Correlation size versus linear regression coefficient size

In a couple of work projects, I've used linear regression to predict a single dependent variables from many independent variables. In a few cases, the strength of the correlation between some particular independent variable and the dependent variable didn't seem to match the relative size of the coefficient learned in the model for that independent variable. I asked Claude about it. Here's the conversation:

ME:

I'm using linear regression to predict a single variable from many variables. What might it mean if (for example) a particular independent variable correlates relatively strongly with the dependent variable but has a relatively small (i.e., close to zero) coefficient in the trained regression model? And what might it mean if the reverse is true: if the independent variable has a relatively low correlation (i.e., close to zero) with the dependent variable but has a relatively large coefficient in the model?

CLAUDE:

There are a few possible explanations for those situations with linear regression models:

  • If an independent variable has a high correlation with the dependent variable but a low coefficient in the model, it likely means that variable is highly collinear (redundant) with other independent variables. The effect of that variable is being "explained away" by the other collinear variables.
  • If an independent variable has a low correlation with the dependent variable but a high coefficient, it suggests that variable has a meaningful effect on the dependent variable when controlling for the other independent variables. The relationship only emerges after accounting for the other variables.
  • The coefficients in a linear regression model represent the independent contribution of each variable after controlling for all other variables. So high correlations between independent variables can differ from the coefficients which measure unique effects.
  • Issues like overfitting can also cause coefficients to poorly reflect the true underlying relationships. The model may be tailored too closely to the particular sample data.

So in summary, correlations show the overall bivariate relationships, while coefficients capture the specific isolated contribution accounting for the other variables. Discrepancies can arise due to collinearity, suppressor effects, overfitting, and other reasons. Carefully examining the analysis and data is needed to understand the cause.