Small Sample (N=32), using simple regression instead of multiple regression to test Hypothesis? As part of my thesis I have proposed hypotheses:


*

*Perceived Supervisor Support is positively related to Employee Engagement

*Perceived Organization Support is positively related to Employee Engagement

*Perceived HRM Practices are positively related to Employee Engagement
I have done the survey in my company and have received 32 valid responses. Correlation analysis provides significant results. Now to test for hypothesis I was using multiple regression but my beta values are all statistically insignificant, whereas when I do simple regression for each individual driver with Employee Engagement I get statistically significant results.
My question is will using simple regression valid approach to prove these hypotheses? If yes can i quote a reference which will justify this approach in this case? Is there another way I can prove these hypotheses?
 A: First, as @whuber mentioned, you have the problem of selective response. This really dwarfs all other problems and makes any results dubious. Even if you have about 60% of the population in your sample, it could be way off.  However if you, as per one of your comments, say "these results only apply to these respondents" then you don't need inference at all.  
Second, your three independent variables are almost certainly colinear. You need to deal with that, even if you solve the nonresponse issue. One way to do this is to use ridge regression.
Third, doing three simple regressions is OK, but it answers a different question than doing one multiple regression.  Specifically, in a multiple regression, the effect of each IV is estimated controlling for the other IVs.  In simple regression there is no control. 
A: 
Significant predictors become non-significant in multiple logistic regression

sounds like a duplicate to your question. For example, (2) might be an issue to you.

Now, obviously, the two variables are strongly related, as you need to be older to have more experience. Hence, the two variables basically "compete" for explaining the status, which may, especially in small samples, 

Your sample size is small, already noted by @whuber. Your variables sound like strongly positively correlated to each other.
