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In statistics, a paired difference test is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders. Specific methods for carrying out paired difference tests are, for normally distributed difference t-test (where the population standard deviation of difference is not known) and the paired Z-test (where the population standard deviation of the difference is known), and for differences that may not be normally distributed the Wilcoxon signed-rank test. In addition to tests that deal with non-normality, there is also a test that is robust to the common violation of homogeneity of variance across samples (an underlying assumption of these tests): this is Welch's t-test, which makes use of unpooled variance and results in unusual degrees of freedom (e.g. df' = 4.088 rather than df = 4). The most familiar example of a paired difference test occurs when subjects are measured before and after a treatment. Such a "repeated measures" test compares these measurements within subjects, rather than across subjects, and will generally have greater power than an unpaired test. ==Use in reducing variance== Paired difference tests for reducing variance are a specific type of blocking. To illustrate the idea, suppose we are assessing the performance of a drug for treating high cholesterol. Under the design of our study, we enroll 100 subjects, and measure each subject's cholesterol level. Then all the subjects are treated with the drug for six months, after which their cholesterol levels are measured again. Our interest is in whether the drug has any effect on mean cholesterol levels, which can be inferred through a comparison of the post-treatment to pre-treatment measurements. The key issue that motivates the paired difference test is that unless the study has very strict entry criteria, it is likely that the subjects will differ substantially from each other before the treatment begins. Important baseline differences among the subjects may be due to their gender, age, smoking status, activity level, and diet. There are two natural approaches to analyzing these data: * In an "unpaired analysis", the data are treated as if the study design had actually been to enroll 200 subjects, followed by random assignment of 100 subjects to each of the treatment and control groups. The treatment group in the unpaired design would be viewed as analogous to the post-treatment measurements in the paired design, and the control group would be viewed as analogous to the pre-treatment measurements. We could then calculate the sample means within the treated and untreated groups of subjects, and compare these means to each other. * In a "paired difference analysis", we would first subtract the pre-treatment value from the post-treatment value for each subject, then compare these differences to zero. If we only consider the means, the paired and unpaired approaches give the same result. To see this, let be the observed data for the pair, and let . Also let , and denote, respectively, the sample means of the , the , and the . By rearranging terms we can see that : where ''n'' is the number of pairs. Thus the mean difference between the groups does not depend on whether we organize the data as pairs. Although the mean difference is the same for the paired and unpaired statistics, their statistical significance levels can be very different, because it is easy to overstate the variance of the unpaired statistic. The variance of is : where and are the population standard deviations of the and data, respectively. Thus the variance of is lower if there is positive correlation within each pair. Such correlation is very common in the repeated measures setting, since many factors influencing the value being compared are unaffected by the treatment. For example, if cholesterol levels are associated with age, the effect of age will lead to positive correlations between the cholesterol levels measured within subjects, as long as the duration of the study is small relative to the variation in ages in the sample. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Paired difference test」の詳細全文を読む スポンサード リンク
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