The paired t-test

The paired t-test also known as the one-sample t-test was also developed by Cosset. This test is primarily used for the analysis of data arising from within-patient designs, although we also see it applied when comparing a baseline value with a final value within the same treatment group. 

Consider a two-period, two-treatment cross-over trial in asthma comparing an active treatment (A) and a placebo treatment (B) in which the following PEF (1/min) data, in terms of the value at the end of each period, were obtained (Table 4.1). 

Patients 1 to 16 received treatment A followed by treatment B while patients 17 to 32 received treatment B first. 

The final column above has calculated the A — B differences and as we shall see, the paired t-test works entirely on the column of differences. Again we will follow through several steps for the calculation of the p-value for the A versus B comparison  

Let p be the population mean value for the column of differences. The null and alternative hypotheses are expressed in terms of this quantity  

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As can be seen from the analysis in Table 11.3, the paired t-test indicates that the effect of AC512 on the constitutive activity is significant at the 99% level of confidence (p<0.01 and AC512 is an inverse agonist and does decrease the constitutive receptor activity of calcitonin receptors). 

Example 7. The -test using samples of differing composition (the paired t-test). 

The two-sample t-test is only applicable, if the data are collected from analysing the same sample several times. If the data come from the analysis of different samples where the differences between the samples are much greater than the differences between the two methods or the two laboratories, it is necessary to apply the paired t-test. 

Comments as above for the between-patient designs apply also for the within-patient designs and in many cases the best approach will be to focus on a sequence of pairwise comparisons using the paired t-test. 

For the paired t-test, the standard deviation of the within-patient differences for the primary endpoint needs to be specified and again, the level of effect to be detected. 

In the paired t-test setting it is the normality of the differences (response on A — response on B) that is required for the validity of the test. The log transformation on the original data can sometimes be effective in this case in recovering normality for these differences. In other settings, such as ANOVA, ANCOVA and regression, log transforming the outcome variable is always worth trying, where this is a strictly positive quantity, as an initial attempt to recover normality. 

This test is the non-parametric equivalent of the paired t-test. Recall from Section 11.3 that the paired t-test assumes that the population of differences for each patient follows the normal shape. If this assumption is violated then the paired t-test does not apply although, as with the unpaired t-test, the paired t-test is fairly robust against modest departures from normality. 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

To see if there is a significant difference between the methods, we use the paired t test. First, column D computes the difference (d,) between the two results for each sample. The mean of the 11 differences ( d = — 2.491) is computed in cell D16 and the standard deviation of the 11 differences (,vd) is computed in cell D17. 

HI Now we use a built-in routine in Excel for the paired t test to see if the two methods in Problem 4-14 produce significantly different results. Enter the data for Methods 1 and 2 into two columns of a spreadsheet. Under the TOOLS menu, select DATA ANALYSIS. If DATA ANALYSIS does not appear, select ADD-INS. Select ANALYSIS TOOLPACK, click OK, and DATA ANALYSIS will be loaded into the tools menu. In the DATA ANALYSIS window, select t-Test Paired Two Sample for Means. Follow the instructions of Section 4-5 and the routine will print out a variety of information including tcafcu ated (which is labeled t Stat) and ftable (which is labeled t Critical two-tail). You should reproduce the results of Problem 4-14. 

This test is employed to estimate whether an experimental mean, x, differs significantly from the true value of the mean, p. This test, commonly known as the Most, has several possible variations the standard t-test, the paired t-test, and the t-test with nonequal variance. The computation of each test is quite simple, but the analyst must ensure that the correct test procedure is used. 

Wade et al. reported the use of a novel statistical approach for the comparison of analytical methods to measure angiotensin converting enzyme [peptidyl-dipeptidase A] activity, and to measure enalaprilat and benazeprilat [8]. Two methods were used to measure peptidyl-dipeptidase A, namely hippuryl histidyl leucine (HHL-method) [9], and inhibitor binding assay (IBA method) [10]. Three methods were used to measure enalaprilat, namely a radioimmunoassay (RIA) method [11], the HHL method, and the IBA method. Three methods were used to measure benazeprilat (then active metabolite of benazepril) in human plasma, namely gas chromatography-mass spectrometry (GC-MS method) [12], the HHL method, and the IBA method, and were statistically compared. First, the methods were compared by the paired t test or analysis of variance, depending on whether two or three different methods were under comparison. Secondly, the squared coefficients of variation of the... 

The paired t-test - comparing two related sets of measurements... 

Demonstrate that the paired t-test has greater power than the two-sample t-test when dealing with paired data... 

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

The paired t-test is only applicable to naturally paired data... 

Where data is genuinely paired, the paired t-test is likely to be considerably more powerful than the two-sample test and should be employed. 

In the paired t-test we calculate the change in a measured end-point for each individual and the test expects these differences to form a normal distribution. If this condition is not met, the Wilcoxon paired samples test can be used instead. 

Results are presented as mean SEM with n> 3. Statistical significance is assessed by means of the paired t-test. 

If the volume values from treatment and control probes come, for example, from the same patient or, as in DICE experiments, from the same gel, then there is a dependency between the resulting values. Hence, the t-test for paired samples, which has even a higher statistical power than the t-test for independent samples, can be used here. The term power is explained below. For the paired t-test the difference d between the treatment and control value is used. 

Figure 3.13 Power-function of the paired t-test using three replicates dotted tine), four replicates (dashed line) and five replicates (solid line) under the assumption that the standard deviation of the spots differences is s = 0.2.

Linnet K. Limitations of the paired t-test for evaluation of method comparison data. Chn Chem 1999 45 314-5. 

The paired t test uses the same type of procedure as the normal t test except that we analyze pairs of data. The standard deviation is now the standard deviation of the mean difference. Our null hypothesis is //q = Aq, where Aq is a specific... 

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