The first thing you can do is, for example, interpret $\hat{\beta_2}$ as the estimated effect of $sex$ on the logit of the quantile you're looking at.
$\exp\{\hat{\beta_2}\}$, similarly to "classic" logistic regression, is the odds ratio of median (or any other quantile) outcome in males versus females. The difference with "classic" logistic regression is how the odds are calculated: using your (bounded) outcome instead of a probability.
Besides, you can always look at the predicted quantiles according to one covariate. Of course you have to fix (condition on) the values of the other covariates in your model (like you did in your example).
By the way, the transformation should be $\log(\frac{y-y_{min}}{y_{max}-y})$.
(This is not really intended to be an answer, as it's just a (poor) rewording of what it's written in this paper, that you cited yourself. However, it was too long to be a comment and someone who doesn't have access to on-line journals could be interested anyway).
For models with more than one discrete outcome, there are several versions of logit models (e.g. conditional logit, multinomial logit, mixed logit, nested logit, ...). See Kenneth Train's book on the subject: http://eml.berkeley.edu/books/choice2.html
For example, in conditional logit, the outcome, $y$, is the car chosen by an individual, and there may be, say $J$ cars to choose from and car $j$ has attributes given by $x_j$. Then suppose that individual $i$ receives utility $u_{ij} = x_j \beta + \varepsilon_{ij}$ from chosing car $j$, where $\varepsilon_{ij}$ is distributed type I extreme value. Then the probability that car $j$ is chosen is given by
$$ \Pr(y=j) = \frac{\exp(x_j \beta)}{\sum_{k=1}^J \exp (x_k \beta)}$$
In this model, $u_{ij}$, form a ranking of the alternatives. We are searching for parameters, $\beta$, so that this ranking conforms with the observed choices we see people making. E.g. if more expensive cars have lower market shares all else equals, then the coefficient on price must be negative.
Economists interpret $u$ as a latent "utility" of making each choice. In microeconomics, there is a considerable body of work on utility theory: see e.g. https://en.wikipedia.org/wiki/Utility.
Note that there is no "threshold" parameter here: instead, when one utility becomes greater than the previously greatest, then the consumer will switch to choosing that alternative.
Therefore, there cannot be an intercept in $x_j \beta$: if there were, this would just scale up the utility of all the available options, leaving the ranking preserved and the choice unchanged.
Best Answer
The diagonal and vertical lines of points are caused by the same individuals, which are those with values of the 0 for
hwt.Let's consider the second plot first. That line of points is concentrated at
hwt = 0, but there is some variability in the predicted logit. That is because those individuals have different values ofist, which produces different values of the logit given the estimated regression coefficients. There is nothing strange about this.The first plot may look strange, but it's actually a result of the same phenomenon. Consider all those units with zeroes for
hwtand think about what the regression equation looks like for them. Becuasehwt = 0, all that is left is a linear relationship betweeniwtand the logit, and that is reflected in that diagonal line. (Note this kind of thing would happen if you had a clustering of values at the exact same value ofhwt, even if that value was not zero). More specifically, if the estimated regression equation looked like $\text{logit}(p) = b_0 + b_1 (\text{iwt}) + b_2 (\text{hwt})$, see what the equation looks like for those withhwt = 0: all we have is $\text{logit}(p) = b_0 + b_1 (\text{iwt})$, which is a perfectly straight diagonal line.To verify this, add
aes(fill = hwt == 0)to yourgeom_point()call. You will see that those on the diagonal line in the first plot correspond exactly to those on the vertical line in the second plot, being those withhwt = 0.