Statistics of repeated measurements

In chapter 2 I introduced the statistics of repeated measurements. Here I describe how these statistics are incorporated into a quality control program. In a commercial operation it is not always feasible to repeat every analysis enough times to apply t tests and other statistics to the results. However, validation of the method will give an expected repeatability precision (sr), and this can be used to calculate the repeatability limit (r), the difference between duplicate measurements that will only be exceeded 5 times in every 100 measurements. 

J. C. Miller and J. N. Miller, Basic Statistical Methods for Analytical Chemistry. A Review. Part 1. Statistics of Repeated Measurements. Part 2. Calibration and Regression Methods, Analyst, 113(1988) 1351 116(1991)3. 

The problems caused by possible outliers in regression calculations have been outlined in Sections 5.13 and 6.9, where rejection using a specified criterion and non-parametric approaches respectively have been discussed. It is clear that robust approaches will be of value in regression statistics as well as in the statistics of repeated measurements, and there has indeed been a rapid growth of interest in robust regression methods amongst analytical scientists. A summary of two of the many approaches developed must suffice. 

Miller, J. N., and J. C. Miller. "Statistics of Repeated Measurements," in J. N. Miller and J. C. Miller, Statistics and Oiemometrics for Analytical Chemistry, 4th ed. Upper Saddle River, NJ Prentice Hall, 2000,80-90. 

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would rest on such a large number of repeat measurements that the result would be extremely expensive and perhaps out of date.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.)... 

There are a fairly complex set of statistical techniques, which go under the heading of repeated measures ANOVA, that do not summarise the serial measurements for each subject as mentioned above, but leave them separated as they are. These methods then provide p-values relating to a comparison of the set of... 

Given two sets of repeated measurements, the question of whether the data come from populations having equal variances might arise. This is tested by calculating the Fisher F statistic, which is defined as... 

All uncertainty estimates start with that associated with the repeatability of a measured value obtained on the unknown. It is neither required for the sake of quality control, nor could it always be economically justified, to make redundant determinations of each measured value, such as would be needed for complete statistical control. Repeat measurements of a similar kind under the laboratory s typical working conditions may have given satisfactory experience regarding the range of values obtained under normal operational variations of measurement conditions such as time intervals, stability of measurement equipment, laboratory temperature and humidity, small disparities associated with different operators, etc. Repeatability of routine measurements of the same or similar types is established by the use of RMs on which repeat measurements are made periodically and monitored by use of control charts, in order to establish the laboratory s ability to repeat measurements (see sect, entitled The responsible laboratory above). For this purpose, it is particularly important not to reject any outlier, unless cause for its deviation has been unequivocally established as an abnormal blunder. Rejection of other outliers leads a laboratory to assess its capabilities too optimistically. The repeatability in the field of a certified RM value represents the low limit of uncertainty for any similar value measured there. 

Theoretical probability identifies the possible outcomes of a statistical experiment, and uses theoretical arguments to predict the probability of each. Many applications in chemistry take this form. In atomic and molecular structure problems, the general principles of quantum mechanics predict the probability functions. In other cases the theoretical predictions are based on assumptions about the chemical or physical behavior of a system. In all cases, the validity of these predictions must be tested by comparison with laboratory measurements of the behavior of the same random variable. A full determination of experimental probability, and the mean values that come from it, must be obtained and compared with the theoretical predictions. A theoretical prediction of probability can never be tested or interpreted with a single measurement. A large number of repeated measurements is necessary to reveal the true statistical behavior. 

They did not recall uniform amounts of information about the five situations, however, and there were distinct differences in how much they remembered. Table 7.3 provides some summary statistics about students recollections. A multivariate test of repeated measures shows that the mean numbers of nodes recalled for the five situations differed significantly, F(4, 22) = 7.96, p <. 01. That this difference is not attributable just to the differing numbers of potential nodes that could be learned can be seen in Table 7.3, where the number of nodes found in instruction is given as well as the proportion of these nodes that were recalled on average by the students. If students had some fixed propensity to remember each detail they encountered in instruction, they should be more likely... 

Why do we bother with means and standard deviations Because these two statistics tell us a great deal about the data and the population from which they come. A mean of a number of repeated measurements of the concentration of a test solution is an estimate of the concentration of the test solution and the sample standard deviation gives a measure of the random scatter of the values obtained by measurement. Together with the appropriate units they represent the result. This information is not necessarily the answer to What is the concentration of the test solution and how sure are you of that... 

In some cases we do not have the luxury of repeated measurements of a single test material, but do have one-off measurements of a number of different test materials performed by two methods. The two methods can be compared by considering the results of each pair of one-off measurements. This is possible as for a particular test material measured by each method the difference in the result should be zero if the two methods give equivalent results. For a number of analyses of different materials any pair of materials is the same and so the mean of the differences can be tested against zero. If the two methods give equivalent results within measurement uncertainty the difference between results on the same material by each method should be zero. In a paired /-test, therefore, the mean xd and standard deviation sd of the differences are calculated and a /-statistic determined from equation 3.5 with /x = 0 ... 

The probable random error in the mean of a set of repeated measurements can be determined statistically. 

We cannot judge the statistical significance of the results from these figures alone, as some measure of the reproducibility of the method is required. If the standard deviation of repeated measurements is o then the standard deviation for the estimate of each effect is o/Vl2. What we need is an estimate of a. 

The problem of designing a statistical experiment with repeated measures has been extensively studied in the DoE literature. Repeated measures implies that experimental units or subject will be used more than once (i.e. at two or more periods of time) [2], Consequently, any potential model for the response variable in terms of the factors considered in the experiment will need to contain parameters for unit or subject effects, period or time effects and possible carryover effects. Many studies of repeated measures involve observations over time (or space) and the evolution of response is often of special importance [53]. Because the same unit is producing several successive responses, those that are closer together will tend to be more closely related in other words, a previous result is playing a role on the ensemble of the response variable realization. Therefore, in such cases, these relationships must also be included in the model. 

Another property of the sampling distribution of the mean is that, even if the original population is not normal, the sampling distribution of the mean tends to the normal distribution as n increases. This result is known as the central limit theorem. This theorem is of great importance because many statistical tests are performed on the mean and assume that it is normally distributed. Since in practice we can assume that distributions of repeated measurements are at least approximately normally distributed, it is reasonable to assume that the means of quite small samples (say >5) are normally distributed. 

In the predictable case all outcomes of repeated measurements will be equal to each other and to the mean value. The standard deviation will equal zero. In the statistical case the outcomes will vary and the standard deviation will be nonzero. 

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