Tests for the Regression Equation

Correlation analysis gives us the correlation coefficient which is a measure of the strength and the direction of the linear association between the variables. This information can be used to decide the suitability of model calibration using a linear regression analysis. The square of the correlation coefficient may be thought of as the percentage of the total variation in y that is explained by the association of y and x. Hence, for R = 1, all the variation is explained by the linear association between the two variables. In this case, all the observations will lie on a straight line of slope $\sigma_{y}^{}$/$\sigma_{x}^{}$, passing through the point ($\mu_{x}^{}$,$\mu_{y}^{}$).

Another measure used to evaluate the goodness of fit is the standard deviation of the errors $\sigma_{e}^{}$, defined as

$$\sigma_{e}^{2} {\frac{1}{\nu}} \sum_{i=1}^{n} \hat{y}_{i}^{}$$ (9)

where $\hat{y}_{i}^{}$ = ax_i + b and $\nu$ represents the number of degrees of freedom. The number of degrees of freedom is equal to the sample size minus the number of unknowns estimated in by the regression procedure. In this case, we have the slope and the intercept as the unknowns, so $\nu$ = n - 2. If $\sigma_{e}^{}$ is very small, we attribute a high reliability to the results of the regression analysis.

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