Analytical Confidence bounds

In this contribution we showedhow confidence bounds of reliability growth processes can be generated using a Monte Carlo simulation approach. Our results are compatible with other analytical confidence bound concepts. We found that the rank distributions at certain time horizons are log-normal distributed. Using the confidence bound concept it is possible to make statistically sound decisions regarding a current development project or the development process itself. 

This graphical method, illustrated in Fig. 6, is a variation of the standard deviation-based approaches discussed previously. In this approach, the 95% confidence bounds are obtained for the best-fit, regression-based, calibration line. The upper 95% confidence bound, extrapolated to zero analyte concentration, reflects the statistical limit as it relates to calibration bias and imprecision, and establishes the LOD in terms of response. Mathematically, this response LOD is converted to a concentration LOD by inputting the response LOD into the lower 95% confidence bound and by solving for concentration. Graphically, this can be accomplished as shown in Fig. 6. 

Regarding confidence bounds for reliability growth models there are at least two analytical approaches namely the confidence bounds according to Crow and according to the Fisher matrix. 

In this equation the confidence interval Xupper,xu,u,er) for a predicted value of jto is calculated. The parameter is the slope of the regression line, t is the Student t statistic, s is the residual mean square error, Sxx is the sum of squares of the X values from the mean of the calibration x values, and n is the total number of calibration samples. The values from these confidence limits place upper and lower bounds for the analyte concentration, jtq. at all points along the calibration range. The best confidence is found near the mean of the data, whereas lower confidence is found at the extremes. 

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