In regression, when 2 parameters are correlated and added to a model separately, how likely is it that one parameter will be a significant predictor of the response variable while the other is not? To me this seems unlikely but I've encountered it in a publication. Based on my understanding and experience of multi-collinearity, if one variable is highly correlated with another, both should be influential in a model. Under what conditions could this occur?
Context: For pedagogical purposes I am replicating an GLMM analysis from a 2004 Science paper that used observational data which was posted with the paper. They are interested in causal interpretations of 2 parameters that are correlated (Pearson's=0.67, p<0.01). They state that the 1st parameter is significant in their regression (beta=0.24,SE=0.05,p<0.01) while the other is not (b=0.02,SE=0.07,p=0.51). The language of the paper implies that they ran two separate models, one with each predictor on its own (but see edit below). A model with just the 1st predictor also has a lower AIC than a model with just the 2nd predictor (335.9 vs 341.0).
I can replicate the much of the analysis, including the correlation, the coefficients of the 1st parameter, and qualitatively the difference in AIC.
Beyond not being able to replicate their analysis (perhaps I am missing some modeling detail), I don't understand how they could have two correlated predictors yet end up with only one being significant in the model.
**EDIT:**The paper implies that they are reporting coefficients for two separate models that had only one or the other predictor. However, the coefficients that they report possibly came from a model that had both predictors in at the same time. I can now replicate the beta and SE for the 2nd parameter. They appear to have ignored issues of multi-colliearity in running and interpreting their models.
I am still wondering if my original intuition about their being a problem with the reported values is generally correct, or are their conditions where two correlated variables will behave differently in a fitted model if entered in separately.