Non-inferiority Trials in Medicine:

Last updated: June 2026

Author: Michele D. Pierri

Reading time: 15–20 minutes

In clinical research, we are used to thinking that a new drug, device, or procedure should be tested by asking a simple question:

Is it better than what we already have?

This is the logic of a superiority trial. A new treatment is compared with an existing treatment, placebo, or standard care, and researchers try to demonstrate that the new option produces better outcomes.

But many modern clinical trials — especially those involving new drugs, medical devices, anticoagulants, antibiotics, interventional cardiology devices, surgical technologies, or minimally invasive procedures — are not designed to prove superiority. Instead, they are designed to prove non-inferiority.

At first glance, this may sound suspicious. Why would we accept a new treatment that is not clearly better? Is this just a convenient way for companies to get a product approved without proving that it improves patient outcomes?

The answer, though, is less clear-cut. Non-inferiority trials are scientifically legitimate, but they are also easy to misunderstand, and in some cases deliberately misused.

Superiority, non-inferiority, and equivalence: three different questions

The first mistake is to confuse three different concepts.

A superiority trial asks:

Is the new treatment better than the control?

A non-inferiority trial asks:

Is the new treatment not unacceptably worse than the control?

An equivalence trial asks:

Are the two treatments similar enough in both directions?

These are not interchangeable.

A non-inferiority trial does not prove that two treatments are identical. It does not prove that the new treatment is equivalent to the old one. It only tries to exclude that the new treatment is worse than the standard treatment by more than a pre-specified clinically acceptable amount. This amount is called the non-inferiority margin (Δ). Regulatory and reporting guidance documents (e.g., FDA guidance; the CONSORT extension for non-inferiority/equivalence) emphasize that this margin must be defined and justified before results are known.

Why use a non-inferiority design at all?

Non-inferiority trials are often used when an effective standard treatment already exists.

Imagine a disease for which a proven treatment reduces mortality. In that situation, it may be unethical to compare a new drug with placebo, because patients in the placebo arm would be denied effective therapy. This is one of the classic reasons for using an active-control non-inferiority trial, where the new treatment is compared against the accepted standard.

But there is another important reason.

A new treatment may not be more effective than the old one, but it may offer other advantages:

• fewer adverse effects;
• simpler administration;
• no need for laboratory monitoring;
• shorter hospital stay;
• lower invasiveness;
• better patient adherence;
• lower cost;
• easier logistics;
• improved quality of life.

For example, a new oral anticoagulant may not reduce thromboembolic events more than warfarin, but it may avoid repeated INR monitoring. A new antibiotic may not cure more infections than the standard drug, but it may require fewer daily doses. A new device may not reduce mortality more than an established procedure, but it may be less invasive or easier to implant.

Cardiology offers the clearest example of this logic at scale: the low-risk TAVI trials were designed against SAVR as non-inferiority studies, and the 2025 ESC/EACTS guidelines translated that evidence into a lowered age threshold for the transcatheter approach.

In these cases, the clinical question is not necessarily “Is the new treatment better?”. It may be:

Can we accept a small possible loss of efficacy in exchange for other meaningful benefits?

That, at least, is the setting where these trials make genuine clinical sense.

A simple numerical example

Suppose we are testing a new device against a standard device. The endpoint is a negative outcome, such as major complications within 30 days.

The standard device has a complication rate of 5%.

The new device has a complication rate of 6%.

The absolute difference is:

6% − 5% = +1%

So the new device appears slightly worse.

But before the trial started, investigators defined a non-inferiority margin of 2%. This means that the new device would be considered clinically acceptable if it did not increase complications by more than 2 percentage points.

Now imagine the 95% confidence interval for the difference is:

+0.2% to +1.8%

The entire confidence interval is below the non-inferiority margin of +2%.

So the study may conclude:

The new device is non-inferior to the standard device.

But notice something important: the confidence interval is entirely above zero. That means the new device is also statistically worse than the standard device, although still within the pre-defined acceptable margin.

This is not a contradiction. It reflects the fact that non-inferiority is not the same as superiority. A treatment can be statistically worse and still be considered non-inferior if the difference is smaller than the accepted clinical margin.

