Repeated measurement, replication

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.)... 

Random deviations (errors) of repeated measurements manifest themselves as a distribution of the results around the mean of the sample where the variation is randomly distributed to higher and lower values. The expected mean of all the deviations within a measuring series is zero. Random deviations characterize the reliability of measurements and therefore their precision. They are estimated from the results of replicates. If relevant, it is distinguished in repeatability and reproducibility (see Sect. 7.1)... 

The coefficients a,- are estimated from the results of experiments carried out according to a design matrix such as Table 5.9 which shows a 23 plan matrix. The significance of the several factors are tested by comparing the coefficients with the experimental error, to be exact, by testing whether the confidence intervals Aai include 0 or not. The experimental error can be estimated by repeated measurements of each experiment or - as it is done frequently in a more effective way - by replications at the centre of the plan (so-called zero replications ), see Fig. 5.2. 

In Fig. 2.11, an example of a biplot is shown (Rudnitskaya et al. 2009b). From the inspection of the graph, it is possible to get information about the repeatability of the measurements (replicate analyses on wine samples), the discrimination among samples (wines of three vintages), the intercorrelation between variables (responses of potentiometric sensors), and their discriminatory importance. 

As I have shown, the response given by the model equation (3.5) has an error term that includes the lack of fit of the model and dispersion due to the measurement (repeatability). For the three-factor example discussed above, there are four estimates of each effect, and in general the number of estimates are equal to half the number of runs. The variance of these estimated effects gives some indication of how well the model and the measurement bear up when experiments are actually done, if this value can be compared with an expected variance due to measurement alone. There are two ways to estimate measurement repeatability. First, if there are repeated measurements, then the standard deviation of these replicates (s) is an estimate of the repeatability. For N/2 estimates of the factor effect, the standard deviation of the effect is... 

Measure the concentration of analyte in several identical aliquots (portions). The purpose of replicate measurements (repeated measurements) is to assess the variability (uncertainty) in the analysis and to guard against a gross error in the analysis of a single aliquot. The uncertainty of a measurement is as important as the measurement itself, because it tells us how reliable the measurement is. If necessary, use different analytical methods on similar samples to make sure that all methods give the same result and that the choice of analytical method is not biasing the result. You may also wish to construct and analyze several different bulk samples to see what variations arise from your sampling procedure. 

When fewer than about 100 measurements of the same type are needed, the use of control charts becomes impractical. A few repeat measurements made within the routinely encountered range of relevant values is sufficient to estimate the repeatability of a single measurement. Difficulty arises only when a measurement type or procedure is inordinately time-consuming or costly to replicate. Relevant examples are the measurement of an unusual trace constituent in a sample of minimal size, and a lengthy isotope dilution mass-spec-trometric determination. The analyst is then required to depend on general experience of reliability of a method and would be wise to estimate the uncertainty with special care. 

Mass spectrometer precision was determined by making repeated measurements on a gas sample prepared by combustion of carbon isotope reference material NBS-22. The standard deviation of the mean derived from 10 consecutive measurements of this gas was 0.02 %o The error associated with the combustion and purification procedure was measured by replicate combustions of the NBS-22 reference material, which resulted in a standard deviation of 0 12 %o for five samples ITius, the overall precision associated with the mass spectrometric measurement of vs PDB was 0.12 %o, or in absolute terms, 1.3 ppm. Most of the error clearly was associated with the combustion and sample handling process. Since sealed-tube combustions have been shown to produce theoretical recoveries of carbon (33). these small errors most likely arise from handling the CO2 after it is released from the sample tube ... 

The third approach is to use experimental methods to assess the error structure. Independent identification of error structure is the preferred approach, but even minor nonstationarity between repeated measurements introduces a significant bias error in the estimation of the stocheistic variance. Dygas emd Breiter report on the use of intermediate results from a frequency-response analyzer to estimate the variance of real and imaginary components of the impedance. Their approach allows assessment of the variance of the stochastic component without the need for replicate experiments. The drawback is that their approach cannot be used to assess bias errors and is specific to a particular commercial impedance instrumentation. Van Gheem et have proposed a structured multi-sine... 

