Total 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 ... 

How to best describe this broadening we expect to occur One way is by analogy to random error in measurements. We know or assume there is a truly correct answer to any measurement of quantity present and attempt to determine that number. In real measurements there are real, if random, sources of error. It is convenient to talk about standard deviation of the measurement but actually the error in measurement accumulates as the sum of the square of each error process or variance producing mechanism or total variance = sum of the individual variances. If we ignore effects outside the actual separation process (e.g. injection/spot size, connecting tubing, detector volume), this sum can be factored into three main influences ... 

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). 

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. 

To form die Sum of Squares for the main effect of the factor S, we square the S totals, sum these squares, and divide by the number of original individuals forming each S total, and subtract from this the correcting factor, i.e. 

The Total Sum of Squares is the difference between the sum of the squares of the original individuals and the correcting factor, i.e. 

The Between Pairs within Blends term is given directly by squaring the pair totals and dividing by the number of individuals therein, and subtracting the correcting factor and the Blend sum of squares, i.e. 

In a confounded experiment the computation proceeds exactly as though the experiment had not been confounded, with the sole difference that we do not compute those interactions which we have confounded. Instead, we take the block totals, square them, divide by the number of treatments occurring in each block, and subtract the usual correcting factor (grand total squared over grand total number of observations). The degrees of freedom for this sum of squares is of course one less than the number of blocks. We would get an identical result if we calculated the sum of squares for the interactions being confounded, and then pooled them. 

When the average centred variable matrix (X — X) is used, the matrix (X - X) (X - X) contains the sums of squares and the crossproducts of the variables. Since the means of each variable have been subtracted, the elements in (X - X) (X - X) are related to the variances and the covariances of the variables over the set of N compounds. The total sum of squares is equal to the sum of the eigenvalues. The variation described by a component is proportional to the sum of squares associated with this component, and this sum of squares is equal to the corresponding eigenvalue. It is usually expressed as a percentage of the total sum of squares and is often called "explained variance", although this entity is not corrected for the number of degrees of freedom. Percent "explained variance" by component "j" is therefore obtained as follows ... 

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

The total sum of squares is sometimes called the corrected total sum of squares, to emphasize that its values refer to deviations from the grand average. 

The total variance, expressed as the sum of squares of deviations from the grand mean, is partitioned into the variances within the different groups and between the groups. This means that the sum of squares corrected for the mean, is obtained from... 

For the total sum of squares, Eq. (2.45) is again valid, that is, the total sum of squares corrected for the mean is obtained as sum... 

Partitioning of the variances of a linear regression follows the scheme given in Table 6.1 and Figure 6.1 by the appropriate sums of squares the total variance of the y values, SS, adds up from the sum of squares of the mean, SSj, and the sum of squares corrected for the mean, SS . 

After integration over the total volume of an isothermal reactor, Eq. a yields the various ffow rates Fj at the exit of the reactor, for which Vr/(,Fcjh )o has a certain value, depending on the propane feed rate of the experiment. If Eq. a is integrated with the correct set of values of the rate coefficients ki...kg the experimental values of Fj should be matched. Conversely, from a comparison of experimental and calculated Pj the best set of values of the rate coefficients may be obtained. The fit of the experimental Fj by means of the calculated ones, Fj, can be expressed quantitatively by computing the sum of squares of deviations between experimental and calculated exit flow rates, for example. These may eventually be weighted to account for differences in degrees in accuracies between the various Fj so that the quantity to be minimized may be written, for n experiments ... 

As before, the between-row, between-column and total sums of squares are calculated. Each calculation requires the term T /nrc (where n is the number of replicate measurements in each cell, in this case 2, r is the number of rows and c is the number of columns). This term is sometimes called the correction term, C. Here we have ... 

In both the sequential mode and the total mode a PRESS is calculated for the final model with the estimated number of significant components. This is often re-expressed as (the cross-validated R ), which is (I — PRESS/SS), where SS is the sum of squares of Y corrected for the mean. This can be compared with R Y = (I — RSS/SS), where RSS is the residual sum of squares. In models with several Y variables one also obtains R Y and Ql, for each Y variable y . 

A total of 185 emission lines for both major and trace elements were attributed from each LIBS broadband spectrum. Then background-corrected, summed, and normalized intensities were calculated for 18 selected emission lines and 153 emission line ratios were generated. Finally, the summed intensities and ratios were used as input variables to multivariate statistical chemometric models. A total of 3100 spectra were used to generate Partial Least Squares Discriminant Analysis (PLS-DA) models and test sets. 

The constraint of minimum negativity (Howard, 1981) applies to data for which it is known that the correctly restored function should be all positive. For our formulation, we want to find the coefficients of v(k) that best satisfy this constraint. These coefficients will be those that minimize the negative deviations in the total function u(k) + v(k). The sum of the squared values of the negative deviations is given by... 

ANOVA calculations are straightforward in this example and are easily expanded to situations in which there are higher numbers of categories. The first ANOVA quantity we calculate is C, the correction factor. C is simply the square of the sum of all the individual values, divided by N, the total number of values (Lxijkif /N, where i = 1 - 2 (analysts), j = 1 - 2 (days). A = 1 — 3 (specimens), and 1=1—2 values for each specimen for each analyst on each day. Table 8 demonstrates the calculation of this and the other calculated ANOVA quantities. The quantity A is calculated as the sum of the squared individual values The ANOVA quantity for each classi-... 

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

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