How to read the confidence interval in a non-inferiority trial (quick guide)

Most non-inferiority conclusions are CI-based. The key is: which side of the CI must stay within the margin depends on how the effect is defined.

• If the endpoint is an undesirable event (lower is better) and you analyze a risk difference (new − control), non-inferiority is typically shown when the upper bound of the CI is below Δ.
• If the endpoint is a desirable event (higher is better) and you analyze a difference (new − control), non-inferiority is typically shown when the lower bound of the CI is above −Δ.
• If you use ratios (risk ratio, hazard ratio), the same logic applies but with a ratio margin (e.g., HR < 1.25). Always check the paper’s estimand and margin definition.

If this sounds pedantic, it is not: many misinterpretations come from not being explicit about direction and scale.

The non-inferiority margin: the most important number in the trial

The whole trial depends on one number: the non-inferiority margin (Δ).

This is the largest loss of efficacy (or increase in harm) that investigators are willing to accept.

If the margin is too strict, the trial may require a very large sample size and may fail to show non-inferiority. If the margin is too generous, almost any treatment can look acceptable.

That is where things get difficult.

Suppose the standard treatment has a 5% event rate. A new treatment has a 7% event rate. That is a relative increase of 40%:

from 5% to 7%

If the non-inferiority margin is set at +2%, the new treatment may still be declared non-inferior.

But would clinicians and patients really accept a 40% relative increase in complications?

Maybe yes, if the new treatment has major advantages. Maybe no, if those advantages are marginal. The answer is rarely obvious in advance.

This is why the margin cannot be only a statistical decision. It must be a clinical decision.

Regulators often discuss margin choice in terms of two related concepts:

• M1: the estimated effect of the active control compared with placebo, based on historical evidence;
• M2: the largest clinically acceptable loss of that effect.

Put simply: the non-inferiority margin should not be so wide that it allows the new intervention to lose most (or all) of the benefit that made the standard treatment worth using in the first place.

Why non-inferiority trials can be attractive to sponsors

Non-inferiority trials can be scientifically appropriate. But they can also be commercially attractive.

If a company knows that its new drug or device is unlikely to be better than the existing standard, a superiority trial may be risky. A non-inferiority trial offers another path: the product does not need to prove that it is better on the primary endpoint. It only needs to prove that it is not worse beyond the accepted margin.

That logic holds, but only when the new product actually delivers on those advantages.

The risk is that the marketing message becomes:

“As effective as the standard treatment.”

When the more accurate interpretation may be:

“Not shown to be unacceptably worse than the standard treatment, according to a margin chosen before the trial.”

These two statements sound similar, but they are not the same.

Why non-inferiority is not automatically easier

It is tempting to say that non-inferiority trials are “easier” because they do not require superiority.

But statistically and methodologically, a good non-inferiority trial is not necessarily easier. In some ways, it is more fragile.

In a superiority trial, poor adherence, imprecise measurements, protocol deviations, or crossover between groups usually make it harder to detect a true difference. They push the trial toward a neutral result.

In a non-inferiority trial, the same problems can be dangerous. If everything blurs the difference between treatments, the new treatment may appear “not much worse” simply because the trial was not good enough to detect meaningful differences.

This is why the concept of assay sensitivity is so important.

Assay sensitivity means that the trial is capable of distinguishing an effective treatment from an ineffective or less effective one. In a placebo-controlled trial, this can often be assessed directly. In an active-control non-inferiority trial, it is harder to verify, because there is usually no placebo group. The trial must rely on the assumption that the active control would have performed as expected if placebo had been included.

The “constancy assumption”

Another hidden assumption is the so-called constancy assumption.

This means that the benefit of the standard treatment observed in older trials is assumed to still apply in the current trial.

But medicine changes.

Patients change. Diagnostic criteria change. Background therapies improve. Surgical and interventional techniques evolve. Event rates decline. Follow-up becomes different. Endpoints are redefined.

A standard treatment that showed a large benefit decades ago may have a smaller absolute benefit today because baseline risk is lower. If the historical treatment effect is no longer valid, the non-inferiority margin may become questionable.

