Aesop, Marginality Principle, Interactions and Children's Books.
In the past few months / years I unfortunately found myself reading less and less. There are only two categories of books that I still regularly read these days: children's books and academic papers. This post is about an unlikely mashup of works from these two categories.
The marginality principle. Sometimes, when we investigate the effect of a certain treatment in a certain population, we might have reasons to believe this effect differs for specific categories within that population. For example, we might think the effect is stronger for younger people. In such cases, one simple option is to include an interaction term in your statistical model, that will tell you about the combined effect of age and treatment. It is often said that if you include interactions, tough, then you absolutely must include the main effects as well. For example, if you want to include an interaction between age and treatment in your model, then you must also include age and treatment separetely in the model as independent factors. This recommendation dates back from the work of Nelder in the 70s, mainly based on theory around design of experiments, but it is often extended to any situation in which a regression model is used. I always found this very general recommendation quite cryptic and never understood why it should always apply. Then I recently read a paper from Tim Morris, Maarten van Smeden and Tra Pham that was precisely about this: the Marginality principle revisited. If you are interested in a technical discussion of this issue with examples, then I recommend you to give that one a read. The objective of this post is a much less ambitious one, instead: I will talk about how their paper came to my mind while I was reading a story from an Aesop collection to my daughter (yeah, I know this is sad, don't tell me!).
The donkey and the load of salt.
Aesop fables are still to this day an amazing read for children. My daughter loves them and one of her favourites is the story of the Donkey and the Load of Salt. Here is one version of it that I found online, just slightly redacted replacing "Ass" with "Donkey", to avoid being accused of clickbaiting:
A Merchant, driving his Donkey homeward from the seashore with a heavy load of salt, came to a river crossed by a shallow ford. They had crossed this river many times before without accident, but this time the Donkey slipped and fell when halfway over. And when the Merchant at last got him to his feet, much of the salt had melted away. Delighted to find how much lighter his burden had become, the Donkey finished the journey very gayly.
Next day the Merchant went for another load of salt. On the way home the Donkey, remembering what had happened at the ford, purposely let himself fall into the water, and again got rid of most of his burden.
The angry Merchant immediately turned about and drove the Donkey back to the seashore, where he loaded him with two great baskets of sponges. At the ford the Donkey again tumbled over; but when he had scrambled to his feet, it was a very disconsolate Donkey that dragged himself homeward under a load ten times heavier than before.
The moral of the story is:
the same measures will not suit all circumstances.
For somebody that thinks about statistics the whole day for a living, this has an immediate connection to interactions in statistical models: the same treatment will not have the same effect for everyone. Now, imagine the Donkey was a very clever one, and that they wanted to use the best statistical methods available before drawing conclusions about this phenomenon. Their statistical model would have some measure of physical effort as the outcome, and potentially three explanatory variables: type of load (salt vs sponges), type of behavior (purposedly falling in the river or not) and their interaction. Under the standard marginality principle, both the type of load and the type of behavior would have to be included as main effects if including the interaction between the two. But is this a meaningful model?
"It depends".
As with most statistical questions, I strongly believe that the refrain "it depends" should apply to the question "should we always add main effects if adding interactions?". In the example I just gave you, it clearly makes no sense to me to include a main effect for purposedly falling in the river (the effect is surely going to be negligible) and even more so a main effect for the type of load: if the donkey is carrying 10 kg of salt or 10 kg of sponges it shouldn't make any difference per se. The only thing that matters is the interaction between these two. Of course this refers to something that is mentioned in Tim and colleagues' paper: what is a main effect and what is not is often relative. Also, what makes sense to include or not in the model, depends on the specific circumstances. There is no reason whatsoever to remotely believe that "falling" in the river or carrying one specific load should have any effect on the effort the donkey will make. So there is not, in my opinion, any justification for including these two main effects in a model. In this specific example, only the interaction makes sense and is of interest.
So what?
You may be wondering what is the point I am trying to make here. Should we just forget about the marginality principle? Is it fine to just include interactions and remove main effects from a model? I am definitely not advocating for anything like that. The work from Nelder is surely important, and in his own settings, safe to say following the marginality principle might be the best thing to do. But, generalisations often arise, and generalisations often are far from the best thing in statistics. In order to choose what is the best model to choose in a certain situation, a careful consideration of the specific problem should be made. In most cases, including both main effects and interactions will probably be the best approach, but this "marginality principle" should not just be a recipe to blindly apply to all situations.
The moral of this story is:
the same Marginality Principle will not suit all circumstances.
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