21 Interpreting Coefficients
21.1 Interpreting your models (they won’t interpret themselves!)
So, multilevel models sure are great, but they can also make interpretations much more challenging. You’ve done OLS regression, so you have an understanding of how to interpret regression coefficients. However, adding additional levels means that some of our interpretations also need to change. This document is intended to provide a brief guide to how to do that.
Coefficients and indices at various levels of the model
But before we even start, we need to talk about how we use different coefficients and letters at different levels of the model. There isn’t a single convention for how to do this, but we’ll try to be consistent at least in this class.
We’ll distinguish between two basic types of models, those that are multilevel and not longitudinal, and those that are longitudinal.
As a canonical example of the first type, let’s consider the model we use in class, namely
\[\begin{aligned} mathach_{ij} &= \beta_{0j[i]} + \beta_{1j[i]}SES_i + \varepsilon_i, \\ \beta_{0j} &= \gamma_{00} + \gamma_{01}sector_j + u_{0j},\\ \beta_{1j} &= \gamma_{10} + \gamma_{11}sector_j + u_{1j},\\ \varepsilon_i &\sim Normal(0, \sigma^2_\varepsilon) \\ \begin{pmatrix} u_{0j}\\ u_{1j}\\ \end{pmatrix} &\sim N \begin{bmatrix} \begin{pmatrix} 0\\ 0 \end{pmatrix}\!\!,& \begin{pmatrix} \sigma^2_0 & \rho\sigma_0\sigma_1\\ \rho\sigma_0\sigma_1 & \sigma^2_1 \end{pmatrix} \end{bmatrix} \end{aligned}\]
Here are the features of the model to attend to. When referring to students (or other first-level units), we will use \(i\) as a subscript. \(X_i\) will indicate a measurement taken for the \(i\)th student. When referring to schools (or other second-level units), we will \(j\) as a subscript. \(X_j\) will indicate a measurement taken for the \(j\)th school. When we expand these models to include third-level units (e.g., districts), we will use the subscript \(k\) for these units. I don’t intend to go past that, although we could. When we introduce cross-classified models (i.e., models will non-nested hierarchies) we’ll pick subscripts that are intended to be evocative.
We’ll also try to be consistent when using coefficients. We’ll use the letter \(\beta\) (beta) to indicate regression coefficients measured at the first level. We’ll use the letter \(\gamma\) (gamma) to indicate regression coefficients measured at the second level. Eventually we’ll use the letter \(\xi\) (xi, or ksi) to indicate regression coefficients measured at the third level.
When we subscript regression coefficients, we’ll need a number of subscripts equal to the level of the model at which this coefficient has been entered. The first subscript will indicate the level-1 coefficient with which this particular coefficient is associated, the second subscript will indicates the level-2 coefficient with which it is associated, and so on. This means that each coefficient will have a number of subscripts equal to the level of the model. As a really complicated example, if a coefficient is labeled as \(\xi_{021}\), this indicates that the coefficient is the first slope coefficient (the 1 at the end) in a model for the second level-2 slope coefficient (the 2 in the second position) in a model for the level-1 intercept. Similarly, the first subscript in a random effect will indicate the level-1 coefficient with which it is associated, and the second will indicate the level-2 coefficient with which is is associated. Random effects will always have one fewer subscript than the coefficients at that level. As you can imagine, subscripts quickly get out of hand as we introduce more and more levels to a model.
We’ll use \(\sigma^2_p\) to indicate the variance of the level-2 residual for the \(p\)th random effect (starting at 0 for the intercept). I’m not yet sure how to do the subscripting at level-3, and for now am hoping to just wing it. The correlation between the \(p\)th and \(q\)th random effects will be subscripted \(pq\), and correlations will always be identified with a \(\rho\) (rho, not p).
Longitudinal models are similar, except for the subscripting. I’ll always (probably) subscript the first level with \(t\), for time. The second level will become \(i\) (assuming that we’re looking at growth in students or other individuals), followed by \(j\) for the third level (we probably won’t include a fourth level).
Interpreting fixed effects
Okay, that was complicated, although I think writing down definitions and rules is often more challenging than applying them. Now let’s practice some interpretations, going back to our model.
