Have population, use inferential statistics? Also, non-normal dependent variable, what to do? Background: We are looking at parental leave in Iceland. We are particularly interested in whether the economic crisis and the resulting changes in parental leave legislation affected the time taken for parental leave.
We have reason to believe that the effect of the crisis / new laws will be different for mothers and fathers (who have equal rights to a leave), will depend on income and education, and that there might be an interaction between factors (e.g., that the length of leave for fathers would be independent of income before the new laws, but would start to depend on income after the laws were passed).

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*We actually have not just a sample, but the entire population (about 50000 kids). Do we even need inferential statistics at all? Can we just describe the results either numerically or graphically, because whatever difference there is, that is the actual difference in the population?


*If we do need to do inferential statistics, then we have some potential problems:
First, the dependent variable (length of leave) is not even close to normally distributed. Instead it is multimodal, e.g., people are likely to take 0 days, 30 days, 60 days, 90 days etc. but not, e.g., 3 days or 34 days. I cannot transform this distribution to look anything like it is normally distributed.
I initially considered using some kind of non-parametric test that looks at differences in medians, but the problem is that the medians might actually always be close to same (e.g., 90 days) but the distribution is changes nonetheless.
I then considered binarizing the dependent variable (e.g., takes less than standard leave vs. takes standard leave or more). This would allow me to use logistic regression and the weirdness of the distribution would be gone. I am fine with this.
However, I am interested not just in main effects (e.g. main effect of time and main effect of income) but also in interactions (e.g., the interaction between time and income). I am not sure how to deal with interactions in logistic regression, especially since I might have to treat the factors as categorical (e.g., I am not expecting the length of leave to, say, linearly increase or decrease with time -- I am expecting a curvilinear relationship between length of leave and time).
What to do?
I mainly use SPSS for analysis, in case that is relevant.
 A: While you may have a population, the central question is whether it's actually the population about which you wish to make statements (at least ones relating to population effects/differences). If it isn't, you might still in many cases treat it like a sample. If it is actually the target population, there's no need for statistical inference, just describe the differences. 
Frequently, for example, it turns out that the wish for example, is to say something that might be relevant in the near future, or to guide policy, which suggests a notional (and perhaps not physically realizable) population somewhat different from the observed. In that case there might arguably be a reason to continue with statistical inference.
The dependent variable itself is not assumed to be normal in ordinary regression; the conditional distribution is. Even when the assumptions are reasonable, bimodal/multimodal marginal distributions are common when there are two or more groups with different means. 
You should check your normality assumption by examining residuals, not the raw response.
You talked about medians but worried the medians might be close even though the distributions might differ overall. You might consider quantile regression for a representative set of quantiles.
Dichotomizing your response is not generally regarded as a good idea. [However, interactions (whether categorical$\times$categorical, categorical$\times$continuous or continuous$\times$continuous) work much the same way in logistic regression as they do in ordinary regression; their effect on the linear predictor, in particular, is understood in almost the same way.]
A: You write "I then considered binarizing the dependent variable (e.g., takes less than standard leave vs. takes standard leave or more). This would allow me to use logistic regression and the weirdness of the distribution would be gone. I am fine with this."
That sounds like a good starting point. You can definitely include interactions in a logistic regression. For example, you could predict "took a leave of length >= threshold" using indicators for income, gender, education, an indicator for whether we are in a timeperiod after the new laws have been passed, interactions of that indicator with income, interactions of that indicator with gender...  actually if you are allowed to share coefficients, it would be pretty cool if you could run that model and post the results here.
Instead of an indicator for whether the new laws have been passed, you could have year-specific effects -- and these could be interacted with income, gender, etc if you have enough data.
If you have data on where the person lives, you could run a hierarchical logistic regression in which you have location effects that are modeled as coming from some common distribution. 
