Group sum of squares

In a like manner, within-group sums of squares and cross products are calculated as... 

The quantities SSB and MSB are usually referred to as the between groups sum of squares and mean square for between groups, respectively. Eq. (1.119) is not suitable for practical calculations so it is transformed into the following expression ... 

The amount of variability among the k groups is important to our hypothesis testing. This is referred to as the group sum of squares and can be denoted as... 

Within samples By exactly the same logic that we saw for the within-groups sums of squares, we can calculate these df as ... 

Dudoit et al. (2002) introduced the ratio of between-group to within-group sums of squares (BW ratio). The BW ratio for a gene j, BW(y), is defined as... 

Figure 9.22 shows the di9 4 step9.m file that contains the code for Multivariate Analysis of Variance, MANOVA (i.e., testing the null hypothesis that the group means are all the same for the n-dimensional multivariate vector, and that any difference observed in the sample stress is due to random chance). The group means must lie in a maximum of dfb-dimensional space, where dfb is the degree of freedom of B matrix. B is the between-groups sum of squares and cross products matrix (see MATLAB online help for manoval). MATLAB manoval will take care of this maximum dimension, dfb. Because d = 1 as calculated by manoval, we cannot reject the hypothesis that the means lie in a 1-D subspace. 

Si = -xgm) sum of squares of residuals relative to the appropriate group mean Xmean, (1.31)... 

Since T and therefore also tr(T) is constant, minimizing tr(W) is equivalent to maximizing tr(B). It can be shown that tr(B) is the sum of squared Euclidean distances between the group centroids. 

Mi A Zagreb group parameter = sum of square of degrees over all vertices... 

Body weight gains from the period immediately preceding each consumption measurement were used, since these were less correlated. For each variable and at each time, the sums of squares for group differences were divided into four meaningful contrasts ... 

In turn, the sum of squares between groups (bg) is found from... 

The sum of squares within group (wg) is then the difference between the last two figures, or... 

If we consider the case where K treatments are being compared such that K = 1,2,K, and we let Xik and Yik represent the predictor and predicted values for each individual i in group k, we can let Xk and Yk be the means. Then, we define the between-group (for treatment) sum of squares and cross products as... 

To estimate the variance components for the random effects model, we also computed the group means regression. The sum of squared residuals from the LSDV estimator is 444,288. The sum of squares from the group means regression is 22382.1. The estimate of a,.2 is 444,288/93 = 4777.29. The estimate of a 2 is 22,382.1/2 - (1/20)4777.29 = 10,952.2. The model is then reestimated by FGLS using these estimates ... 

In this equation, the term SSW is refereed to as the the sum of squares within groups or error sum of squares. The quantity SSW when divided by the appropriate degrees of freedom J(I-l) is referred to as the mean square or error mean square and is denoted by MSW- As Eq. (1.114) is not particularly convenient for calculation purposes, it can be presented in the more usable form ... 

We need to become familiar with the topic of analysis of variance, often abbreviated ANOVA, in order to test the null hypothesis (H0) p1 = p2 = " = where k is the number of experimental groups, or samples. In the ANOVA, we assume that a = a2 = = ol, and we estimate the population variance assumed common to all k groups by a variance obtained using the pooled sum of squares (within-groups SS) and the pooled degree of freedom (within-groups DF) ... 

There are many different methods for the task of fitting any number of parameters to a given measurement [14-16], We can put them into two groups (a) the direct methods, where the sum of squares is optimized directly, e.g., finding the minimum, similar to the example in Figure 7.4, and (b) the Newton-Gauss methods, where the residuals in r or R themselves are used to guide the iterative process toward the minimum. 

Sum of squares of characters multiplied by the number of operations in each class equals the order of the group... 

An alternative method is based on the investigation of the Jacobian of the kinetic system of odes, J = df/dc. A species may be considered redundant if its concentration change has no significant effect on the rate of production of important species. An element of the normed Jacobian (3 ln/,)/(3 In Cy) shows the fractional change of the rate of production of species i caused by the fractional change of the concentration of species j. The influence of the change of the concentration of species i on the rate of production of an A-membered group of important species can be taken into account by the sum of squares of normalized Jacobian elements,... 

The next step is to calculate the 3 x 3 matrix of pooled within groups sums of (mean corrected) squares and cross products (SSCP) by the formula ... 

These two sums of squares are used to obtain the between-groups variation and the within-groups variation. The error sum of the squares is related to the individual group variances by... 

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