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If I understand things correctly, we want to look at $log(p/(1-p))$ since we want the independent variables to be able to take on any real value. But what if you know your independents can never be negative? Wouldn't it make more sense to simply look at $p/(1-p)$ in that case?

(if so, how would the syntax for such a model look in R?)

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    $\begingroup$ This question seems to confuse the roles of independent and dependent variables. In logistic regression, the response is a Bernoulli variable with parameter $p$. The logit link is one way to model the limited range of $p$, which can extend only from $0$ to $1$. There are otherwise no restrictions on independent variables, just as there are no restrictions on the IVs in ordinary regression. It would therefore appear that this question concerns a problem that just doesn't exist. Have you perhaps mis-typed "independent" for "dependent" or vice versa? $\endgroup$ – whuber Oct 20 '14 at 18:57
  • $\begingroup$ What I mean is that we want the range of $\alpha + \beta^t X$ to be between $\pm \infty$. If we know we only need it to go from 0 to infinity, do we need to take the log of the probability ratio? $\endgroup$ – Benjamin Lindqvist Oct 20 '14 at 19:06
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    $\begingroup$ I don't follow that. Given $X$ (the design matrix of values of the independent variables), the set of all possible $\alpha+\beta^\prime X$ is going to be all real numbers no matter what. Would you be able to explain your motivation for this question in non-mathematical terms? In those terms, what quantity should be non-negative? $\endgroup$ – whuber Oct 20 '14 at 19:09
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Let's assume that we somehow know a priori that the logit function is (or should be) non-negative. This would imply that

$$\ln \left(\frac {p}{1-p}\right) > 0 \Rightarrow \frac {p}{1-p} > 1 \Rightarrow p>\frac 12$$

So in effect we restrict the possible values of the underlying unknown probability. Does this reflect the case at hand?

Technically, what does it mean for the logit function $g(\mathbf x_i) = \alpha + \mathbf x_i'\beta$ to be non-negative? Restricting the range of the $X$'s is not enough -after all, the unknown coefficients may be negative. So, our estimation procedure will be a constrained optimization one, i.e. we will obtain the maximum likelihood estimators for the coefficients under the set of constraints $g(\mathbf x_i) = \alpha + \mathbf x_i'\beta \geq0 \; \forall i$.

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I am also confused about your question here. The reason we model the log-odds in logistic regression is to map the dependent variable onto the real line. Why do you want $\alpha + \beta^tX$ to be positive?

If you really want to restrict the $\alpha + \beta^tX$ to the positive real line, one possible solution is to use $-log(p)$ as your response, though I am not sure why you would want do this and how you would do interpretation. But since $p\in(0, 1)$, then $log(p)\in(-\infty, 0)$, thus$-log(p)\in(0, \infty)$ as you wish.

Note that even your independent variables are always positive, $\alpha + \beta^tX$ can still be negative, since the estimates of $a$ and $b$ could be negative, unless you really want to constrain them to be positive.

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