# Appropiate model for $Y \sim f_1(X) + f_2(X) + f_3(X) + ...$, $Y \in [0,1]$

I wish to model the following relationship $$Y \sim af_1(X) + bf_2(X) + cf_3(X) + ..., Y \in [0,1]$$ where $Y$ is a response (a percentage divided by 100) and $X$ is a covariate which I want to model using those $f$-basis functions. The study's interest lies in the effect that covariate $X$ has on $Y$.

Based on what I have gathered from google, perhaps Beta regression? Presumably zero-inflated since I have $Y = 0$ at times?

Or something else, like some sort of GLM/GAM model? Any recommendations?

EDIT: I forgot to mention: apart from the fixed covariate $X$ effect, I intend to include some random effects.

• How do these percentages arise? The data-generating mechanism is crucial to giving a coherent answer to this question. Apr 23, 2017 at 15:55
• Are you simultaneously trying to estimate the $f_i$ or not?
– whuber
Apr 23, 2017 at 16:41

$\hat{Y_i} = \frac{exp(\sum_i a_i f_i(X_i))}{1+exp(\sum_i a_i f_i(X_i))}$
• And this naturally leads to a appliction of gams with a binomial family and a logit-link. I would suggest to use the mgcv package in R and to something like: gam($Y$~s($x_1$)+s($x_2$)+...,family=binomial(link = "logit")). If you are interested in a varying coefficent model of the style $Y=x_1b(z_1)+x_2b(z_2)+...$ the mgcv package also provides some nice implementations. May 4, 2017 at 14:50