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How is variability measured in Linear Regression?

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The overall variability of a regression model is quantified by the parameter ? (after estimated), which is a measure of the average deviation of the model’s predictions from the actual target values. A larger ? indicates that the model’s predictions are further away from the actual values on average, indicating a poorer fit. The estimate for the residual standard error is found by 

The variance and standard error of the beta hats provides a measure of the variability in the coefficient estimates. A large standard error indicates less precision in the estimate and all else equal, results in a lower chance of that coefficient being significant. In matrix form, the variance and standard error of the coefficients is given by the following formula, which produces a pxp matrix where the variance terms are on the diagonal and the covariances on the off-diagonals.

The point estimate and standard error provide the components needed to conduct a hypothesis test to determine the statistical significance of a regression coefficient. A t-test statistic is formed by t = / SE(), which follows a t-distribution with n-2 degrees of freedom. The p-value can then be computed using the test-statistic and the quantiles of the t-distribution, and if it is below the predetermined significance level, the null hypothesis that the actual value of the coefficient is 0 can be rejected, meaning that predictor has a nonzero effect on the outcome. 

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