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 ttest (where the population standard deviation of difference is not known) and the paired Ztest (where the population standard deviation of the difference is known), and for differences that may not be normally distributed the Wilcoxon signedrank test. In addition to tests that deal with nonnormality, 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 ttest, 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 posttreatment to pretreatment 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 posttreatment measurements in the paired design, and the control group would be viewed as analogous to the pretreatment 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 pretreatment value from the posttreatment 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 Y_{i1}, Y_{i2} be the observed data for the i^{th} pair, and let D_{i} = Y_{i2} − Y_{i1}. Also let D, Y_{1}, and Y_{2} denote, respectively, the sample means of the D_{i}, the Y_{i1}, and the Y_{i2}. By rearranging terms we can see that

\bar{D} = \frac{1}{n}\sum_i (Y_{i2}Y_{i1}) = \frac{1}{n}\sum_iY_{i2}  \frac{1}{n}\sum_iY_{i1} = \bar{Y}_2  \bar{Y}_1,
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 D is

\begin{array}{ccl} {\rm var}(\bar{D}) &=& {\rm var}(\bar{Y}_2\bar{Y}_1)\\ &=& {\rm var}(\bar{Y}_2) + {\rm var}(\bar{Y}_1)  2{\rm cov}(\bar{Y}_1,\bar{Y}_2)\\ &=& \sigma_1^2/n + \sigma_2^2/n  2\sigma_1\sigma_2{\rm corr}(Y_{i1}, Y_{i2})/n, \end{array}
where σ_{1} and σ_{2} are the population standard deviations of the Y_{i1} and Y_{i2} data, respectively. Thus the variance of D 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.
Power of the paired Ztest
Suppose we are using a Ztest to analyze the data, where the variances of the pretreatment and posttreatment data σ_{1}^{2} and σ_{2}^{2} are known (the situation with a ttest is similar). The unpaired Ztest statistic is

\frac{\bar{Y}_2  \bar{Y}_1}{\sqrt{\sigma_1^2/n + \sigma_2^2/n}},
The power of the unpaired, onesided test carried out at level α = 0.05 can be calculated as follows:

\begin{array}{lcl} P\left(\frac{\bar{Y}_2  \bar{Y}_1}{\sqrt{\sigma_1^2/n + \sigma_2^2/n}} > 1.64\right) &=& P\left(\frac{\bar{Y}_2  \bar{Y}_1}{S} > 1.64\sqrt{\sigma_1^2/n + \sigma_2^2/n}/S\right)\\ &=& P\left(\frac{\bar{Y}_2  \bar{Y}_1\delta+\delta}{S} > 1.64\sqrt{\sigma_1^2/n + \sigma_2^2/n}/S\right)\\ &=& P\left(\frac{\bar{Y}_2  \bar{Y}_1\delta}{S} > 1.64\sqrt{\sigma_1^2/n + \sigma_2^2/n}/S  \delta/S\right)\\ &=& 1  \Phi(1.64\sqrt{\sigma_1^2/n + \sigma_2^2/n}/S  \delta/S), \end{array}
where S is the standard deviation of D, Φ is the standard normal cumulative distribution function, and δ = EY_{2} − EY_{1} is the true effect of the treatment. The constant 1.64 is the 95th percentile of the standard normal distribution, which defines the rejection region of the test.
By a similar calculation, the power of the paired Ztest is

1  \Phi(1.64  \delta/S).
By comparing the expressions for power of the paired and unpaired tests, one can see that the paired test has more power as long as

\sqrt{\sigma_1^2/n + \sigma_2^2/n}/S = \sqrt{\frac{\sigma_1^2+\sigma_2^2}{\sigma_1^2+\sigma_2^22\sigma_1\sigma_2\rho}} > 1 ~~\text{where} ~~ \rho := {\rm corr}(Y_{i1},Y_{i2}).
This condition is met whenever \rho, the withinpairs correlation, is positive.
A random effects model for paired testing
The following statistical model is useful for understanding the paired difference test

Y_{ij} = \mu_j + \alpha_i + \epsilon_{ij}
where α_{i} is a random effect that is shared between the two values in the pair, and ε_{ij} is a random noise term that is independent across all data points. The constant values μ_{1}, μ_{2} are the expected values of the two measurements being compared, and our interest is in δ = μ_{2} − μ_{1}.
In this model, the α_{i} capture "stable confounders" that have the same effect on the pretreatment and posttreatment measurements. When we subtract to form D_{i}, the α_{i} cancel out, so do not contribute to the variance. The withinpairs covariance is

