Atlas
statminds
NonparametricThe underlying model family class (e.g. GLM, linear model, categorical matrix, log-linear).Parametric ReferenceStatistical methods that assume a specific probability distribution family (typically normal).12-stage workflow
Jonckheere-Terpstra Test
Nonparametric test for ordered alternatives across 3+ independent groups (tests for monotonic trend)..
G1
association
G2
description
Visual Overview Dashboard
1
What is it?
Jonckheere-Terpstra Testevaluates if group medians follow a predefined monotonic order (e.g. Dose 1 < Dose 2 < Dose 3).
2
When to use it
- Ordered Hypothesis: Hypothesized ordered pattern across groups.
- Nonparametric scale: Ordinal or non-normal data violating parametric trends assumptions.
3
Core Idea
Performs all possible pairwise cross-comparisons. Computes the proportion of pairs matching the expected trend:
4
Hypotheses
H0: Group medians are identical (M1 = M2 = M3)
Ha: Medians follow ordered trend (M1 <= M2 <= M3, with at least one strict inequality)
5
How it works
- Conduct pairwise comparisons for all combinations where i < j.
- Count how many times score in higher group exceeds lower group.
- Sum counts to yield Jonckheere-Terpstra J statistic.
- Test J statistic against expected null mean and variance.
6
Assumptions
📊 Ordinal Scale: Data values can be ordered.
👤 Independence: Independent subjects across groups.
7
Effect Size
Standardized J statistic represents a non-linear correlation coefficient (similar to Kendall's Tau) indicating monotonic trend strength.
8
Quick Example
| Dose Group | n | Median Score |
|---|---|---|
| Control | 8 | 12.2 |
| Low Dose | 8 | 18.5 |
| High Dose | 8 | 24.4 (Significant trend) |
Interactive Sandbox
Jonckheere-Terpstra Ordered Trend Laboratory
Slide Group 2 and 3 offsets to observe ordered pairwise separation.
Presets
Group observations by rank level (G1: amber, G2: blue, G3: green)Values mapped on Y-axis (scale: 30 to 70)
Calculations Output
| Metric | Value |
|---|---|
| Jonckheere J Stat | 119 |
| Expected J under H0 | 96 |
| Z-statistic | 3.416 |
| p-value | 0.0007 |
Statistical Verdict
✅ Significant Ordered Trend
Group medians follow hypothesized ordered sequence (J = 119, p = 0.0007). Reject H0.
01Hypothesis test logic
Hypotheses
Pragmatic null and alternative hypotheses defined in mathematical notation.
A hypothesis is a question sharpened to a point. Ambiguity is the enemy of inference.
Logic CoreNull · H₀
H₀: No ordered trend (all distributions identical or randomly ordered)
Alternative · Hₐ
Hₐ: Ordered trend exists (e.g., Group1 ≤ Group2 ≤ Group3, with at least one strict inequality)
Why it matters directional (one-tailed)
Tests for ORDERED alternatives (monotonic trend) across groups. More powerful than Kruskal-Wallis when a priori ordering is expected (e.g., dose-response: low < medium < high). IMPORTANT: Like other rank tests, only tests medians when distribution shapes are similar (Divine et al., 2018); otherwise tests stochastic ordering.
02Model diagnostics
Assumptions
The core mathematical criteria needed to ensure that statistical testing remains unbiased and valid.
Build your analysis on rock, not sand. Verify the mathematical foundation before building the model.
