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I have a dataset with counts of preterm births for five years by county. I am running analyses to see whether contaminant water concentration levels for each county for each year predict preterm births. To do this, I am using a Poisson regression model, with a random effect for county and dummy variables for each year. I also have two lagged variables for contaminant water concentration. The first is group centered, and the second is grand-mean centered at level 2, with the goal of separately estimating the within and between county effects. I also have a county-level demographic control for age.

My main question is, why do I get very different results for contamination levels when I include an offset versus when I do not, and how should I interpret these differing results?

Running the regression on the raw counts, I get the following output for fixed effects:

Model 1 without offset

I interpret this to mean that, within counties, there is not a significant relationship between contaminant concentration and the log expected count of preterm births (b = .006), but there is a significant positive relationship between these two variables at the county level, such that, for a one unit increase in the contaminant at the county level, the expected log count of preterm birth increases by .66 (alternatively, the expected (exponentiated) count is 1.93).

However, this model does not adjust for the differing number of women in each county. From what I’ve read, it makes sense to add an offset term to address this issue. If I add an offset, logged number of childbearing women within each county, results for the focal predictor are much different:

Model 2 with offset

Now it appears that the county-level relationship between contaminant concentration and preterm birth is in the opposite direction, and nearly significant so. From what I understand, this relationship would be interpreted in terms of the expected log rate, rather than log count.

My questions are: Am I misunderstanding something about how to specify this type of model? I'm not trained in applying epidemiological-type models, so I'm not confident that I didn't make a mistake somewhere.

What do these two differing results mean for my research question? Can I say that the apparently positive relationship between contaminant levels and preterm births didn’t hold (and even reversed) after adjusting for the number of childbearing women? Should I just disregard the first model? The model with the offset term seems more appropriate, if correctly specified.

I am using SPSS and GENLINMIXED, but I think the question is not software-specific. I'm happy to provide any further detail, and I’m hoping for any help you can provide. Thanks.

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    $\begingroup$ One might expect population growth and pollution to go hand in hand as economies develop (environmental Kuznets curve), so by omitting CBW from the model, the pollution variables are confounded, since more births leads to more pre-term births (even if pollution has not effect itself). $\endgroup$
    – dimitriy
    Dec 15, 2020 at 23:21
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    $\begingroup$ I also think the interpretation of $\exp(0.66)=1.93$ is not quite right. That is a multiplicative effect, which is almost a doubling from a one unit increase. $\endgroup$
    – dimitriy
    Dec 15, 2020 at 23:35

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I think you're understanding things correctly, and the second analysis is indeed the more reasonable one, assuming the counties differ substantially in the numbers of women of childbearing age (though knowing the total number of pregnancies would be better). Rates would typically be considered the way to go in this kind of situation.

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