This is particularly relevant in areas such as cardiology, oncology, infectious diseases, intensive care, and surgery, where standards of care evolve rapidly. It is not always clear, when designing a new trial, whether the historical reference still holds.

A second example: when the margin changes everything

Imagine a trial comparing a new antibiotic with a standard antibiotic.

Clinical cure rate:

• standard antibiotic: 90%
• new antibiotic: 86%

Difference:

−4 percentage points

The new antibiotic is less effective.

Now consider two possible non-inferiority margins.

Scenario A: margin = 5%

The new antibiotic loses 4 percentage points. This is within the allowed margin.

Conclusion:

Non-inferior.

Scenario B: margin = 3%

The new antibiotic loses 4 percentage points. This exceeds the allowed margin.

Conclusion:

Not non-inferior.

The data are identical. The conclusion changes only because the margin changes.

This is why the non-inferiority margin is not a technical detail. It is the foundation of the entire study.

Common pitfalls (what can make a non-inferiority trial misleading)

• Comparator issues: the “standard” treatment may be suboptimal (dose, timing, operator expertise), making non-inferiority easier to show.
• Adherence and crossover: non-adherence tends to make groups look artificially similar (dangerous in non-inferiority).
• Endpoint choices: composite endpoints can hide clinically important trade-offs.
• Over-interpretation: “non-inferior” is sometimes communicated as “equivalent” or “just as good,” which is not what the design proves.

What readers should look for

When reading a non-inferiority trial, especially one involving a new commercial product, the most important questions are:

1) Was a non-inferiority design justified?

Was there already an effective standard treatment? Would placebo have been unethical? Does the new treatment offer plausible advantages other than efficacy?

2) Was the margin clinically reasonable?

Do not just ask whether the margin was pre-specified. Ask whether it was clinically acceptable. A margin can be statistically convenient and clinically unacceptable.

3) Was the comparator appropriate?

The control treatment must be the real standard of care, used at the correct dose, with correct timing, and under appropriate conditions. A weak comparator makes the new treatment look better.

4) Were both intention-to-treat and per-protocol analyses reported?

In superiority trials, intention-to-treat analysis is usually conservative. In non-inferiority trials, it may not be. Non-adherence and crossover may make treatments look artificially similar. For this reason, both intention-to-treat and per-protocol analyses are often important, and consistency between them strengthens the conclusion.

5) Was superiority also tested (and was it pre-specified)?

Sometimes a trial first shows non-inferiority and then tests superiority. This can be valid if properly pre-specified and statistically controlled. But post-hoc claims of superiority should be treated cautiously.

6) Are the claimed advantages real?

If the new treatment is slightly less effective but safer, cheaper, simpler, or better tolerated, it may be clinically valuable. But those advantages must be demonstrated, not merely suggested.

The practical interpretation

A non-inferiority result should never be read as:

“The two treatments are the same.”

A better interpretation is:

“The trial did not show that the new treatment is worse than the standard by more than the pre-specified acceptable margin.”

That sentence is longer, less attractive, and less marketable.

But it is more accurate.

Conclusion

Non-inferiority trials are not inherently problematic. They are often necessary and ethically appropriate when an effective treatment already exists and placebo would be unacceptable. They can help identify treatments that are slightly less effective but safer, simpler, less invasive, or more acceptable to patients.

They do, however, require careful reading.

The question worth asking is not only whether non-inferiority was demonstrated. The more important one is:

Was the amount of possible inferiority clinically acceptable?

In other words, the entire interpretation depends on the margin.

A well-designed non-inferiority trial can support good clinical decision-making. A poorly designed one can turn “not clearly worse” into a misleading impression of “just as good.”

For clinicians, researchers, and patients trying to make sense of trial reports, the practical takeaway is straightforward:

Whenever a study says “non-inferior,” always ask: non-inferior by how much, compared with what, and in exchange for which real benefit?

FAQ

Does “non-inferior” mean “equivalent”?

No. “Non-inferior” means the trial excluded that the new treatment is worse than the control by more than the pre-specified margin. “Equivalent” requires showing differences are small in both directions.