Integrity Shield5
Assumptions
4
Critical / High Severity
How to check
Quick
Inspect variable type; verify data has meaningful ordering. For continuous data with non-normal distribution or outliers, JT is appropriate alternative to linear trend test
Rigorous
For continuous data: check normality per group (Shapiro-Wilk). If p < .05 in any group, nonparametric JT justified. For ordinal: verify ordered categories with consistent interpretation
If violated
If nominal (no ordering) → use Kruskal-Wallis or chi-square test. If continuous and normal → use linear contrast in ANOVA (more powerful). If binary → use Cochran-Armitage trend test
kruskal wallisone way anova
How to check
Quick
Verify groups have clear ordering (e.g., low-medium-high dose, none-mild-moderate-severe disease). Ordering must be specified BEFORE analysis, not after seeing data. Minimum 3 groups required
Rigorous
Check study design: was ordering hypothesis pre-registered or theory-driven? If ordering chosen post-hoc after seeing data (e.g., reordering groups to maximize trend), this inflates Type I error and invalidates test. Confirm ≥3 groups (for 2 groups, use Mann-Whitney)
If violated
If no natural ordering → use Kruskal-Wallis (tests any difference, not trend). If only 2 groups → use Mann-Whitney U. If continuous IV → use Spearman correlation or linear regression. If ordering is post-hoc → must use Kruskal-Wallis (JT is invalid when ordering chosen after seeing data)
kruskal wallismann whitneyspearman correlation
How to check
Quick
Design review: verify each subject appears in exactly one group. Check for clustering (sites, families, therapists). No within-subject repeated measures
Rigorous
Confirm randomization scheme (if experimental). Check for clustering indicators (site ID, family ID). Verify no subject appears in multiple groups. If longitudinal, check for autocorrelation
If violated
If repeated measures (same subjects across ordered conditions) → use Page test for ordered alternatives with repeated measures. If clustered → use GEE or mixed models with ordered factor. If matched sets → use Friedman test with post-hoc trend analysis
friedmangeelinear mixed model
How to check
Quick
Verify ordering was hypothesized before analysis (in protocol, pre-registration, or based on established theory). Post-hoc ordering after seeing data inflates Type I error
Rigorous
Check pre-registration, study protocol, or theoretical justification for ordering. If ordering was determined by exploratory analysis (e.g., ordering groups by observed medians to maximize trend), JT is invalid—must use Kruskal-Wallis with appropriate correction
If violated
If ordering is post-hoc (chosen after seeing data): CANNOT use JT (inflated Type I error). Instead: (1) Use Kruskal-Wallis for omnibus test; (2) If significant, use post-hoc pairwise tests with adjustment; (3) For next study, can test this ordering a priori. Never reorder groups to maximize trend statistic
kruskal wallis
How to check
Quick
Visual inspection: boxplots, density plots per group. Compare IQRs (should be similar). Check if distributions have same variance and skewness across groups
Rigorous
Levene's test for equal variances across groups. Compare skewness coefficients (difference <0.5). If shapes differ substantially, JT tests stochastic ordering, not median trend
If violated
If shapes differ: JT tests stochastic ordering (distribution shift), NOT specifically median trend. Report as 'ordered trend in distributions' not 'ordered trend in medians'. For pure median trend test when shapes differ, use quantile regression with ordered factor. For mean trend with non-normality, use permutation test with linear contrast
quantile regression
03Residual Forensics
Diagnostics
Checking residual plots and indices to examine model deviations and ensure standard error integrity.
Trust, but verify. The outliers often hold more truth than the averages.
System HealthEssential checks
- Boxplots showing ordered groups (visual trend inspection)
- Check for monotonic pattern (medians increase/decrease across ordered groups)
- Verify distribution shapes are similar
Recommended checks
- Group medians and IQRs (should show monotonic pattern)
- Scatter plot with group medians overlaid
- Levene's test for homogeneity of variance
- Density plots overlaid by group (check shape similarity)
- Effect size plot (medians with CIs across ordered groups)
04Live Instances
Applied Minds
Review concrete study examples, data layout guidelines, and copy executable syntax scripts.
Theory is the map. Practice is the terrain. Simulation bridges the gap.
Applied WisdomExample 01
Yoga Dose-Response on Stress (Ordered 4-Group Design)
Research question: Does yoga practice frequency show dose-response relationship with stress reduction? Design: 4 independent groups with ordered yoga frequency (Never, 1x/week, 3x/week, Daily), n=30 per group. Outcome: Perceived Stress Scale (PSS, 0-40, lower = less stress, ordinal). JT tests ordered hypothesis: Never ≥ 1x/week ≥ 3x/week ≥ Daily.
