Appendix W — SPSS Tutorial: Nonparametric Tests

This tutorial walks through each nonparametric procedure covered in Section 19.1–Section 19.10 using SPSS. Each section maps directly to the corresponding chapter section, follows the same dataset (core_session.sav), and uses the same examples and statistics reported in the chapter.

TipBefore you start

Open core_session.sav in SPSS. Verify that the following variables are present and correctly coded:

Variable	Type	Values
sex_category	Nominal	1 = Female, 2 = Male
group	Nominal	1 = Control, 2 = Training
time	Ordinal	1 = Pre, 2 = Mid, 3 = Post
rpe_6_20	Scale	6–20
strength_kg	Scale	kg
sprint_20m_s	Scale	seconds
function_0_100	Scale	0–100
balance_errors_count	Scale	count

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W.1 Part A: Chi-Square Tests

W.1.1 A1 — Goodness-of-Fit Test

Use this when you have one categorical variable and want to test whether the observed frequencies match a theoretical distribution (e.g., equal proportions).

Step 1: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → Chi-square…

Step 2: Specify the variable

Move sex_category into the Test Variable List box.

Step 3: Set expected values

Under Expected Values, leave All categories equal selected (this tests for a 50/50 split between females and males).

• If you had a non-equal expected distribution (e.g., 60% female), select Values and enter each expected frequency one at a time using the Add button.

Step 4: Run

Click OK.

Step 5: Read the output

SPSS produces two tables:

Expected and Observed Frequencies — shows the observed (N) and expected counts for each category. Verify that no expected count is below 5.

Test Statistics — reports Chi-Square, df, and Asymptotic Sig. (2-tailed). Record these for your APA write-up.

Output value	Meaning
Chi-Square	The χ² test statistic
df	Degrees of freedom (k − 1)
Asymptotic Sig. (2-tailed)	The p-value

For the sex distribution example, SPSS should report χ²(1) = 0.27, p = .607.

Step 6: Effect size

Cramér’s V for a goodness-of-fit test with two categories equals:

\[V = \sqrt{\frac{\chi^2}{n}}\]

SPSS does not report V for the goodness-of-fit test. Use the Statistical Calculators appendix to compute it from the χ² and N values in the output.

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W.1.2 A2 — Test of Independence (Two Categorical Variables)

Use this when you have two categorical variables and want to test whether they are associated (e.g., sex × group).

Step 1: Open the dialog

Analyze → Descriptive Statistics → Crosstabs…

Step 2: Assign variables

• Move sex_category to the Row(s) box.
• Move group to the Column(s) box.

Step 3: Request statistics

Click Statistics…, check Chi-square and Phi and Cramer’s V, then click Continue.

Step 4: Request expected counts

Click Cells…, check Expected under Counts, then click Continue.

Step 5: Run

Click OK.

Step 6: Read the output

Crosstabulation table — examine the expected counts column. All cells must have expected counts ≥ 5 for the chi-square approximation to be valid. If any cell is below 5, SPSS will warn you with a footnote; use Fisher’s exact test instead (available in the same output under “Fisher’s Exact Test” when the table is 2 × 2).

Chi-Square Tests table — use the Pearson Chi-Square row. Record:

Output row	What to report
Pearson Chi-Square	χ²(df, N = …) = [Value], p = [Asymp. Sig.]
df	Degrees of freedom
Asymp. Sig. (2-tailed)	p-value

Symmetric Measures table — record Phi (for 2 × 2 tables) or Cramer’s V (for larger tables) as the effect size.

For the sex × group example: χ²(1, N = 60) = 4.31, p = .038, φ = .268.

WarningFisher’s exact test

When any expected cell count is below 5, SPSS automatically prints Fisher’s exact test p-value in the Chi-Square Tests table (for 2 × 2 tables only). Report Fisher’s exact p instead of Pearson chi-square in those cases, and note in your write-up that Fisher’s exact test was used due to small expected frequencies.

