Neurociencia y neuroeducación

Multilevel modelling was used in the analyses contained in each empirical results chapter.

Multilevel models extend single level regression models (which assume that observations are uncorrelated) and are used for analysing data with complex patterns of variability as observed in nested, or hierarchically structured, data (Snijders & Bosker, 2012).

Multilevel models take into account the potential non-independence of observations within groups. In the context of this project, multilevel models account for the fact that individuals selected randomly from within one country were more likely to have similar quality of life compared to individuals that were selected randomly from across all 13 countries. They also enable the investigation of between group variability and the factors associated with this variability (Diez Roux, 2002). Without taking into account these sources of variability, the efficacy of the estimates is decreased and the standard errors may be underestimated (Maas & Hox, 2004). It should be noted that, in multilevel models, the groups in the sample are considered to be a random sample from a (real or hypothetical) population of groups and therefore inference can be made to that

population of groups (Snijders & Bosker, 2012). Thus, in this study the 13 countries were theoretically viewed as a sample of countries within continental Europe.

5.5.1.1 Multilevel structure of the data

The data were considered to consist of a two-level hierarchy containing individuals (micro

‘level 1’ units) nested within countries (macro ‘level 2’ units). Therefore, variability was considered to be present both between individuals and between countries.

5.5.1.2 Fixed and random effects

Multilevel models contain both fixed and random effects and are therefore often referred to as mixed effects models (Snijders & Bosker, 2012). Fixed effects refer to regression coefficients (intercepts or covariate effects) that are not allowed to vary randomly across higher level units (Diez Roux, 2002). On the other hand, random effects are regression coefficients (intercepts or covariate effects) that are allowed to vary randomly across higher level units. For example, in this study where individuals were nested within countries in a random intercept multilevel model, country effects can be thought to vary randomly around an overall mean within a normal distribution of potential country effects. Thus, in a random intercept model, the mean quality of life in each country is allowed to vary, but if other covariates are included (for example education level) their association with quality of life is assumed to be the same (fixed) in each country. A random slope model allows the effect of the covariates to vary across groups. Figure 5.1 illustrates the difference between random intercept and random slope models.

Figure 5.1: Demonstration of random intercept (left) and random slope (right) multilevel models showing the hypothetical effect of education level on quality of life in different countries (represented by the lines)

5.5.1.3 Number of level two units

A limitation of using multilevel models here was the low number of countries, or level two units. However, the 13 countries included were considered enough to allow the use of multilevel modelling. It is recommended that if the number of higher level groups is less than 10, a single level model with country fixed effects should probably be used (Snijders

& Bosker, 2012). However, there is no agreed number of higher level units that is

considered sufficient (Maas & Hox, 2004). Models containing less than 50 level two units have been shown to produce reliable estimates of the regression coefficients, their standard errors, and the variance components. However, biased estimates of the between country variance may result (Maas & Hox, 2005). As the key results of this project were the regression coefficients and not the level two statistics, the benefits of using multilevel modelling were considered to outweigh this risk. However, when results relating to the level two variances are reported, these should be interpreted with caution.

Advice was also sought from two statisticians with expertise in multilevel modelling to confirm the appropriateness of this strategy and sensitivity analysis was also conducted using single level regression models.

A number of papers are also published using SHARE data which use multilevel modelling, with as few as 11 countries (Brandt et al., 2012; Hank, 2010; Reinhardt et al., 2013).

Often, multilevel models are estimated using maximum likelihood, which is the default estimation method in Stata. Maximum likelihood estimation seeks parameter values that, given the data and the choice of model, produce predicted values which are most comparable to the observed values (Baayen et al., 2008). Snijders & Bosker (2012) recommend the use of restricted maximum likelihood estimation (REML), a variation of maximum likelihood estimation that is more precise and produces less biased standard errors for mixed effects models that contain a low number of higher level groups.

Therefore, all multilevel models were estimated using REML.

The low number of countries prevented the optimal use of random slope models (Brandt et al., 2012). However, as this thesis was focused on the overall association between life course socioeconomic position and quality of life and second, differences between welfare regimes, random slope models were not required. The analysis strategy for investigating the influence of the welfare regime is described further in section 5.6.

5.5.1.4 The intraclass correlation

An advantage of using multilevel models is that the total variance in an outcome at the individual level can be partitioned into the variance occurring within, as well as between groups (Diez Roux, 2002). Therefore, in this project the total variance in the mean quality of life across individuals was decomposed into the variance observed within and between

countries. The intraclass correlation is a measure of the internal homogeneity of the level-two units (in this case the country level) according to the outcome variable (Snijders

& Bosker, 2012). Here, it can be understood as the proportion of the total variance in quality of life that was accounted for by the country level, or the correlation between the quality of life of two randomly selected individuals from the same randomly chosen country. To calculate the intraclass correlation the population variance between the country units is divided by the total variance, which is composed of the population between-country variance plus the population within-country variance (Snijders &

Bosker, 2012).

The intraclass correlation usually varies from 0 to 1, where a value of 0 represents no more variation in quality of life between countries than would be expected by chance. A value of 1 means that all of the variance is due to the grouping variable, thus all

individuals within a country would share the same quality of life once the country level variance has been accounted for (Merlo et al., 2005). Hence, an intraclass correlation of 0.1 means that 10% of the variance in quality of life is at the country level. The first step in generating the intraclass correlation is to calculate an empty ‘null’ model containing only a random intercept for each country and no explanatory variables. In the empty model only the intercepts are allowed to vary across groups, therefore the outcome variable (quality of life) is the sum of the general mean quality of life, a random effect at the group (country) level, and a random effect at the individual level (Diez Roux, 2002;

Snijders & Bosker, 2012). Likelihood ratio tests (also known as deviance tests) can also be used to compare the model fit using a multilevel regression model compared to a single level regression model (Baayen et al., 2008).