And then do a multivariate analysis for each exposure that was found to be significant?

Multivariate: because one wants to know how good the variables jointly predict the

dependent variable, while taking into account the interrelatedness of the predictors.

Include only variable with p < 0.10: Seemingly they wished to make the analysis

less complex by reducing the number of variables beforehand. And/or maybe they

wanted to circumvent the problem that ususally one would demand > 20 or so cases

for each predictor variable in the analysis. So with a small sample it looks desirable

to have only a few variables (the pre-testing does not solve the problem, though, since

it introduces bias not accounted for in subsequent the multiple regression model, but

often those who use the pre-selection procedure are not aware of this).

Wouldnt doing a multivariate analysis on ALL of the exposures in the beginning SAVE an extra step of not doing the bivariate analysis?

That's the alternative. One may run short of subjects and/or the model is too complex,

but nevertheless it is an alternative.

Kind regards

K.