SO5005 Chap.7 Paired and Independent Comparisons
Paired and Independent Comparisons
Dependence comes from design
Measurements from the same person, two eyes from one participant or deliberately matched units are paired. Observations from unrelated participants in separate groups may be independent.
Paired numerical analysis works with one within-pair difference, while independent analysis compares group distributions.
Name the target quantity
A paired t procedure concerns the mean difference. Signed-rank and rank-sum procedures use ranks and should not be described casually as universal tests of medians.
Report group or difference summaries, the estimated contrast and uncertainty, and define the subtraction order before interpreting the sign.
What this chapter covers
- 01
Sources of dependence
- 02
Within-pair differences
- 03
Paired t procedures
- 04
Signed-rank procedures
- 05
Independent group comparisons
- 06
Direction and subtraction order
Worked example · free
Compute paired differences
- 2Use one difference per participant.
- 2Sum and divide by five.
- 2Interpret the sign from the stated order.
Key terms
- Paired Difference
- A directional subtraction between linked measurements from the same or matched unit.
- Independent Groups
- Comparison groups whose observations come from separate unrelated units under the design.
- Signed-rank Test
- A rank-based paired procedure applied to within-pair differences under its assumptions.
Paired and Independent Comparisons FAQ
Are two columns automatically paired?
No. Pairing comes from a meaningful link between observations, such as repeated measures on the same person or deliberate matching. Equal column lengths alone do not create dependence, and unrelated groups should not be analysed as pairs.
Which distribution matters for a paired t procedure?
The relevant distribution is the set of within-pair differences because each difference is the analysed observation. Looking only at the two original marginal distributions can miss skewness or influential values in the quantity the test actually uses.
Is Mann-Whitney always a test of medians?
No. It compares rank distributions and has a simple location interpretation only under additional shape conditions. State the target contrast and inspect distributions rather than automatically translating every rank comparison into a median difference.
Assessment move
Create tiny datasets and label the observational unit before any calculation. For paired data, print each difference and verify the subtraction order. For independent data, summarise each group and explain why observations can reasonably be treated as unrelated. Draw a line between every pair that belongs to one participant or matched unit.
If a value has no meaningful partner, investigate the missing pair rather than quietly treating it as independent. Calculate after minus before and then before minus after on the same data; confirm that the sign reverses while the evidence against a zero difference is unchanged. This exposes direction errors that software cannot prevent.
For paired mean analysis, inspect the distribution of differences, not only the original columns. For rank-based work, write what is being ranked and avoid promising a median comparison unless the required distributional conditions are credible. For independent groups, build a table with sample size, centre, spread and missing count for each group before calculating a contrast.
State how recruitment or allocation supports independence and whether unequal variability matters for the chosen method. Create examples where the same variable is paired in one design and independent in another, such as two eyes from one person versus one eye from different people. The variable name does not determine dependence; the data-generating structure does.
Before reporting, write the contrast in symbols or plain language, including the reference order. Read the numerical sign back into that sentence. Finish with one limitation that follows from the design rather than from the test. This routine makes the analysis auditable and prevents a correct calculation from becoming a reversed or over-broad conclusion.
Build one example with a missing member of a pair and decide whether the remaining value contributes to the paired analysis, a descriptive table or neither. Compare complete-pair and available-observation counts explicitly. Practise explaining why two eyes from one participant violate independence even when measured separately, using shared biology as the mechanism.
For matched observational data, list what the matching variables can and cannot balance. Draw individual lines for repeated measurements and compare the visual pattern with the mean difference. End by writing a result where the numerical sign is correct but the clinical direction is counterintuitive, then resolve it through the outcome scale and subtraction order.
Read the final sentence without the table and confirm that its direction, unit and population remain unambiguous to someone who did not perform the calculation. Correct any ambiguity before adding contextual explanation. Direction comes before interpretation. Repeat the stated reference every time the comparison appears in prose.
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