GroupsNever · 1x/week · 3x/week · Daily
Total n120
Outcome ScalePSS stress score (0-40, ordinal/continuous)
# Jonckheere-Terpstra Test: Yoga dose-response on stress
library(clinfun)
library(tidyverse)
library(rstatix)
set.seed(2025)
data <- data.frame(
yoga_frequency = factor(rep(c("Never", "1x/week", "3x/week", "Daily"), each=30),
levels=c("Never", "1x/week", "3x/week", "Daily")), # Ordered!
stress_pss = c(
round(rnorm(30, 28, 6)), # Never: high stress
round(rnorm(30, 24, 5.5)), # 1x/week: moderate-high
round(rnorm(30, 19, 5)), # 3x/week: moderate-low
round(rnorm(30, 14, 4.5)) # Daily: low stress
)
)
data$stress_pss <- pmin(pmax(data$stress_pss, 0), 40) # Bound 0-40
# Visualize ordered trend
ggplot(data, aes(x=yoga_frequency, y=stress_pss, fill=yoga_frequency)) +
geom_boxplot() +
stat_summary(fun=median, geom="line", aes(group=1), color="red", size=1) +
stat_summary(fun=median, geom="point", color="red", size=3) +
labs(title="Dose-Response: Yoga Frequency → Stress(Ordered Trend)",
subtitle="Red line shows median trend",
x="Yoga Practice Frequency(Ordered)", y="Perceived Stress(PSS)") +
theme_minimal() + theme(legend.position="none")
# Descriptive stats by group
data %>%
group_by(yoga_frequency) %>%
summarise(n=n(), Median=median(stress_pss), IQR=IQR(stress_pss))
# Jonckheere-Terpstra Test
# Convert ordered factor to numeric for clinfun::jonckheere.test
yoga_numeric <- as.numeric(data$yoga_frequency) # 1=Never, 2=1x/week, 3=3x/week, 4=Daily
jt_result <- jonckheere.test(data$stress_pss, yoga_numeric, alternative="decreasing")
print(jt_result)
# Alternative: use DescTools::JonckheereTerpstraTest
# Effect size: Kendall's tau for trend
tau <- cor.test(yoga_numeric, data$stress_pss, method="kendall")
cat("\nKendall's tau(trend effect size):", round(tau$estimate, 3), "\n")
cat("Interpretation: |tau| = .10 (small), .30 (medium), .50 (large)\n\n")
# Compare to Kruskal-Wallis (non-directional)
kw_result <- kruskal.test(stress_pss ~ yoga_frequency, data=data)
cat("Kruskal-Wallis(non-directional) p =", round(kw_result$p.value, 4), "\n")
cat("JT(directional trend) p =", round(jt_result$p.value, 4), "\n")
cat("JT is more powerful when trend hypothesis is correct\n\n")
# APA Report
cat("=== APA Report ===\n")
cat(paste0(
"A Jonckheere-Terpstra test was conducted to test the a priori hypothesis of ",
"an ordered dose-response relationship between yoga practice frequency and stress. ",
"Groups were ordered: Never ≥ 1x/week ≥ 3x/week ≥ Daily practice. There was a ",
"significant decreasing trend in stress across increasing yoga frequency, ",
"JT = ", round(jt_result$statistic, 2), ", p < .001, Kendall's tau = ",
round(tau$estimate, 2), " (large effect). Median stress scores decreased monotonically: ",
"Never(Mdn=28), 1x/week(Mdn=24), 3x/week(Mdn=19), Daily(Mdn=14), supporting ",
"a dose-response relationship between yoga frequency and stress reduction."
))Interpretation Blueprint
JT statistic = 9875, p < .001, Kendall's tau = -.68 (large). Significant ordered trend: stress decreases monotonically with increasing yoga frequency. Daily practice (Mdn=14) showed lowest stress, Never (Mdn=28) highest. JT more powerful than Kruskal-Wallis (p=.002 vs p<.001) because it leverages a priori ordering. Supports dose-response hypothesis.
05Tactical Pivots
Alternatives
Structured fallback pathways for choosing alternative tests when normality or slopes requirements fail.
When the path is blocked, pivot. Rigor is not rigidity; it is the intelligent adaptation to reality.
Adaptive StrategyMeasurement Precision Ladder Ideal · Ordered Groups x Continuous
Ratio
Maintain JT logic. Exploit the rank-order of populations to maximize trend-detection power.
Peak SignalInterval
Ideal for Dose-Response Scales. Ensure the 'Direction' of groups matches your clinical theory.
Standard PrecisionNominal Groups
Abandon JT. Use Kruskal-Wallis if categories lack a natural, logical order (e.g., Hospital A vs B).
Model CollapseTemporal Trajectory Audit Static Ordered Comparison
Static Dose
Cross-sectional trend.
Stay with Jonckheere-Terpstra. The most powerful distribution-free trend audit.
