Within-sample variation

A common example where ANOVA can be applied is in interlaboratory trials or method comparison. For example, one may wish to compare the results from four laboratories, or perhaps to evaluate three different methods performed in the same laboratory. With inter-laboratory data, there is clearly variation between the laboratories (between sample/treatment means) and within the laboratory samples (treatment means). ANOVA is used in practice to separate the between-laboratories variation (the treatment variation) from the random within-sample variation. Using ANOVA in this way is known as one-way (or one factor) ANOVA. 

The Within-Sample Variation (Within-Treatment Variation)... 

In many cases, the method variance will be known from replicate measurements of a single laboratory sample. Under this circumstance, can be computed from measurements of for a series of laboratory samples, each of which is obtained from several gross samples. An analysis of variance (see Section 7C) can reveal whether the between-samples variation (sampling plus measurement variance) is significantly greater than the within-samples variation (measurement variance). 

To attempt to sort out some of these issues with sourcing native copper, Ron Hancock and colleagues at the SLOWPOKE nuclear reactor facility at the University of Toronto decided to use the multielement capability of neutron-activation analysis. They started with 43 samples from 19 collections of native copper and 23 samples of copper from archaeological contexts but which were believed to be reworked metal of European origin. Eive other samples of copper from artifacts from known contexts but of unknown source were also included. To maximize the possibility that the provenience postulate would be true, they included 27 elements in their analytical procedure and got useful data for 22 of them. They also analyzed 14 subsamples from the same specimen to assess within sample variation and three modem copper samples for comparison. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

Fairhead et al. (2005b) examined small-scale within-thallus variations in the phlorotannin contents of D. anceps and D. menziesii and also compared individuals collected from different sites at Anvers Island as well as from the upper and lower limits of their depth distributions at those sites. Thirteen defined thallus parts within each individual were sampled, and although there was considerable variation within individuals at that spatial scale, there were no consistent patterns. When fine scale parts within the individuals were grouped, the main axis of D. anceps had significantly lower phlorotannin levels than did lateral branches. In D. menziesii, the distal-most tips of the lateral branches had significantly higher phlorotannin levels... 

The raw data (baseline corrected at 1600 nm) for all of the plastic samples plotted in Figure 4.18 reveals significant within class" variation (the offset for the different plastic types was added for clarity). However, none of the samples have fear tures that appear to be unusual given that the reflectance spectra are of xmwashed plastic containers. The decision is to leave all samples in for further analysis. 

The spectra for the 29 training set samples are shown in Figure 4.50 (the baseline is corrected at 1600 nm and the classes are offset for claritjO Ideally, the spectra for each sample within a class would overlay and the features would-be different between the classes. In Figure 4.50, there appears to be significant within-class variation, which is addressed in the next section. No unusual samples are observed, but this finding is reevaluated after preprocessing is applied. 

Although the within-class variations have been reduced significantly, one or two samples in each class are still somewhat different from the others in the class. However, they appear to be members of their respective classes (i.e., the shapes of the spectra are approximately the same). Furthermore, because there are so few samples in most clas.ses. it is not reasonable at this point to exclude any sa.mple from the an.alysis. 

We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent sampling variation in the sampling process. The confidence interval provides us with a compromise rather than trying to pin down precisely the value of the mean p or the difference between two means — p2> for example, we give a range of values, within which we are fairly certain that the true value lies. 

Precision is the degree of agreement between a series of measurements obtained from multiple sampling of the same homogeneous sample under prescribed conditions. Precision may be considered at three levels repeatability (precision under the same operating conditions over a short interval of time), intermediate precision (precision within-laboratory variations different days, different analysis, different equipment, etc.), and reproducibility (precision between laboratories, collaborative studies, usually applied to standardization of methodology). 

Intermediate Precision. Intermediate precision expresses within-laboratory variation and is generally performed on different days using different analysts, equipment, and sample preparations. This test may not be applicable if the laboratory has only one workstation. Additionally, this test may not be appropriate for automated workstations that are operating under the same environment and controls within a laboratory. This assumption is made on the basis that the automated workstations are identical (i.e., same configuration, same software and hardware) and that they have been suitably qualified and maintained to a consistent standard and operate under a similar climatic environment. The influence of the analyst is reduced to the preparation of solvents, and this should be covered by the robustness studies. 

A CSF sample was analyzed 11-fold. The within-run variation coefficient ranged from 1 to 3.5% with two exceptions tryptophan (5%) and methionine (7%), which partially coeluted. The interassay coefficients of variation were calculated from a series of 11 analyses over a 7-month period. The median CV was 8% only taurine, arginine, and glutamate had CVs slightly in excess of 10%. The recovery of added amino acids to three CSF samples ranged form 83% (taurine) to 101% (isoleucine). Most recoveries were between 90 and 100%. At the lower end of the concentration range for CSF, a level of 1 pmol/1 can be safely detected. 

In analyzing a lot with random sample variation, you find a sampling standard deviation of 5%. Assuming negligible error in the analytical procedure, how many samples must be analyzed to give 95% confidence that the error in the mean is within 4% of the true value Answer the same question for a confidence level of 90%. 

To demonstrate the ability to evaluate inter-sample variations, tablet groups from two different manufacturers in an over-the-counter (OTC) pain relief medication were compared. Pure acetaminophen, aspirin, and caffeine samples are obtained in either tablet form or powder compact and included within the same FOV as the tablets to provide simultaneous reference materials for the tablet samples. The tablets and pure components were arranged as shown in Plate la. This FOV contains 20 tablets and the... 

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