My question is certainly quite basic for statisticians!
Var2 are highly correlated with a poor $R^2$.
Var3 is another variable that we will use in a regression (standard linear model (Gaussian error distribution, OLS estimator)) as a response variable.
Var2 are the explanatory variables of this regression where the interaction effect is not computed. Let's call this regression
Is it possible that
Var3 (highly significantly) but not
Var2 (not significant at all)?
Let's assume that the causation is the following:
Var3 Is it plausible that both
Var1 are associated with highly significant p-value in
In a given species of fish, there is a very significant relation between the size of the fishes and the color of the fishes. But the linear model that gives this very significant p-value does not explain much of the total variance in fish color and size.
When performing a regression of depth on fish size and fish color (without the interaction term), I get two highly significant p-values (for the two explanatory variables).
Can I infer that both the size of the fishes and the color of the fishes "influence separately" the depth at which the fishes are?
Or maybe depth has an effect on the color of the fishes (because the diet is different for example) and the color of the fishes influenced the size of the fishes (because fishes that are bright need be big to escape predators or something like that). In such case, the size of the fishes would be associated to a significant p-value (in the regression of depth) only because it is correlated to the other explanatory variable which is the color of the fishes.