Confidence limits for slope and intercept

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

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. 

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). 

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. 

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. 

An approach for analyzing data of a quantitative attribute that is expected to change with time is to determine the time at which the 95% one-sided confidence limit for the mean curve intersects the acceptance criterion. If analysis shows that the batch-to-batch variability is small, it is advantageous to combine the data into one overall estimate by applying appropriate statistical tests (e.g., p-values for level of significance of rejection of more than 0.25) to the slopes of the regression lines and zero-time intercepts for individual batches. If it is inappropriate to combine data from several batches, the overall shelf life should be based on the minimum time a batch can be expected to remain within the acceptance criteria. 

If it is assumed that / is constant for a given series, a plot of (1/DF) vs. (k U]/ Rp) should have a slope of / and an intercept of k r/kp at (ka[I]/Rp) = 0. The benzoyl peroxide-initiated polymerization data were analyzed in this manner. There was considerable experimental scatter in these plots and therefore, the data are given (Table VII) as the least squares slopes and intercepts as well as their 95% confidence limits. The average values of k r/kp, ranging from 0.8 to 1.1 X... 

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