Why not just run a superiority trial?

Because in many settings placebo is unethical, and the goal is to trade a small potential loss of efficacy for meaningful advantages (safety, simplicity, cost, quality of life).

What is the non-inferiority margin (Δ)?

It is the maximum difference considered clinically acceptable. The entire conclusion depends on how Δ is chosen and justified.

References

• Food and Drug Administration. (2016). Non-inferiority clinical trials to establish effectiveness: Guidance for industry. U.S. Department of Health and Human Services. https://www.fda.gov/media/78504/download
• International Council for Harmonisation. (2000). ICH E10: Choice of control group and related issues in clinical trials.
• Kaul, S., & Diamond, G. A. (2006). Good enough: A primer on the analysis and interpretation of noninferiority trials. Annals of Internal Medicine, 145(1), 62–69.
• Piaggio, G., Elbourne, D. R., Pocock, S. J., Evans, S. J. W., Altman, D. G., & CONSORT Group. (2012). Reporting of noninferiority and equivalence randomized trials: Extension of the CONSORT 2010 statement. JAMA, 308(24), 2594–2604.
• Schumi, J., & Wittes, J. T. (2011). Through the looking glass: Understanding non-inferiority. Trials, 12, Article 106.
• Snapinn, S. M. (2000). Noninferiority trials. Current Controlled Trials in Cardiovascular Medicine, 1(1), 19–21.

Non-Parametric Statistics in Medicine:

A Practical Guide for Skewed, Ordinal, and Small Clinical Datasets

Last updated: May 2026

Author: Michele D. Pierri

Reading time: 15–20 minutes

Introduction

Clinical data rarely behave like the textbook examples we encounter in introductory statistics courses. Length of stay tends to be right-skewed. Biomarkers such as C-reactive protein, ferritin, D-dimer, troponin, or NT-proBNP frequently show extreme values, sometimes by an order of magnitude. Pain scores and functional classes are bounded, discrete, partly ordinal. And pilot studies, which most of us run at some point in our careers, often involve small samples where a single outlier can drag the mean in a misleading direction.

For these reasons, non-parametric statistics should not be viewed as a secondary or somehow inferior form of analysis. In many clinical scenarios, they are simply the most coherent statistical choice available.

The key idea is straightforward:

Non-parametric tests are useful when the clinical variable is ordinal, markedly skewed, affected by outliers, or when the assumptions required by classical parametric tests are not clinically or statistically convincing.

That said, non-parametric statistics should not be reduced to the simplistic rule:

“If the Shapiro-Wilk test is significant, use a non-parametric test.”

This rule is incomplete and, in our clinical experience, sometimes misleading. The real question is not only whether the data are normally distributed. The real question is whether the statistical method matches the clinical structure of the data, the study design, and the interpretation we want to make.

In this article we will walk through four simulated clinical scenarios:

1. Pain before and after treatment: Wilcoxon signed-rank test
2. Length of stay in two independent groups: Mann-Whitney U test
3. Biomarker levels across three clinical groups: Kruskal-Wallis test
4. Repeated ordinal clinical scores over time: Friedman test

The examples are intentionally simple. Their purpose is not to replace a full statistical analysis plan but to show how non-parametric thinking works in realistic medical situations.

Quick cheat sheet (clinical workflow)

• Summarize skewed continuous variables as median (IQR); ordinal variables as median (IQR) or category counts when appropriate.
• Match design to test: paired → Wilcoxon; 2 independent groups → Mann-Whitney; >2 independent groups → Kruskal-Wallis; repeated measures (ordinal/ranked) → Friedman.
• Report beyond p-values: an effect size with confidence interval (or at least an effect size), and a clinically interpretable difference (e.g., median change).
• Plan post-hoc: if Kruskal-Wallis or Friedman is significant, use corrected pairwise comparisons (e.g., Holm or Benjamini-Hochberg).

1. What Does “Non-Parametric” Mean?

Parametric tests, such as the t-test or ANOVA, usually rely on assumptions about the distribution of the data or the distribution of model residuals. They typically focus on means and standard deviations.

