# Logistic regression when only TRUE values are observed, but a control sample exists

I'm involved with a creel survey. We have reason to believe that anglers are keeping the larger fish they catch, or at least that a relationship exists between the size of caught fish and the probability of retention.

We have the potential for two datasets:

• A creel sample consisting of the lengths of angler-kept fish (obtained by interviewing anglers)
• A "control" sample consisting of the lengths of fish that our crew catches using similar methods.

Let's make a couple of assumptions:

• The creel sample is unbiased and representative of the fish that anglers keep
• The control sample is unbiased and representative of the fish that are subject to capture by anglers.

It seems to me that a logistic relationship "should" exist between fish size and retention probability. Instead of breaking lengths into bins and treating this as a categorical problem, I'd love to be able to use something like logistic regression as a simple way of modeling the relationship. The obvious problem is that we only have kept=TRUE fish in the creel sample.

Is there a way of incorporating the control sample data that will end up with a valid (and interpretable) logistic regression? If I treat the control sample as having an associated response variable kept=FALSE, will the slope coefficient at least be meaningful?

• This appears a standard "selection bias" problem in observational studies, certain techniques like IPTW or double robust estimators exist to address such cases. That said, can you please state what is the final research question? Jan 29, 2019 at 21:30
• (I added some tags that I think they are relevant, please feel free to edit/remove them if you think other options are more relevant) Jan 29, 2019 at 21:32
• Is there a minimal length based on which fish should be kept? Say, all fish less than 20cm in length should be NOT be kept and all fish greater than or equal to 20cm in length should be kept? If there is no such limit, there is nothing wrong with anglers keeping the larger fish. But if there is, then some of the fish retained by anglers will be less than 20 cm in length. So can't you just compare the (true) proportion of fish less than 20cm against 0, since you know your crew would not keep any fish less than 20cm in length? Jan 30, 2019 at 2:41
• But this all may depend on species of fish - different species may be subjected to different size limits? Jan 30, 2019 at 2:42