Conceptually, it is useful to view the multilevel regression model as a hierarchical system of regression equations. For example, assume that we have data from J classes, with a different number of pupils nj in each class. On the pupil level, we have the outcome variable ‘popularity’ (Y), measured by a self-rating scale that ranges from 0 (very unpopular) to 10 (very popular). We have two explanatory variables on the pupil level: pupil gender (X₁: 0=boy, 1=girl) and pupil extraversion (X2, measured on a selfrating scale ranging from 1–10), and one class level explanatory variable teacher experience (Z: in years, ranging from 2–25). There are data on 2000 pupils in 100 classes, so the average class size is 20 pupils.  To analyze these data, we can set up separate regression equations in each class to predict the outcome variable Y using the explanatory variables X as follows:

Y_ij=β_0j+ β_1j*X_1ij+β_2j*X_2ij+e_ij 

or

popularity_ij=β_0j+ β_1j*gender_ij+β_2j*extraversion_ij+e_ij

In this regression equation, β_0j is the intercept, β_1j is the regression coefficient (regression slope) for the dichotomous explanatory variable gender (i.e., the difference between boys and girls), β_2j is the regression coefficient (slope) for the continuous explanatory variable extraversion, and e_ij is the usual residual error term. The subscript j is for the classes (j=1…J) and the subscript i is for individual pupils (i=1…n_j). The difference with the usual regression model is that we assume that each class has a different intercept coefficient β_0j, and different slope coefficients β_1j  and β_2j. This is indicated in the equation attaching a subscript j to the regression coefficients. The residual errors e_ij are assumed to have a mean of zero, and a variance to be estimated. Most multilevel software assumes that the variance of the residual errors is the same in all classes. Different authors (cf. Goldstein, 2011; Raudenbush & Bryk, 2002) use different systems of notation. This book uses σ_e² to denote the variance of the lowest level residual errors. 

The figure to the left  shows a single level regression line for a dependent variable Y regressed on a single an explanatory variable X. The regression line represents the predicted values ŷ for Y, the regression coefficient b₀ is the intercept, the predicted value for Y if X=0. The regression slope b1 indicates the predicted increase in Y if X increases by one unit.

Since in multilevel regression the intercept and slope coefficients vary across the classes, they are often referred to as random coefficients. Of course, we hope that this variation is not totally random, so we can explain at least some of the variation by introducing higher-level variables. Generally, we do not expect to explain all variation, so there will be some unexplained residual variation. In our example, the specific values for the intercept and the slope coefficients are a class characteristic. In general, a class with a high intercept is predicted to have more popular pupils than a class with a low value for the intercept. Since the model contains a dummy variable for gender, the value of the intercept reflects the predicted value for the boys (who are coded as zero). Varying intercepts shift the average value for the entire class, both boys and girls. Differences in the slope coefficient for gender or extraversion indicate that the relationship between the pupils’ gender or extraversion and their predicted popularity is not the same in all classes. Some classes may have a high value for the slope coefficient of gender; in these classes, the difference between boys and girls is relatively large. Other classes may have a low value for the slope coefficient of gender; in these classes, gender has a small effect on the popularity, which means that the difference between boys and girls is small. Variance in the slope for pupil extraversion is interpreted in a similar way; in classes with a large coefficient for the extraversion slope, pupil extraversion has a large impact on their popularity, and vice versa.

The figure to the right presents an example with two groups. The panel on the left portrays two groups with no slope variation, and as a result the two slopes are parallel. The intercepts for both groups are  different. The panel on the right portrays two groups with different slopes, or slope variation. Note that variation in slopes also has an effect on the difference between the intercepts!  Across all classes, the regression coefficients β_0j, … β_2j are assumed to have a multivariate normal distribution. The next step in the hierarchical regression model is to explain the variation of the regression coefficients β_0j, … β_2j  by introducing explanatory variables at the class level, for the intercept:

β_0j=ϒ₀₀+ ϒ₀₁Z_j+u_0j

and for the slopes:

β_1j=ϒ₁₀+ ϒ₁₁Z_j+u_1j

β_2j=ϒ₂₀+ ϒ₂₁Z_j+u_2j

 

the equation for the intercept  predicts the average popularity in a class (the intercept β_0j) by the teacher’s experience (Z). Thus, if ϒ₀₁ is positive, the average popularity is higher in classes with a more experienced teacher. Conversely, if ϒ₀₁ is negative, the average popularity is lower in classes with a more experienced teacher. The interpretation of the equations of the slopes is a bit more complicated. The first equation states that the relationship, as expressed by the slope coefficient β_1j, between the popularity (Y) and the gender (X) of the pupil, depends upon the amount of experience of the teacher (Z). If ϒ₁₁ is positive, the gender effect on popularity is larger with experienced teachers. Conversely, if ϒ₁₁ is negative, the gender effect on popularity is smaller with more experienced teachers. Similarly, the second equation  states, if ϒ₂₁ is positive, that the effect of extraversion is larger in classes with an experienced teacher. Thus, the amount of experience of the teacher acts as a moderator variable for the relationship between popularity and gender or extraversion; this relationship varies according to the value of the moderator variable.  The u-terms u_0j, u_1j and u_2j in the equations  are (random) residual error terms at the class level. Note that in the above equations the regression coefficients ϒ are not assumed to vary across classes. They therefore have no subscript j to indicate to which class they belong. Because they apply to all classes, they are referred to as fixed coefficients. All between-class variation left in the β coefficients, after predicting these with the class variable Z_j, is assumed to be residual error variation. This is captured by the residual error terms u_j, which do have subscripts j to indicate to which class they belong.

Our model with two pupil level and one class level explanatory variables can be written as a single complex regression equation by substituting equations 2.3 and 2.4 into equation 2.1. Substitution and rearranging terms gives:

Y_ij=ϒ₀₀+ ϒ₁₀X_1ij+ ϒ₂₀X_2ij+ϒ₀₁Z_j + ϒ₁₁X_1ijZ_j+ ϒ₂₁X_2ijZ_j + u_1jX_1ij+u_2jX_2ij +u_0j+e_ij

or

popularity_ij=ϒ₀₀+ ϒ_10*gender_ij+ ϒ_20*extraversion_ij+ϒ_01*experience_j + ϒ_11*gender_ij*experience_j+ 

ϒ_21*extraversion_ij*experience_j + u_1j*gender_ij+u_2j*extraversion_ij +u_0j+e_ij