Repeated Trends
Trajectory growth.
Pivot to Page's Trend Test to model within-subject rank progression over time.
Adaptive Technical Safeguards · adaptive safeguards
no directional order
- Kruskal-Wallis — Return to the omnibus rank audit if the trend hypothesis is violated.
normal distributions detected
- Linear Contrast ANOVA — Reclaim higher efficiency by utilizing raw scores and ordered mean-comparisons.
extreme ties
- Monte Carlo JT — Resample the null distribution to calculate exact significance for discrete ordered scales.
06Adjusted Comparisons
Post-hoc
Group mean comparisons and correction controls (e.g. Tukey HSD, Bonferroni) to protect against Family-Wise Error Rates.
The omnibus test opens the door; post-hoc analysis explores the room.
Forensic DetailAdjusted Comparisons
Post-hoc pairwise tests defined for this model.
Interpretation Guidelines
Jonckheere-Terpstra assumes an order. Post-hoc forensics should verify that the order was respected—JT loses authority if the data 'Wiggles' instead of 'Trends'.
07Standardized scale impact
Effect Size
Understanding effect sizes (e.g., Cohen's d, Partial Eta-Squared) and clinical impact benchmarks.
Significance is noise. Magnitude is the signal. Measure the impact, not just the probability.
Impact MagnitudeRange: -1 to +1. Small: .10, Medium: .30, Large: .50 (Cohen, 1988 adapted). Measures strength of monotonic association between ordered groups and outcome
Z-score version of JT statistic. Standardized for sample size. Larger |Z| = stronger trend
Range: -1 to +1. Similar to Kendall's tau but based on ranks. Slightly larger in magnitude than tau for same data
Recommended Metric: Kendall's tau-b for trend (more interpretable, robust to ties)
Small
0.2
0.2
Medium
0.5
0.5
Large
0.8
0.8
0.50
Kendall's tau-b for trend (more interpretable, robust to ties)
Recommended Measure
4
Available Metrics
ReportUse Kendall's tau-b for trend (more interpretable, robust to ties) to represent clinical impact magnitude.
08Statistical Power
Sample Size
Guidelines for minimum sample requirements and power analysis parameters.
An underpowered study is an ethical failure. Respect the data by collecting enough of it.
Power ProtocolFloor Requirements
At least 5 observations per group. For k=3 groups, minimum n=15 total. Smaller samples: use exact permutation version
| Effect Size | Parameters | Required n |
|---|---|---|
| Small Effect | Kendall's tau = .10 | n ≈ 300 total |
| Medium Effect | Kendall's tau = .30 | n ≈ 90 total |
| Large Effect | Kendall's tau = .50 | n ≈ 45 total |
G*Power StrategyJT is more powerful than Kruskal-Wallis when ordered hypothesis is correct (~10-30% gain). Use simulation-based power analysis or approximate via Spearman correlation power
09APA narrative blueprint
Reporting
How to compile statistical results into publication prose matching APA and journal style guides.
Data does not speak for itself. It requires a translator. Be clear, be precise, be honest.
Narrative ArcReusable template
A Jonckheere-Terpstra test was conducted to test the a priori hypothesis of an ordered increasing/decreasing trend in outcome across k ordered groups. Groups were ordered: list ordering, e.g., Group1 < Group2 < Group3. State assumption checks: 'Distributions had similar shapes' OR 'noted for stochastic ordering interpretation'. There was a significant/non-significant increasing/decreasing trend, JT = X.XX, p = .XXX, Kendall's tau = .XX interpret effect size. If significant: Median outcome increased/decreased monotonically across groups: list medians. Conclude with interpretation in research context.
Essential statistics to report
- JT statistic or standardized Z-score
- p-value
- Effect size (Kendall's tau or Spearman's rho)
- Direction of ordering tested (increasing/decreasing)
- Medians (or medians + IQRs) for each ordered group
- Statement confirming a priori ordering hypothesis
- Justification for using JT over Kruskal-Wallis
10Exhibit Builder
Manuscript Lab
Copy standard summary tables and forensic reporting grids to outline analysis details.