Non-parametric tests, by contrast, require fewer distributional assumptions. Many of them work by replacing raw values with their ranks.

Consider, instead of comparing the exact length of stay values:

5, 6, 7, 8, 35

5, 6, 7, 8, 35

A rank-based method asks where each patient stands relative to the others. This simple reframing makes the method far less sensitive to extreme values.

It does not mean, however, that non-parametric tests have no assumptions at all. They still require appropriate study design, independent observations when required, and correct pairing when the data are paired. They also answer specific statistical questions that are not always identical to their parametric counterparts.

One particularly important point deserves emphasis:

The Mann-Whitney U test is often described as a test comparing medians, but this is strictly valid only under specific distributional conditions. More generally, it tests whether values from one group tend to be larger or smaller than values from another group.

This distinction matters a great deal in clinical writing, even if it is routinely glossed over.

Scenario 1: Pain Before and After Treatment

Clinical question

A group of patients with chronic low back pain receives a new analgesic treatment. Pain is measured using a Visual Analogue Scale (VAS) from 0 to 10, both before and after treatment.

The question is:

Did pain decrease after treatment?

Why a non-parametric test may be appropriate

VAS scores are bounded between 0 and 10. They are frequently not normally distributed. Anyone who has collected VAS data in clinic knows the typical clustering at specific values (0, 5, 8, or 10). The data are also paired: each patient has a before-treatment and after-treatment value.

The appropriate non-parametric test is the Wilcoxon signed-rank test.

It is the non-parametric analogue of the paired t-test, although the interpretation is not exactly the same. It evaluates whether paired differences are symmetrically distributed around zero.

Practical assumptions to check

• Correct pairing (each “before” matches the same patient “after”).
• Differences are roughly symmetric around a central value (often 0). If differences are extremely skewed, consider alternative approaches such as the sign test or a model-based strategy.

Basic Python example

import numpy as np
import pandas as pd
from scipy.stats import wilcoxon

np.random.seed(42)

n = 40

pain_before = np.clip(np.random.normal(loc=7.0, scale=1.4, size=n), 0, 10)
improvement = np.random.gamma(shape=2.0, scale=0.8, size=n)
pain_after = np.clip(pain_before - improvement, 0, 10)

data_pain = pd.DataFrame({
    "patient_id": range(1, n + 1),
    "pain_before": pain_before,
    "pain_after": pain_after
})

stat, p_value = wilcoxon(data_pain["pain_before"], data_pain["pain_after"])

print(data_pain.head())
print(f"Wilcoxon statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

import numpy as np
import pandas as pd
from scipy.stats import wilcoxon

np.random.seed(42)

n = 40

pain_before = np.clip(np.random.normal(loc=7.0, scale=1.4, size=n), 0, 10)
improvement = np.random.gamma(shape=2.0, scale=0.8, size=n)
pain_after = np.clip(pain_before - improvement, 0, 10)

data_pain = pd.DataFrame({
    "patient_id": range(1, n + 1),
    "pain_before": pain_before,
    "pain_after": pain_after
})

stat, p_value = wilcoxon(data_pain["pain_before"], data_pain["pain_after"])

print(data_pain.head())
print(f"Wilcoxon statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

Interpretation

If the p-value falls below the chosen significance threshold, we conclude that the post-treatment pain scores differ from the pre-treatment scores in a statistically meaningful way.

Statistical significance, however, is not enough. A reduction of 0.3 points on a VAS scale may be statistically significant in a large sample yet clinically trivial. For this reason the analysis should always report the median change together with its interquartile range.

A reasonable reporting sentence might read:

Pain decreased after treatment, with median VAS changing from 7.1 before treatment to 5.3 after treatment. The reduction was statistically significant according to the Wilcoxon signed-rank test.

Scenario 2: Length of Stay in Two Independent Groups

Clinical question

A cardiac surgery team wants to compare postoperative length of stay between patients with preoperative anemia and patients without preoperative anemia.

The question is:

Is postoperative length of stay different between the two groups?