{\rm cov}(Y_{i1}, Y_{i2}) = {\rm var}(\alpha_i).
This is nonnegative, so it leads to better performance for the paired difference test compared to the unpaired test, unless the α_{i} are constant over i, in which case the paired and unpaired tests are equivalent.
In less mathematical terms, the unpaired test assumes that the data in the two groups being compared are independent. This assumption determines the form for the variance of D. However when two measurements are made for each subject, it is unlikely that the two measurements are independent. If the two measurements within a subject are positively correlated, the unpaired test overstates the variance of D, making it a conservative test in the sense that its actual type I error probability will be lower than the nominal level, with a corresponding loss of statistical power. In rare cases, the data may be negatively correlated within subjects, in which case the unpaired test becomes anticonservative. The paired test is generally used when repeated measurements are made on the same subjects, since it has the correct level regardless of the correlation of the measurements within pairs.
Use in reducing confounding
Another application of paired difference testing arises when comparing two groups in a set of observational data, with the goal being to isolate the effect of one factor of interest from the effects of other factors that may play a role. For example, suppose teachers adopt one of two different approaches, denoted "A" and "B", to teaching a particular mathematical topic. We may be interested in whether the performances of the students on a standardized mathematics test differs according to the teaching approach. If the teachers are free to adopt approach A or approach B, it is possible that teachers whose students are already performing well in mathematics will preferentially choose method A (or viceversa). In this situation, a simple comparison between the mean performances of students taught with approach A and approach B will likely show a difference, but this difference is partially or entirely due to the preexisting differences between the two groups of students. In this situation, the baseline abilities of the students serve as a confounding variable, in that they are related to both the outcome (performance on the standardized test), and to the treatment assignment to approach A or approach B.
It is possible to reduce, but not necessarily eliminate, the effects of confounding variables by forming "artificial pairs" and performing a pairwise difference test. These artificial pairs are constructed based on additional variables that are thought to serve as confounders. By pairing students whose values on the confounding variables are similar, a greater fraction of the difference in the value of interest (e.g. the standardized test score in the example discussed above), is due to the factor of interest, and a lesser fraction is due to the confounder. Forming artificial pairs for paired difference testing is an example of a general approach for reducing the effects of confounding when making comparisons using observational data called matching.^{[1]}^{[2]}^{[3]}
As a concrete example, suppose we observe student test scores X under teaching strategies A and B, and each student has either a "high" or "low" level of mathematical knowledge before the two teaching strategies are implemented. However, we do not know which students are in the "high" category and which are in the "low" category. The population mean test scores in the four possible groups are

A

B

High

\mu_{HA}

\mu_{HB}

Low

\mu_{LA}

\mu_{LB}

and the proportions of students in the groups are

A

B

High

p_{HA}

p_{HB}

Low

p_{LA}

p_{LB}

where p_{HA} + p_{HB} + p_{LA} + p_{LB} = 1.
The "treatment difference" among students in the "high" group is μ_{HA} − μ_{HB} and the treatment difference among students in the "low" group is μ_{LA} − μ_{LB}. In general, it is possible that the two teaching strategies could differ in either direction, or show no difference, and the effects could differ in magnitude or even in sign between the "high" and "low" groups. For example, if strategy B were superior to strategy A for wellprepared students, but strategy A were superior to strategy B for poorly prepared students, the two treatment differences would have opposite signs.
Since we do not know the baseline levels of the students, the expected value of the average test score X_{A} among students in the A group is an average of those in the two baseline levels:

E\bar{X}_A = \mu_{HA}\frac{p_{HA}}{p_{HA}+p_{LA}} + \mu_{LA}\frac{p_{LA}}{p_{HA}+p_{LA}},
and similarly the average test score X_{B} among students in the B group is

E\bar{X}_B = \mu_{HB}\frac{p_{HB}}{p_{HB}+p_{LB}} + \mu_{LB}\frac{p_{LB}}{p_{HB}+p_{LB}}.
Thus the expected value of the observed treatment difference D = X_{A} − X_{B} is

\mu_{HA}\frac{p_{HA}}{p_{HA}+p_{LA}}  \mu_{HB}\frac{p_{HB}}{p_{HB}+p_{LB}} + \mu_{LA}\frac{p_{LA}}{p_{HA}+p_{LA}}  \mu_{LB}\frac{p_{LB}}{p_{HB}+p_{LB}}.
A reasonable null hypothesis is that there is no effect of the treatment within either the "high" or "low" student groups, so that μ_{HA} = μ_{HB} and μ_{LA} = μ_{LB}. Under this null hypothesis, the expected value of D will be zero if

p_{HA} = (p_{HA}+p_{LA})(p_{HA}+p_{HB})
and

p_{HB} = (p_{HB}+p_{LB})(p_{HA}+p_{HB}).
This condition asserts that the assignment of students to the A and B teaching strategy groups is independent of their mathematical knowledge before the teaching strategies are implemented. If this holds, baseline mathematical knowledge is not a confounder, and conversely, if baseline mathematical knowledge is a confounder, the expected value of D will generally differ from zero. If the expected value of D under the null hypothesis is not equal to zero, then a situation where we reject the null hypothesis could either be due to an actual differential effect between teaching strategies A and B, or it could be due to nonindependence in the assignment of students to the A and B groups (even in the complete absence of an effect due to the teaching strategy).
This example illustrates that if we make a direct comparison between two groups when confounders are present, we do not know whether any difference that is observed is due to the grouping itself, or is due to some other factor. If we are able to pair students by an exact or estimated measure of their baseline mathematical ability, then we are only comparing students "within rows" of the table of means given above. Consequently, if the null hypothesis holds, the expected value of D will equal zero, and statistical significance levels have their intended interpretation.
See also
References

^ Rubin, Donald B. (1973). "Matching to Remove Bias in Observational Studies". Biometrics (International Biometric Society) 29 (1): 159–183.

^ Anderson, Dallas W.; Kish, Leslie; Cornell, Richard G. (1980). "On Stratification, Grouping and Matching". Scandinavian Journal of Statistics (Blackwell Publishing) 7 (2): 61–66.

^ Kupper, Lawrence L.; Karon, John M.; Kleinbaum, David G.; Morgenstern, Hal; Lewis, Donald K. (1981). "Matching in Epidemiologic Studies: Validity and Efficiency Considerations". Biometrics (International Biometric Society) 37 (2): 271–291.
External links

Relative Measurement and Its Generalization in Decision Making Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors The Analytic Hierarchy/Network Process

Sequence Comparison Evaluation (google cache)

Deseng.ryserson.ca
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