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W.2 Part B: Spearman Rank-Order Correlation

Step 1: Open the dialog

Analyze → Correlate → Bivariate…

Step 2: Select variables

Move strength_kg and sprint_20m_s (post-test values only — filter or select cases first if needed) to the Variables box.

Step 3: Select Spearman

Under Correlation Coefficients, uncheck Pearson and check Spearman. (You may keep Pearson checked if you wish to compare both.)

Step 4: Run

Click OK.

Step 5: Read the output

The Correlations table reports:

Output value	What to report
Correlation Coefficient	ρ
Sig. (2-tailed)	p-value
N	Sample size

For the strength vs. sprint example: ρ(58) = −.71, p < .001.

NoteFiltering to a single time point

To analyze post-test data only, use Data → Select Cases → If condition is satisfied, enter time = 3 (if 3 = post), click Continue, then OK. Run the correlation. When finished, return to Select Cases and choose All cases to restore the full dataset.

Step 6: Effect size

Spearman ρ is itself the effect size. Report it alongside the p-value using the benchmarks: |ρ| = .10 (small), .30 (medium), .50 (large).

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W.3 Part C: Mann-Whitney U Test

Step 1: Filter to post-test only (if comparing groups at one time point)

Data → Select Cases → If condition is satisfied → enter time = 3 → Continue → OK

Step 2: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → 2 Independent Samples…

Step 3: Assign variables

• Move rpe_6_20 to the Test Variable List box.
• Move group to the Grouping Variable box.
• Click Define Groups…, enter 1 for Group 1 and 2 for Group 2, then click Continue.

Step 4: Select the test

Under Test Type, ensure Mann-Whitney U is checked.

Step 5: Run

Click OK.

Step 6: Read the output

Ranks table — shows the mean rank and sum of ranks for each group. Higher mean rank = higher scores on the outcome variable.

Test Statistics table — record:

Output value	What to report
Mann-Whitney U	U
Wilcoxon W	The larger sum of ranks (not needed for APA)
Z	Standardized test statistic
Asymp. Sig. (2-tailed)	p-value

For the RPE × group example: U = 501, p = .031.

Step 7: Effect size — rank-biserial r

SPSS does not directly report rank-biserial r in Legacy Dialogs. Use the formula:

\[r = \frac{U_1 - U_2}{n_1 \times n_2}\]

where \(U_1\) and \(U_2\) are obtained from the Ranks table (Mean Rank × n for each group gives the U values). Use the Statistical Calculators appendix to compute r from the U and group sizes.

Alternatively, use Analyze → Nonparametric Tests → Independent Samples (the newer dialog) and request Effect Size in the Settings tab — SPSS will report r directly.

For the RPE example: r = −.34 (medium effect).

TipMedian reporting for Mann-Whitney

APA style recommends reporting the median (not the mean) alongside Mann-Whitney U. Run Analyze → Descriptive Statistics → Explore, add rpe_6_20 as a Dependent List and group as a Factor, to get medians and IQRs for each group.

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W.4 Part D: Wilcoxon Signed-Rank Test

Step 1: Restructure data (if in long format)

If your dataset is in long format (one row per participant per time point), you need wide format for the Wilcoxon test (one row per participant, with separate columns for pre and post). Use:

Data → Restructure… → Restructure selected variables into cases or vice versa → follow the wizard to pivot function_0_100 from long to wide, creating function_pre and function_post columns.

Step 2: Filter to training group only

Data → Select Cases → If condition is satisfied → enter group = 2 → Continue → OK

Step 3: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → 2 Related Samples…

Step 4: Assign variables

• In the Test Pairs area, click function_pre then function_post and move them as a pair into the pair list using the arrow button.

Step 5: Select the test

Under Test Type, ensure Wilcoxon is checked.

Step 6: Run

Click OK.

Step 7: Read the output

Ranks table — shows the number and mean rank of negative ranks (scores that decreased), positive ranks (scores that increased), and ties.

