Correction for the mean

Figure 4.12 Magnification of the region near Lq in Figure 4.11. L is the Lorentz point (the Lane point corrected for the mean refractive index). A is one of the tie-points selected on the dispersion surface, and o and are the deviation parameters at that point (shown on branch 2)...

The mean response can be subtracted from each of the individual responses to produce the so-called responses corrected for the mean. This terminology is unfortunate because it wrongly implies that the original data was somehow incorrect responses adjusted for the mean might be a better description, but we will use the traditional terminology here. It will be convenient to define a matrix of responses corrected for the mean, C. 

This matrix may be used to calculate another useful sum of squares, the sum of squares corrected for the mean, SS, sometimes called the sum of squares about the mean or the corrected sum of squares. 

The sum of squares corrected for the mean has n - degrees of freedom associated with it. 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSf (sometimes called the sum of squares due to regression, SS ) the sum of squares of residuals, SS, and the sum of squares corrected for the mean, (or the total sum of squares, SSj, if there is no Pq term in the model). 

It is important to realize that an / or r value (instead of an or value) might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of 55 has been explained by the factors. Although most regression analysis programs will supply both R (or r) and R (or r ) values, researchers seem to prefer to report the coefficients of correlation R and r) simply because they are numerically larger and make the fit of the model look better. 

Use the C matrix of Equation 9.5 to calculate the sum of squares corrected for the mean, SScorr> for the nine responses in Section 3.1 (see Equation 9.6). How many degrees of freedom are associated with this sum of squares ... 

From the mean of the data set () to the response itself (j>w). This is a distance that has been corrected for the mean. 

Although the partitioning of the total sum of squares into a sum of squares due to the mean and a sum of squares corrected for the mean may be carried out for any data set, it is meaningful only for the treatment of models containing a / 0 term. In effect, the /30 term provides the degree of freedom necessary for offsetting the responses so the mean of the corrected responses can be equal to zero. 

It is important to realize that R might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of SScori has been explained by the factors. 

Apart from the mentioned one, the analysis of variance is also frequently used where the total sum of squares SST is corrected for the mean sum of squares SSM-Such a sum of squares is called the total corrected sum of squares SStc=SSt SSM-Analysis of variance of the linear regression with total corrected sum of squares is given in Table 1.69. 

If k is the wavevector of the incident wave in vacuum, then K is the magnitude of k after correction for the mean inner potential Uq of the crystal. The fact that K is different from k means that the crystal has a refractive index, and it is clear that the mean refractive index n must be related to Uq. Direct measurements show that n is of the order of 1 x 10 In the dynamical theory of electron diffraction, which we develop in this chapter, we formally make all the Fourier components Kg complex quantities, that is, we replace Kg by Vg + iV. The full physical significance of the procedure will become clear in due course, but for the moment it will be helpful to consider the consequences of making Kq complex. If Kq is complex, then the mean refractive index n must also be complex, and so we write n = n +in". 

The first and fourth terms in the dispersion equations represent the difference between the square of the wavevectors Kq and Kg inside the crystal and the square K, which is the vacuum value k corrected for the mean inner potential. If there is no difference, there is no unique solution. Thus, the refractive index for Kq and Kg waves must be different from the average refractive index. This is the crux of the dynamical theory. 

If 2-fold dilutions are used, the quantities Xp, x , [ip], and may be obtained from Table 3. The products [xy] are calculated as Sxy — (SxSy/n), where Sxy indicates the sum of the products of each log plasma dilution with its corresponding clotting time in working units, and SxSy/n is the correction for the mean and is the product of the sum of the log dilutions (from Table 3) and the sum of the clotting times n is the number of plasma dilutions tested. 

Corrected for the mean of solvent contribution and normalised to an exact concentration based on the exact sample weights. 

Correction for the mean Subtraction of the grand mean from each measurement result in ANOYA. This quantity is also known as the mean corrected value. (Section 4.4)... 

Resample the modified residuals (denoted e ) and holding the predictor variables as fixed, generate the bootstrap dependent variable as Y = f(x 0) + e. Sometimes instead of resampling the modified residuals, e is resampled from ei — e, e2 — e,... en — e instead. In the linear model the mean residual is zero so this is equivalent to bootstrapping the modified residuals. But, the mean residual is not necessarily equal to zero in the nonlinear case so correcting for the mean is necessary. 

---