Corrected sum of squares

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. 

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. 

Corrected sum of squares See total sum of squares. (Section 4.4) Cross-classified system In a multiway ANOYA when the measurements are made at every combination of each factor. (Section 4.8) Degrees of freedom The number of data minus the number of parameters calculated from them. The degrees of freedom for a sample standard deviation of n data is n — 1. For a calibration in which an intercept and slope are calculated, df=n — 2. (Sections 2.4.5, 5.3.1) Dependent variable The instrument response which depends on the value of the independent variable (the concentration of the analyte). (Section 5.2)... 

Total sum of squares, SST (also corrected sum of squares) In ANOYA the number arising from the sum of the squares of the mean corrected values. (Section 4.4)... 

Square each mean-corrected value and then sum them all to give the total sum of squares also known as the corrected sum of squares ... 

The various sums of squares (this term always refers to corrected sums of squares that is, it is the sums of squares of deviations about some average value) may be calculated most easily in the following order ... 

As a check, the sum of the eight squared standardised contrast values should equal the corrected sum of squares of the nine results (i.e. standard deviation x 8), provided all eight contrasts are orthogonal. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

The model itself can be tested against the sum of squared residuals c2=4.01. If, as a first approximation, we admit that intensities are normally distributed (which may not be too incorrect since all the values seem to be distant from zero by many standard deviations), c2 is distributed as a chi-squared variable with 5 — 3 = 2 degrees of freedom. Consulting statistical tables, we find that there is a probability of 0.05 that a chi-squared variable with two degrees of freedom exceeds 5.99, a value much larger than the observed c2. We therefore accept to the 95 percent confidence level the hypothesis that the linear signal addition described by the mass balance equations is correct, o... 

There are different ways of testing whether there is continuing improvement of the fit or whether the progress is finished and, hopefully, the best minimal ssq has been reached. In the progress of the iterations, the shifts 8p, as well as the sum of squares, usually decrease continuously. Thus both could be inspected for constancy. The most common and intuitively correct test is the constancy of the sum of squares, as indicated in Figure 4-35. 

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 ... 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

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. 

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