Row	Meaning
Negative Ranks	Participants whose post-score was lower than pre-score
Positive Ranks	Participants whose post-score was higher than pre-score
Ties	No change

Test Statistics table — record:

Output value	What to report
Z	Standardized test statistic
Asymp. Sig. (2-tailed)	p-value
Exact Sig. (2-tailed)	Use for n < 25

SPSS reports the Wilcoxon W statistic in the background, but the Z and p are what you report. The W = 20 in the chapter was obtained directly from the smaller rank sum.

Step 8: Effect size — rank-biserial r

\[r = \frac{Z}{\sqrt{n}}\]

where Z is from the Test Statistics table and n is the number of non-tied pairs. Use the Statistical Calculators appendix or compute directly. For Z = −4.38, n = 30: r = −4.38/√30 = −.80.

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W.5 Part E: Kruskal-Wallis Test

Step 1: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → K Independent Samples…

Step 2: Assign variables

• Move balance_errors_count to the Test Variable List box.
• Move time to the Grouping Variable box.
• Click Define Range…, enter Minimum: 1 and Maximum: 3, then click Continue.

Step 3: Select the test

Under Test Type, ensure Kruskal-Wallis H is checked.

Step 4: Run

Click OK.

Step 5: Read the output

Ranks table — shows mean rank for each time point (group).

Test Statistics table — record:

Output value	What to report
Kruskal-Wallis H	H(df)
df	Degrees of freedom (k − 1)
Asymp. Sig.	p-value

For the balance errors example: H(2) = 2.09, p = .351 (non-significant).

Step 6: Post-hoc tests (if H is significant)

If Kruskal-Wallis is significant, run pairwise Mann-Whitney U tests for all group combinations. Apply Bonferroni correction: divide α (.05) by the number of pairwise comparisons [k(k−1)/2] to get the corrected threshold.

For k = 3: corrected α = .05/3 = .017. Report only pairs with p < .017 as significantly different.

Alternatively, use Analyze → Nonparametric Tests → Independent Samples (new dialog) → Settings tab → Multiple Comparisons to request Dunn’s test automatically.

Step 7: Effect size

SPSS does not report a Kruskal-Wallis effect size in Legacy Dialogs. Use the rank-based η²:

\[\eta^2_H = \frac{H - k + 1}{n - k}\]

where H is the test statistic, k is the number of groups, and n is the total sample size. This value can be computed from the output in the Statistical Calculators appendix.

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W.6 Part F: Friedman’s Test

Friedman’s test requires wide format data — one row per participant, with one column for each repeated condition.

Step 1: Restructure to wide format (if needed)

Follow the same restructuring approach described in Part D to create rpe_pre, rpe_mid, and rpe_post columns. Filter to the training group (group = 2).

Step 2: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → K Related Samples…

Step 3: Assign variables

Move rpe_pre, rpe_mid, and rpe_post into the Test Variables box.

Step 4: Select the test

Under Test Type, ensure Friedman is checked. Optionally also check Kendall’s W — SPSS will compute it automatically.

Step 5: Run

Click OK.

Step 6: Read the output

Ranks table — shows mean rank for each condition (time point).

Test Statistics table — record:

Output value	What to report
N	Sample size
Chi-Square	χ²(df)
df	Degrees of freedom (k − 1)
Asymp. Sig.	p-value
Kendall’s W	Effect size (if requested)

For the training group RPE example: χ²(2) = 4.46, p = .107, W = .07 (non-significant).

Step 7: Post-hoc tests (if Friedman is significant)

Run pairwise Wilcoxon signed-rank tests for all condition pairs with Bonferroni correction. For three conditions, corrected α = .05/3 = .017.

NoteKendall’s W interpretation

W ranges from 0 (no concordance — ranks are essentially random across participants) to 1 (perfect concordance — all participants rank conditions in the same order). Benchmarks: .10 = weak, .30 = moderate, .50 = strong concordance. Report W alongside χ² whenever Friedman’s test is used.

