I want to figure out how to do a logistic regression (alternatively GLM w/ logit link) where some featured affect the output linearly (additively) while others affect it multiplicatively.
That is, in standard logistic regression, you have:
$$
P(y)=\sigma(ax_1+bx_2+cx_3+dx_4)
$$
Where P(y) is the probability that $y=1$, and $\sigma(*)$ is the sigmoid/logit link function. What I want to do is:
$$
P(y)=\sigma(ax_1x_2^b+cx_3+dx_4)
$$
Note that here essentially $x_2$ scales $x_1$. Also, note that while the regressors, $\{x_i\}$ are continuous, the output variable $y$ is binary, so log transforms wont work.
Also, to give more info on the specific problem, I am trying to predict a rats behavior. My regressors are position, speed, and 3 others. I believe that speed does not independently/additively contribute to the output, but modulates/scales the effects of position. I do not know the strength of this scaling, although I believe it should range from either nonexistant (i.e. the $b=0$ in above equation) to linear (i.e. $b=1$ in above equation).
Please let me know ways to approach this.
Best Answer
This is not linear in the parameters (given $b$ and $c$ it is, but if you're estimating $b$ and $c$ from the data as well, it's nonlinear), so instead of being a generalized linear model it's a generalized nonlinear model. Alternatively if you fix $a$,$d$ and $e$ it's linear in $b$ and $c$, using a non-standard link. Such structure can be exploited.]
It's perfectly possible to write the log-likelihood and so to use standard optimization methods* to identify the MLEs and even their standard errors.
* (or indeed to combine ideas from typical solution methods from GLMs and nonlinear least squares to solve the problem iteratively)
[There is a generalized nonlinear modelling package for R, (
gnm) ... see here. The overview vignette is here]