Regression through the Origin

Under such conditions the estimated regression line. Equation 6.1, reduces to 

The option to use this model is often available in statistical computer packages, and for manual calculations the arithmetic is reduced compared with the full linear regression discussed above. A caveat should be made, however, since forcing the line through the origin assumes that the measured blank value is free from experimental error and that it represents accurately the true, mean blank value. 

For the nickel data from Table 6.1, using Equation 6.19, = 0.094, and the 

Sometimes the regression model is linear and is known to go through the origin at the point (0,0). An example may be the regression of dose against area under the curve (AUC). Obviously when the dose of 

Regression through the origin is presented here because of a number of peculiarities to the model, some of which may be unfamiliar to pharmacokineticists. First, the residuals may not necessarily sum to zero and a residual 

Equally, terms on the right hand side of Eq. (2.27) can be viewed as variability due to the regression line and variability around the regression line. Clearly, a good model is one where SSregression SSE. Assuming that the residuals are independent and normally distributed with mean 0 and variance a2, an F-test can be computed to test the null hypothesis that 0 = 0, 

Under the null hypothesis, F is distributed as an F-distribution with p, n-p degrees of freedom. If F FP n P a the null hypothesis is rejected. This is called the analysis of variance approach to regression. The power of this approach comes in when multiple covariates are available (see Multiple Linear Regression later in the chapter). The F-test then becomes an overall test of the significance of the regression model. 

One of the most commonly used yardsticks to evaluate the goodness of fit of the model, the coefficient of determination (R2), develops from the analysis of variance of the regression model. If SStotai is the total sum of squares then 

For the nickel data from Table 1, using Equation (19), b = 0.094, and the sum of squares of the deviations, SSd, is 0.00614. This value is greater than the computed value of SSd for the model using data not corrected for the blank, indicating the poorer performance of the model of Equation (18). 

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Accordingly, the Cammarata-Yau model is a regression through the origin defined as... 

Step 1 Estimate the population serial correlation coefficient, P, with the sample correlation coefficient, r. It requires a regression through the origin or the (0, 0) points, using the residuals, instead of y and x, to find the slope. The equation has the form... 

Using a hand calculator, find the slope of the linear regression line that passes through the origin and best satisfies the points... 

The use of the following statistical characteristics of the test set was also recommended [33] (1) correlation coefficient between the predicted and observed activities, (2) coefficients of determination (predicted versus observed activities R, and observed versus predicted activities Rq), (3) slopes fc and k of the regression lines through the origin. Thus, we consider a QSAR model predictive if the following conditions are satisfied [33] ... 

Eq. 12-12 implies that the logarithm of the ratio [A],/[A]0 yields a straight line through the origin with slope -k. Thus, if data from kinetic experiments are plotted as in Fig. 12.2, we can both check whether the reaction is first order in [A] and determine the rate constant k using a linear regression analysis. We note that in the case of first-order kinetics, the half-life, tm, of the compound (i.e., the time in which its concentration drops by a factor of 2) is independent of concentration and equal to ... 

The difference in elevation computed using the equation in Currie et al (2005) and Equation (5) ranges from about +150 to -84 m in the range of the fit. Polynomial regression constrained to pass through the origin of the median of the distribution yields the relationship... 

The data for the high concentration range (0, = 8-16 mmol, i.e. I = 0.18 - 0.09 ), can also be fitted with a linear curve. This curve, however, does not pass through the origin (figure 9.11), in contrast with both the modelling curves and the experimental curve at low concentration. Linear regression results in equation (4) ... 

Slopes (k) of regression lines through the origin were calculated according to Eq. (1) ... 

Fig. 1. Confidence limits and prediction intervals for immunological comparisons of five proteins. For each protein the heavy central line is the regression line through the origin relating immunological distance in the microcomplement fixation test to percent difference in amino acid sequence, and the shaded region portrays the 95% confidence limits for that line. The outer two lines in each graph define the boundaries of the intervals for one prediction made at the 90% level of confidence.

If the mathematical model indicates that the regression will go through the origin, e.g., in the case of the yield from a reactor as a function of time, the calculations can obviously be simplified over those shown in Eq. (19). The slope expression is obtained by differentiating the linear equation and setting the sum of squares of the derivative at a minimum. Terms in the slope expression may be simplified in the following manner ... 

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