The model is divided in two parts. The “Measurement part” is the part where the latent variables are observed through “Functionings”, being endogenous observed variables. The second part is called the “Structural Part”, where the unobserved variables are explained by exogenous factors. Both parts contain uncorrelated error terms, between individuals and between them. This is a Multiple Indicators and Multiple Causes (MIMIC) model.
The model can be written as follows for an individual:
yi=Λfi+εi
fi=Bxi+ui
where yi is a vector (P × 1) containing all indicators, fi is a vector (M × 1) containing the latent variables, xi is a vector (Q×1) containing the explanatory variables, εiandui are vectors of errors of respective dimensions (P×1) and (M
× 1). The indicators as well as the exogenous variables are observed; while the latent factors are unobserved and will be estimated.
The matrix of coefficientsΛ, of dimension (P×M), give magnitudes of the impact of one unit of latent factor on observed endogenous variables. The matrix B, of dimension (M×Q), measures the impact of exogenous factors on latent variables.
We have the following assumptions:
E(εi) =0, E(ui) =0
V(εi) =Φ, V(ui) =Ψ
C(εi,ui) =0
The MIMIC model we will use differs from a Structural Equation model (SEM) in that there is no causality between latent factors but we allow only correlation between them. In other words, if we wanted to have a SEM, the latent factors should be included on the right hand of the equation in the Structural Part of the model, in order to allow for simultaneity, which would be:
fi=Afi+Bxi+ui
In our particular case, indicators are not continuous but ordinal variables, so the measurement part will become:
2.6. The Model
The vectory∗i represent latent responses, which are continuous variables that give us the link to the observed indicators. Cj represent the number of categories of ordinal variable j andτj is a vector of thresholds corresponding to the variable j.
We can rewrite the expression above as follows:
yij=
The theoretical expressions of the expected values ofy∗i conditioned onxi are:
E(y∗i|xi) =ΛBxi
V(y∗i|xi) =ΛΨΛ′+Φ
In general, Φ is a diagonal matrix since it is assumed that the entire correlation between indicators is captured by the latent variable.
Figure 2.2 shows our empirical model. Rectangles on the left contain all signifi-cant exogenous variables (from “AGE” to “DEFTY”)6that correspond toxiin the equations and represent the resources that determine the capabilities. The latter are drawn in the middle of the graph in circles,Autonomy in self care is the latent
6We decided not to include all exogenous variables in order to have less saturated graph, but the excluded variables were mentioned before.
variable F1 and Participation in the life of the household is the latent variable F2 that correspond to fi in the equations. It should be noted that we tested all exogenous variables in the model for both latent variables but we only included arrows when the exogenous variables had a significant effect on the capability.
On the right of the graph we can see the functionings (from“F T OIL′′to“F T RAV′′), measuring the capabilities F1 and F2 and correspond to yi in the equations. It ought to be remembered that, in contrast to the exogenous variables, the set of functionings was already defined for each latent variable, so there is no reason why they should cross. One notes that for each capability the arrow going to the first functioning has the number 1 just above. This means that for these respective functionings, loading coefficients were fixed at one for identification reasons.
The remaining circles represent error terms (from e1 to e11) for each one of the observed endogenous variables in the entire model that correspond to εi in the measurement equations. For all the disturbances there is a number 1 above the arrow as well, meaning that there is no coefficient estimate for these terms, as can be seen in the theoretical model. Circles with an arrow pointing to F1 and F2, in other words u1 and u2, are error terms associated with both capabilities respectively that correspond touiin the structural equations.
2.6. The Model
Figure 2.2: Path diagram with significant variables
2.7 Identification of the model
Before running the estimation we need to be sure that the model is identified.
If the number of parameters is the same as the number of equations given by the empirical moments, the model is just identified. If there are more equations than unknown parameters then the model is over-identified; and if there are more unknown parameters than equations, the model is under-identified (not identified).
T-rule (Bollen ,1989) is a necessary but insufficient condition for identification which relies on the number of known and unknown parameters in the model. The formula is given by:
t≤ (1
2 )
(P+Q)(P+Q+1)
wheretis the number of all unknown parameters to be estimated,Pis the number of endogenous observed variables andQis the number of exogenous variables.
In our case, we have 11 observed dependent variables, so P= 11; and originally 42 independent variables7, soQ= 42. The number of total unknown parameters, denoted as t, is equal to 118 for the first variant and to 140 for the second and third variant. This can be detailed as follows:
1. Number of coefficients for dependent variables8: 9.
2. Number of coefficients for independent variables: 84.
3. Number of variances or covariances for the structural part: 3.
4. Number of thresholds: for the first variant this is equal to 22 and for the second and third variant this is equal to 44.
As indicated, the T-rule is largely satisfied in our case, with a left hand side equal to 118 for the first variant and 140 for the second and third variants and a right hand side equal to 1431.
