These are two very different approaches. In

y_i = b0 + b1 x1_i + b2 x2_i + e_i

b1 is an estimate of the relationships between x1 and y CONDITIONAL on x2. In

y_i = b0 + b1 x1_i + e_i

 it is not conditional. They estimate different things so will often have different signs. Which to use (if either) depends on why you are doing these analyses. This will be covered in most intro stats textbooks.

Jatin  -

The sign for the coefficient in a simple regression would appear to tell us whether x and y are positively or negatively correlated, and I have had cases in practice where that sign changed once another regressor or regressors were placed into a multiple regression.  I think my problem was too much collinearity.  Whatever the interaction due to multiple regressors, if it is enough to change a sign (which may not take as much if the correlation was weak to begin with), then you might want to treat this as a collinearity or other interaction problem. 

Perhaps you could compare possible multiple regression models with each other by comparing estimates of prediction errors, or to include bias, use cross validation.

But as far as a "true" sign for a coefficient, I'd think of the sign in the simple regression case as "true," but that does not mean you should use it in multiple regression if interactions cause an estimated coefficient to be of a different sign.  Instead, it means you have a variable interaction problem, and may need to change your multiple regression model.  (In deciding what to do, you should consider what is reasonable in the world of your subject matter theory.  What models make sense?) 

Cheers - Jim 

PS -  I see from Daniel's response that this may not actually be a "problem," but I think looking at estimated variances of the prediction errors, and using cross validation, and looking at subject matter theory, as I noted above, is still what you want to do to judge whether or not it is a problem in your case. 

I would think the true signs for your regression coefficients are those from the multiple regression. In simple regression you are assuming that the variable depends only on one variable, and if regression coefficients of the other variables are different from 0, then those variables also affect the response.

Generally speaking, a possitive value of a coefficient says that a change in the variable x is augmenting the value of variable y. A negative value means that when x increases, y decreases. In the results of your experiments you can see if some x increase cause tha y increases or decreases its value and this will help you decide which is the correct sign of the variable coefficient.

First, I assume that you have real data of some sort. I agree that multicollinearity  may be at least part of the problem. I would try a ridge regression on your data( see Lander,  Jared, R for Everyone, Chapt. 19. This is probably the easiest and cheapest way to go. I would also see the sections in Kutner et al, Applied Linear Statistical Models, 5th ed. on multicollinearity and ridge regression. Good luck

If given the choice between multiple simple linear regressions and one multiple linear regression, I would always choose the multiple linear regression. However, I would be interested to know what the VIF is for each term in your multiple regression model.

You may have a data set that shows "Simpon's Paradox". While it is not related to Homer Simpson, it can make you think, "D'Oh". I've seen this issue many times. I can't think of a simple data set where it really happened though.

I do have a data set where I took a pair calibration curve verification standards and ran them A-B-A-B... When I finished, it looked like CCV B had a higher concentration. When I ran a multiple linear regression analysis using CCV standard (A or B) and run position from my run sequence, CCV B had a lower concentration. It turned out the instrument I used increased the response of the CCV by 2% on every run. Once I eliminated the effect of Run Position, the data showed that CCV B was a lower conc. If I assumed that the instrument was in "Good working order", I would conclude CCV B was a higher conc.     

Thank you very very much every body. Your responses helped me.