# Regression with sigmoid link for diminishing returns?

I am working with some researchers who would like to see how a dozen or so "life inputs" can affect a measure of happiness. My feeling is that treating these as additive, as in a regular regression might be a bit ambitious. Surely, once one input has had an effect, the space for another to improve things has to be reduced? We can't just keep making you happier and happier and happier. But searching this site, the only options I can find regarding diminishing returns involve things like squared predictors, which only allow each predictor in isolation to diminish in its strength.

I know the "real" answer is these are interactions, but to model the interactions would be crazy complex, and it occurred to me that a much simpler model would be one like this:

$$y = f(\mathbf X\beta)$$

where is $$f$$ is some kind of sigmoidal function, such as

$$f(z) = k_1\mathrm{sign}(z)\mathrm{log}(1+k_2|z|)$$

There will also be random effects and autocorrelated residuals, just to add to the fun (!), but I think that could all be handled in R using the nlme package.

My question, though, is this: just doing this crazy sigmoidal nonlinear thingy off my own back sounds a bit like I'm charging off into the Wild West of my own ideas. But this also sounds like a really obvious idea. Does it already exist under a different name? Or is there already an established approach for dealing with this problem?

EDIT

A little more info on the project, while trying to keep sufficient mystery not to disclose information that isn't mine to disclose...

Happiness in this case is measured using a validated psychological scale (I think the total has a range of around 60 possible values). Each IV captures the frequency with which a given activity was engaged in (this is purely observational). The design is somewhat-longitudinal in that each participant is measured each week for a few weeks (though not for a long period of time). So, rather than viewing it as really "longitudinal", I would think of it as a few samples from each participant, but where there is likely some autocorrelation (though we have discussed and agreed with think that autocorrelation is likely in the residuals, rather than the activities leaving a lasting impact). I'm fairly happy (I think?) with most aspects of the design, but I just feel like treating the effects of each activity as additive seems possibly problematic. One of the academics I'm working with is a fan of stepwise regressions, but I'm not, so I'm keen to create as clean a model as possible so we can avoid his impulses in that direction if possible!

• Maybe you can tell us about the kind of data & variables you will have? How do you measure happiness? Sample size? Experimental design (since you mentioned autocorrelation, I suppose longitudinal)? – kjetil b halvorsen Jun 11 at 22:40
• @kjetilbhalvorsen Thanks for chiming in! I've added a little more info -- is that enough? I'm trying to keep it anonymous-ish, but I think that's the sort of extra colour you're talking about? As I say, my main concern is that treating a dozen or more things each as additive to a scale of happiness feels like asking the IVs to have a fight... – justme Jun 11 at 23:59