One can identify the model in two steps. First, the measurement part as if it were a factor analysis and then the structural part as a classic model considering the latent variables as observed. If each part is identified then the whole model is identified, which, without going into detail, happens to the case here.
7We count each of the categories that were included in the model as a variable with a coefficient to estimate.
8Since we fixed at 1 one of the coefficients for each latent variable, this would be equal to the number of observed endogenous subtracted by 2.
2.8. Model estimation
2.8 Model estimation
Estimation methods for these type of models have been the object of intensive research. The nature of the variables and the number of latent variables, among other factors, are some of the elements which influence the choice of the estimator and numerical algorithm.
Our model is estimated according to works of Muth´en (1978, 1983, 1984). The best approach when having categorical variables is the robust weighted least squares that takes the diagonal weight matrix to estimate the parameters and then uses the full weight matrix to compute chi square test and standard errors (WLSMV).
The difference with WLS is since, we are using a diagonal matrix the residuals will be closer to zero. (Muth´en, DuToit and Spisic, 1997).
The model is estimated in a three steps procedure:
1. Estimation of vectors of thresholdsτ, conditional expectationsE(y∗i|xi), con-ditional variancesV(y∗i|xi)and all correlations two by two by using Maximum Likelihood (ML) with a probit link, since it is assumed that the underlying latent response is normally distributed. All these elements will be included in a vector denoted as ρ. In the presence of covariates the correlations are estimated by the correlation based approach (Muth´en, 1984), otherwise it uses the classical polychoric correlations.
2. Estimation of a consistent asymptotic covariance matrix of the moments cal-culated a step above, denoted asG.
3. Minimization of the following fitting function:
[ˆρ−σ(θ)]′G−1[ˆρ−σ(θ)]
whereσ(θ)represent the theoretical counterpart ofρand θ is a vector con-taining all unknown parameters in the model.
One of the consequences of the assumptions on the functionings was an unbalanced proportion of individuals in their categories. It could have several implications when estimating the whole model, including biased estimators and underestimated standard errors misleading the choice of relevant variables of our model9.
Nevertheless, Forero, Maydeu Olivares and Gallardo Pujol (2009) studied bias and variances in estimations of Structural Equation Modeling with simulations consid-ering multiple cases. Important conclusions were drawn, giving four possible cases
9When calculating thresholds in the first step of the three step estimator (WLSMV), their estimates would be imprecise and we would not predict accurately the number of individuals in the categories.
leading to biased and imprecise estimations: sample size, binary items, unbalanced proportions of categorical variables and number of indicators. Our only concern is the unbalanced proportions since all other cases fulfill the required conditions.
Besides, the authors found evidence (with samples of more than 2000 individuals) that none of the issues mentioned above need any particular treatment.
Notwithstanding, let us note that there are a number of assumptions in this model that are not tested: appropriateness of the choice of the binary response model, linearity and additivity. We will not go into further details about these aspects as our intention in the present chapter is simply to operationalize the capability approach by using a MIMIC model.
Once the parameter estimation is complete we proceed with the estimation of la-tent factors, giving both capabilities for each individual. These estimates can be obtained by using the Empirical Bayes estimator or maximizing the posterior likeli-hood10(Krishnakumar, 2008; Krishnakumar and Nagar, 2008; Skrondal and Rabe-Hesketh, 2004). The Empirical Bayes estimator can be written as:
fˆi=Bxi+Ψ′Λ′(ΛΨΛ′+Φ)−1(y∗i −ΛBxi)
This estimator has minimal variance and is derived from the multivariate normal distribution of y∗i andfi.
2.9 Estimation Results
The model was estimated by using the whole sample (AD plus Non-AD), in other words, persons having Alzheimer’s disease as well as those who do not have Alzheimer’s disease. The same model cannot be used only for Alzheimers since the variability within this group is too small to run the entire theoretical framework (function-ings, exogenous variables) of the study, which would need to be modified for such purposes. Nonetheless, once we estimate the model with all individuals we can separate our results according to whether or not the individuals have AD, and then proceed to compare both groups in terms of the estimated capabilities (factor scores).
2.9.1 1
stvariant
Let us remember that this variant is constructed by using Assumption 1 and As-sumption 2 (weakest assumptions). Table 2.4 and Table 2.5 present the results for measurement equations for both capabilities for the fist variant.11
10Both methodologies give similar results.
11For all tables: *,** and *** denotes significance at 10%, 5% and 1% levels, respectively.
2.9. Estimation Results Standardized coefficients are given in each table in order to compare the magnitudes of the parameters. They are obtained by multiplying the value of the coefficient by the standard deviation of the explained variable and dividing it by the standard deviation of the explanatory variable. For example, let us denoteyas the explained variable,xas the explanatory variable, andbthe standardized coefficient ofxony.
