# How to interpret: a variable has a highly significant effect in a model one, however, a second model without the variable has similar R2? [closed]

I am confused.

How to interpret the following results? A variable has a highly significant effect in a model one, however, a second model without the variable has similar (non-different) R2?

Thus, the variable has an effect on dependent variable, but this does not explain much of the variance?

• What do you mean by “non-different”? Also, how many observations do you have? Small differences can be highly significant when large sample sizes make your hypothesis test extremely sensitive. – Dave Jun 30 at 10:23
• With non-different, I mean that there are overlapping confidence intervals. R2 model with variable: 0.15 (CI: 0.11-0.20). R2 model without the variable: 0.13 (CI: 0.09-0.20). These are retrieved using the function bayes_R2(modelname). Plus yes, I have large dataset, about 5000 subjects. – st4co4 Jun 30 at 12:06

For example, let's consider correlated variables, $$X_1$$ and $$X_2$$ with relationship $$X_1 = k \cdot X_2$$.
If you train your model including two variables with linear regression, you would get this linear equation: $$y = \tilde{\alpha}_1 X_1 + \tilde{\alpha}_2 X_2 + \tilde{\varepsilon}$$
where $$\tilde{\varepsilon}$$ is residuals.
With the formula, $$X_1 = k \cdot X_2$$, the equation is converted into this form: \begin{align} y & = \tilde{\alpha}_1 \cdot k \cdot X_2 + \tilde{\alpha}_2 X_2 + \tilde{\varepsilon} \\ & = (\tilde{\alpha}_1 k + \tilde{\alpha}_2)X_2 + \tilde{\varepsilon} \end{align}
What this equation means that even if you train your model only with the $$X_2$$, you get the same residuals. And if you have the same residuals, you would get the same $$R^2$$.