Intercept confidence limits

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. 

Intercept 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. 

CL is inserted in Eqs. (2.18) and (2.19), with k - oo, and using the + sign. The intercept of the horizontal with the lower confidence limit function of the regression line defines the limit of quantitation, jcloq, any value above which would be quoted as 2f(y ) t s ... 

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... 

The horizontal intercepts the lower confidence limit function twice, i.e., if n is small, ires is large, and all calibration points are close together this can be guarded against by accepting Xloq only if it is smaller than... 

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). 

Figure 4.26. Shelf-life calculation for active components A and B in a cream see data file CREAM.dat. The horizontals are at the j = 90 (specification limit at t = shelflife) resp. y = 95% (release limit) levels. The linear regression line is extrapolated until the lower 90%-confidence limit for Kfl = a + h x intersects the SLs the integer value of the real intersection point is used. The intercept is at 104.3%.

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

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... 

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. 

Miller and Miller, pp. Ill and 112 in reference [4] cite the following Equations for calculation of the intercept (a) confidence limits. 

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. 

In this calculation it has been assumed that the value of data points in the table can be used, thus making it unnecessary to repeat the experiments many times. Furthermore, if the distribution is considered to be Gaussian, then the confidence limits for the estimates of the gradient and intercept of the fitted line can be determined. For this type of distribution it can be shown 8 that 95 per cent of the readings will lie within 2ffof the mean. Hence it is possible to say with 95 per cent certainty (or with 95 per cent confidence limits) that the equation of the fitted line is ... 

This approach uses the upper confidence limit of the calibration intercept a0 derived from nc measurements of calibration samples [DIN 32 645] ... 

Stability data (not only assay but also degradation products and other attributes as appropriate) should be evaluated using generally accepted statistical methods. The time at which the 95% one-sided confidence limit intersects the acceptable specification limit is usually determined. If statistical tests on the slopes of the regression lines and the zero-time intercepts for the individual batches show that batch-to-batch variability is small (e.g., p values for the level of significance of rejection are more than 0.25), data may be combined into one overall estimate. If the data show very little degradation and variability and it is apparent from visual inspection that the proposed expiration dating eriod will be met, formal statistical analysis may not be necessary. 

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