We would interpretb as the change in standard units ofy when there is a change of one standard unit of x.
Also, for identification reasons, loadings from the first indicators in each of the measurement equations are fixed at one. The corresponding variable in the first measurement equation isbathe (FTOIL) and in the second measurement equation ispreparing meals (FREPAS).
Autonomy in self care (F1) andParticipation in the life of the household (F2) have positive and very significant effects on all their respective indicators. This is one of the results that we expected to see since for a greater capability the individual will tend to choose a more valuable outcome and, as we can see, the model verifies this assumption. The strongest impact of an increase of F1 is for dress (Standardized
= 0.918) and of an increase of F2 is fordaily cleaning (Standardized = 0.923). The lowest impact of an increase of F1 is foreat and drink (Standardized = 0.809) and of an increase of F2 is for administrative affairs (Standardized =0.796).
Table 2.4: Measurement equation for F1 (1stvariant)
Estimate Std. Error Standardized t value Pr(>|t|)
FTOIL 1.000 - 0.908 -
-FSHABI 1.024 0.020 0.918 51.508 ***
FMEDIC 0.821 0.027 0.824 30.817 ***
FNOURR 0.929 0.023 0.878 40.974 ***
FMB 0.793 0.042 0.809 19.009 ***
FSERTOI 0.924 0.029 0.876 31.976 ***
Table 2.5: Measurement equation for F2 (1stvariant)
Estimate Std. Error Standardized t value Pr(>|t|)
FREPAS 1.000 - 0.918 -
-FTMEN 1.011 0.014 0.923 70.179 ***
FCOURS 0.917 0.015 0.883 62.859 ***
FADMIN 0.751 0.016 0.796 47.174 ***
FTRAV 0.934 0.014 0.891 65.968 ***
Table 2.6 and 2.7 present results for structural equations for variant 1. For both capabilities age (AGE) and gender (GENDER) are retained. The older the in-dividual, the lower the capability in both cases; which is not a surprising result.
Gender, on the contrary, has opposite signs, negative for F1 and positive for F2.
It means that Autonomy in self care decreases when the individual is male, all things being equal, whileParticipation in the life of the household increases. The last result could seem unanticipated a priori because in the literature women are supposed to mobilize their resources in the life of household more than men. In our case, we found thatParticipation in the life of household ismore valuedby men than women. According to our results, men express difficulties and a desire for aid in self care more often than women, while this trend would be reversed in the Participation in the life of the household. This may be due to the fact that women can be more sensitive to difficulties associated with household activities than men, meaning that their options are less valued. Women would be more demanding than men and they would like tasks to be completed to a higher standard. We are not saying that men perform these activities better but rather that they feel more satisfied with their options.
On the one hand Household size (INDMEN) has a negative effect on both capa-bilities. On the other hand, variables of number of children alive12 (CENF1 and CENF2) have positive effects, the first being higher than the second. It means that having other people living in the house is not a factor that converts to the advantage of the individual (less available space, more things to do for the others, so probably “worse life conditions”) while having living children is a clear resource that increases the set of choices. Let us stress that we are not evaluating a “re-source consumption model”, where having more means being better, we are using a social approach (household composition, social effects). Based on our results we can not say that household size is a good predictor of the quantity and quality of aid, an idea that has been discussed in the literature (Snyder 2001). For example, given the average age of circa 74, if one of the individuals from our sample still has children living with him or her then there is a high probability that the children also have difficulties.
The frequency of visits of family members (F1RENC) was not significant in the model. We first included it in the model as originally in the database13and when we found that it was insignificant we decided to create dichotomous variables for each one of the categories and we estimated once again the model with the new variables. The conclusions were the same, all of them were insignificant. The variable living in couple (COUPLE) plays a role only for theParticipation in the life of the household; its effect is positive and very significant as one might expect.
12Let us remind that this variable is split in two: 1-2 children and 2-4 children as explained above.
13F1RENC is an ordered categorical variable.
