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I am doing a regression project on some medical data using SAS. I used forward selection, backward selection, stepwise selection, and the LASSO, and all procedures gave me the same subset of variables.

However, when I run the reduced model, for some reason I have a higher MSRes and a lower adjusted $R^2$ than I had for the full model.

Assumptions have been met and there is no multicollinearity.

Is this normal? It seems a bit weird to me.

Thanks!

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R-squared will always increase as you increase the number of parameters in your model but that does not necessarily mean that it is the best model. You may want to look at other criteria like AIC, BIC, deviance, etc.

You may be trading some predictive power in order to avoid overfitting.

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