Slope confidence limits

Standard Error of Estimate Syx = 0.4328 Slope Confidence Limits Method 1... 

Slope confidence limits at selected confidence level are ... 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

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.

To.xicity values for carcinogenic effects can be e.xprcsscd in several ways. The slope factor is usually, but not always, the upper 95th percent confidence limit of the slope of the dose-response curve and is e.xprcsscd as (mg/kg-day). If the extrapolation model selected is the linearized multistage model, this value is also known as the ql. That is ... 

Because the slope factor is often an upper 95 percentile confidence limit of the probability of response based on experimental animal data used in tlie multistage model, tlie carcinogenic risk estimate will generally be an upper-bound estimate. Tliis means tliat tlie EPA is reasonably confident tliat tlie true risk will not exceed the risk estimate derived tlirough use of tliis model and is likely to be less than tliat predicted. 

The model of simple competitive antagonism predicts that the slope of the Schild regression should be unity. However, experimental data is a sample from the complete population of infinite DR values for infinite concentrations of the antagonist. Therefore, random sample variation may produce a slope that is not unity. Under these circumstances, a statistical estimation of the 95% confidence limits of the slope (available in most... 

FIGURE 6.6 Schilcl regression for pirenzepine antagonism of rat tracheal responses to carbachol. (a) Dose-response curves to carbachol in the absence (open circles, n = 20) and presence of pirenzepine 300 nM (filled squares, n = 4), 1 jjM (open diamonds, n=4), 3j.lM (filled inverted triangles, n = 6), and 10j.iM (open triangles, n = 6). Data fit to functions of constant maximum and slope, (b) Schild plot for antagonism shown in panel A. Ordinates Log (DR-1) values. Abscissae logarithms of molar concentrations of pirenzepine. Dotted line shows best line linear plot. Slope = 1.1 + 0.2 95% confidence limits = 0.9 to 1.15. Solid line is the best fit line with linear slope. pKB = 6.92. Redrawn from [5],... 

Figure 2.6. A graphical depiction of a significant and a nonsignificant slope (slopes si, = 4.3 0.5 (A) resp. -0.75 1.3 (B)). If a horizontal line can be fitted between the confidence limits an interpretation X = f y ) is impossible. It suffices in many cases to approximate the curves by straight lines (C).

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Slope b is close to zero and/or ires is large, which in effect means the horizontal will not intercept the lower confidence limit function, and... 

It is important to realize that for the typical analytical application (with relatively few measurements well characterized by a straight line) a weighting scheme has little influence on slope and intercept, but appreciable influence on the confidence limits of interpolated X(y) resp. Y(x). 

Display key results number of points N, intercept a, slope b, both with 95 % confidence limits, coefficient of determination r, residual standard deviation. 

The details differ, however. By convention, instead of plotting the mean, /x0, as a function of n, the sum of the data, which has a theoretical value of n /x , is used. Clearly this line will slope upward with a slope of /x , instead of being horizontal, as will the data plot. The rest of the conceptual picture is the same, however. As we saw previously in reference [3], the slope of the line represented by n /x is paralleled by the confidence limits for the sum of the data, as represented by the equations in that... 

Y as a function of a change in X. These include, but are not limited to correlation (r), the coefficient of determination (R2), the slope (, ), intercept (K0), the z-statistic, and of course the respective confidence limits for these statistical parameters. The use of graphical representation is also a powerful tool for discerning the relationships between X and Y paired data sets. 

A method and Worksheet for computing the confidence limits for the slope and intercept of a data set at user-selected confidence levels. 

Comparison of Goodness of Fit Statistics for Linear Regression Part 4 - Confidence Limits for Slope and Intercept... 

For this chapter we continue to describe the use of confidence limits for comparison of X, Y data pairs. This subject has been addressed in Chapters 58-60 first published as a set of articles in Spectroscopy [1-3]. A MathCad Worksheet ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) provides the computations for interested readers. This will be covered in a subsequent chapter or can be obtained in MathCad format by contacting the authors with your e-mail address. The Worksheet allows the direct calculation of the f-statistic by entering the desired confidence levels. In addition the confidence limits for the calculated slope and intercept are computed from the original data table. The lower limits for the slope and the intercept are displayed using two different sets of equations (and are identical). The intercept confidence limits are also calculated and displayed. 

The confidence limits for the slope and intercept may be calculated using the Student s t statistic, noting Equations 61-27 through 61-30 below. 

The slope (fej confidence limits are computed as shown in Equations 61-27 through 61-30. 

Miller and Miller, pp. 110 and 111 in reference [4], cite the following equations for calculation of the slope (b) confidence limits. 

---