Dealing with the paired data

Paired data contains values which fall normally into pairs and can therefore be expected to vary more between pairs than within pairs. The pairing is to reduce the variability. After the pairing, The between-subject variability will be eliminated. If pairing is effective it will reduce variability enough to justify the effort involved to obtain paired data.

There are many practical examples of paring. In clinical trial, crossover design is a special case of the pairing where the same subject receive more than one treatment. If all subjects receive treatment A, then treatment B, it can still be called crossover design (single sequence cross over design). In Epidemiology field, the case-control study is typically paring. There are terms 1:1 matched case-control, and 1:m matched case-control. In education, we can do the paring to compare the scores before and after the training;......

When outcome measures are continuous variable (such as drug concentration), without considering the covariates, analysis of paired data can be implemented by using paired t-test which can be easily performed using SAS PROC UNIVARIATE (calculate the difference for each pair, then run PROC UNIVARIATE) or SAS PROC TTEST (without calculating the difference first). Suppose x1 and x2 are paired variables,

proc ttest;
 paired x1*x2;
run;

If the normality assumption is questionable, the non-parametric tests (sign test and Wilcoxon signed rank sum test) can be used. UCLA's Statistical Consulting Services web site provided examples for these tests.

In more complicated situation (such as crossover design) or if we have to do the modeling to include the covariates, mixed model needs to be used. SAS PROC MIXED can implement the mixed model easily. See SAS/Stat User's Manual for PROC MIXED. In a research paper titled "Detection of emphysema progression in alpha 1-antitrypsin deficiency using CT densitometry; Methodological advances", I actually dealt with the paired data using so called 'random coefficient model'.


When outcome variable is discrete data, the easiest example is McNemar test. McNemar's test is performed if we are interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a case-control study) or two outcome variables from a single group.

In more complicated situation or if the covarites need to be included in the model, 'conditional logistic regression' needs to be employed. 'Conditional logistical regression' can be implemented using SAS Proc Logistic or SAS Proc PHREG. See following links for detail descriptions.

• A Tutorial on Logistic Regression by Ying So in SAS

• Condition Logistic Regression using SAS Proc PHREG procedure by David Brown