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Predictors of being admitted to the Emergency Department for a suicide attempt

I want to examine the predictors of being admitted to the Emergency Department for a suicide attempt. Would it be better to consider all other ED admits as my controls, or to consider all other injury admits as my controls?

Well as you likely know you may have strong selection bias related to these being failed suicide attempts, so you miss out on successful attempts. Also generalizability may be limited to comparable regions. Ignoring these concerns, I wonder if you have possible subgroups based on mechanism. This is definitely not my area of research, but I have a paper in press on poisonings, where intentional poisonings used convenience agents. Given this I wonder if patient characteristics may be heterogeneous between mechanisms, so grouping all suicides together may lose some finite information (age, time of day/week)? Though you may find macro generalizable characteristics via grouping altogether (e.g., # previous encounters). Just my 2 cents. I am interested to see what others suggest.

I am also not sure if you are actually interested in the question as posed. Or at least, I am not sure how the answer would be helpful to a clinician or a patient or public health. Also, I am going to assume that you are interested in determining factors that cause admission to the ED for a suicide attempt, rather than just predict them.
First, as the other answer noted, why are you interested in failed suicide attempts that are admitted to hospital? If guns are more effective for suicide and males use guns more often than females, your analysis may conclude that being female is a risk factor for suicide attempt admissions to hospital. This is not incorrect, but how does it help for any prevention program? I guess you could argue that it might help plan for treatment programs, which can only be applied to those that survive. Is this what you are interested in?
Second, I was taught that there are no case-control studies, just cohort studies analyzed from a case-control perspective. Taking this perspective can help you think about what the control group should be. If we were going to design a cohort study, we would gather an entire population. In a case-control study, the best practice would be to begin with such a theoretical cohort in mind that serves as the framework for a nested case-control study.
Beginning with a full cohort, the causal risk factor profile will depend on the population we choose. Now let’s think about your control group. If you pick “injuries admitted to the ED”, then your full cohort will end up including only those with suicide attempts and injuries, and exclude all other people in society. When you compare for differences in exposures, you will be saying that suicide attempts admitted are more or less likely for an exposure than people who are admitted for injury. Therefore, let’s say domestic violence has the same causal effect for admissions for suicide attempt and admissions for injury. If that were the case, your study would find a coefficient for risk ratio of 1 for domestic violence even though it is a risk factor compared to the general population.
Finally, if you are actually interested in risk factors for all suicide attempts instead of those just admitted to hospital, matching on hospital admission is problematic. If you draw a causal diagram, you will see the hospital admission is a collider for suicide attempts and every cause of hospital admission. Therefore, there will be a non-causal association between suicide attempt and every risk factor for suicide attempt that also causes hospital admission, such as mental health disorder, domestic violence, injury, and so on.
From a more general perspective of case-control studies and what to adjust for in the analysis, I would recommend the article by Mansournia et al. Matched designs and causal diagrams. International Journal of Epidemiology 2013;42:860–869 to help understand the effects of relationships between confounders and the matching variable used for the control group. However, this paper does not directly address your issue where the “exposures” (risk factors) are causes of the disease (suicide attempt) and the collider (hospital admission).