Paired data

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  

Essential Statistics for the Pharmaceutical Sciences Philip Rowe 

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

1 Does a weight-loss drug really work  

Subject number Weight after placebo (kg) Weight after active (kg) Change in weight (active-placebo) (kg) 

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Significance testing for comparing two mean values is divided into two categories depending on the source of the data. Data are said to be unpaired when each mean is derived from the analysis of several samples drawn from the same source. Paired data are encountered when analyzing a series of samples drawn from different sources. 

Statistical test for comparing paired data to determine if their difference is too large to be explained by indeterminate error. 

In a study involving paired data the difference, d[, between the paired values for each sample is calculated. The average difference, d, and standard deviation of the differences, are then calculated. The null hypothesis is that d is 0, and that there is no difference in the results for the two data sets. The alternative hypothesis is that the results for the two sets of data are significantly different, and, therefore, d is not equal to 0. 

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... 

The most recendy developed model is called UNIQUAC (21). Comparisons of measured VLE and predicted values from the Van Laar, Wilson, NRTL, and UNIQUAC models, as well as an older model, are available (3,22). Thousands of comparisons have been made, and Reference 3, which covers the Dortmund Data Base, available for purchase and use with standard computers, should be consulted by anyone considering the measurement or prediction of VLE. The predictive VLE models can be accommodated to multicomponent systems through the use of certain combining rules. These rules require the determination of parameters for all possible binary pairs in the multicomponent mixture. It is possible to use more than one model in determining binary pair data for a given mixture (23). 

While a t-test can be used to determine if the means of two samples can be considered to come from the same population, paired data sets are more powerful to determine difference. 

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 r-test yields insights only for the particular (addi-... 

In engineering practice we are often faced with the task of fitting a low-order polynomial curve to a set of data. Namely, given a set of N pair data, (y , x ), i=l,...,N, we are interested in the following cases,... 

The data were statistically analyzed using the SOLO Statistical System (BMDP Statistical Software, Inc., Los Angeles, CA) on a personal computer. Differences between groups were tested by the Mann-Whitney test or a paired t-test in cases where paired data sets were tested. Possible relationships were studied with (multiple) linear regression using least-square estimates. 

Paired-data performance, involving comparison of predicted and observed values for exact locations in time and space. This may be a more rigorous test than needed for many purposes timing differences can have severe impacts on the statistical comparison. 

Time and space integrated, paired-data performance. Spacially and/or temporally integrated data can be compared to analogous model predictions, such as daily or monthly averages or totals. This can circumvent some of the timing problems noted in (J ). 

Statistical measures for the paired-data, and integrated paired-data performance tests noted above are essentially identical. 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

Y as a function of a change in X. These include, but are not limited to correlation (r), the coefficient of determination (R2), the slope (, ), intercept (K0), the z-statistic, and of course the respective confidence limits for these statistical parameters. The use of graphical representation is also a powerful tool for discerning the relationships between X and Y paired data sets. 

Thus a comparison of the correlation coefficient between two or more sets of X, Y data pairs cannot be adequately performed unless the standard deviations of the two data sets are nearly identical or unless the correlation coefficient confidence limits for the data sets are compared. In summary, if one Set A of X, Y paired data has a correlation of 0.95, this does not necessarily indicate that it is more highly correlated than a second Set B of X, Y paired data with a correlation of say 0.90. The meaning of this will be described in greater detail later. 

To begin, the following summation notation may be used to calculate the slope (kj) of a linear regression line given a set of X, Y paired data (equation 61-23). 

The mean measured activity per unit surface area are shown for airways and bifurcations separately in Table II. These data are for those segments which contained only airway lengths bifurcations. The results are given as the number of particles which deposit per cm2 for 10 particles which enter the trachea. This assumes that the particle and activity distributions are equivalent. For the 0.2 and 0.15 ym particles the surface density at the bifurcations is greater than that along the airway lengths at p <. 01 when the paired data are compared by a one tailed t-test. 

An additional approach to handling paired data is to assess the degree of correlation between the pairs. The data can be presented as a graph in which one axis is used for the results obtained by one method and the other axis for the results of the same samples obtained by the other method. If each sample analysed gave an identical result by both methods then a characteristic graph would result (Figure 1.2(a)). The closeness of the fit between all the points and... 

Graph 8.5 Paired data for the proportion of DCA (percentage of total bile acids) in bile before and during Octreotide treatment (SOOpg/day for 8 months). Data taken from reference 18. 

Because there was no method for doing a meta-analysis with crossover designs, we developed a method (a variation on Hedges method for uncrossed designs) with suitable modification for paired data (12). 

When we wish to test for a zero mean difference in matched paired data the appropriate test is a one-sample or paired t-test. For such data the values of two quantities whose comparison is of interest are both made on the same individual, and both measurements are repeated on many individuals. For example, in an experiment heart rate was measured in 20... 

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

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

Use with paired data Correct procedure Possible, but a poor choice - lacks power... 

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