Should i exclude random effects from a model if the random effect itself has missing data?

Question

I have found a number of questions relating to missing data in fixed effects and how packages like lme4 handle this in mixed models. For example, here. However, I am keen to hear people's advice for situations where the missing data is in the random effect itself? If the random effect itself has too many missing values does this preclude you from using it as a random effect, and should you exclude it form your model altogether?

For example, I have a dataset of bird captures at a single site over a 20 year period. During each field trip birds are captured, weighed, sexed, measured, etc. Field trips occur at the same location 1-6 times annually. For the first 10 years of this data no information was collected on individual bird identification, and then progressively over the next 10 years more and more birds were banded with an individually identifiable band. So, the dataset consists of repeated measurements on the same individual birds through time (acknowledging that some birds persisted in the data for a short time and others for a long time).

I would now like to look at the association between rainfall (predictor variable) and body condition (outcome variable). In a mixed model context it would be important to include individual bird identification as a random intercept in this model. However, as described above, individual bird identification has about 60% missing values, with birds only being individually identifiable for about 40% of observations. What should one do in this situation? Although, individual bird identity should technically be included as a random intercept, I would argue it has too much missing data and should be excluded. Keen to hear how others would approach this?

Answer 1

Sorry for my late response. I see two different situations.

First, suppose all the unidentified birds of the first ten years are also observed during the last ten years, meaning there are no data of completely unidentified birds (probably that is not true, as birds died or left the area during the first ten years). But if this situation would hold, I guess that imputation of bird ID could theoretically be possible, but I wonder if it would be feasible if you have many birds. And such imputation may be done with the R package MICE, as Robert Long said in his comment.

Second, suppose part or many of your first-ten-years birds are not recaptured during the second ten year period. My guess is that it is impossible to impute completely new bird ID values for such data. I browsed the MICE book by Stef van Buuren (https://stefvanbuuren.name/fimd/) but there is no example given for such a situation. Look at paragraph 7.3 of that book. Also the Goldstein book referred does not give a clue in this respect, as far as I can tell.

Maybe the following (stupid) idea is worth considering. Cameron and Miller (2015) show a formula for adjusting the std. errors of linear regression coefficients estimated by OLS, in case you "suffer" from clustered data. That is, use OLS (instead of a random effects approach that you were asking about) and then adjust the std. errors of your regression coefficients for the fact that the data are clustered within birds. The formula (6) they show reads:

$\tau_k \simeq 1 + \rho_k \rho_e (\bar{N_g}-1)$

$\tau_k$ gives the inflation factor for the variance (squared std. error) of the regression coefficient of independent variable $k$ in your model equation; that variance should be multiplied by $\tau_k$. $\rho_k$ is the (intra-class) correlation of two observations of independent variable $k$ taken at random from the same bird. $\rho_e$ is the (intra-class) correlation of two error terms taken at random from the same bird. These two correlations can be estimated for the second-ten-year observations only. But maybe it makes sense to assume that both correlations also hold for the unidentified birds of the first ten years. $\bar{N_g}$ is the average "group-size" meaning the average number of observations taken from the same bird. This average is unknown for birds only observed during the first set of ten years; but also for those which are recaptured in the second period of ten years, because you do not know how often these birds were observed during the first ten year period. So, the "group-size" is known only for those birds which were first observed during to later years. The formula shows that the larger the group size is, the higher the inflation is. So, if I were to guess the average group-size, I would like it to be relative large, to be on the safe side for the inflation factor. However, Cameron and Miller also mention that, in case the group-sizes differ, it is better to use the following quantity instead of $\bar{N_g}-1$ in the above formula:

$V[N_g] / \bar{N_g} + \bar{N_g} - 1$

where $V[N_g]$ is the variance of the group-sizes across birds. Again this value is difficult to determine, but maybe an educated guess is somehow possible. Note that the more the group-sizes differ across birds, the higher the inflation factor should be!

The funny thing is that, given (estimates) of the group-sizes and the two correlations mentioned, you could apply a OLS regression model and then next inflate your std. error to "take care" of the clustered nature of your data. So you do not have to know the bird ID as long as the estimates of the two correlations and group-sizes can be trusted for both ten year time periods.

Cameron and Miller paper see https://jhr.uwpress.org/content/50/2/317.short