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What does it mean if my adjusted r-squared decreases between my bivariate and multivariate regression models? is this a bad or good thing?

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The $R^2_{adj.}$ is a version of the $R^2$ which is adjusted for multiple regressors. Whereas the $R^2$ rises the more regressors you add, even when there are irrelevant, the $R^2_{adj.}$ stays robust.

$$R^2=1-\frac{SSR}{SST};\quad R^2_{adj.}=1-\frac{SSR/(n-p-1)}{SST/(n-1)}$$

Hence, when you add more regressors to your bivariate model, and then your $R^2_{adj.}$ decreases, the model tends to deteriorate and you might call it a "bad thing."

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  • $\begingroup$ does this mean the bivariate model is better? Z is the third regressor or a confounding variable. $\endgroup$
    – Louis Rich
    Commented Dec 17, 2020 at 19:00
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    $\begingroup$ @LouisRich Yes, I would consider that model with the higher $R^2_{adj.}$ better, because it explains more variance in the dependent variable. Or put the other way round, the added regressors contribute nothing but "blur" to the model. $\endgroup$
    – jay.sf
    Commented Dec 17, 2020 at 19:05
  • $\begingroup$ that's perfect, thank you so much. So would this mean that Z is not a confounding variable? and would provide support for hypothesis in the bivariate model? $\endgroup$
    – Louis Rich
    Commented Dec 17, 2020 at 19:07
  • $\begingroup$ @LouisRich I would'nt conclude that solely from the $R^2_{adj.}$. The effect of the $X$ variable is probably more important (does it change considerable? does it change sign? does it get insignificant?). $\endgroup$
    – jay.sf
    Commented Dec 17, 2020 at 19:21
  • $\begingroup$ HI X in the bivraite model 1.2052, STD error 0.1603, p value= 2.58e-12 ***. $\endgroup$
    – Louis Rich
    Commented Dec 17, 2020 at 19:25

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