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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.

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    $\begingroup$ Please give more information, e.g., how do you know the two effects are power relationships? If you don't really have that much prior knowledge (and I'm not sure how you would) you may want to relax the form using for example regression splines. $\endgroup$ – Frank Harrell Nov 24 '15 at 4:13
  • $\begingroup$ What's $P$ in your notation? $\endgroup$ – Glen_b -Reinstate Monica Nov 24 '15 at 5:46
  • $\begingroup$ @FrankHarrell I updated the desciption. hope it helps. $\endgroup$ – DankMasterDan Nov 25 '15 at 19:05
  • $\begingroup$ Your formulation of the model does not follow from the experimental design, and the model does not follow the hierarchy principle so the arbitrary origin of the covariates will greatly affect the parameters estimates for all parameters in the model. Regression splines with tensor spline interaction are usually better choices, although this will involve more parameters. But unless the subject matter dictates a specific equation I don't know how to do otherwise. $\endgroup$ – Frank Harrell Nov 25 '15 at 20:32
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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]

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