Linear regression limitations

Application of IP and NCS in conjunction with specification tolerance limits enables to substantiate acceptance criteria for linear regression metrological characteristics (residual standard deviation, correlation coefficient, y-intercept), accuracy and repeatability. Acceptance criteria for impurity influence (in spectrophotometric assay), solution stability and intermediate precision are substantiated as well. 

Confidence limits are also drawn on Figure 2.15(a) to give boundaries of Cpi for a given q determined from the analysis, which are within 95%. The relationship between q and Cp is described by a power law after linear regression giving ... 

Light filters for colorimeters, see Filters, optical Limiting cathode potential 509 see also Controlled potential electro-analysis Linear regression 145 Lion intoximeter 747 Liquid amalgams applications of, 412 apparatus for reductions, 413 general discussion, 412 reductions with, (T) 413 zinc amalgam, 413 Liquid ion exchangers structure, 204 uses, 204, 560... 

We will explore the two major families of chemometric quantitative calibration techniques that are most commonly employed the Multiple Linear Regression (MLR) techniques, and the Factor-Based Techniques. Within each family, we will review the various methods commonly employed, learn how to develop and test calibrations, and how to use the calibrations to estimate, or predict, the properties of unknown samples. We will consider the advantages and limitations of each method as well as some of the tricks and pitfalls associated with their use. While our emphasis will be on quantitative analysis, we will also touch on how these techniques are used for qualitative analysis, classification, and discriminative analysis. 

Figure 2.9. The confidence interval for an individual result CI( 3 ) and that of the regression line s CLj A are compared (schematic, left). The information can be combined as per Eq. (2.25), which yields curves B (and S, not shown). In the right panel curves A and B are depicted relative to the linear regression line. If e > 0 or d > 0, the probability of the point belonging to the population of the calibration measurements is smaller than alpha cf. Section 1.5.5. The distance e is the difference between a measurement y (error bars indicate 95% CL) and the appropriate tolerance limit B this is easy to calculate because the error is calculated using the calibration data set. The distance d is used for the same purpose, but the calculation is more difficult because both a CL(regression line) A and an estimate for the CL( y) have to be provided.

Figure 4.9. Product deterioration according to technicians A (left) and B (right) using the same analytical method. Technician A s results are worthless when it comes to judging the product s stability and setting a limit on shelf life. The bold circles indicates the batch 1 result obtained by technician B this turns out to be close to the linear regression established for batch 2, suggesting that the two batches degraded at the same rate.

Situation A cream that contains two active compounds was investigated over 24 months (incomplete program if today s ICH standards are applied, which require testing at 0, 3, 6, 9, 12, 18, and 24 months). The assays resulted in the data given in file CREAM.dat. Program SHELFLIFE performs a linear regression on the data and plots the (lower) 90% confidence limit for the regression line. For each full time unit, here months, it is determined whether this CL drops below levels of y = 90% resp. y = 95% of nominal. Health authorities today require adherence to the 90% standard for the end-of-shelf-life test, but it is to be expected that at least for some products the 95% standard will be introduced. 

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%.

Calculate linear regression and display graph points, regression line, upper and lower 95% confidence limits CL for regression line... 

Purpose Calculate the intersection of two linear regression lines and estimate the 95% confidence limits on the intersection coordinate. (See Fig. 2.19.)... 

Purpose Perform a linear regression analysis over the selected data points display and print results, do interpolations, determine limits of detection. 

The data are also represented in Fig. 39.5a and have been replotted semi-logarithmically in Fig. 39.5b. Least squares linear regression of log Cp with respect to time t has been performed on the first nine data points. The last three points have been discarded as the corresponding concentration values are assumed to be close to the quantitation limit of the detection system and, hence, are endowed with a large relative error. We obtained the values of 1.701 and 0.005117 for the intercept log B and slope Sp, respectively. From these we derive the following pharmacokinetic quantities ... 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

Comparison of Goodness of Fit Statistics for Linear Regression Part 3 - Computing Confidence Limits for the Correlation Coefficient... 

Workman, J. and Mark, H., Chemometrics in Spectroscopy Comparison of Goodness of Fit Statistics for Linear Regression - Part 3, Computing Confidence Limits for the Correlation Coefficient, Spectroscopy 19(7), 31-33 (2004). 

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