Multiple regression with more than 20 predictors I have a numerical dependent variable, and many independent variables. Most of my independent variables are dummy variables, but I have some categorical and numerical variables, too. I tried forward and backward model selection in R, but R returns my model empty! When I run separate simple regressions, there seems to be a significant relationship between my independent and dependent variables!
My question is: Am I going to have biased results if I run separate simple regressions with each variable, and then run a multiple regression with all the significant variables?
 A: Yes, that will lead to bias. 
The question is: Why do you need to select variables in the first place? People often seem to assume that this is just required for some reason, but it's not clear that it ever is. It may well be the case that there are some variables in the original (full) model that are not significant.  But this is just fine.  There is no problem if your model includes some non-significant variables.  There is no need to remove those non-significant variables from the model. Doing so harms your model, whereas leaving them in does not. Forward and backward selection, simply put, cannot help with finding out which variables are 'actually significant'. Which variables are significant or not is what is reported in the original full model.  
A: Adding to Gung's excellent answer:
Some reasons that you might need (or want) to eliminate variables


*

*Collinearity (although there are other solutions to that problem)

*Overfitting -- if your don't have enough data. There are various rules of thumb; one common one is that you need 10 observations for every independent variable.


Some specific reasons for keeping nonsignificant variables (beyond what Gung listed)


*

*A small effect is interesting.  Sometimes theory predicts a large effect and you find a small one. E.g. if you find a tribe of people where men and women are the same height, then sex will show as a nonsignificant variable. But very interesting!

*It's involved in an interaction. There are very few cases where you want to include an interaction but not the main effects.

*It mediates an effect

*It is the main variable you are interested in

