How to estimate the change in prevalence of a trait in the population from non-random samples? Suppose I am interested in the prevalence of drug use in a population over time, but I am unable to conduct audits (sample some transactions at random and check to see if drugs were exchanged). I do have data on seizures, needles or vials found in the park, as well as crime hotline reports, but these are not perfect because the policing technology, drug delivery methods, and reporting norms change, so these cannot reliably tell me about changes over time. If all these were immutable, I could implement a dynamic factor model where  all these variables are a function of one or more unobserved latent factors.
How is this problem usually handled? Are there any statistical procedures that can identify how the latent factor changes over time?
 A: For the seizure data (arrests?) do you have unique identifiers to tell if a person has had drugs seized multiple times? For a cross-sectional measurement of the prevalence I've seen capture-recapture methods used with arrest data (Bouchard, 2007; Rossmo & Routledge, 1990). I'm not familiar with applying these techniques over time, but I am sure it has been broached in other fields (see Utilization of Capture-Mark-Recapture for the Study of Recruitment and Population Growth Rate (Pradel, 1996) for one quick example I was able to find.)
Another option criminologists are fond of are vital statistics, which I believe they do have classifications for particular over-doses. Emergency room visits are used sometime as well - but that takes some insider information from the hospital itself (the vital stats are public).
These ultimately have the same biases though in changes in reporting practices that any of your measures do. Even vital statistics appear to be subject to the prejudices of the examiners (Noymer et al., 2011). This is partly why surveys are popular, and a few ones for cities in the US do exist, e.g. the ADAM series or the DAWN series from ICPSR. So if you are lucky you could see if your city is available for comparison to the other measures.