Why a non-parametric test may be appropriate

Length of stay is rarely normally distributed. Most patients are discharged after a relatively short period. A handful of patients, however, remain hospitalized for considerably longer, often because of complications, frailty, infections, or prolonged rehabilitation needs.

The net result is a right-skewed distribution, sometimes severely so.

In this scenario the two groups are independent. The appropriate non-parametric test is the Mann-Whitney U test.

Practical assumptions to check

• Independence between groups (no repeated measurements of the same patient across groups).
• The variable is at least ordinal or continuous, and comparable across groups.

Basic Python example

import numpy as np
import pandas as pd
from scipy.stats import mannwhitneyu

np.random.seed(42)

n_non_anemic = 55
n_anemic = 50

los_non_anemic = np.random.lognormal(mean=1.8, sigma=0.35, size=n_non_anemic)
los_anemic = np.random.lognormal(mean=2.05, sigma=0.45, size=n_anemic)

data_los = pd.DataFrame({
    "group": ["Non-anemic"] * n_non_anemic + ["Anemic"] * n_anemic,
    "length_of_stay": np.concatenate([los_non_anemic, los_anemic])
})

stat, p_value = mannwhitneyu(
    data_los.loc[data_los["group"] == "Non-anemic", "length_of_stay"],
    data_los.loc[data_los["group"] == "Anemic", "length_of_stay"],
    alternative="two-sided"
)

print(data_los.groupby("group")["length_of_stay"].median())
print(f"Mann-Whitney U statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

import numpy as np
import pandas as pd
from scipy.stats import mannwhitneyu

np.random.seed(42)

n_non_anemic = 55
n_anemic = 50

los_non_anemic = np.random.lognormal(mean=1.8, sigma=0.35, size=n_non_anemic)
los_anemic = np.random.lognormal(mean=2.05, sigma=0.45, size=n_anemic)

data_los = pd.DataFrame({
    "group": ["Non-anemic"] * n_non_anemic + ["Anemic"] * n_anemic,
    "length_of_stay": np.concatenate([los_non_anemic, los_anemic])
})

stat, p_value = mannwhitneyu(
    data_los.loc[data_los["group"] == "Non-anemic", "length_of_stay"],
    data_los.loc[data_los["group"] == "Anemic", "length_of_stay"],
    alternative="two-sided"
)

print(data_los.groupby("group")["length_of_stay"].median())
print(f"Mann-Whitney U statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

Interpretation

The Mann-Whitney U test evaluates whether observations in one group tend to be larger than observations in the other group. In our example, if the anemic group has longer length of stay and the p-value is significant, the result supports the hypothesis that preoperative anemia is associated with prolonged hospitalization. This is, incidentally, consistent with several registry studies on cardiac surgery cohorts.

The result should always be reported with medians and interquartile ranges, not just a p-value.

A suitable reporting sentence:

Length of stay was longer in anemic patients than in non-anemic patients. Data are reported as median and interquartile range, and between-group comparison was performed using the Mann-Whitney U test.

Scenario 3: Biomarker Levels Across Three Clinical Groups

Clinical question

Suppose we want to compare postoperative C-reactive protein levels across three groups of patients:

1. No complication
2. Minor complication
3. Major complication

The question is:

Do postoperative inflammatory marker levels differ across complication severity groups?

Why a non-parametric test may be appropriate

Biomarkers such as CRP are notoriously skewed. Some patients show extremely high values because of infection, systemic inflammation, tissue injury, or postoperative complications.

When we have more than two independent groups, the non-parametric analogue of one-way ANOVA is the Kruskal-Wallis test.

Practical assumptions to check

• Independent groups (each patient contributes to one group only).
• The test detects distributional differences across groups; interpreting it as a comparison of “median differences” is safest only under additional shape assumptions.

