Significant predictors become non-significant in multiple logistic regression When I analyze my variables in two separate (univariate) logistic regression models, I get the following:
Predictor 1:    B= 1.049,    SE=.352,    Exp(B)=2.85,    95% CI=(1.43, 5.69),    p=.003
   Constant:    B=-0.434,    SE=.217,    Exp(B)=0.65,                            p=.046

Predictor 2:    B= 1.379,    SE=.386,    Exp(B)=3.97,    95% CI=(1.86, 8.47),    p<.001
   Constant:    B=-0.447,    SE=.205,    Exp(B)=0.64,                            p=.029

but when I enter them into a single multiple logistic regression model, I get:
Predictor 1:    B= 0.556,    SE=.406,    Exp(B)=1.74,    95% CI=(0.79, 3.86),    p=.171
Predictor 2:    B= 1.094,    SE=.436,    Exp(B)=2.99,    95% CI=(1.27, 7.02),    p=.012
   Constant:    B=-0.574,    SE=.227,    Exp(B)=0.56,                            p=.012

Both predictors are dichotomous (categorical). I have checked for multicollinearity.
I am not sure if I have given enough info, but I cannot understand why predictor 1 has gone from being significant to non-significant and why the odds ratios are so different in the multiple regression model. Can anyone provide a basic explanation of what is going on?
 A: There is no particular reason why this should not happen. Multiple regression asks a different question from simple regression. In particular, multiple regression (in this case, multiple logistic regression) asks about the relationship between the dependent variables and the independent variables, controlling for the other independent variables. Simple regression asks about the relationship between a dependent variable and a (single) independent variable.
If you add the context of your study (e.g., what are these variables?) it may be possible to give more specific responses. Also, given that all three variables in your case are dichotomies, you could present us with the data pretty easily... there are only 8 lines needed to summarize this:
\begin{array}{llll}
DV    &IV1    &IV2    &{\rm Count}  \\
A     &A      &A      &10  \\
A     &A      &B      &20
\end{array}
etc.      
A: There are several reasons (none of which are specifically related to logistic regression, but may occur in any regression).


*

*Loss of degrees of freedom: when trying to estimate more parameters
from a given dataset, you're effectively asking more of it, which
costs precision, hence leads to lower t-statistics, hence higher
p-values.

*Correlation of Regressors: Your regressors may be related to each
other, effectively measuring something similar. Say, your logit
model is to explain labor market status (working/not working) as a
function of experience and age. Individually, both variables are
positively related to the status, as more experienced/older (ruling
out very old employees for the sake of the argument) employees find
it easier to find jobs than recent graduates. 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,
result in both variables "losing", as none of the effects may be
strong enough and sufficiently precisely estimated when controlling
for the other to get significant estimates. Essentially, you are
asking: what is the positive effect of another year of experience
when holding age constant? There may be very few to no employees in
your dataset to answer that question, so the effect will be
imprecisely estimated, leading to large p-values.

*Misspecified models: The underlying theory for t-statistics/p-values
requires that you estimate a correctly specified model. Now, if you
only regress on one predictor, chances are quite high that that
univariate model suffers from omitted variable bias. Hence, all bets
are off as to how p-values behave. Basically, you must be careful to
trust them when your model is not correct.
