I'm confused about the interaction of binary predictor values in multiple linear regression. Here's an example to illustrate my problem. Say that I want to investigate the relationship between life expectancy (response) and whether someone works the night shift and has a pet (both binary 0 or 1 predictors). Add to this that I want to control for the effect of gender, because I know that being female increases your life expectancy, so I put male/female in as a binary predictor as well. So, my model is: life expectancy (continuous)=night shift+pet+gender.
Now, through the nature of my data, I have male respondents who code for both pet presence and absence, but all females have pets (this is obviously a bad study design - but ignore this for now and assume this is the only data I have). For the night shift, all combinations of absence/presence of night shift and gender exist. Now in my regression, I find a significant negative coefficient for having a night shift - and then interpret this to mean that night shift lowers life expectancy, irrespective of gender. This part is straightforward. But for pets, I find no significant effect. However, my model could really only compare pet/no pet in the male category, because all females have pets. So my interpretation here is that males do not increase life expectancy from having a pet, but the model cannot tell us how this affects life expectancy in females. In fact, the potential positive effect of a pet for a female may be entirely swallowed up by the gender variable, since all females have pets.
My questions are:
Is there in general a problem with running a model of this kind, more specifically I wonder if this counts as non-independence of the predictor values (all females have pets)?
Is my interpretation correct?