# Dealing with poorly estimated/missing explanatory variable values in GLMs

Context

I am using generalised linear models to analyse some ecological data looking at the relationship between the population density of moth larvae and the prevalence (%) of viral mortality in the previous year (i.e. testing for "delayed density-dependent mortality"). The data are from a number of different sites, sampled across years.

Problem

The data are very sparse at times, and so in some sites at some years the density of individuals is very low or 0. Therefore, for these cases % disease mortality is either poorly estimated (e.g. with 2 individuals you can only have 0, 50 or 100% disease mortality values), or cannot be estimated if there were 0 individuals.

Questions

1) When % disease mortality estimates are based on very small sample sizes, although the estimates are all that is available to work with, are there any useful statistical techniques to help explore how variation in these explanatory variable values/their poor accuracy might affect the results of the model? e.g. I have recently been learning some Bayesian methods, and I am aware that there may be some simulation approaches that may help, although I've only encountered these so far in the context of missing dependent variable data. I am also very dimly aware of the existance of sensitivity analyses etc. Any pointers as to which methods might be useful would greatly appreciated.

2) When sample size is 0, is it valid to use 0% as an estimate of disease mortality? Obviously statistically if there is no data you cannot estimate anything, but is it defensible to argue that as the estimate is a measure of the prevalence of disease, if there are no individuals around then there can be no disease, and so 0% is a valid value? i.e. from a biological point of view can it be argued that the estimate is attempting to measure how 'much' disease was around in the previous year, and if there are no individuals then there was by defintion no disease, and 0% is valid? Similarly to question 2, if any statistical techniques may help in coping with this 'missing' data, please point me in the right direction.

Thank you very much for any help.

• When the sample size is $0$, the disease has $100\%$ prevalence because--by your very argument--the absence of disease has $0\%$ prevalence :-).
– whuber
Mar 19, 2014 at 14:19
• I don't think I can follow your argument/logic. Are you able to state this in a different way? Mar 21, 2014 at 14:15
• Whether you count cases of mortality or not doesn't matter, because they are two equivalent ways of representing the same outcome. Your argument, applied to counting the number of survivors (rather than the number of deaths), implies that with a sample size of 0 you should use 0% as your estimate of survivorship. The obvious inconsistency between this and the 0% mortality claim made in (2) shows that your argument is inherently flawed.
– whuber
Mar 21, 2014 at 14:27

## 1 Answer

Joe Ibrahim and co-authors had a number of good papers about ten years ago about missing covariates in generalized linear models (see the technical paper and an overview). In the former, they propose is a hybrid of the frequentist EM algorithm and Bayesian Gibbs sampling producing a Monte Carlo EM. In the latter, they compare it to other methods such as fully Bayesian, multiple imputation, etc. (Ibrahim is one of the leaders on computational Bayesian techniques who published a book on this; he expresses his thoughts very clearly, although the technical content is dense).

Alternatively, survey statisticians have been developing what they call small area estimation models specifically for the situations when the sample size in any given cell is small or zero. SAE models combine direct estimates from the data with model predictions based on some sort of (generalized linear) mixed model formulation with cells as random effects. With sufficient sample sizes, SAE models automatically lean towards the direct estimates; with small cell sizes, SAE models automatically lean towards the model-based estimates. In the latter case, SAE models admit that there may be biases due to model specification, so they incorporate the expected squared bias into the MSE estimates.

Arguably, these methods may have to be combined with the measurement error literature, as you are using predictors measured with error. Luckily, you can quantify the variance of that error, and that simplifies things quite a bit.