Table 1: Jonckheere-Terpstra Test for Ordered Group Trends
| Group | Median | Mean Rank | J (Statistic) | z | p (Trend) |
|---|---|---|---|---|---|
| Low | 45 | 32.4 | 1845 | 4.12 | < .001 |
| Med | 52 | 45.1 | — | — | — |
| High | 65 | 57.5 | — | — | — |
Note. N = 90. Predictor: Training Intensity (Low < Med < High).
p < .001Powerful Trend Detection. Unlike Kruskal-Wallis, J-T explicitly audits the 'Ladder Effect', making it far more powerful for dose-response hypotheses.
Header glossary
The Step-Wise Count. Sums the number of times a person in a higher group ranks higher than someone in a lower group.
Linear Median Probability. Proves that as intensity increases, performance medians rise in a systematic ladder.
11Algorithmic Logic
Command Center
Syntax libraries and function parameters for executing calculations in stats packages.
Code is the modern laboratory. Clean execution ensures reproducible discovery.
Execution Engine# 1. Execute Jonckheere-Terpstra Test for ordered alternatives
clinfun::jonckheere.test(x, g, alternative = 'two.sided')Library stack
R
clinfun
Python
scipy.stats
Elite Forensic Strike
Jonckheere-Terpstra assumes a specific monotonic ordering. If the trend is non-monotonic (e.g. U-shaped), JT loses power and Kruskal-Wallis should be used.
# Run pairwise Wilcoxon post-hoc comparisons to locate the trend shift
rstatix::wilcox_test(df, score ~ group, p.adjust.method = 'bonferroni')12The Over-adjustment Trap
Common Mistakes
Analytical caveats and corrections to maintain modeling integrity.
Wisdom is learning from the failures of others. Anticipate the error before it occurs.
Defensive LogicWhy it's wrong
Like all rank-sum tests (Kruskal-Wallis, Mann-Whitney), JT tests stochastic ordering of distributions, NOT specifically medians, unless groups have similar distribution shapes. If Group A is wider/more skewed than Group B, significant JT means ordered distribution shift, not necessarily ordered medians. This is the same issue Divine et al. (2018) clarified for Mann-Whitney. The null is 'no ordered trend in distributions', NOT 'no ordered trend in medians' (except when shapes similar). See one_way_anova.json common mistake #7 for analogous issue.
The correction
ALWAYS check distribution shapes: boxplots, Levene's test, compare skewness. If shapes SIMILAR → interpret as ordered median trend. If shapes DIFFER → report as 'ordered trend in distributions' or 'stochastic ordering'. For pure median trend test when shapes differ, use quantile regression with ordered predictor. Reference Divine et al. (2018).
Why it's wrong
JT REQUIRES a priori ordering hypothesis. If you order groups by observed medians (e.g., reordering to maximize trend statistic), this capitalizes on chance and severely inflates Type I error. Post-hoc ordering is data snooping—you're 'cheating' by using data to create hypothesis, then testing same data. This can turn non-significant Kruskal-Wallis into 'significant' JT.
The correction
Ordering MUST be specified before analysis (pre-registration, protocol, theory). If ordering is post-hoc: CANNOT use JT—must use Kruskal-Wallis. Document ordering justification: 'Groups were ordered based on dose levels (0mg < 10mg < 20mg) as specified in study protocol'. For exploratory analysis, use Kruskal-Wallis; save JT for confirmatory study with a priori hypothesis.
Why it's wrong
JT is for DIRECTIONAL hypothesis (ordered trend). If you have no a priori ordering or testing any difference (not specifically trend), Kruskal-Wallis is correct test. Using JT without ordered hypothesis: (1) inflates Type I error if ordering is wrong, (2) loses power if true differences are non-monotonic (e.g., quadratic: low and high different from medium).
The correction
Use decision rule: (1) A priori ordered hypothesis (e.g., dose-response) → JT (more powerful for trend); (2) No ordered hypothesis or exploratory → Kruskal-Wallis (tests any difference). Example: 'JT was used to test pre-specified hypothesis of increasing effect with dose' vs 'K-W was used for exploratory comparison of treatment groups'.
Why it's wrong
p-value indicates significance, not magnitude of trend. With large n, trivial trends can be 'significant'. Effect size quantifies strength of ordering: is trend strong (tau=.60) or weak (tau=.12)? Essential for interpreting practical importance and comparing across studies.
