Repeated-measures

If the normalized method is used in addition, the value of Sjj is 3.8314 X 10 /<3 , where <3 is the variance of the measurement of y. The values of a and h are, of course, the same. The variances of a and h are <3 = 0.2532C , cf = 2.610 X 10" <3 . The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for h indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute. 

Example 2 Calculation of Error with Doubled Sample Weight Repeated measurements from a lot of anhydrous alumina for loss on ignition established test standard error of 0.15 percent for sample weight of 500 grams, noting V is the square of s.e. Calculation of variance V and s.e. for a 1000 gram sample is... 

The rows represent the type of measurement (e.g., compositions, flows, temperatures, and pressures). The columns represent streams, times, or space position in the unit. For example, compositions, total flows, temperatures, and pressures would be the rows. Streams I, 2, and 3 would be columns of the matrix of measurements. Repeated measurements would be added as additional columns. 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

Precision Variation about the mean of repeated measurements of the same pollutant concentration expressed as one standard deviation about the mean. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

The precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

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

The variation observed when repeated measurements of the same parameter on the same specimen are taken with the same device. 

C. —To every measurable property of a system there corresponds a linear operator in Stf. If the measurement of the property corresponding to the operator L is performed on a system always initially prepared in the normalized state Q at time t, the mean value of the result of a series of such repeated measurements is... 

More often than not, measurements are accompanied by two kinds of error. A systematic error is an error present in every one of a series of repeated measurements. An example is the effect of a speck of dust on a pan, which distorts the mass... 

Individuals with hereditary low plasma cholinesterase levels (Kalow 1956 Lehman and Ryan 1956) and those with paroxysmal nocturnal hemoglobinuria, which is related to abnormally low levels of erythrocyte acetylcholinesterase (Auditore and Hartmann 1959), would have increased susceptibility to the effects of anticholinesterase agents such as methyl parathion. Repeated measurements of plasma cholinesterase activity (in the absence of organophosphate exposure) can be used to identify individuals with genetically determined low plasma cholinesterase. 

The title implies that in this first chapter techniques are dealt with that are useful when the observer concentrates on a single aspect of a chemical system, and repeatedly measures the chosen characteristic. This is a natural approach, first because the treatment of one-dimensional data is definitely easier than that of multidimensional data, and second, because a useful solution to a problem can very often be arrived at in this manner. 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Figure 1.9. A large number of repeat measurements x,- are plotted according to the number of observations per x-interval. A bell-shaped distribution can be discerned. The corresponding probability densities PD are plotted as a curve versus the z-value. The probability that an observation is made in the shaded zone is equal to the zone s area relative to the area under the whole curve.

Narrow limits any statement based on a statistical test would be wrong very often, a fact which would certainly not augment the analyst s credibility. Alternatively, the statement would rest on such a large number of repeat measurements that the result would be extremely expensive and perhaps out of date. 

In everyday analytical work it is improbable that a large number of repeat measurements is performed most likely one has to make do with less than 20 replications of any detemunation. No matter which statistical standards are adhered to, such numbers are considered to be small , and hence, the law of large numbers, that is the normal distribution, does not strictly apply. The /-distributions will have to be used the plural derives from the fact that the probability density functions vary systematically with the number of degrees of freedom,/. (Cf. Figs. 1.14 through 1.16.)... 

After having characterized a distribution by using n repeat measurements and calculating x ean and Sx, an additional measurement will be found within... 

Figure 1.25. The number of measurements n that are necessary to obtain an error probability p of Xmean exceeding L is given on the ordinate. The abscissa shows the independent variable Q in the range L - 3 Sx. .. T in units of Sx- It is quite evident that for a typical p of about 0.05, Xmean must not be closer than about 0.5 standard deviations Sx from L in order that the necessary number of repeat measurements remains manageable. The enhanced line is for p -0.05 the others are for 0.01 (left), 0.02, and 0.1 (right).

The FDA mandates that of all the calibration concentrations included in the validation plan, the lowest jc for which CV < 15% is the LOD (extrapolation or interpolation is forbidden). This bureaucratic rule results in a waste of effort by making analysts run unnecessary repeat measurements at each of a series of concentrations in the vicinity of the expected LOD in order to not end up having to repeat the whole validation because the initial estimate was off by + or - 20% extrapolation followed by a confirmatory series of determinations would do. The consequences are particularly severe if validation means repeating calibration runs on several days in sequence, at a cost of, say, (6 concentrations) x (8 repeats) x (6 days) = 288 sample work-ups and determinations. 

In the previous sections of Chapter 2 it was assumed that the standard deviation obtained for a series of repeat measurements at a concentration X would be the same no matter which x was chosen this concept is termed homoscedacity (homogeneous scatter across the observed range). 

One performs so many repeat measurements at each concentration point that standard deviations can be reasonably calculated, e.g., as in validation work the statistical weights w, are then taken to be inversely proportional to the local variance. The proportionality constant k is estimated from the data. 

The chosen weighting model should also be applied to a number of repeat measurements of a typical sample. The resulting GOF figure is used as a benchmark against which those figures of merit resulting from parameter fitting operations can be compared. (See Table 1.26.) The most common... 

The means resulting from several (k) repeat determinations on every of (j) samples and/or sample preparations must comply several such sample averages Xmean, would go into the over-all average Xmean,total-In the simplest case oi j = Ijk - 2 (duplicate determinations on each of two sample work-ups), both sample averages xji and Xj2 would have to comply. It could actually come to pass that repeat measurements, if foreseen in the SOP, would in this way not count as individual results this would cut the contribution of the measurement s SD towards the repeatability by v/2, x/3, etc. For true values /x that are less than 2-3 a from the limit, the OOS-risk would be reduced. 

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).

Table 4.32. The Joint OOS-Risk [%] Associated with n Repeat Measurements...

---