Almost all measurements are subject to some random error. This is the inherent variability of the process of measurement. Even something as apparently straightforward as measuring the length of a table with a tape measure will have some variability. At some level, repeated measurements will not be identical. If the readings are unbiased they will be evenly spread around the mean result and, if the tape measure is accurate, the mean result of a set of replicates will be very close to the true result. 

We have seen that variability in toxicity testing can arise from repeat measurements made within a laboratory and also between laboratories. In reality, the variability seen between laboratories is a consequence of both within- and be-tween-laboratory sources of variability, and both are also subject to the within-test variability referred to earlier, as evident from differences between test replicates. Research based on a series of acute aquatic toxicity tests (Whitehouse el al., 1996) shows that variation between laboratories is higher than that between repeat tests in the same laboratory. This, in turn, accounts for more variability than that seen between replicates within a test. Similar findings are evident from the work of others in connection with the introduction of whole-effluent toxicity tests in the USA (e.g. Warren-Hicks and Parkhurst, 1992 Fulk, 1995). Over the years, a number of authors have examined variability in aquatic toxicity testing. Typically these describe variability in terms of the coefficient of variation (standard deviation divided by the mean) in EC50 or LC50 values that is achieved when the same toxicant is tested several times (or by several laboratories) using the same method. Table 2.3 summarises the results of a review of published data. 

In many instances, the normal (Gaussian) distribution best describes the observed pattern, giving a symmetrical, bell-shaped frequency distribution (p. 274) for example replicate measurements of a particular characteristic (e.g. repeated measurements of the end-point in a titration). 

Automated methods are more reliable and much more precise than the average manual method dependence on the technique of the individual technologist is eliminated The relative precision, or repeatability, measured by the consistency of the results of repeated analyses performed on the same sample, ranges between 1% and 5% on automated analyzers. The accuracy of an assay, defined as the closeness of the result or of the mean of replicate measurements to the true or expected value (4), is also of importance in clinical medicine. 

Random error is associated with every measurement. To obtain the last significant figure for any measurement, we must always make an estimate. For example, we interpolate between the marks on a meter stick, a buret, or a balance. The precision of replicate measurements (repeated measurements of the same type) reflects the size of the random errors. Precision refers to the reproducibility of replicate measurements. 

A lack-of-fit procedure, which is straightforward, can be used in this situation. However, it requires repeated measurements (i.e., replication) for... 

Perform true replication, not just repeated measurements in the same experiment. 

Figure 7.2 shows another problem—that of inadequate sample points within the x,s. The large gaps between the x,s represent unknown data points. If the model were fit via a polynomial or piecewise regression with both replication and repeated measurements, the model would still be inadequate. This is because the need for sufficient data, specified in step 1, was ignored. 

Another type of problem occurs when repeated measurements are taken, but the study was not replicated (Figure 7.3). The figure depicts a study that was replicated five times, and each average repeated measurement plotted. That a predicted model based on the data from any single replicate is inadequate and unreliable is depicted by the distribution of the replicates. 

Represents each of five replicated experiments with repeated measurements... 

Precision of a chemometric method refers to the reproducibility of the method. For quantitative chemometric methods, it is important to test both the instrument and method precision. Instrument precision is done by repeating measurements on the same sample method precision is the closeness of replicate sample measurements while intermediate precision can be evaluated by running the same samples with different analysts on different days. 

There is a clear distinction between replications and repeated measurements. The latter refers to taking several measurements from a single occurrence of a phenomenon. To fully understand what a true replicate is, we need to understand the term experimental unit. A true replicate is simply a replicated experimental unit. An experimental unit is defined as the unit that is directly exposed and randomly assigned to the treatment independent of other units (Kuehl, 2000). In our mouse diet example, the treatments we want to study and compare are the two diets. The units to which... 

---