LVM with parametric indirect effects

M2a η =Sex+Inc+Age Tab 7.7

M2b η =Sex+Inc+Age+Sex∗Inc+ Tab 7.8

η =Sex∗Age+Inc∗Age

7.3.3 M3a η =f(Age) Fig 7.7, Tab 7.9

M3b η =Sex+Inc+f(Age) Fig 7.8, Tab 7.10

7.3.4 M4a η =fspatial(Reg) Fig 7.9, Tab 7.11

M4b η =Sex+Inc+f(Age) +fspatial(Reg) Fig 7.10/7.11, Tab 7.12 7.3.5

M5a η =Sex∗f(Age) Fig 7.12, Tab 7.13

M5b η =Inc∗f(Age) Fig 7.13, Tab 7.14

M5c η =Inc+Sex∗Inc+Sex∗f(Age)+ Fig 7.14/7.15, Tab 7.15

η =fspatial(Reg)

Table 7.5: Overview of the predictors in the structural equation of all estimated models with one latent variable in Section 7.3.

this respect, parameters 3 and 4 stand out. Indicator 3 (Retirement) has the highest factor loading with a communality of 0.69 because this question regarding the old-age provision aims very closely at the idea of the latent construct. The question related to parameter 4 (Emergency), on the other hand, hints into a slightly different direction than the questions belonging to the other 4 indicators, thus its factor loading is the lowest.

Furthermore, the mean values almost perfectly agree with the mode values indicating a symmetric distribution of the factor loading samples. This fact, the narrow standard deviation, and 10%- and 90%-quantiles demonstrate the significance of the factor loading parameters, and strongly support the idea of a latent construct being responsible for the variation of the five indicators. As expected, the highest factor loading of indicator 3 shows the highest absolute standard deviation.

In order to check the basic validity of those estimates, a classic factor analysis was carried out. The results showed communalities which were 10%–35% below the estimates of our model. The reason for this inferior result lies in the simplification of treating the ordinal indicators as metric ones. In particular, the communality of binary indicator 1 with a value of 35% below the correct estimate confirms that assuming binary indicators to be metric produces very weak results.

7.3.2

LVM with parametric indirect effects

Now, the classic factor analysis model is extended by introducing indirect parametric co- variates modifying the latent construct. Two models are analyzed – the predictor of the first model M2a contains the categorical covariates Sex and Inc, and the metric covariate Age which is treated as a parametric and hence linear effect; the second model M2b ad-

7.3 Model estimations with one latent variable 121

ditionally includes some interactions between those variables. The predictors of the two models are defined as

ηi =γ1·Sex2i+γ2·Inc2i+γ3·Inc3i+γ4·Inc4i+γ5·Agei, (M2a) and

ηi =γ1·Sex2i+γ2·Inc2i+γ3·Inc3i+γ4·Inc4i+γ5·Agei+ (M2b)

γ6·Sex2iInc2i+γ7·Sex2iInc3i+γ8·Sex2iInc4i+γ9·Sex2iAgei

γ10·Inc2iAgei+γ11·Inc3iAgei+γ12·Inc4iAgei.

Standard dummy coding is used – for example Inc3i is set to 1 if observationi belongs to the third income category, otherwise Inc3i is set to zero. Estimates of the factor loadings and parametric indirect effects are summarized in Table 7.7 for M2a, and in Table 7.8 for M2b.

Let us start with the discussion of model M2a. First of all, it is conspicuous that the estimates of all factor loadings are slightly lower than for the pure factor analysis with- out indirect covariates. However, this does not mean that this model is inferior to the traditional factor analysis. In a traditional factor analysis model, the indicators response solely determines the value of the latent variable. In any model with covariates however, the covariates additionally influence the value of the latent variable. Specifically, the nor- mally distributed latent variable is not centered statically at zero like in a traditional factor analysis, but the mean of the distribution equals the value of the predictor in the struc- tural part of the model. Thus, the latent variables cover a broader range of values and thus exert a greater influence on the variability of the indicators, even if the factor loadings are slightly lower. In a sense, the covariates explain some of the correlation among the indicators. Using covariates therefore allows more detailed statements on the dependence structure of the latent variable.

We proceed with the discussion of the estimates of the parametric indirect effects. First of all, females seem to have leaning towards a strong state because the mean of the latent factor is by 0.359 lower than for males. Furthermore, the covariate income exerts a very strong influence on the latent construct. With increasing income, the mean of the latent factor increases considerably, for example about 1.098 for a person in income category 4 compared to the reference category 1. This effect can be explained by the fact that people with high incomes generally show a higher inititiative of their own and a higher readiness to take risks than people with lower incomes. In addition, big earners make high monetary contributions to the social system without getting an adequate service in return. Finally, increasing age has a negative influence on the mean of the latent factor, hence older people tend to prefer a stronger state taking care of the social services. At this point, it is doubtful if the assumption of a parametric linear effect of age on the latent construct is justified. This question is further examined in the next section where the effect of age is modeled by a smooth nonparametric function.

The parameter estimates of model M2b including interactions between the covariates show slightly different results than model M2a. The effect for females (Sex2) is even more nega- tive, and the pure income effect seems to be less pronounced than in the model excluding interactions. However, these effects are compensated by the positive effects of the interac- tion between females and age (Sex2Age), and the positive interaction between income and age (Inc2Age, Inc3Age, Inc4Age). The probably most useful interaction between females and income shows that for higher income categories, females have a bias towards a state provision system compared to males in the same income category.