Incomplete Pairs in a Paired T Analysis

Why Incomplete Pairs Need Special Attention

A paired study records two measurements that belong together: for example, a participant’s measurement before and after an intervention. The analysis uses the within-participant difference, so both measurements must be observed for that participant. If one measurement is missing, that participant does not supply a calculable difference for an ordinary paired t analysis.

This has a practical consequence: the sample size for the paired analysis is not necessarily the number of participants recruited, nor the total number of measurements recorded. It is the number of participants with both measurements observed. Counting that denominator correctly matters because it determines which differences enter the analysis, and it affects the standard error and degrees of freedom.

Definition: A complete pair is a participant’s pair of measurements for which both values are observed. If participant \(i\) has measurements \(x_{i1}\) and \(x_{i2}\), their difference \(d_i=x_{i1}-x_{i2}\) can be calculated only when both measurements are available. The sample size for an ordinary paired t analysis is the number of complete pairs, denoted here by \(n\).

Which Observations Contribute?

The paired analysis first reduces each complete pair to one difference. A participant with a missing first measurement or a missing second measurement has no observed difference. Their one available value is still a recorded measurement, but it cannot be used to calculate that participant’s change.

For a standard complete-pair analysis, calculate the mean and sample standard deviation from the observed differences only. If there are \(n\) complete pairs, the standard error of the mean difference is \(SE=s_d/\sqrt{n}\), where \(s_d\) is the sample standard deviation of those \(n\) differences. A paired t statistic for testing a hypothesized mean difference \(\mu_{d,0}\) is

t=\frac{\bar{d}-\mu_{d,0}}{s_d/\sqrt{n}}, \qquad df=n-1.

Here, \(\bar{d}\) is the sample mean of the complete-pair differences, and \(df\) is the degrees of freedom. The formulas use the number of differences, not the number of measurement values. The direction of the difference must also be kept consistent: if differences are defined as first measurement minus second measurement, every included pair must use that order.

Worked Example: Counting Complete Pairs and Recorded Values

In a hypothetical study, eight volunteers are scheduled for a measurement before and after a session. The recorded data are shown below. Identify the complete pairs, the number of differences, and the total number of measurements actually recorded. Define each difference as before minus after.

Volunteer	Before	After	Complete pair?	Difference (before − after)
A	12	10	Yes	2
B	15	14	Yes	1
C	11	12	Yes	−1
D	13	Missing	No	Not available
E	10	9	Yes	1
F	Missing	11	No	Not available
G	14	13	Yes	1
H	12	Missing	No	Not available

Volunteers A, B, C, E, and G have both measurements, so the analysis has five complete pairs and five differences: \(2, 1, -1, 1, 1\). D, F, and H each have only one observed value and contribute no difference.

There are eight volunteers and ten measurements from the five complete pairs. D, F, and H contribute one additional recorded measurement each, so the total is \(10+3=13\) recorded measurements. Thus, the data contain eight participants, 13 recorded values, and five usable paired differences. These counts describe different things and should not be substituted for one another.

For illustration, the mean of the five observed differences is \( \bar{d}=(2+1-1+1+1)/5=0.8 \) measurement units. This is a summary of the complete pairs; the three single measurements cannot be added to the difference calculation.

Using the Complete-Pair Sample Size in a Test

The paired t procedure treats the differences as the observations in a one-sample t analysis. Before using it, the relevant assumptions concern those differences: pairs should be independent of one another, the complete pairs should support inference to the population of interest, and the distribution of differences should be approximately normal when the number of complete pairs is small, without severe outliers. A larger number of pairs makes the t procedure less sensitive to nonnormality, but it does not correct dependence or an unrepresentative set of complete pairs.

Missingness deserves attention alongside those conditions. The calculations below describe the observed complete-pair differences. If participants with missing measurements tend to have different changes from those with complete data, the observed mean difference may not represent the mean change for everyone originally recruited. A paired t calculation alone cannot determine what the missing differences would have been or remove bias caused by selective missingness.

Worked Example: Testing a Mean Difference With One Missing Follow-Up

In a hypothetical study, six participants have a baseline symptom score and are scheduled for a follow-up score. One follow-up value is missing. For the five complete pairs, the baseline-minus-follow-up differences, in score points, are \(1, 2, 0, 3, 4\). Suppose the complete pairs are independent, their differences are reasonably modeled by a normal distribution, and the complete-pair group is suitable for the stated inference. Test \(H_0:\mu_d=0\) against \(H_a:\mu_d\ne 0\).

Only five participants have both scores, so \(n=5\), not six. The mean difference is

\bar{d}=\frac{1+2+0+3+4}{5}=\frac{10}{5}=2.0 \text{ score points}.

The deviations from the mean are \(-1, 0, -2, 1, 2\), whose squared values sum to \(1+0+4+1+4=10\). Therefore,

s_d=\sqrt{\frac{10}{5-1}}=\sqrt{2.5}=1.581 \text{ score points}, \qquad SE=\frac{1.581}{\sqrt{5}}=0.7071 \text{ score points}.

