Regression lines

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

The sum of squares of differences between points on the regression line yi at Xi and the arithmetic mean y is called SSR... 

If the data set is Puly nomial and the enor in y is random about known values of a , residuals will be distr ibuted about the regression line according to a normal or Gaussian distribution. If the dishibution is anything else, one of the initial hypotheses has failed. Either the enor dishibution is not random about the shaight line or y =f x) is not linear. 

For the nine substituents m- andp-methyl, p-fluoro, m- and p-chloro, m- and p-bromo, and m- and p-iodo, using the results for nitration carried out at 25 °C in nitromethane or acetic anhydride - (see tables 9.1, 9.5), a plot of logjoA/ j against cr+ produced a substituent constant p = —6-53 with a standard deviation from the regression line i = 0-335, 2 correlation coefficient c = 0-975. Inclusion of... 

The most commonly used form of linear regression is based on three assumptions (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x. Because we assume that indeterminate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. 

Normal calibration curve for the hypothetical data in Table 5.1, showing the regression line. 

Three replicate determinations are made of the signal for a sample containing an unknown concentration of analyte, yielding values of 29.32, 29.16, and 29.51. Using the regression line from Examples 5.10 and 5.11, determine the analyte s concentration, Ca, and its 95% confidence interval. 

Fig. 6. Dose—response regression line for mortaUty data (represented by x) expressed by log-probit plot.

Fig. 9. The two materials, A and B, have overlapping 95% confidence limits at the LD q level. Because the slopes of the dose—mortahty regression lines for both materials are similar, there is no statistically significant difference in mortahty at the LD q and LD q levels. Both materials may be assumed to be lethahy equitoxic over a wide range of doses, under the specific conditions of the test.

Fig. 10. Two materials, A and B, have statistically similar LD q values but, because of differences ia the slopes of the dose—mortaUty regression lines, there are significant differences ia mortaUty at the LD q and LD jq levels. Material A is likely to present problems with acute overexposure to large numbers of iadividuals ia an exposed population when lethal levels are reached. With Material B, because of the shallow slope, problems may be encountered at low...

Calculations of the confidence intervals about the least-squares regression line, using Eq. (2-100), reveal that the confidence limits are curved, the interval being smallest at Xj = x. 

Abstracted from the compilation by Jaffe, where original references may be found. Value of log k on the least-squares regression line where a = 0 the time unit is seeonds. 

Table 7-2 lists 15 reactions whose rates are correlated by the Hammett equation. Besides the reaction constant p, the table gives a value for k° (from the fitted line), which provides all the information needed to estimate k for any member of the series, if the corresponding o is available, by means of Eq. (7-24). Note that kP in Table 7-2 is generally not identical to the experimental value of k for the ct = 0 member of the series, because this experimental point may deviate from the regression line. 

There are instances where it is important to know if a given regression line is linear. For example, simple competitive antagonism should yield a linear Schild regression (see Chapter 6). A statistical method used to assess whether or not a regression is linear utilizes analysis of covariance. A prerequisite to this approach is that there... 

There are methods available to test whether or not two or more regression lines statistically differ from each other in the two major properties of lines in Euclidean space namely position (or elevation) and slope. This can be very useful in pharmacology. An example is given later in the chapter for the comparison of Schild regressions (see Chapter 6). 

The procedure for determining possible differences in position of regression lines is given in Table 11.13a. In contrast to the analysis for the slopes, these data indicate... 

The determination of errors in the slope a and the intercept b of the regression line together with multiple and curvilinear regression is beyond the scope of this book but references may be found in the Bibliography, page 156. 

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