I encountered a strange thing while performing a regression.

When I computed the variables for the regression I discovered some of them had a low Cronbach's Alpha reliability. So I decided to remove a few items from the questionnairs that they were computed from. Still, I computed both versions of the variables. E.g, one variable might have been computed from items 1-6 and the other might have been computed from items 1,2,4,6 (lets call these double v's). These variables had Pearson correlations of about 0.85.

Now, none of my predicting variables had a significant correlation with predicted variable. So I performed a backword regression, with all of my variables, expecting at least one version of the double v's to be left out of the model. This resulted in a significant regression with a good R square (although not all predictors were significant). However, the model included double v's - and for more than one variable. Unsurprisingly, VIF's levels were pretty high (although none were higher than 10) so I manually performed an enter regression with only one version of the double v's. This resulted in a non-significant regression with a very low R square, no matther which version of the double v's I included inside the model. Admittedly, my sample is quite small - only 41 observations with around 10 predictors.

So, I have two question:

1. How is this even possible?

2. Given all of the above, does it make sense to use the regression with the high R square?

Thank you for reading all of this!