Test for Paired Data

Suppose that a researcher is interested in certain responses (e.g., the blood pressure, or the time to respond to a tone) of subjects under two distinct conditions (e.g., before and after being affected by the consumption of alcohol). The pairs of responses

$\displaystyle (X_1,Y_1), \ldots, (X_n,Y_n)

are measured for $ n$ experimental units under these conditions, say ``Condition 1'' and ``Condition 2''.

Measured for Condition 1: $ X_1,\ldots,X_n$ in

Measured for Condition 2: $ Y_1,\ldots,Y_n$ in

The pair of measurements must be specified from the data sheet below:

We first have to take the difference $ Z_i = X_i - Y_i$ of measurements for $ i=1,\ldots,n$ if not given. The new variable ``Difference'' is introduced in order to eliminate the effect of experimental units (i.e., subjects). Then the hypothesis test can be stated with the following alternative hypothesis:

$ H_A:\hspace{0.05in} \mu = \mu_1 - \mu_2$

where $ \mu_1$ and $ \mu_2$ are the mean values of measurement under Condition 1 and 2, respectively. In fact, the test becomes that of inference on mean with respect to the population mean $ \mu$ of difference.

The appropriateness of t-test is ensured by either (a) an approximate normality of the sample distribution of difference, or (b) an adequately large sample size $ n$, say $ n \ge 30$. Here the estimate ``est.difference'' is the sample mean $ \bar{Z}$ of $ Z_1,\ldots,Z_n$.

When we find it difficult to justify the normality assumption for the t-test, we may consider its ``nonparametric'' alternative, called the Wilcoxon signed-rank test. The signed rank sum test is based on signed ranks. First the ranks are assigned from the smallest nonzero absolute value (rank 1) to the largest absolute value (the largest rank). Then the sign is added to each rank associated with its original value being positive (positive sign) or negative (negative sign).

Here the null hypothesis becomes ``the distribution of the difference is symmetric about zero.'' Then the test procedure determines whether it can be rejected in favor of the alternative hypothesis stated above. The p-value of test indicates the significance of the test: p-value < $ \alpha$ suggests an evidence against the null hypothesis, and therefore, the null hypothesis can be rejected in favor of the alternative hypothesis.

The test procedure also computes the confidence interval ``(lower.bound,upper.bound)'' for the estimate with the confidence level . In case of the Wilcoxon test, ``est.difference'' is viewed as the difference of locations, and calculated from the sample median of the Walsh averages $ \dfrac{Z_i + Z_j}{2}$, $ i,j = 1,\ldots,n$ (which is also known as the Hodges-Lehmann estimator).