PSY2041 Chap.9 Non-parametric Tests
Non-parametric Tests
Non-parametric Tests use ranks to answer association and difference questions when raw-score distances or parametric conditions are unsuitable. Spearman correlation addresses monotonic association. Wilcoxon signed-rank addresses paired differences or deviations from a reference, while Wilcoxon rank-sum compares two independent groups.
The choice begins with the research question and pairing, not a mechanical reaction to one normality test. Ranking changes the scale of the claim and does not remove all assumptions.
What this chapter covers
- 01
Parametric and non-parametric approaches compared conceptually
- 02
Ranks as preserved order with reduced dependence on raw spacing
- 03
Spearman correlation for monotonic association
- 04
Wilcoxon signed-rank for paired differences
- 05
Wilcoxon rank-sum for independent groups
- 06
Absolute differences, signs, ties and rank sums
- 07
Reporting conclusions in terms consistent with ranks
Worked example · free
Construct signed ranks from paired differences
- 1Take absolute differences: 2, 1, 4 and 3.
- 1Rank them from smallest to largest: 2 receives rank 2, 1 rank 1, 4 rank 4 and 3 rank 3.
- 1Restore the original signs to obtain signed ranks +2, -1, +4 and -3.
- 1The positive-rank sum is 2 + 4 = 6.
- 1The negative-rank magnitude is 1 + 3 = 4; the tiny example is descriptive only.
Key terms
- Rank transformation
- Replacement of raw values with their ordered positions in a sample.
- Monotonic association
- A relationship that consistently moves in one direction without needing a straight-line form.
- Spearman correlation
- A coefficient describing monotonic association through ranked values.
- Signed-rank test
- A rank procedure for paired differences or differences from a reference value.
- Rank-sum test
- A rank procedure comparing observations from two independent groups.
- Tied ranks
- Shared ordered positions assigned according to the procedure when observations are equal.
Non-parametric Tests FAQ
Are non-parametric tests assumption-free?
No. They often reduce reliance on distributional form and raw-score spacing, but still require conditions about independence, pairing, ordering and the research question. Their results also concern ranks or ordered distributions rather than automatically reproducing a mean-based claim.
How do signed-rank and rank-sum differ?
Signed-rank uses linked differences from paired observations or a reference. Rank-sum compares two independent groups. The methods can sound similar because both use ranks, but pairing determines which design is appropriate.
When should I use Spearman instead of Pearson?
Spearman suits a monotonic association expressed through ranks, including ordinal variables or situations where raw-score linearity is unsuitable. Pearson targets linear association in raw standardised values. Inspect the research question and plot before choosing.
Exam move
Use a two-question tree: association or difference, then paired or independent. Place Spearman, signed-rank and rank-sum on the correct branches. Practise signed ranks with zero differences and ties after mastering the basic sequence. For every method, write the closest parametric question and then explain what ranking changes.
Avoid the vague justification that data are not normal; state the scale, shape, outlier or form issue and the rank-based claim. In reports, use language about ordered association or group ranks rather than silently switching back to means. Check that the observation unit and pairing are correct before any computation. Draw the data structure for every scenario before selecting a method.
A scatter of two variables points toward an association question; linked before-and-after rows indicate pairing; separate group columns indicate independence. Change one design feature at a time and trace the branch that moves. For signed-rank practice, calculate differences with a declared subtraction order, remove zeros, rank absolute values, average ties, restore signs and sum positive and negative ranks.
Keep raw differences and signed ranks in different columns so one cannot be mistaken for the other. For Spearman correlation, sketch monotonic linear, monotonic curved and direction-reversing patterns. Explain why ranks can capture the first two but a single coefficient can obscure the third. For rank-sum, identify the observational unit and check that no participant contributes to both groups.
Compare each rank procedure with its closest mean-based question, stating what information about raw spacing is surrendered. Write result sentences about ordered association or relative group position unless an estimated raw-unit quantity is also available. Close each response with remaining conditions such as independence, appropriate pairing, monotonic form or similarly shaped group distributions.
Ranking can reduce sensitivity to extreme distances, but it cannot repair biased sampling, dependence or a question that does not match the design. For rapid recall, write the three method names on blank cards and add question, structure, ranking action and reporting language from memory. Correct the structure first whenever an answer is wrong.
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