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W.7 Part G: Sign Test (One-Sample Nonparametric)

The sign test is available in SPSS via the Legacy Dialogs menu. Use it when comparing a single set of scores against a hypothesised median, or as a more conservative alternative to the Wilcoxon signed-rank test when the distribution of differences is severely asymmetric.

Step 1: Open the dialog

Analyze → Nonparametric Tests → Legacy Dialogs → 2 Related Samples…

Step 2: Assign variables

Create a constant column in your dataset equal to the hypothesised median (e.g., add a variable benchmark with the value 75 for every participant). Then pair your outcome variable (e.g., function_post) with benchmark in the Test Pairs list.

Step 3: Select the test

Under Test Type, uncheck Wilcoxon and check Sign.

Step 4: Run

Click OK.

Step 5: Read the output

Frequencies table — shows the count of negative differences (score below benchmark), positive differences (score above), and ties (exactly equal to benchmark). Ties are excluded from the test.

Test Statistics table — records:

Output value	What to report
Exact Sig. (2-tailed)	p-value (use Exact for small n; Asymptotic for large n)

Report the p-value alongside the count and proportion of positive signs and a 95% binomial confidence interval for the proportion (compute using the Statistical Calculators appendix).

TipWhen to use Exact vs. Asymptotic significance

For samples smaller than n = 25, use the Exact Sig. row, which is based on the binomial distribution. For larger samples, the Asymp. Sig. row (normal approximation) is appropriate. SPSS computes both automatically.

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W.8 Part H: Scheirer-Ray-Hare Test (Nonparametric Factorial ANOVA)

SPSS does not have a dedicated Scheirer-Ray-Hare procedure. The test is performed by: (1) manually ranking the outcome variable, (2) running a standard factorial ANOVA on those ranks, and (3) converting each sum of squares to an H statistic.

Step 1: Rank the outcome variable

Transform → Rank Cases…

• Move your outcome variable (e.g., strength_kg) into the Variable(s) box.
• Under Assign Ranks to, select All cases together (not within groups).
• Leave the ranking method as Ranks (1 = smallest).
• Click OK. SPSS creates a new variable (e.g., Rstrength_kg) containing the ranks.

Step 2: Run a standard two-way ANOVA on the ranks

Analyze → General Linear Model → Univariate…

• Set the ranked variable (Rstrength_kg) as the Dependent Variable.
• Add both factors (e.g., sex_category and group) to the Fixed Factor(s) box.
• Click Options…, check Descriptive statistics and Estimates of effect size, then click Continue.
• Click OK.

Step 3: Record the sums of squares

From the Tests of Between-Subjects Effects table, record:

Value needed	Where to find it
SS for Factor A	Type III SS row for Factor A
SS for Factor B	Type III SS row for Factor B
SS for A × B interaction	Type III SS row for the interaction
SS total	Sum of all SS rows (or compute as SS_corrected_model + SS_error + SS_corrected_total)
N	From Descriptive Statistics table

Step 4: Compute the H statistics

\[MS_{total} = \frac{SS_{total}}{N}, \quad H_A = \frac{SS_A}{MS_{total}}, \quad H_B = \frac{SS_B}{MS_{total}}, \quad H_{AB} = \frac{SS_{AB}}{MS_{total}}\]

Each H statistic follows a chi-square distribution with df equal to the number of levels of that factor minus one (or the product of the two factors’ df for the interaction). Obtain the p-value from the Statistical Calculators appendix chi-square table, or use the SPSS chi-square probability function:

Transform → Compute Variable… → set p_value = 1 - CDF.CHISQ(H_value, df)

NoteART-ANOVA as a more powerful alternative

The Aligned Rank Transform ANOVA (ART-ANOVA) is a more powerful and better-validated nonparametric factorial procedure, especially for detecting interaction effects. It is available in R via the ARTool package (function art()). If SPSS is your only option, the Scheirer-Ray-Hare procedure described here is an acceptable approximation for main effects; interpret interaction effects with caution given the test’s limited power.

