The two-part approach: a factor mixture model

The observed variables take non-negative values and show a strong floor effect with a high number of zero values, referred to as ‘zero-inflation’. Zero values in this study carry

5.7 Analytical strategy 163

information about non-ownership, which is essential for understanding the composition of a wealth portfolio. In this case, treating zero as ‘proxies for negative or missing values’ ignores the meaning of these zeros (Olsen and Schafer, 2001) and leads to misrepresentation of the data or to failure to identify unobserved heterogeneity (Kim and Muthén, 2009; McLachlan and Peel, 2000). A sensible analytical approach is a two-part approach in which the model consists of the binary and the continuous parts. This approach is analogous to the Hurdle model, which is found to outperform other approaches such as the zero-inflated Poisson (ZIP) model (Miller, 2007).

Several studies have implemented the two-part approach using FMM (Kim and Muthén, 2009; Muthén, 2006, 2008) with an estimation strategy developed by Olsen and Schafer (2001). In the study of school children’s aggressive behaviours, Kim and Muthén (2009) use a latent class model to represent the binary part of the data and a class-specific factor structure to represent the continuous part; this resulted in a 2-factor and 2-class model. They conclude that it provides new insight into children’s aggressive behaviour; the class model distinguishes groups of children with different likelihoods of engaging in aggressive behaviour, while the class-specific factor model measures the level of behavioural aggressiveness (Kim and Muthén, 2009).

The notions of ‘ownership’ and ‘intensity’ of each of the balance sheet items discussed in Section 5.5.1 can be studied similarly to those outlined by Kim and Muthén (2009). Construction of the two-part data is as follows. The wealth indicators for individual i(yi)

are decomposed into two parts – (a) a dichotomous part denoted by ui, for ownership of

asset or debt categories, and (b) a continuous part denoted by vi, representing the value of

holding (‘intensity’) given the ownership. The continuous part is log-transformed to follow an approximately normal distribution. The dichotomous part is

u_i=    1, if yi> 0 0, if yi= 0 , (5.1)

5.7 Analytical strategy 164

while the continuous part is,

vi=    log(yi), if yi̸= 0 (irrelevant), if yi= 0 . (5.2)

The FMM is estimated in three stages, as outlined by (Kim and Muthén, 2009). The first step concerns only the dichotomised part of the data (ui) using LCA. The second stage

contains only the continuous part (vi) which is conditioned on the binary part; only individuals

with positive values in each of the balance sheet items contribute to this part of the model. The last step combines the binary and continuous parts.

Latent class analysis for the binary part of the data

LCA is used when the latent construct of interest is assumed to be categorical, such as saver types. LCA is a model-based technique to identify and describe multiple subpopulations that share characteristics within a population. The modelling exercise focuses on assigning individuals to specific groups. It is assumed that individuals who are randomly selected from the population would belong to only one group (Collins and Lanza, 2010). The idea behind the LCA method is that, if a group of individuals share attitudes towards an object/issue, their responses to survey questions about the object/issue would share a pattern. These similarities in the response patterns help identifying subgroups of individuals who share characteristics unique to the subgroups (Bartholomew et al., 2008). Following notations from Collins and Lanza (2010), an LCA model can be written as

P(U = u) = C

∑

∏

∏

where ujis the response pattern for an item j, rjrepresents the response category of item j.

The response for item j is uj. γcrepresents the probability of belonging to class c, which is

termed posterior probability, while ρj,rj|c represents the probability of observing response rj

given the class membership c for item j, which is referred to as conditional item probability. The sum of rcfor the C number of classes is 1, and the sum of ρj,rj|cfor all categories of item

5.7 Analytical strategy 165

jis 1. That is, the classes are independent and mutually exclusive. It is important to note that LCA assumes all individuals in the same class have the same conditional item probability and that their similarities are sufficiently accounted for by belonging to the same class (i.e. local independence assumption) (Bartholomew, Knott and Moustaki, 2011; Bartholomew et al., 2008; Skrondal and Rabe-Hesketh, 2004). Logistic regression is used as a link function. Putting the conditional item probabilities for all items together by saver type and comparing them across classes are key to distilling information unique to each class. Lower (higher) probabilities indicate that the characteristic of the item is less (more) likely to occur in the particular group membership.

