Exposure in mediation model

[Daniela Kaiser]

Hello,

I'm trying to use the mediate command to run a mediation model with a Poisson outcome model. My understanding of Poisson models is that one needs to include an exposure term to capture the size of the population at risk for the outcome modeled. In my case, the outcome is a count of child maltreatment reports, so the exposure would be the size of the child population.

I haven't found a way to include this exposure term in either the outcome or the mediator model, even though it is possible to specify that the outcome and/or mediator models should be Poisson.

Thank you!

[Erik Ruzek]

Hi Daniela,

I do not see an exposure option in the mediate command either. But such an option may not be needed, depending on your data. The exposure is not necessarily about capturing the size of the population at risk for the outcome, although in some circumstances that might need to be accounted for as an offset. Following advice from Clyde Schechter in other Statalist threads (see links below), think of the Poisson model as predicting a rate. The numerator is the number of times something happened and the denominator (in your case) is the amount of time by which that event could happen. The exposure is the denominator. Accordingly, in your case, if the children were of different ages (years, months, or however you clock it), then you would likely include your age variable as an exposure. However, if all children were measured at 5 years of age, then they all have the same denominator and an exposure is not needed.

As noted, there are a number of threads on Statalist about exposures, with Clyde Schechter and others providing some good guidance on how to understand them. See the following:

https://www.statalist.org/forums/forum/general-stata-discussion/general/1693727-poisson-regression-adjusted-to-exposure-time
https://www.statalist.org/forums/forum/general-stata-discussion/general/1569043-xtpoisson-with-exposure-varname-how-to-select-the-appropriate-varname
https://www.statalist.org/forums/forum/general-stata-discussion/general/895616-how-to-account-for-population-p-in-poisson-regression-on-municipal-level-data-y-p-or-offset-p

[Daniela Kaiser]

Hi Erin ruzek, thank you so much for your reply! Apologies, I should have given more detail on my data. The child maltreatment data is not all for one unit; my units are counties, each of which have their own counts of maltreatment reports and different child population sizes. My outcome is not each child maltreatment report but rather the count of reports made in each county. I'm using county dummies in the outcome and mediator models to estimate a within-county fixed effects model. Given that the outcome is a raw count of reports by county, my sense is that I need to somehow include a measure of county child population, to function as a denominator.

[Leonardo Guizzetti]

Just some thoughts on this, but I'm not an expert with -mediate- or causal mediation methods myself. "Exposure" here has two meanings which I think might be getting conflated.

1) In the Poisson regression model, -exposure- is specifically used for a covariate that has been log-transformed and it's coefficient fixed to 1. (Offset is the same thing, but on the raw coefficient scale, while Stata handles the transformation and constraint for you with either option.) This is often used to estimate rates or rate densities, as Erik mentions. Exposures are not necessary or required in a general sense when using Poisson models, however there are cases when they are useful. This would be appropriate to your context because you have aggregate counts at the county level of number of maltreatment reports whose exposure could be normalized to the county population (exposure).

2) Exposure in the causal mediation sense is altogether different. This is the variable of interest which you hypothesize has causal paths to your outcome through a mediator. In the -mediate- documentation, this is the treatment variable.

Unfortunately, I don't see that the -mediate- command is compatible with sense #1. I also don't see that the -mediate- syntax allows constraints that would let you mimic the effect of an exposure/offset variable.

It may be that you might need to look at a two-stage type of mediation estimator and therefore you would need to program the estimates needed by hand.

[Daniela Kaiser]

Thank you Leonardo Guizzetti! I want to make sure I understand your recommendation - would a two-stage mediation estimator entail estimating two models separately, one with the predictor estimating the mediator, and another one with the predictor estimating the outcome?

[Leonardo Guizzetti]

Thank you Leonardo Guizzetti! I want to make sure I understand your recommendation - would a two-stage mediation estimator entail estimating two models separately, one with the predictor estimating the mediator, and another one with the predictor estimating the outcome?

I think it's usually 3 models: outcome of interest predicted by treatment; outcome of interest predicted by mediator; and sometimes mediator predicted by treatment, depending on how you do your causal mediation analysis. I would still be open to hearing form others more experienced in the area as I'm not confident this is necessarily what you want to do, as I said in #4.

[Erik Ruzek]

Building on Leonardo Guizzetti's excellent suggestions, the causal mediation approach can be implemented in structural equation modeling programs. In Daniela's case, she could use gsem in Stata given the count variable outcome and ability to specify exposure variables. Valente et al. (2020) have a nice paper on software implementations for causal mediation analysis. They highlight Mplus, but this can be done similarly in gsem. See also Rijnhart et al. (2020) for more discussion as well as gsem code in the Supplementary Materials for estimating the model and the various direct and indirect effects.