Regression sum of square

If replicates are not available a test based on the ratio of the regression sum of squares to the residual sum of squares can be applied ... 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

There are many cases of interest to kineticists in which the necessity for the inclusion of terms in nonlinear rate models must be tested. This may be done approximately by again calculating the regression sum of squares with and without the added term. Kabel and Johanson (Kl), for example, considered the model... 

Referring back to the ordinary least squares regression, we now compute the mean squared residual, 1911.9275/50 = 38.23855. Then, we compute v = (1 /38.23855)t",2 for each observation. In the regression of v on a constant, Xx, and X2. the regression sum of squares is 145.551, so the chi-squared statistic is 145.551/2 = 72.775. We reach the same conclusion as in the previous paragraph. In this case, the degrees of freedom for the test are only two, so the conclusion is somewhat stronger. 

Then, one determines the amount of variability among the Yt values that results from there being a linear regression this is termed the linear regression sum of squares. 

The value of the regression SS will be equal to that of the total SS only if each data point falls exactly on the regression line. The scatter of data points around the regression line is defined by the residual sum of squares, which is calculated from the difference in the total and linear regression sums of squares ... 

SSE is the error (or residual) sum of squares. This is the quantity represented by Equation 1-13, that is, the equation we want to minimize. It is the sum of the differences squared between the observed Y s and the estimated (or computed) Y s. Figure 1-4 shows the differences between the Y s and Y s. SSR is the regression sum of squares and measures the variation f the estimated values, Y, about the mean of the observed Y s and Y. Figure 1-5 shows the differences between Y s and Y. The regression sum of squares is the sum of the squared differences. 

The regression mean squares, MSR, is an estimate of the variance of the estimates, Y, and since the regression sum of squares has one degree of freedom, the regression mean squares is... 

The remaining sum of squares (usually much larger) represents that part of the data that is explained by the model. This is the regression sum of squares, SS/ c r< defined by ... 

The regression sum of squares SS qk is associated with Vj = p - 1 degrees of freedom, where p is the number of coefficients in the model, because the constant term has already been subtracted in adjusting the total sum of squares. In this example, where p =2, there is 1 degree of freedom. 

The regression sum of squares is associated with 3 degrees of freedom, equal to the number of coefficients in the model (except for the constant term already accounted for in calculating about the mean). The mean regression... 

In particular, we shall see that the regression sum of squares can also be subdivided, each part being associated with certain terms in the model - a sum of squares for the linear model, another for the quadratic and interaction terms. The statistical significance of the different terms in the model can then be assessed. 

The formulation has a significant effect on the response. The only factor influencing the response is the concentration of bile salt. However we also see that there is almost as much residual variation (residual sum of squares) as there is variation explained by the model (regression sum of squares). The systematic differences between subjects (block sum of squares) are of similar importance. Note that the block effects sum to zero (compare with the presence-absence model for qualitative variables in chapter 2). 

In the ANOVA table, the regression sum of squares has been divided into the part associated with the first-order terms (with 2 degrees of freedom) and that which is explained by the second-order terms (with 3 degrees of freedom). 

In addition to separating the regression sums of squares associated with the linear and quadratic terms (and also cubic terms in the case of a third-order model), the sums of squares for individual coefficients may be isolated in the same way and significant interaction and quadratic terms may be identified. Table 5.6 lists the values and the significances of the coefficients of the second degree model. 

ANOVA of the regression with partition of the regression sum of squares... 

The regression sum of squares is the total sum of squares explained by the model. The model consists of first-order terms, square terms and first-order interactions and the regression sum of squares can be divided among these three sets of terms as follows. For simplicity we will consider all the second-order terms together. We fit the first-order model ... 

Of the total adjusted sum of squares, 28% is accounted for by the first-order terms alone, and 75% by all the terms of the second-order model, first-order, square, and interaction. The residual sum of squares of the first order model includes the sum of squares for the second-order terms. So the difference between the residual sums of squares for the two models is the second-order regression sum of squares... 

Not only are the models for mixtures different from those for independent variables, but the analysis of variance also shows certain differences. TTiere is no constant term in the model equation, but the regression sum of squares should be calculated with respect to the mean term. Since we have decided to use a logarithmic transformation of the response, it will be the mean logarithm of the solubility or the... 

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