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According to the algorithm for the backward stepwise selection from the book ISLR which is shown below:

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says that we need to choose the model among the $k$ models by having a smallest RSS or highest $R^{2}$, while other resources tells that for choosing the predictor/s to be included in the model based on their $p$-values as stated here.

Can someone tell what is the right process?

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These are two different methods, each with advantages and disadvantages.

The method in ISLR is more computationally intensive. It fits 1+p(p+1)/2 models, where p is the number of predictors. The other backward selection method fits at most p+1 models.

On the other hand, the method in ISLR is more statistically sound. In general, the practice of comparing p-values to each other is frowned upon by statisticians. See principle number 5 in the ASA's Statement on p-Values: https://www.tandfonline.com/doi/pdf/10.1080/00031305.2016.1154108?needAccess=true

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