Where is traditional inference really relevant? I have been occupied with a fairly simple question regarding ordinary inference procedures where my own, and many others, practice feels slightly uncomfortable. We know that the purpose of ordinary inferential methods is to deal with the situation wehre we don't have full knowledge of the population. The populaton parameters are estimated by means of a sample and some smart way to use the data from the sample. The variation and uncertainty introduced by taking a sample is also taken into account.
But we also seem to use the same methods where there is no sample, but rather some census or at least some kind of "full" sample. Let's take an example:


*

*The proportion of women representatives in the city council is fixed, at least at this particular moment. There is no need to calculate a confidence interval for that proportion, or to see if there is a significant difference between the proportion of men and women. These numbers are both known and fixed, they could be compared, but I can se no real need for inferential methods.

*Still, if the question was slightly more complicated, as for example whether there is any relationship between gender/age/time in the council/... and income, I would be less hesitant to jump on the inference wagon. The numbers are there, they could be put into the number crunching machine, and both coefficient estimates in a regression model and their p-values wouls turn up. And I could easily fall into interpreting these as if the background was a random sample.


How do we end up here? In neither case, there is no random sample, but still many of us would be prepared to use common inferential procedures. So my questions are fairly general:


*

*Is it relevant at all to use inferential statistics in cases like this? 

*If so, are there any particular features in a situation where we can say that inference is relevant and valid? 


I have encountered reasoning like 'this census can be regarded as a sample in time' or similar, but that usually seems farfetched and hypothetical.
Besides your general thoughts, it would be interesting to get suggestions for literature on this topic. Someone must have thought about this before.
Robert
 A: 
I have encountered reasoning like 'this census can be regarded as a sample in time' or similar, but that usually seems farfetched and hypothetical.

Actually, I'd say the whole issue hinges on this point. If you consider the current council body to be your population of interest, the usual justification for statistical inference is gone. But if you consider the population to be anybody who could be in the council—in the past, the future, or hypothetical counterfactuals—then the current council body is only a sample from this population (albeit only a convenience sample).
This reasoning goes for whatever you might want to know about the data, whether it's the distribution of one variable or a relationship between several variables.
Consider, for example, the proportion of women in the council.


*

*If the current council body is the population, then the sample proportion is, exactly, the population proportion. Constructing a confidence interval or something would make no sense, because what would you be estimating? You already know the population proportion. Nor would hypothesis testing make sense, since you can tell whether any null hypothesis is true or not by inspection of the data.

*If the actual population is anybody who could be in the council, though, you could construct a confidence interval for the proportion of women in this population.


Similarly, suppose you want to know the relationship between age and income of council members.


*

*If the current council body is the population, then a scatterplot of age and income says all there is to be said on the matter. A linear model might be useful as a sort of data-reduction technique, a way of summarizing all those data points (like how the mean of a random variable summarizes the whole distribution of the variable), but it clearly cannot tell you anything new. It would make no sense to say that the correlation is "significantly" different from zero, because you don't have a mere estimate of a correlation, you have the exact correlation, and you can see whether or not it's zero.

*On the other hand, again, if the actual population is anybody who could be in the council, you could estimate or test hypotheses about the population correlation as usual.

A: The difference between your two examples is that you want to understand/predict a variable that is income in your second example. 
So in the first example you said 'I know something' and in the second you wanted to infer a variable (with a regression for example) => you end up there because you asked a very different question : knowing a set of variables age, time etc, can I infer the income ?
If you had added the variable income to infer given only gender like in example 1, it would have become an inferential problem like example 2.
You said there is no random sample. Well in both cases there is. You should consider thinking as a Bayesian :) .
