Straight residual variance

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

The degree of dispersion of the points around the straight line provides an estimate of the variance. Accordingly, the estimate of the residual variance of the regression is given by s yjC... 

Following Mandel," if the null hypothesis cannot be rejected, z.e. the straight line is a correct model and the alternative does not improve its fit, both the numerator and denominator estimate the true pure residual variance, the experimental Fisher-Snedecor s test will be lower than the critical value for a given probability level, and 1 and —3 dof for the numerator and denominator (in case a polynomial other than the quadratic one is considered, n — k dof should be considered instead of — 3, with k the order of the polynomial plus 1). Otherwise, the alternative hypothesis must be accepted z.e. the numerator contains structured variance which is larger than the pure residual variance. 

An F test to compare the variances associated with the lack of fit and the pure error allows us to decide whether the straight line model is eompatible with the experimental data. If the experimental value is greater than the tabulated one i.e. we can reject the null hypothesis) we should conclude that the model is inadequate." In this case a visual inspection of the residuals will help identify the cause of the problem. 

Mandel s test was summarized as a comparison of the residual standard deviation of the linear model with that of the nonlinear mode . Such a definition results in the well-known conceptual Fischer-Snedecor s F test (Fexperimentai = s ylx,Mn / s y x,nof), whcrc s Stands for standard error of the regression, lin for straight line model and non for non-linear model (here a quadratic one, although this is not mandatory). Note that although the term variance should be used senso stricto instead of standard error, this is not relevant for our discussions here. The definition was then resolved to eqn A 1.1 below (numbered 51 in ref. 8) ... 

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