# An example: educational attainment in local labour markets in Norway

Let us end the theoretical part of the module by reviewing an example of a multilevel analysis that contains all the points we have covered so far. This will hopefully help to clarify the concepts and the interpretation of multilevel models.

The theoretical point of departure is modernization theory. The main hypothesis is that the degree of modernization of the local labour market will affect the status attainment process, including educational attainment. Two testable derived hypotheses follow from this:

In labour markets dominated by the (traditional) primary sector, the level of individual attainment will be lower than in more modern labour markets.

The stronger the traditional sector, the stronger the effect of social background on individual attainment.

The data stem from a 10 per cent sample of the Norwegian 1960-1980 censuses. Data for the analysis of status attainment were available for a subsample of men and women aged 30-36 in 1980. The total sample size at level 1, the individual level, was about 40,000. The 19 Norwegian counties (shown below) constitute level 2 as proxies for the local labour markets.

The dependent variable is child’s education (ED) measured in years. The model below includes three explanatory variables: mother’s education, father’s education and occupational status. The black arrows indicate the effects of level 1 and the blue ones the effects of one level 2 variable, the percentage of people in the county who are employed in the primary sector. This is the indicator of modern versus traditional labour markets.

Figure 3.6. Counties in Norway

Figure 3.7. Model for explaining a child's education

Let us look at the formalization of the model, first in separate equations for the two levels, then in a single equation.

Level 1 (the micro model):

ED_{ij} = β_{0j} + β_{1}Med_{ij} + β_{2j}Fed_{ij} + β_{3}Focc_{ij} + e_{ij}

Level 2 (the macro model):

β_{0j} = β_{0} + β_{4}Primary_{j} + u_{0j}

β_{2j} = β_{2} + β_{5}Primary_{j} + u_{2j}

One equation version:

ED_{ij} = (β_{0} + β_{4}Primary_{j} + u_{0j}) + β_{1}Med_{ij} + (β_{2} + β_{5}Primary_{j} + u_{2j})Fed_{ij} + β_{3}Focc_{ij} + e_{ij}

Simplified, with random part in the second line:

ED_{ij} = β_{0} + β_{1}Med_{ij} + β_{2}Fed_{ij} + β_{3}Focc_{ij} + β_{4}Primary_{j} + β_{5}Fed_{ij}Primary_{j} + u_{0j} + u_{2j}Fed_{ij} + e_{ij}

Before we present the estimated parameters, let us take try to achieve a deeper understanding of the random coefficient for father’s education. In the figure below, separate OLS estimates are presented for the coefficient in each of the 19 counties

Figure 3.8. The OLS estimates for the coefficient in all counties

The regression coefficients vary from around 0.17 to 0.37, and they are positively related to the primary sector percentage, although the pattern seems to deviate slightly from linearity.

Let us present the estimates and interpret the coefficients:

E^{^}D_{ij} = 11.277 + 0.498Med_{ij} + 0.372Fed_{ij} + 0.026Focc_{ij} - 0.056Primary_{j} + 0.016Primary_{j}Fed_{ij}

All individual level variables are grand-mean centred. This implies that the regression constant, 11.277, can be interpreted as the predicted mean years of education for counties, with the other variables at their mean values. An increase of 10 per cent in primary employment is expected to decrease the predicted mean years of education by about half a year (10*-0.056=-0.56). Mother’s and father’s education are measured in levels and have a roughly equal overall effect (not shown), while the effect of father’s occupation is small. The effect of father’s education in the above equation is not the overall or main effect of the variable, but the effect when the primary percentage is set to zero. Those who do not fully understand this can go back to the review of interaction effects in the OLS regression part at the beginning of the module. Thus, the effect of father’s education (FED) is 0.372 when the primary percentage is set to zero. The coefficient of the interaction term, 0.016, shows that the effect of father’s education increases by that amount for each per cent increase in primary sector employment. Thus, an increase of 10 per cent in the primary sector will increase the effect of father’s education by 0.16, to 0.53. Both the direct effect of primary sector employment on the intercept and the interaction effect lend support to the two hypotheses.