Paired -Test

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 amount of calcium in different samples of milk powder (in mg of calcium per g of milk powder) were analyzed by two methods, one employing extraction followed by analysis using atomic absorption spectroscopy, the other using a complexometric titration method. The results of nine analyses are shown in table 3.4. 

Determine whether the two analytical methods give equivalent analytical results. 

As we have a number of one-off measurements, we will use a paired /-test. 

For each pair of data the difference is calculated (see spreadsheet 3.7 and figure 3.9). 

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Fig. 2. Percent variation of beta rhythm in cirrhotic patients after a 1-week period of treatment with rifaximin (200-400 mg t.i.d.) or placebo. The relative beta power of the EEG decreased in the rifaximin-treated group (Wilcoxon paired test Z = 2.1, p = 0.03), but not in the placebo-treated group (Z = 1.07, p = nonsignificant) (from Del Piccolo et al. [40]).

Figure 15 Rotational (tangential) flow ( Ua 1 as a function of stirring rate (co) for the basket using 900 mL (filled circles) and 500 mL (open circles). Mean SD, n = 6, P<0.001, paired /-test lateral position S2. Source From Ref. 10.

Model-independent techniques compare data pairs observed at corresponding time values, where time is only a class effect, as in a paired -test or in an ANOVA. A data-poor set of only two or three observations, originating from routine quality control of an immediate-release dosage form, cannot be treated other than model independent. 

Set up the analysis of variance table as shown. The number of degrees of freedom is one less than the level of each factor For interactions, the number of degrees of freedom is the product of the degrees of freedom for each factor. For replication, the number of degrees of freedom is given by the number of pairs tested. The mean square is the sum of squares divided by the degrees of freedom. 

Consider the requirement for comparing two methods covering a number of sample types or matrices. Here the paired /-test is very useful. In the following example, eight different samples have been analysed by both methods with one determination each. 

The GECE sensors were used for lead determination in real water samples suspected to be contaminated with lead obtained from water suppliers. The same samples were previously measured by three other methods a potentiometric FIA system with a lead ion-selective-electrode as detector (Pb-ISE) graphite furnace atomic absorption spectrophotometry (AAS) inductively coupled plasma spectroscopy (ICP). The results obtained for lead determination are presented in Table 7.1. The accumulation times are given for each measured sample in the case of DPASV. Calibration plots were used to determine the lead concentration. GEC electrode results were compared with each of the above methods by using paired -Test. The results obtained show that the differences between the results of GECE compared to other methods were not significant. The improvement of the reproducibility of the methods is one of the most important issues in the future research of these materials. 

Data are analyzed by a paired /-test with vector treated and contralateral uninjected eyes serving as determinants. 

The critical value of t at the 95% confidence limit is 2.26, so we accept the null hypothesis that there is no significant difference in the accuracy of the two methods. The paired /-test is a common type of test to use, as it is often the availability of test objects or specimens that is the critical factor in analysis. 

There are many cases where we are faced with two columns of measured values and, as with previous examples, we want to see whether values are generally higher in one column than the other. Thus far, the situation is familiar. However, the data in the two columns may be related - they form natural pairs. In that case, the paired /-test provides a superior alternative to the two-sample t-test. An example follows ... 

We know from Chapter 6 that data variability is a spoiler for t-tests and the large SDs for the weights in the first two columns of Table 12.1 are a prime contributor to the non-significant outcome of the two-sample /-test. It would be attractive to be able to base a statistical test solely on the column ofweight changes, as these are considerably less variable. This is exactly what the paired /-test does. 

The calculation of a paired /-test is based on the changes in weight. The actual weights are, of course, necessary in order to calculate the changes, but once this has been done, only the changes are used. 

To perform a paired /-test we simply calculate a 95 per cent Cl for the mean among the weight changes (using final column of Table 12.1) and check to see whether the interval includes the value posited by the null hypothesis (zero). 

When we initially applied a two-sample /-test to the first two columns of data in Table 12.1 (Section 12.2), the result was non-significant (P = 0.155), but the result of the paired /- test was clearly significant (P < 0.001). This is a typical example of the greater power of the paired /-test. 