2.9. Estimation Results
Table 2.6: Structural Equation for F1 (1stvariant)
Estimate Std. Error Standardized t value Pr(>|t|)
AGE -0.011 0.002 -0.068 -5.456 ***
GENDER -0.081 0.034 -0.029 -2.351 **
INDMEN -0.113 0.020 -0.065 -5.505 ***
CENF1 0.131 0.037 0.048 3.546 ***
CENF2 0.111 0.041 0.037 2.691 ***
TDHDOM 0.149 0.022 0.102 6.881 ***
TLIEU3 0.267 0.068 0.072 3.908 ***
TRANS1 0.127 0.038 0.046 3.301 ***
TRANS5 0.105 0.036 0.036 2.909 ***
TRANS6 0.346 0.064 0.091 5.371 ***
STOC 0.040 0.015 0.030 2.616 ***
RAL7 -0.194 0.022 -0.074 -8.856 ***
CLIC -0.321 0.080 -0.044 -4.017 ***
BCONC 0.063 0.018 0.045 3.475 ***
BLIMI 0.410 0.024 0.377 17.005 ***
BSANTE 0.131 0.020 0.094 6.546 ***
BSAVOIR 0.044 0.017 0.036 2.630 ***
BTEMPS 0.067 0.017 0.044 3.897 ***
BVIEQ 0.193 0.017 0.144 11.204 ***
DEFTY -0.096 0.016 -0.083 -6.162 ***
Regarding the set of variables representing the Mobility and access to in-frastructure, moving from home (TDHDOM), access to shops or local services (TLIEU3) and public transportation (TRANS6) have all positive impacts on both capabilities. The remaining variables for transportation walking (TRANS1) and car (TRANS5) only favor theAutonomy in self care, whileaccess to public services (TLIEU2) and access to supermarkets (TLIEU4) turned out to be significant and have positive effects forParticipation in the life of the household. Also, the bigger thesize of urban area(TUU), the lower the value of this last capability. This vari-able is not a measure of diversity, it is a measure of proximity, so it only means that people living in smaller areas where everything is closer have advantages leading to greater capabilities all other things being equal.
It is interesting to note that having access to key places and being able to move will increase at least one of the capabilities. These findings fit Sen’s theory very well and one of the main goals in this work is to be able to accurately operationalize his approach. When somebody can go out of his or her house by themselves and access the places s/he wants to visit via any form transport, then he or she has more freedoms and opportunities than someone who is not in the same situation.
Economics factors such as tenure (STOC) and Equivalent income (RUC) have a positive impact in at least one capability. However, standardized coefficients are among the lowest ones, specially for the income variable. This result is interesting because it shows thatEconomic Capitalis not, by far, the most important factor that determines the choices of individuals and reinforces the importance of using
Sen’s approach in our framework.
We observe that variables that encompass social policy (Personal Autonomy Bene-fits (RAL7) andKnow the Local center of gerontological information and coordina-tion(CLIC)), have negative values. Two explanations arise from this phenomenon.
On the one hand, since our analysis focuses on capabilities and not functionings, it could mean that people who have access to these resources are not able to take advantage of them. Thus, the individual would not benefit from a greater set of choices. People who receive Personal Autonomy Benefits are highly dependent and, as a result, would feel unsatisfied with their situation and would always ask for more help. On the other hand, in the case of the first variable, some individuals may not realize that they receive this allowance. In this case the coefficient will be biased, explaining why we obtained a negative value. Even if this is a plausible situation, since there is no empirical evidence, we decided to keep this variable in our model. Our results therefore suggest that better social policy implementation is needed. It should also be noted that the proportion of individuals who receive Personal Autonomy Benefits and those who know the Local center of gerontological information are both very small.
Table 2.7: Structural Equation for F2 (1stvariant)
Estimate Std. Error Standardized t value Pr(>|t|)
AGE -0.021 0.002 -0.129 -12.863 ***
GENDER 0.259 0.028 0.091 9.340 ***
COUPLE 0.122 0.032 0.043 3.774 ***
INDMEN -0.041 0.019 -0.023 -2.103 **
CENF1 0.167 0.031 0.060 5.462 ***
CENF2 0.115 0.034 0.037 3.405 ***
TUU -0.011 0.004 -0.025 -2.608 ***
TDHDOM 0.237 0.018 0.158 12.902 ***
TLIEU2 0.165 0.055 0.044 2.984 ***
TLIEU3 0.158 0.061 0.042 2.592 ***
TLIEU4 0.093 0.055 0.024 1.686 *
TRANS6 0.113 0.039 0.029 2.901 ***
RUC 0.025 0.014 0.016 1.769 *
STOC 0.058 0.012 0.042 4.705 ***
RAL7 -0.088 0.023 -0.033 -3.869 ***
CLIC -0.232 0.070 -0.031 -3.322 ***
BLIMI 0.368 0.016 0.327 23.346 ***
BSANTE 0.149 0.017 0.104 8.640 ***
BSAVOIR 0.054 0.013 0.044 4.190 ***
BTEMPS 0.029 0.015 0.018 1.927 **
BVIEQ 0.190 0.015 0.138 12.818 ***
DEFTY -0.112 0.013 -0.094 -8.821 ***
Let us note that neither of the cultural variables were relevant. Thus, neither education (highest degree, DIP14) nor social status (occupational category, TCMCS) have an effect on the capabilities. This is not really surprising since in our sample
2.9. Estimation Results most people did not have a high school diploma, so the behavior of this variable is very homogenous.
Health variables were almost all significant and had the strongest impact on both capabilities according to the standardized coefficients. Concentrating (BCONC),
Health variables were almost all significant and had the strongest impact on both capabilities according to the standardized coefficients. Concentrating (BCONC),