Interpretation of transformed ratio variable

[Daniel Assen]

Dear all,

I ran a meta-regression in which I have an explanatory variable which is the amount of months since an event. Other studies transform this variable into a ratio variable: (months-1)/months so I transformed it using this formula as well. The coefficient on this variable = 0.967 as can be seen in the screenshot. How can i interpret and logically explain this coefficient on the effect size? Thanks in advance

Best,
Daniël

[Nick Cox]

I would turn this around: why use this transformation unless you know that it has a helpful interpretation?

[Daniel Assen]

Nick Cox good question. Since the values of my variable months differ quite a lot (ranging from 2 to 840) a linear functional form is not the most useful functional form I think. When I perform the meta regression with the linear function of months it also isn't significant. When I use this ratio functional form it is, next to that R-squared goes up, so I think this is a better fit. The only thing is that the interpretation is not that clear. When (months-1)/months goes up by 1, effect size goes up with .937, but what can I say about each additional month? Thanks for your reply.

[Nick Cox]

I think you're close to answering your own question. I am not familiar with your field or its literature, so I don't have anything useful to add.

[Clyde Schechter]

When (months-1)/months goes up by 1, effect size goes up with .937, but what can I say about each additional month?

The problem is, of course, that (months-1)/months never goes up by 1. It's a monotone increasing function of months, and when months = 1 the ratio is 0, and it is always < 1 for any finite positive value of months. It approaches 1 as months goes to infinity.

And you can't say much about "each additional month" because it depends on how many months it is additional to. If we start at months = 1, then adding 1 month gives us months = 2, and the ratio is (2-1)/2) = 0.5. So the ratio has increased from 0 to 0.5, which is associated with an increase of 0.5*.967 = 0.484 in the outcome variable. But if we start at, say months = 100, then a one-month increment takes months to 101, and the ratio goes from (100-1)/100 to (101-1)/101, i..e. from 0.99, to 0.99009901. The resulting change in ratio of 0.00009901 is then associated with an increase of 0.00009901*.967 = .00009574 in the outcome.

Basically the (months-1)/months transform is a way of modeling a rapidly diminishing return to increases in months. But, by its very nature, it becomes meaningless to speak of an effect of "each individual month." You could perhaps instead calculate expected outcome values at a series of values of months and then plot a graph to show the relationship between them.

[Daniel Assen]

Clyde Schechter Yes that could be useful. Thank you for the helpful explanation