Where data form natural pairs (as in the current example), it is very frequently the case that individuals who have the highest values under the first set of experimental conditions also have the highest values under the alternative conditions. As a result, the variability in the two original sets of data is greater than that among the individual changes that occur. This is the reason why the paired /-test is more powerful (often much more powerful) than the two-sample test. 

The whole logic of the paired /-test that we just performed was founded on the fact that we could calculate a change for each individual participant in the experiment and then use those changes to calculate the rest of the test. With the theophylline/ rifampicin experiment (Table 6.1), the data also consisted of two columns of data, but in that case there was no pairing. The first figure in the first column and that in the second column were derived from different individuals and it would have made no sense to start calculating the difference between these two figures and then move down to the second number in each column and so on. 

It is perhaps worth noting that the enhanced power of the paired /-test is dependent upon the data being genuinely paired. The paired test has no superiority... 

If data is not naturally paired, the use of paired test is unjustified (and entirely pointless). 

The paired /-test is just a special application of the 95 per cent Cl for the mean and the requirement for normally distributed data, described in Chapter 5, applies in this case too. Just be aware that the confidence interval is calculated using the column of individual changes in weight (etc.), so it is these that need to be normally distributed. The subjects placebo and active treated weights could be skewed or bimodal or any other horrible distribution - that would not matter. We just need the changes to be normal. As previously, we do not expect small samples to form perfect classic normal distributions, but if the column of individual changes shows extreme signs of nonnormality, we should not trust a paired /-test (see Chapter 18 for possible solutions). 

Chapters 8-10 explained sample size calculations, practical significance and onesided tests in detail and the point was made that these were general concepts that could be applied to a wide variety of tests. This section briefly reviews their application to the paired /-test. 

The factors that influence necessary sample size for a paired /-test are very similar to those encountered with the two-sample test, but notice that we take into account variability among the individual changes in the end-point, not the two sets of initial observations. Below are reasonable values for the three relevant factors for our experiment on the effect of the weight-loss drug. 

Wilcoxon paired samples test equivalent to the paired /-test ... 

Without permeation data for at least two temperatures, it is not possible to predict the magnitude of the temperature effect for any given chemical/ material pair. Tests must be performed at the maximum (worst case) temperature and/or over the temperature range of the anticipated application of the clothing. 

Subcase b2 This case, called the paired /-test , is often done when two test procedures, such as methods A and B, are applied to the same samples, for instance when validating a proposed procedure with respect to the accepted one. In practicular, an official content uniformity" 3 assay might prescribe a photometric measurement (extract the active principle from a tablet... 

After testing each of the four individual products, three pairs of products were tested. Formaldehyde release when multiple products were in the same chamber was measured as above. The three pairs tested were particle board/plywood, particle board/insulation, and particle board/carpet. 

One small subclade contained several isolates of H. thompsonii (ARSEF 2464, 3323, and on a linked subclade, ARSEF 257), a pathogen of citrus rust mite, as well as other insects and nematodes (50). Of the four isolates of H. thompsonii in the study, ARSEF 256 and 2464 were PCR-positive with all five primer pairs, while ARSEF 257 and 3323 were positive with three of the five primer pairs tested. 

Long-term behavior of insulation systems can be estimated after the temperature index has been determined by the impregnated twisted pair test (IEC 172, criterion breakdown voltage) or helical coil test (IEC 1033, Method B, criterion bond strength). The two tests lead to differing results that do not correlate and the end user of the system enameled wire/impregnating resin has to decide which test is applicable for his electrical appliance. It is important to choose a suitable combination of materials and to ensure close cooperation between manufacturers of electrical insulating materials and those who later process them. 

Fe(III) Tap, waste water Chloroform UV-Vis 29.0 pg L-1 C-FIA with merging zones [465] membrane for phase separation different ion pairs tested... 

When there are a number of repeated analyses of the same material then do a means /-test. When there are many different test materials with a single measurement performed, then do a paired /-test. (Sections 3.8, 3.9)... 

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