Regression with additive and multiplicative predictors

#1
Hi,

we are analyzing pollutants [µg/kg] in mussels with regression models. Let's say we have only the two predictors (1) "year" and (2) "month". However, the effect of "year" on the outcome is assumed to be additive, i.e., in avergage a constant amount of poullutants is added each year, independent of the total concentration. However, the effect of "month" is rather multiplicative, since it depends on seasonal mussel growth. I.e., the pollutant concentration changes relative to the total amount.

In case every predictor would be additive in nature, I would use the identity-link function. If all predictors are multiplicative, I would use the log-link function or log transform the outcome with a id-link function. But what can I do with this "mixed" situation??

Thank you very much in advance
 
#3
Hi hlsmith, do you mean by this multiple measurements in time at the same sites? Yes, actually the design is a little bit more complicated then described above: We have multiple years and multiple sites, in each year we measure (at always the same five sites) four times at winter, spring, summer and autumn. Differences between years and sites are assumed to be additive (i.e. independent of the already existing pollutant concentration in mussels), the impact of the time of year is assumed to be multiplicative (relative change of the existing pollutant concentration).
 

hlsmith

Omega Contributor
#4
I not well versed in time series but they have tests for seasonality. I wonder if time series literature can help or if you can just make the IV a categorical variable? Sorry I don't have a better suggestion!
 
#5
Hi, thank you nevertheless. You mean if I treat each variable as categorical, it doesn't matter if its additive or multiplicative? That may be right if we would consider only main effects. Unfortunately, we are also considering interaction terms between additive and multiplicative acting varibales, and in this case it does matter if we consider them as additive or multiplicative: If e.g. predictors X and Y are categorical and multiplicative in nature, and we (wrongly) use an ID-link function, we would measure a biased interaction effect. And we have the same problem is one is additve and one is multiplicative... I wonder if it is possible to use a link function which is applied only to a selected set of predictors...
 

hlsmith

Omega Contributor
#6
OK. I had wondered if you were referring to interaction terms. I get what your question is now. So you have potentially two interaction terms, one on the additive scale and one on the multiplicative scale. I haven't heard of a case of this before. I will have to stew on this, but I didn't think you needed to use a different link function in logistic reg if you have either types, you just use odds or probabilities generated. Which I would guess you may be able to use the same model but interpret it two ways. Check out the following and I will think about it later today if I have time. Vander Weele has a wealth of information in this area. It may help if you write out your model.


You now got my attention. Though, I may get confused by your repeat measures.


https://cdn1.sph.harvard.edu/wp-content/uploads/sites/603/2013/03/InteractionTutorial.pdf
 

hlsmith

Omega Contributor
#7
Mmercker,

It has been a few years since I played around in this area. I believe if the the outcome is rare you probably could run logistic reg then report both interaction terms on the additive scale then on the multiplicative scale (two tables of results). I couldn't see the harm in doing it that way. However I am not well versed in longitudinal procedures to point out any relevant concerns.
 
#8
Hlsmith,

thank you very much for both, the very interesting and extensive interaction-document dealing with exactly this topic, as well as for your suggestions. As always: Great help!

Last question: What do you think about using (1) the additive model (i.e., the ID-link function) and (2) the multiplicative model (i.e., the Log-link functon) and selecting the most appropriate via AIC-selection?
 
Last edited:

hlsmith

Omega Contributor
#9
I guess I am not following when you say, "ID-link". So you would be using the Poisson model and getting risks? Is your data prospective, that comes into play when thinking about risks versus odds. If you don't have time to event data, typically you are not suppose to use risks. Though if the outcome is rare, odds approximate risks (e.g., risk: 10/100 versus odds: 10/90).


Well I guess it comes down to whether both interactions are actually present. So you know you have significant or approaching significant interaction for both the additive term and multiplicative term? I have been a little confused about your terms. Can you flat out write the model with terms that you are interested in. Month seems like a subgroup of year. So you have a model with both month and year in it at the same time? I would imagine there could be threat of co-linearity.


If the models are not nested, then AIC can be used to examine fit. So you are saying the only difference would be the link? Then this could be an option if everything above seems fine.