logistic regression for modelling 
I have these data plotted above. The explanatory variable represents intensity levels of ground shaking at different locations in an earthquake, and the response variable represents amounts of compensation awarded to damaged buildings. I want to fit a model to these data so that how much compensation is likely to be awarded can be estimated given an intensity level. Can I use logistic regression here? How should I go about building the model for these data? Thanks in advance. 
 A: Since your response variable (dependent variable) - compensation - is continuous, you wouldn't use logistic regression for this. Logistic regression is normally used to model the underlying probabilities in binomial proportions.
The fact that your predictor (independent variable) has been categorized has no impact on this choice, which is driven by the response variable.
A typical model for data like these might be a GLM. Note that both the mean and the variation change as the predictor changes; the fact that the spread seems to roughly increase with the mean would suggest trying a gamma GLM as a first attempt.
However, the response isn't linear in intensity; you might try a small-order polynomial (such as a quadratic in the log-mean), or you might try splines or some other form of smooth model.
If you have more than a few zeros in the response, you might use a zero-inflated model - essentially to model the probability of any compensation in terms of intensity, and then model the amount of compensation conditional on there being a non-zero amount. (If there are only a few zeros you might simply add a small constant to those few values to allow the gamma fit)
A: I think quantile regression is the way to go here. First, it makes no assumptions about the distribution of residuals. Second, it allows you to model any quantile of the dependent variable. 
