Estimation of random intercept and random slope for singleton cluster in multilevel modeling I am performing some multilevel analyses with the R package lme4.
The study design is longitudinal with the hierachical structure of observations (L1) nested into study participants (L2). I have 215 participants and the number of observations for each participant range from 1 to 5, for a total of 749 observations. Specifically, I have 43 singleton clusters (i.e., participants with one observation only).
I have a few questions concerning the singleton clusters:

*

*How the random intercept and random slope's coefficients are calculated for the singleton clusters?

*To what extent the random effects variances (intercept, slope) and the resulting R-squares are impacted by the singleton clusters?

Thank you
 A: Your first question:
It is OK to have "singletons" in your data, i.e. some levels of the factor id only appear once in the complete data. The random intercept and random slope coefficients for singleton ids are calculated in the same way as if they were no singletons. It might first seem odd, since those singletons add two new columns to the model matrix, while only adding one new row, but the data from the other ids are used, in the shrinkage effect, to find the best common fit for all ids, including the singletons.
Your second question:
As said above, the singletons are used in the same way as other ids, and therefore those singletons will also, in general, improve the fit of the variances of the random effects as well as the coefficient of determination $R^2$ (provided a random effects model is a good approximation for the population). They just don't have that much influence, since they contribute only a single observation.
Maybe a picture helps. Let's consider the single row of a singleton with the slope $s$ and intercept $i$ with all the other coefficients fixed; say it is the $k$th row. Then, the equation for this single row, when we combine $y_k$ and all the constants into a new $y^\prime_k$, has the form:
$$
y_k^\prime = sx_k + i + \epsilon,
$$
i.e. it is a linear equation with two free variables $s$ and $i$ and some independent error $\epsilon$. That would define the straight line (fuzzy because of $\epsilon$) in the following picture:

The concentric ellipses depict the density with the covariance matrix that has been optimized with the other levels (i.e. ids). Now, intuitively, the optimization will find a point near the (blue) line that has the largest density and at the same time slightly adjust the (red) density to accommodate for this new point, just a little bit, given that this covariance matrix is used for all ids.
