller than 0.05 indicate a very good model fit, whereas values larger than 0.10 indicate a poor fit.

The CFI is another widely used fit index. Its computation is based on the comparison between the estimated model and a baseline model, which is more restricted than the estimated one. The values of the CFI range between zero and one, where one represents a perfect fit. Also in this case, a general accepted rule of thumb suggests that values of CFI smaller than 0.90 represent poor model fit (Bollen & Curran, 2006).

In the results section model-fit indices for the best fitting model will be presented alongside the estimated results.

5.2 Growth mixture models

Growth mixture models assume that the sample does not belong to a single homogeneous population, but rather to a mixture of different groups (Muth´en & Shedden, 1999). In some cases, a single trajectory well represents the development of the behavior of interest,

5.2. Growth mixture models

and it can be used for further more complex analysis where, for instance, covariates are included in the model (see Muth´en, 2004). In other situations, however, the researcher might suspect the existence of groups of subjects that share a particular development worth to be represented separately. In fact, either theoretical and statistical assumptions might indicate the necessity of more than a single trajectory in order to represent the given behavior in the sample. An important distinction for the statistical analysis of group differences in developmental processes is between observed and unobserved heterogeneity, i.e. the fact whether the source of these differences is known. Observed heterogeneity uses known and measured information to divide the sample in specific groups and then estimate single trajectory for each of them. This is the case when the researcher suspects that particular characteristics, for instance gender, ethnic, and psychological differences, might influence the development of the behavior under study for those people belonging to each category of the observed independent variable. A straightforward example is the assumption that important differences in the development of drug use behavior can be observed between male and female, suggesting thus the necessity of estimating two independent trajectories, one for each group. In unobserved heterogeneity these groups (or classes) are not directly observed (i.e., are not directly measured as it would be for gender), but are implied in the statistical distribution of the outcome variable. Groups with different trajectories are then statistically inferred from the distribution of the dependent variable, and these models are generally known as mixture models. Mixture techniques applied to latent growth models are known as growth mixture models (GMM), although similar techniques are also widely used (see for instance factor mixture analysis, FMA, for an overview see Muth´en, 2001).

5.2.1 General model

GMM modeling techniques account for the unobserved heterogeneity in the sample by means of subgroups (or classes). Each group is allowed to have its own trajectory (inde-pendently estimated growth parameters), and each individual is probabilistically assigned to a particular group. For each estimated trajectory a completely different equation can be specified, allowing each group to have its own singular trajectory. This is the result of the combination of both continuous and categorical latent variables within the SEM framework, which is also the basis for both LGM and GMM model construction. This idea can be better understood by introducing a categorical latent variable C to the above shown graphical description of a simple LGM for a curvilinear development assuming five time points.