In regression analysis, you'd like your regression model to have
significant variables and to produce a high R-squared value. This
low P value / high R^{2} combination indicates that changes
in the predictors are related to changes in the response variable
and that your model explains a lot of the response variability.

This combination seems to go together naturally. But what if
your regression model has significant variables but explains little
of the variability? It has low P values *and* a low
R-squared.

At first glance, this combination doesn’t make sense. Are the significant predictors still...