The correction
ALWAYS report Kendall's tau or Spearman's rho: 'JT = 4521, p < .001, Kendall's tau = .48 (large effect)'. Interpret: |tau| = .10 (small), .30 (medium), .50 (large). Also report medians for each group to show trend magnitude: 'Medians increased monotonically: Low=12, Med=18, High=24'.
Why it's wrong
JT requires INDEPENDENT groups. For repeated measures (same subjects measured under ordered conditions: pre-treatment, mid-treatment, post-treatment), independence is violated. Using JT inflates Type I error by ignoring within-subject correlation.
The correction
For repeated measures with ordered alternatives: use Page test (nonparametric test for ordered repeated measures). For k=2 repeated conditions: use Wilcoxon signed-rank. For independent groups: use JT. Example: 'Page test was used to test ordered trend across pre-mid-post timepoints (repeated measures)'.
Why it's wrong
JT is designed for 3+ ordered groups. With only 2 groups, JT reduces to Mann-Whitney U test—same result, but unnecessarily complex. Mann-Whitney is simpler, more widely recognized, and provides standard effect sizes (rank biserial).
The correction
For 2 independent groups: use Mann-Whitney U (one-tailed if directional hypothesis). For 3+ ordered groups: use JT. Minimum 3 groups required for meaningful trend test.
Why it's wrong
Despite both testing ordered alternatives, JT is for independent groups (between-subjects) and Page test is for repeated measures (within-subjects). Using wrong test: violates independence assumption (JT on repeated measures) or ignores pairing (Page on independent groups), both invalidate results.
The correction
Verify study design: Independent groups (different subjects per group) → Jonckheere-Terpstra. Repeated measures (same subjects across ordered conditions) → Page test. Always specify: 'JT was used to test ordered trend across independent dose groups' vs 'Page test for repeated measures across timepoints'.
Why it's wrong
JT tests for monotonic trend (always increasing or always decreasing). If true pattern is non-monotonic (e.g., inverted-U: low=10, medium=25, high=15), JT may be non-significant or misleading. Significant JT assumes monotonic pattern—must verify with descriptive statistics.
The correction
ALWAYS plot medians before interpreting JT: boxplots or median trend line. Check if pattern is monotonic. If medians show non-monotonic pattern (e.g., quadratic), JT is inappropriate—use Kruskal-Wallis or polynomial contrasts in ANOVA. Report: 'Visual inspection confirmed monotonic decreasing trend (medians: Never=28, 1x/wk=24, 3x/wk=19, Daily=14), justifying JT'.
13Academic Lineage
References
Scholarly lineage and citation keys grounding the statistical framework.
We stand on the shoulders of giants. Honor the source of the method.
Academic Lineage[1]
Jonckheere, A. R. (1954). A distribution-free k-sample test against ordered alternatives. Biometrika, 41(1/2), 133-145.
Original paper introducing the Jonckheere test for ordered alternatives. Foundation for nonparametric trend tests.
doi: 10.2307/2333011
[2]
Terpstra, T. J. (1952). The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking. Indagationes Mathematicae, 14, 327-333.
Terpstra's extension handling ties. Combined with Jonckheere's work forms the Jonckheere-Terpstra test.
[3]
Divine, G. W., Norton, H. J., Barón, A. E., & Juarez-Colunga, E. (2018). The Wilcoxon–Mann–Whitney procedure fails as a test of medians. The American Statistician, 72(3), 278-286.
CRITICAL: Clarifies rank-sum tests (including JT) test distributions/stochastic ordering, NOT medians, unless shapes are similar. Essential for correct JT interpretation. See one_way_anova.json common mistake #7.
doi: 10.1080/00031305.2017.1305291
[4]
Smith, C., Hancock, H., Blake-Mortimer, J., & Eckert, K. (2007). A randomised comparative trial of yoga and relaxation to reduce stress and anxiety. Complementary Therapies in Medicine, 15(2), 77-83.
Demonstrates dose-response relationship between yoga practice frequency and stress reduction. Basis for Example 1.
doi: 10.1016/j.ctim.2006.05.001
[5]
Bewick, V., Cheek, L., & Ball, J. (2004). Statistics review 10: Further nonparametric methods. Critical Care, 8(3), 196-199.
Accessible overview of JT test, including when to use vs. Kruskal-Wallis and interpretation guidelines.
doi: 10.1186/cc2857
statminds · Jonckheere-TerpstraMind reference · v2.2 · updated 2026-01-1715 of 15 sections