Paired f-test

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

This is an example of a paired data set since the acquisition of samples over an extended period introduces a substantial time-dependent change in the concentration of monensin. The comparison of the two methods must be done with the paired f-test, using the following null and two-tailed alternative hypotheses... 

Subcase b2 This case, called the paired f-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" 5 assay might prescribe a photometric measurement (extract the active principle from a tablet... 

Area n Averaged organic carbon concentration (mg C perl) Difference by paired f-test at 95% confidence level... 

If the data are recorded at corresponding time values, an alternative is to treat them in a way similar to paired differences as in a paired f-test or in an ANOVA, where time is not considered as continuous independent variable but only as a class effect. The result is a model-independent index, which... 

The null hypothesis to be tested states that there is no significant difference between the pairs of results. Using a paired f test, the standard deviation for the difference ((/) between the pairs (sd) is... 

The formula for the t test described in Procedure 1.3 compares the mean of replicate analyses of only one sample but it may be preferable to compare the accuracy over the analytical range of the method. To do this a paired f test may be used in which samples with different concentrations are analysed using both methods and the difference between each pair of results is compared. A simplified example is given in Procedure 1.4. 

Continuous Student s unpaired f-test Student s paired f-test ANOVA Paired ANOVA... 

To perform a paired f-test we simply calculate a 95 per cent Cl for mean individual change and check whether the interval includes zero. 

CH12 THE PAIRED f-TEST - COMPARING TWO RELATED SETS OF MEASUREMENTS... 

Two-sample t-test - means, then difference of the means Paired f-test - differences, then mean of the differences... 

This is summarised in Figure 12.3. The logic is very similar to that for the two-sample f-test, with one crucial difference. For a two-sample f-test, the variability that has to be considered is that among the two sets of weighings (the first two columns of data in Table 12.1). With a paired f-test, what matters is the variability among the individual weight changes (final column of Table 12.1). Therefore, with a paired f-test, the initial... 

Figure 12.3 Factors influencing the outcome of a paired f-test...

Greater problems in the case of data loss If we used an unpaired design, we might find ourselves unable to obtain a measurement from one of our subjects. In that case, we would be left with 15 observations in one column, and only 14 in the other. With a two-sample /-test, we would still be able to use all of the data obtained. However, if we were performing a paired study (and presumably a paired f-test), we would not only lose that data point, but additionally its accompanying paired value would become useless and would have to be discarded. 

The paired f-test is used where data form natural pairs. Its classic use arises when we have observed the same individual (human or otherwise) under two different circumstances (before vs after treatment or placebo treatment period vs active treatment period, etc). 

It is always worth considering the use of a paired experimental design, as it will allow the use of the more powerful paired f-test. However, paired experiments can present practical difficulties that may outweigh this statistical advantage. For a valid paired f-test, the individual changes in the measured parameter should form a normal distribution. 

General statistical methods such as sample size estimation, determination of practical significance and one-sided testing can be applied to the paired f-test in the same manner that we have already seen for the two-sample f-test. 

Figure 17.5 strongly suggests that this treatment is much more likely to lead to an increase than to a decrease in haemoglobin levels, but a formal test is required. No transformation is going to convert this to a normal distribution. With normally distributed changes, we would have used a paired f-test, but with this data we will change to the Wilcoxon paired sample test. 

The efficiency of an experiment can depend crucially on the details of its design. A classic example is whether to design a paired or unpaired study. We saw in Chapter 12 that a paired experiment followed by a paired f-test may be much more powerful than the unpaired equivalent. In some circumstances, the difference is so great that an unpaired design could be predicted to have inadequate power, leading to inevitable failure. In other cases, the difference in power may less dramatic and you could waste your time struggling with the extra practical complexities of a paired design. 

The example demonstrates that all relevant information must be used ignoring the fact that the PM and HPLC measurements for / = 1. .. 5 are paired results in a loss of information. The paired data should under all circumstances be plotted (Youden plot. Fig. 2.1, and Fig. 1.23) to avoid a pitfall it must be borne in mind that the paired f-test yields insights only for the particular (addi-... 

In a pilot clinical trial which was an open design conducted in two public hospitals in Nigeria, 29 volunteers with acute malaria infection were involved in the study. Both clinical and parasite counts were done. The result showed that all the volunteers responded to the phytomedicine with 100% parasite clearance and total remission of clinical signs associated with malaria infection. The volimteers on the control arm (16) were given fansidar. Only 40% of them responded to fansidar. The statistical analysis showed that the difference between the phytomedicine-treated and fansidar-treated groups was profoimd (p <0.0005, paired f-test). 

Mean TSH at R + 0 is significantly different from all other examinations (paired f-test, p < 0.001). 

Parametric statistics (t-test, ANOVA) are by far the most commonly used in studies of sensory-motor/psychomotor performance due, in large part, to their availability and ability to draw out interactions between dependent variables. However, there is also a strong case for the use of non-parametric statistics. For example, the Wilcoxon matched-pairs statistic maybe preferable for both between-group and within-subject comparisons due to its greater robustness over its parametric paired f-test equivalent, with only minimal loss of power. This is important due to many sensory-motor measures having very non-Gaussian skewed distributions as well as considerably different variances between normal and patient groups. 

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