After fitting a linear regression model, whether for inference or prediction, it is crucial to assess the quality of the fit. Key questions include:

  • Have we selected the best set of predictor variables? and

  • Can the model provide reliable predictions with good accuracy?

Key Questions to Ask

1. Is There a Linear Relationship Between the Response and Predictors?

Consider a scenario in multiple linear regression where none of the regression coefficients are significantly different from zero. Such a model would neither be useful for inference nor for prediction. How can we ensure or verify that at least one regression coefficient is significantly different from zero?

To answer this, a hypothesis test is conducted:

  • Null Hypothesis (H0): All regression coefficients are zero.

  • Alternative Hypothesis (Ha): At least one regression coefficient is non-zero.

This test is performed using the F-test, whose p-value is provided by most statistical software like R, Python, SAS, SPSS, and even Excel. If the p-value is low (commonly below 0.05), it indicates that at least one predictor is significantly related to the response. This step is essential whether your goal is inference or prediction.

2. Which Variables Are Important?

Once we establish that at least one predictor is significant, the next step is to identify which specific predictors contribute meaningfully to the response…

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