# Regression with missing covariate data

My question is related to regression with missing covariate data. I have a sample of count data representing number of explosion incidents at each year. First $M$ observations were made by considering one criteria and the last $K$ were made considering two criteria. E.g. for the first $M$ years explosions were recorded if there were some human injuries and then, due to the change of regulations, explosion were recorded if they caused human injuries OR the damage were higher than, say, $D$. I introduce two binary covariates: $X_{1}$ for human injury criteria and $X_{2}$ for damage criteria. So, I get that the first covariate $X_{1}$ is always observed and equal to 1 for the first $M$ years but not for the next $K$ years. The second covariate $X_{2}$ for the first year is not observed at all (so it might be 0 or 1), while for the last $K$ years both covariates are not observed but I know that at least one of them is equal to 1.
At first I though that I just put a Bernoulli distribution on these two covariates and integrate out them. Then I perform Bayesian analysis. So, at first I state the model (for simplicity, assume we have just one covariate) $$Y|\theta_{1},\theta_{2}\sim \mathrm{Poisson} \left (\mathrm{exp}\left [ \theta_{1}X_{1}+\theta_{2} \right ] \right )$$ $$X|p\sim \mathrm{Bernoulli}(p)$$ Then perform integration over $X_{1}$: $$\pi(Y|p,\theta)=\frac{1}{Y!}\left ( \mathrm{exp}\left ( -e^{\theta_{1}+\theta_{2}}+Y\left ( \theta_{1}+\theta_{2} \right ) \right )p + \mathrm{exp}\left ( -e^{\theta_{2}}+Y \theta_{2} \right )(1-p) \right )$$ Having this, I form likelihood, state priors and obtain estimates. As I understand, such approach is a so called "Missing completely at random or MCAR". However I understand that in my problem the missingness model is different from MCAR, but I do not know what model would be more appropriate. Any advise would be appreciated.

• It is hard to say whether you have MCAR data (where the distribution of missingness does not depend on the observed covariate), MAR (Missing At Random; the distribution of missingness depends on observed but not on missing covariates) or MNAR data. For instance, the regulatory change may have decreased the incidence of property damage, given that this was recorded after the change. This article may be helpful: psycnet.apa.org/… Nov 29, 2012 at 14:00
• Well, the impact of new regulations is not known. I assume, that it has no impact, since gas pipelines, for which explosions were recorded, were not renewed, i.e. were the same for the entire observation period. It might have impact on the maintenance, however it is impossible to evaluate for the data that is available to me. Nov 29, 2012 at 14:23
• In that case you probably do have MCAR data. The article I linked to discusses Bayesian approaches to this; I honestly don't know how helpful it will be... Good luck! Nov 29, 2012 at 14:28