For the null value \(\mu_{d,0}=0\), the test statistic and degrees of freedom are

t=\frac{2.0-0}{0.7071}=2.828, \qquad df=5-1=4.

A two-sided t test with \(t=2.828\) and \(df=4\) gives \(p\approx 0.0474\). At the 0.05 significance level, the result provides evidence against a zero mean baseline-minus-follow-up difference among the population represented by the complete pairs. Positive differences correspond to lower follow-up scores. This calculation does not include the participant with the missing follow-up, and the test alone does not establish that the observed change was caused by the study session.

Why the Denominator Changes the Analysis

The effect of incomplete pairs on the standard error follows directly from the formula. If there are \(n\) independent differences with population variance \(\sigma_d^2\), the variance of their sample mean is

\operatorname{Var}(\bar{d}) =\operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^{n}d_i\right) =\frac{1}{n^2}\sum_{i=1}^{n}\operatorname{Var}(d_i) =\frac{n\sigma_d^2}{n^2} =\frac{\sigma_d^2}{n}.

The middle equality uses independence, so the variances add. Taking the square root gives the standard deviation of the sample mean, \(\sigma_d/\sqrt{n}\), which is estimated by \(s_d/\sqrt{n}\). Thus, a smaller number of complete pairs generally means a larger estimated standard error, all else equal. It also means fewer degrees of freedom, because the sample standard deviation is estimated from the \(n\) differences and has \(n-1\) degrees of freedom.

This does not mean that one can recover the lost information by counting a lone measurement as half a pair or by treating the two measurement columns as independent samples. A lone value supplies no within-person difference. Other statistical methods can sometimes use incomplete repeated measurements under additional modeling assumptions, but those are not the ordinary paired t analysis described here.

Worked Example: Reporting the Analysis Denominator

Suppose a hypothetical study enrolls ten participants. Six have both a baseline and follow-up measurement; among these six complete pairs, the baseline-minus-follow-up differences are \(2, 1, 3, 0, 4, 2\) units. The other four participants each have at least one missing measurement. Find the complete-pair mean and state what sample size should be reported for a paired analysis.

The number of differences is six. Their sum is \(2+1+3+0+4+2=12\), so

\bar{d}=\frac{12}{6}=2.0 \text{ units}.

The paired analysis therefore has \(n=6\), not \(n=10\). There are ten enrolled participants, but only six contribute two observed measurements and a difference. If reporting a test based on these differences, the degrees of freedom would be \(6-1=5\). A clear description would say that ten participants were enrolled, six had complete pairs, and the mean baseline-minus-follow-up difference among the six complete pairs was 2.0 units.

That report makes the denominator visible rather than suggesting that all ten participants contributed a change score. It also helps readers judge how much of the enrolled group is represented. Whether the six complete pairs represent all ten participants is a separate question that requires information about the missingness and study design.

Interpreting the Complete-Pair Result

The mean difference from a complete-pair analysis describes the average difference among pairs included in the calculation. Its meaning depends on the chosen subtraction order and measurement units. For example, a positive baseline-minus-follow-up mean indicates that baseline values tend to exceed follow-up values; it does not mean the same thing as a positive follow-up-minus-baseline mean.

When data are incomplete, keep two questions separate:

Calculation: Which participants have both measurements, and what do their differences show?
Scope: Does the complete-pair group provide a sound basis for conclusions about the originally intended population?

The first question is answered by identifying complete pairs and using their differences. The second cannot be settled by the paired t formula. If missingness is related to participants’ values or changes, conclusions for the complete cases may not extend to those with missing measurements. Report the number of recruited participants, the number with complete pairs, and any relevant limitations rather than presenting the paired-analysis sample size as the original enrollment.

Common Mistakes

Using the number enrolled as \(n\). If ten people enroll but only six have both measurements, the paired t sample size is six.
Counting measurement values instead of differences. Six complete pairs contain 12 measurements but provide six differences, so the paired analysis uses \(n=6\), not 12.
Using a single observed measurement as if it were a change. Without the matching measurement, that participant’s difference is unavailable; do not invent or assume a value.
Assuming the complete cases automatically represent everyone. The calculations summarize complete pairs. Generalizing beyond them depends on the study design and how measurements came to be missing.
Changing the subtraction order partway through. Define the difference once and apply that order to every complete pair.

Key takeaway: An ordinary paired t analysis uses one difference for each participant with both measurements observed. Count those complete pairs to determine \(n\), calculate the mean and standard deviation from their differences, and report both the enrolled total and the complete-pair sample size when they differ.

Check Your Understanding

Use the ideas in this tutorial to answer the following questions.

A study enrolls 18 participants. Four have a missing second measurement, and two different participants have a missing first measurement. How many complete pairs are available?
There are seven complete pairs and five additional single measurements. How many differences enter the paired analysis, and how many recorded measurements are there in total?
Why does one observed measurement from an incomplete pair not provide a participant-level difference?
A paired t analysis uses 12 complete pairs. What are the degrees of freedom, and which sample size determines them?
What limitation should be considered if participants with missing follow-up values may have had different changes from participants with complete pairs?