Is it OK to use an original variable & another variable constructed from it in a regression model if there is no multicollinearity? I'm doing binary logistic regression. I want to predict the chance of being in an advanced class. There no multicollinearity among my variables. I have three predictors: 


*

*If you passed the test or not (a binary variable >60 or <=60), 

*The number of correct answers you had on the test (a continuous variable between 0 and 100), 

*The number of times you did a practice test before taking the test officially (a continuous variable). 


There is no multicollinearity, but I'm wondering, can use variables 1 and 2 since 1 is created from 2? (That is, if you had more than 60 good answers on variable 2, your code on variable 1 would be >60.) I'm wondering if I'm not using the same concept twice.
 A: Yes, this is perfectly acceptable.  In effect, you are testing to see if there is a "jump" or level shift associated with crossing the 61-answer threshold for passing the test.  When tests have a minimum score to pass (which pretty much all tests do), they implicitly assume that there is a meaningful difference between the knowledge / ability of a person who scores a 61 and someone who only scores a 60.  However, it is reasonable to imagine that someone whose score is 61 knows about the same amount as someone who scored 60.  Moreover, we might believe that the increase in the amount of knowledge represented by 62 vs. 61 is the same as the increase in knowledge represented by 61 vs. 60 (and 60 vs. 59, etc.).  This possibility is somewhat inconsistent with the test having a hard threshold.  Thus, you might want to test if the threshold is meaningful.  That is what you are doing when you include variable 1 in addition to variable 2.  

For further study, what you have is the simplest possible form of a spline term.  The next step up in complexity would be a linear spline, in which you would have a variable indicating how far past 60 someone's score is (i.e., either 0, or +1, ..., +40).  A linear spline would let you test if increasing scores above the threshold represent a larger increment than the scores below.  You can take these ideas further with various cubic splines and additional thresholds (called "knots").  I provided an introductory discussion of those ideas in my answer here:  What are the advantages / disadvantages of using splines, smoothed splines, and Gaussian process emulators?