As the analysis aims to identify distinctive subgroups in a population, the choice of a number of classes is important for model selection. Unless the additional group significantly improves the model fit, a smaller number of groups is preferred in mixture models. This is because the classes become less distinctive as the number of classes increases (Collins and Lanza, 2010; McLachlan and Peel, 2000). The ‘right’ number of classes is determined by comparing the goodness of fit statistics and interpretability of classes for models with different number of classes, for example, C − 1, C, and C + 1, where C may be determined initially based on the hypothesis. Goodness of fit statistics used include information criteria that penalise increases in the number of parameters such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) (Byrne, 2012; Collins and Lanza, 2010; McLachlan and Peel, 2000). When these criteria point to different models for better fit, more weight is given to the BIC, which is found to be the most stable information criterion for LCA (Nylund, Asparouhov and Muthén, 2007). As there are five hypothesised profiles as shown in Table 5.2, models with 3, 4, 5 and 6 classes are tested to determine the most appropriate number of classes.

Factor analysis for the continuous part of the data

The second step concerns the continuous part of the model and focuses on FA. The motivation behind FA is that the latent quality, which is assumed to be on a continuous scale, can be modelled based on the observed items that are conceptually relevant to the underlying quality

5.7 Analytical strategy 166

(Bartholomew et al., 2008). The factor model in FMM with one-factor is

v_jc= τjc+ λjcηc+ ejc, (5.4)

where vjcare the continuous outcome variables for an item j( j = 1, 2, . . . p) in a latent class

c(c = 1, 2, . . .C). The intercept term τjcand the factor loading λjc can be specified for each

item j depending on class c. As seen in the equation above, Factor (ηc) is a latent factor

which is conditioned on class c and predicts the continuous part of the outcome variable v. Interpretations of these parameters are analogous to standard regression analysis.

FMM for the binary and the continuous parts

A model in the FMM framework consists of two interlinked models: on the left-hand side of Figure 5.1, the binary variables (u1 – u9) are represented by a latent class fc, and the

continuous indicators (v1– v9) by a continuous latent factor fη. The arrows from fcto fη

(straight) and factor loadings (dotted) denote that the factor ( fη) is conditioned on the latent

class ( fc).

The parameters of the factor model (factor loadings and item intercepts) can be fixed to be equal for all latent classes or estimated freely by class membership (see Figure 5.1). It pro- duces four possible parameterisations for the combinations of the factor loadings (restricted or freely estimated) and the item intercepts (restricted or freely estimated). Restricting these parameters to be consistent cross classes suggests that the factor interpretation applies to all classes. In contrast, freely estimating them lead to class-specific factor interpretations.

The hypothesis here is that a latent factor explaining levels of wealth accumulation is consistent regardless of saver type, while savers have different levels of wealth by saver type (conditioning on the class). This hypothesis is tested through comparing the model fit of the four possible model parameterisations. The final model is chosen based on the model goodness of fit criteria as well as the substantive interpretation of the model.

The goodness of fit criteria for model selection for FMM are similar to those of LCA. McLachlan and Peel (2000) suggest using the Bootstrapped Likelihood Ratio Test (BLRT), as

5.7 Analytical strategy 167

Fig. 5.1 A factor mixture model with binary and continuous observed indicators

the LRT statistic obtained in the mixture model violates the regularity conditions assumption (boundary value issue in the null model), which makes it problematic to assume a chi-square distribution. The BLRT statistic, however, cannot be obtained when using a weight and cluster correction in Mplus. Instead, the model selection criteria used include AIC, BIC, adjusted BIC, entropy as well as a number of the parameters.