At the first level, we interpret (almost) exactly as we would in a standard regression model. If we have
\[mathach_{ij} = \beta_{0j[i]} + \beta_{1j[i]}SES_i + \varepsilon_i,\]
then we interpret \(\beta_{0j}\) as the predicted value of \(mathach\) for a student of 0 SES (which represents the grand mean) who is located in school \(j\). Because this is a multilevel model, different schools have different intercepts. Similarly, we can interpret \(\beta_{1j[i]}\) as the expected difference in math achievement associated with a one-unit difference in SES for students in school \(j\). We don’t interpret it, but \(\varepsilon_i\) indicates the difference between what we observed for this student and what we predicted based on her or his school and SES.
We interpret the level-2 units depending on the coefficients they predict. For the school-intercept we have
\[\beta_{0j} = \gamma_{00} + \gamma_{01}sector_j + u_{0j}.\]
We can interpret \(\gamma_{00}\) as the predicted intercept for schools for which \(sector = j\) (i.e., public schools). We can interpret \(\gamma_{01}\) as the predicted difference in school intercepts between Catholic and public schools. Although it’s less common, we can also interpret the residual for school \(j\), \(u_{0j}\), because you can’t tell me what to do. \(u_{0j}\) represents the difference between the observed/inferred intercept for school \(j\) and the predicted intercept.
Turning to the model for the slope, we have
\[\beta_{1j} = \gamma_{10} + \gamma_{11}sector_j + u_{0j}.\]
Here \(\gamma_{10}\) is the predicted slope for SES in public schools, while \(\gamma_{11}\) is the mean difference in slopes between Catholic and public school. Finally, \(u_1j\) is the difference between the slope observed/inferred for school \(j\) and the slope predicted by the model.
We can also interpret these coefficients at the student level. Rewrite the model by substituting \(\beta_{0j} = \gamma_{00} + \gamma_{01}sector_j + u_{0j}\) and \(\beta_{1j} = \gamma_{10} + \gamma_{11}sector_j + u_{1j}\) to obtain
\[\begin{aligned} mathach_i &= \gamma_{00} + \gamma_{01}sector_{j[i]} + u_{0j[i]} + (\gamma_{10} + \gamma_{11}sector_{j[i]} + u_{1j[i]})SES_i + \varepsilon_i \\ &= \gamma_{00} + \gamma_{01}sector_{j[i]} + \gamma_{10}SES_i + \gamma_{11}sector_{j[i]}SES_i + (u_{0j[i]} + u_{1j[i]}SES_i + \varepsilon_i). \end{aligned}\]
Now we can interpret these coefficients as in a typical one-level linear regression model.
\(\gamma_{00}\) is the predicted mean value of \(mathach\) for students of \(SES = 0\) in public schools;
\(\gamma_{01}\) is the predicted difference in \(mathach\) between students of \(SES = 0\) in Catholic schools and similar peers in public schools;
\(\gamma_{10}\) is the predicted difference in \(mathach\) associated with a one-unit difference in SES for students in public schools; and
\(\gamma_{11}\) is the predicted difference in the above difference between students in Catholic schools and students in public schools.
Either interpretation is acceptable, and you should base your decision on how you’re framing your question.
Interpreting variance-covariance parameters
Now we’re going to turn to the variance-covariance matrix for the random offsets, namely
\[\begin{aligned} \Sigma = \begin{pmatrix} u_{0j}\\ u_{1j}\\ \end{pmatrix} &\sim N \begin{bmatrix} \begin{pmatrix} 0\\ 0 \end{pmatrix}\!\!,& \begin{pmatrix} \sigma^2_0 & \rho\sigma_0\sigma_1\\ \rho\sigma_0\sigma_1 & \sigma^2_1 \end{pmatrix} \end{bmatrix} \end{aligned}\]
The variance of a random offset (e.g., \(\sigma_0^2\), the variance of \(u_{0j}\)) represents how variable the coefficient associated with that coefficient is, conditional on the variables in the model. The correlations (e.g., \(\rho\), the only correlation in this model) represent the tendency of the random offsets to covary, i.e., to be associated with each other.