Can we use bounded continuous variables as predictors in regression and logistic regression? I am currently looking at a cheminformatics problem where I am looking at the relationship between chemical structure and reactivity, e.g. how the angle at which two molecules approach each other affects the rate of the subsequent reaction. Obviously, the angle can only vary between 0° and 360°.
This is "quick check" question from a cautious non-statistician. I understand that in regression analysis, the dependent variable must be continuous and unbounded. I just wondered - do continuous predictor variables in regression need to be unbounded too? Instinctively I assume not.   
 A: The condition that dependent variables must be "continuous and unbounded" is unusual: there is no mathematical or statistical requirement for either.
In most regression models we posit that the dependent variable be a linear combination of the independent variables plus an independent random error term of zero mean, approximately and within the ranges attained by, or potentially attained by, the independent variables.  For instance, it would be fine to regress the length of the Mississippi River on time for the period 1700 - 1850 but not to project the regression back, say, a million years or forward 700 years:

In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

(Mark Twain, Life on the Mississippi.)
In the present case it sounds like the angle is an independent variable, not the dependent one, so this question does not even arise.  The problem that arises is that the angle seems to be defined only modulo 360 degrees (actually mod 180).  Actually, the angle is really a latitude and varies from 0 to 180 (or -90 to 90) without "wrapping around" at all.  Really, then, all that matters is how best to express this angle: does the reaction rate vary linearly with the angle or does it vary perhaps with its sine or cosine?  Or maybe its tangent, which is unbounded?  But that matter is addressed with appropriate exploratory analysis, perhaps by some stereochemical considerations, and standard procedures to fit and check models.  Therefore this angular variable neither enjoys nor suffers from any special quality that would distinguish it from other independent variables.
A: With respect to the question in the header
With logistic regression predicting posterior probabilities, the dependent variable (outcome) is both bounded and continuous.
One train of thoughts to arrive at logistic regression in fact is thinking how to construct a regression with limits for the continuous outcome. 


*

*You want e.g. to do a regression directly on the probability

*"Common" regression methods (e.g. linear regression) give you continuous output in the set of real numbers, $\mathbb R$.

*But probabilities are in [0, 1]

*So put a sigmoid transformation into your model to transform $\mathbb R \mapsto [0, 1]$

*If you choose the logistic function $\frac{1}{1 + e^{-x}}$ (a standard choice of a sigmoid), you end up  with logistic regression.


With respect to modeling angles in general I'd like to follow up with another question: how to model cyclic behaviour, how would I tell a model that 359° is almost the same as 0° (regardless of whether the variable is dependent or independent)?
