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I have a data set of reported food-borne illnesses and we're trying to determine what environmental conditions during food cultivation led to high bacterial counts in the food, and thus caused the illnesses. Unfortunately, I only have data of foods that caused confirmed illnesses. I requested that we go back and "randomly" sample from food tags that did not cause a reported illness but am not allowed to do so for various reasons. Even that would have had problems (because just because an illness is not reported doesn't mean it didn't occur), but at least this would have given me some negative observations.

I was originally planning to model these data using a logistic regression but I am stuck at what to do now. Without negative observations, I can only really provide univariate descriptive statistics, right? I'm hoping that someone else has had this problem and perhaps there's some model I haven't heard of before that can handle this. Thank you.

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  • $\begingroup$ If you have a good grasp of what the different foods and environmental conditions in the"universe" are you can use techniques for "Positive and Unlabeled Examples." They do tend to have more of a machine learning than a statistical motivation, though. I believe Charles Elkan has a good paper on the subject. $\endgroup$
    – alex
    Commented Oct 17, 2013 at 19:14
  • $\begingroup$ Alex - thanks for this, I've filed it away for future reference. The hope with this is to eventually make a predictive model and the Elkan paper seems like it might help with that, in the later stages $\endgroup$
    – HFBrowning
    Commented Oct 17, 2013 at 21:00
  • $\begingroup$ You're not allowed to sample non-event cases because there could be error in those samples? Is there no possibility of error in the observed events? There's always error. That's why we bother with statistical inference. $\endgroup$
    – ndoogan
    Commented Oct 17, 2013 at 21:28
  • $\begingroup$ No, it was for non-statistical reasons that I was asked to look for another method. Although if nothing else appears to work as well as what I had planned then that may be persuasive enough to let me get my samples. I need to find out first though. $\endgroup$
    – HFBrowning
    Commented Oct 17, 2013 at 21:40

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Do you have any other information available? previous studies of similar cases? number and types of foods that were not measured? guesses about the process from biologists?

If you can find other existing information then possibly your best bet (other than the simple descriptives) is a Bayesian analysis. But be very careful in how you create your priors and the relationships, without a lot of data it can be easy to over interpret results that are more a reflection of the prior than the data.

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  • $\begingroup$ Unfortunately I know little about Bayesian analysis beyond what it can be used for. I can always start doing some reading though! I'll keep this open for a few more days to see if I get other suggestions $\endgroup$
    – HFBrowning
    Commented Oct 17, 2013 at 21:03
  • $\begingroup$ Looking into it more, your suggestion would have worked well Greg if I had better prior information. Thanks. $\endgroup$
    – HFBrowning
    Commented Oct 19, 2013 at 20:23

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