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W.9 Part I: Quade’s Rank ANCOVA (Nonparametric ANCOVA)

SPSS does not have a dedicated Quade’s test procedure. The most practical SPSS-based approach for nonparametric ANCOVA is to residualise the outcome variable against the covariate using regression, then compare the residuals between groups using Mann-Whitney U.

Step 1: Regress the outcome on the covariate (ignoring group)

Analyze → Regression → Linear…

• Set the post-test outcome (e.g., strength_post) as the Dependent variable.
• Set the covariate (e.g., strength_pre) as the Independent(s) variable. Do not include group here.
• Click Save…, check Unstandardized Residuals, then click Continue → OK.

SPSS saves a new variable (RES_1) containing each participant’s residual — the portion of their post-test score not explained by pre-test.

Step 2: Compare residuals between groups using Mann-Whitney U

Analyze → Nonparametric Tests → Legacy Dialogs → 2 Independent Samples…

• Move RES_1 into the Test Variable List.
• Move group into the Grouping Variable box and define groups (1 and 2).
• Ensure Mann-Whitney U is checked.
• Click OK.

Step 3: Read and report the output

Interpret the Mann-Whitney U test on the residuals exactly as described in Part C. A significant result indicates that, after removing the covariate’s influence, the groups differ on the outcome. Report U, p, and rank-biserial r as the effect size.

WarningLimitations of this approach

The residual-then-Mann-Whitney approach is an approximation of Quade’s rank ANCOVA — it gives similar results but is not identical. For a fully rigorous nonparametric ANCOVA, use R (coin::kruskal_test() with a covariate, or the Rfit package). If your ANCOVA assumptions are only mildly violated, a data transformation (e.g., log of the outcome) followed by standard ANCOVA is often a more defensible and interpretable solution. Always report which approach was used and why.

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W.10 Troubleshooting

“Warning: X cells have expected count less than 5.” For a 2 × 2 table, use Fisher’s Exact Test p-value (shown in the same output row). For larger tables, consider collapsing categories that have few observations.

The Wilcoxon dialog only shows Z, not W. SPSS Legacy Dialogs report the Z approximation. The exact W statistic can be obtained from Analyze → Nonparametric Tests → Related Samples (new dialog), which also reports exact p-values for small samples.

Kruskal-Wallis shows significant overall, but none of the Bonferroni-corrected pairwise comparisons are significant. This can happen due to the difference in sensitivity between the omnibus test and the stricter pairwise threshold. It is not a contradiction. Report the omnibus result, note that no individual pairwise comparisons survived Bonferroni correction, and interpret cautiously.

Friedman test and repeated-measures ANOVA give different conclusions. This can occur with non-normal data. The Friedman test is more appropriate when normality is violated. If in doubt, report both and note the discrepancy with a brief explanation of why you selected Friedman as the primary analysis.

Mann-Whitney U gives U = 0 or U = n₁ × n₂. This indicates perfect separation: all scores in one group are above (or below) all scores in the other group. This is a valid result but warrants inspection for data entry errors and should be described as complete stochastic ordering of one group over the other.

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W.11 Practice Exercises

1. Chi-square independence. Using core_session.sav, run a chi-square test of independence to determine whether sex_category is associated with time (i.e., is the sex distribution stable across measurement occasions?). Report χ², df, p, and Cramér’s V. Verify that all expected cell counts meet the minimum threshold.

2. Spearman correlation. Compute the Spearman correlation between pain_0_10 and function_0_100 using all observations at pre-test. Interpret the sign, magnitude, and statistical significance. How does ρ compare with Pearson r for the same variables?

3. Mann-Whitney U. Compare balance_errors_count at mid-test between the control and training groups. Report U, p, median (IQR) for each group, and rank-biserial r. Is the difference statistically significant? Practically meaningful?

4. Wilcoxon signed-rank. Within the control group, test whether pain_0_10 changed from pre- to post-test. Report Z, p, and rank-biserial r. Does pain change significantly in participants who received no intervention?