Is it legitimate to use factor analysis or clustering before regression My  goal is to make a logistic regression. 
The DV is a yes or no variable, and I already found 3 significant IV in my model. 
The problem is: I have 5 other categorial (yes or no) variables (they are nearly about the same topic) that I think they have an impact in the DV. Unfortunately, none of them were significant in my model.
The question is: Is it legitimate to use a factor analysis or a clustering technique to construct one variable that summarize the information? and then use it as an IV in my model? if yes,can this variable be significant?  and  what kind of  analysis do you suggest? 
Thanks in advance.
 A: A joint test of significance could be the way to go, but to answer your question, Yes -- factor analysis or another dimension reduction method could be used to transform your variables. You would then do your logistic regression on the transformed variables. 
However you want to test out the data reduction on different data from that used to create the transformation, or clusters (or whatever method you are using). You need to build the model in one step; validate in another.
If you really believe that a factor model is appropriate -- in the sense that the response is caused by an underlying factor or factors that are manifested by your predictor variables, then you can do a structural equation model. Classic SEM assumes normal (or at least continuous) variables throughout, but methods exist for categorical variables as well. The assumption is that the categorical variable represents a cut point of an underlying continuous distribution.
SEM is an elegant approach that allows you to account for the structure in the independent variables and their relationship to the response in one model. However it depends strongly on the validity of the model you are using.
A: As alluded to in the comments, it may be sufficient to perform a joint test of significance. The idea of doing so is to identify whether or not excluding all of the variables you have identified as correlated (and were thinking of combining somehow) will impact the model. 
A joint test you can perform is the likelihood ratio test of two models: one model with the variables and another without them. The test will compare the likelihoods of the two models and test if they are different. Here is an example of how you can do this:
I take data off the internet -- you can find some more background on the dataset here:
http://www.ats.ucla.edu/stat/r/dae/logit.htm
Running this sample code
mydata <- read.csv("http://www.ats.ucla.edu/stat/data/binary.csv")
mydata$rank <- factor(mydata$rank)
mylogit <- glm(admit ~ gre + gpa + rank, data = mydata, family = "binomial")
mylogit2 <- glm(admit ~ rank, data = mydata, family = "binomial")
test <- anova(mylogit, mylogit2, test = "LRT")

I get a p-value for whether the two models are significantly different when I look at "test," an object I get when I run the ANOVA on the two models. Here I find that by indiscriminately dropping some of the variables, (GRE and GPA) mylogit2 is significantly different that mylogit.
