Measurement confidence limits

The error curve and the measurement Confidence limit Significant figures Correct values... 

The confidence limits of a measurement are the limits between which the measurement error is with a probability P. The probability P is the confidence level and a = 1 - P is the risk level related to the confidence limits. The confidence level is chosen according to the application. A normal value in ventilation would be P = 95%, which means that there is a risk of a = 5 /o for the measurement error to be larger than the confidence limits. In applications such as nuclear power plants, where security is of prime importance, the risk level selected should be much lower. The confidence limits contain the random errors plus the re.sidual of the systematic error after calibration, but not the actual systematic errors, which are assumed to have been eliminated. 

Since the confidence limits of a repeated measurement are based on the dispersion of the measurement result, they usually are presented as symmetrical limits ... 

Usually there is no opportunity to repeat the measurements to determine the experimental variance or standard deviation. This is the most common situation encountered in field measurements. Each measurement is carried out only once due to restricted resources, and because field-measured quantities are often unstable, repetition to determine the spread is not justified. In such cases prior knowledge gained in a laboratory with the same or a similar meter and measurement approach could be used. The second alternative is to rely on the specifications given by the instrument manufacturer, although instrumenr manufacturers do not normally specify the risk level related to the confidence limits they are giving. 

Frequently the value of the quantity of interest has to be determined indirectly. For example, the determination of the efficiency of any system is based on the measurement of several quantities and some equation relating the measured quantities X, and the final quantity Y under consideration. When the confidence limits of the different measured quantities are known, and the relationship y = f(X,) is known, an estimate for the cumulated confidence limits dy of the final quantity can be determined from... 

The bias error is a quantity that gives the total systematic error of a measuring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The repeatability error consists of the confidence limits of a single measurement under certain conditions. The mac-curacy or error of indication is the total error of the instrument, including the... 

Figure 12.12 demonstrates that the larger the confidence limits of measurement are, the closer to the target value the measurement result must be for symmetrical tolerances. A consequence of this is that if the confidence limits of... 

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available.These usually include, beyond the estimation procedure itself, some basics and worked examples. 

Lower confidence limit (LCL) A statistical procedure to estimate whether the true value is lower than the measured value. 

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

Fig. 5. Comparison of leaf extension rates in two grasses of contrasted cell size and nuclear DNA amount, (a) Brachypodium pinnatum, 2c DNA = 2.3 pg (b) Bromus erectus, 2c DNA = 22.6 pg., Watered prior to measurement O, droughted for 3 weeks prior to measurement. The 95% confidence limits are indicated by the vertical lines.

Assuming for the moment that a large number of measurements went into a determination of a mean Xmean and a standard deviation s, what is the width of the 95% confidence interval, what are the 95% confidence limits ... 

The measurement has noise superimposed on it, so that the analyst decides to repeat the measurement process several times, and to evaluate the mean and its confidence limits after every determination. (Note This modus operandi is forbidden under GMP the necessary number of measurements and the evaluation scheme must be laid down before the experiments are done.) The simulation is carried out according to the scheme depicted in Fig. 1.19. The computer program that corresponds to the scheme principally contains all of the simulation elements however, some simplifications can be introduced ... 

Figure 1.21. Monte Carlo simulation of six groups of eight normally distributed measurements each raw data are depicted as x,- vs. i (top) the mean (gaps) and its upper and lower confidence limits (full lines, middle) the confidence limits CL(s ) of the standard deviation converge toward a = 1 (bottom, Eq. 1.42). The vertical divisions are in units of 1 a. The CL are clipped to +5a resp. 0. .. 5ct for better overview. Case A shows the expected behavior, that is for every increase in n the CL(x,nean) bracket /r = 0 and the CL(i t) bracket a - 1. Cases B, C, and D illustrate the rather frequent occurrence of the CL not bracketing either ii and/or ff, cf. Case B n = 5. In Case C the low initial value (arrow ) makes Xmean low and Sx high from the beginning. In Case D the 7 measurement makes both Cl n = 7 widen relative to the n 6 situation. Case F depicts what happens when the same measurements as in Case E are clipped by the DVM.

The confidence limits thus established indicate the y-interval within which F(x) is expected to fall the probability that this is an erroneous assumption is 100 p% in other words, if the measurements were to be repeated and shghtly differing values for a and b were obtained, the chances would only be 100 p% that a Tis found outside the confidence limits CL(y). Use option (T(x)) in program LINREG. The details of the calculation are found in Table 2.5. 

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). 

Example 57 The three files can be used to assess the risk structure for a given set of parameters and either four, five, or six repeat measurements that go into the mean. At the bottom, there is an indicator that shows whether the 95% confidence limits on the mean are both within the set limits ( YES ) or not ( NO ). Now, for an uncertainty in the drug/weight ratio of 1%, a weight variability of 2%, a measurement uncertainty of 0.4%, and fi 3.5% from the nearest specification limit, the ratio of OOS measurements associated with YES as opposed to those associated with NO was found to be 0 50 (n == 4), 11 39 (n = 5), respectively 24 26 (u = 6). This nicely illustrates that it is possible for a mean to be definitely inside some limit and to have individual measurements outside the same limit purely by chance. In a simulation on the basis of 1000 sets of n - 4 numbers e ND(0, 1), the Xmean. Sx, and CL(Xmean) were calculated, and the results were categorized according to the following criteria ... 

Figure 4.34. The confidence limits of the mean of 2 to 10 repeat determinations are given for three forms of risk management. In panel A the difference between the true mean (103.8, circle ) and the limit L is such that for n = 4 the upper confidence limit (CLu, thick line) is exactly on the upper specification limit (105) the compound risk that at least one of the repeat measurements yi >105 rises from 23 n = 2) to 72% (n = 10). In panel B the mean is far enough from the SLj/ so that the CLu (circle) coincides with it over the whole range of n. In panel C the mean is chosen so that the risk of at least one repeat measurement being above the SLu is never higher than 0.05 (circle, corresponds to the dashed lines in panels A and B).

Purpose Illustrate what happens when a series of measurements is evaluated for mean and standard deviation each time a new determination becomes available the mean converges towards zero and the SD towards 1.0. The CL(Xmean) and the CL(i c) normally enclose the expected values E x) = /r = 0, respectively E Sx) = a =. Due to the stochastic nature of the measured signal, it can happen that confidence limits do not bracket the expected value this fact is highlighted by a bold line. 

The mean, the standard deviation, and the confidence limits of the population at each concentration with multiple measurements are calculated and tabulated. 

The use of confidence intervals is one way to state the required precision. Confidence limits provide a measure of the variability associated with an estimate, such as the average of a characteristic. Table I is an example of using confidence intervals in planning a sampling study. This table shows the interrelationships of variability (coefficient of variation), the distribution of the characteristic (normal or lognormal models), and the sample frequency (sample sizes from 4 to 365) for a monitoring program. 

One measure of the quality of an estimate of an average Is the confidence limits (or maximum probable error) for the estimate. For averages of Independent samples, the maximum probable error Is... 

The result of this analysis provides a measure of the precision of the estimate of the mean plus confidence limits for the estimate. 

Confidence limits for the parameter estimates define the region where values of bj are not significantly different from the optimal value at a certain probability level 1-a with all other parameters kept at their optimal values estimated. The confidence limits are a measure of uncertainty in the optimal estimates the broader the confidence limits the more uncertain are the estimates. These intervals for linear models are given by... 

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

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