Basic Python example

import numpy as np
import pandas as pd
from scipy.stats import kruskal

np.random.seed(42)

n_no = 45
n_minor = 40
n_major = 35

crp_no = np.random.lognormal(mean=3.2, sigma=0.35, size=n_no)
crp_minor = np.random.lognormal(mean=3.6, sigma=0.40, size=n_minor)
crp_major = np.random.lognormal(mean=4.0, sigma=0.45, size=n_major)

data_crp = pd.DataFrame({
    "group": (
        ["No complication"] * n_no +
        ["Minor complication"] * n_minor +
        ["Major complication"] * n_major
    ),
    "crp": np.concatenate([crp_no, crp_minor, crp_major])
})

stat, p_value = kruskal(
    data_crp.loc[data_crp["group"] == "No complication", "crp"],
    data_crp.loc[data_crp["group"] == "Minor complication", "crp"],
    data_crp.loc[data_crp["group"] == "Major complication", "crp"]
)

print(data_crp.groupby("group")["crp"].median())
print(f"Kruskal-Wallis statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

import numpy as np
import pandas as pd
from scipy.stats import kruskal

np.random.seed(42)

n_no = 45
n_minor = 40
n_major = 35

crp_no = np.random.lognormal(mean=3.2, sigma=0.35, size=n_no)
crp_minor = np.random.lognormal(mean=3.6, sigma=0.40, size=n_minor)
crp_major = np.random.lognormal(mean=4.0, sigma=0.45, size=n_major)

data_crp = pd.DataFrame({
    "group": (
        ["No complication"] * n_no +
        ["Minor complication"] * n_minor +
        ["Major complication"] * n_major
    ),
    "crp": np.concatenate([crp_no, crp_minor, crp_major])
})

stat, p_value = kruskal(
    data_crp.loc[data_crp["group"] == "No complication", "crp"],
    data_crp.loc[data_crp["group"] == "Minor complication", "crp"],
    data_crp.loc[data_crp["group"] == "Major complication", "crp"]
)

print(data_crp.groupby("group")["crp"].median())
print(f"Kruskal-Wallis statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

Interpretation

A significant Kruskal-Wallis test tells us that at least one group tends to differ from the others. It does not, on its own, tell us which groups differ.

When the global test is significant, post-hoc pairwise comparisons may be performed, commonly using Mann-Whitney U tests with correction for multiple comparisons. The plan, however, should be prespecified and reported transparently. Adding tests after seeing the data is a recipe for false-positive findings.

A reasonable reporting sentence:

CRP levels differed across complication groups according to the Kruskal-Wallis test. Post-hoc pairwise comparisons were then performed with correction for multiple testing.

Scenario 4: Repeated Ordinal Clinical Scores Over Time

Clinical question

A group of patients is followed after a rehabilitation program. Functional limitation is measured at baseline, 1 month, 3 months, and 6 months using an ordinal score from 1 to 5, where higher values indicate worse functional limitation.

The question is:

Does functional limitation improve over time?

Why a non-parametric test may be appropriate

The score is ordinal. The distance between score 1 and score 2 is not necessarily the same as the distance between score 4 and score 5; in fact, in clinical experience, the jump from 4 to 5 often represents a far more substantial functional deterioration. In addition, the same patients are measured repeatedly over time.

The appropriate non-parametric test in this situation is the Friedman test, the non-parametric analogue of repeated-measures ANOVA for ranked data.

Practical assumptions to check

• Repeated measurements are on the same subjects across time points (complete blocks are ideal).
• The ordinal scale is appropriate for rank-based comparisons. If missingness is substantial, dedicated longitudinal methods may be preferable.

Basic Python example

import numpy as np
import pandas as pd
from scipy.stats import friedmanchisquare

np.random.seed(42)

n = 35

baseline = np.random.choice([3, 4, 5], size=n, p=[0.25, 0.45, 0.30])
month_1 = np.clip(baseline - np.random.choice([0, 1], size=n, p=[0.45, 0.55]), 1, 5)
month_3 = np.clip(month_1 - np.random.choice([0, 1], size=n, p=[0.40, 0.60]), 1, 5)
month_6 = np.clip(month_3 - np.random.choice([0, 1], size=n, p=[0.55, 0.45]), 1, 5)

data_function = pd.DataFrame({
    "patient_id": range(1, n + 1),
    "baseline": baseline,
    "month_1": month_1,
    "month_3": month_3,
    "month_6": month_6
})

stat, p_value = friedmanchisquare(
    data_function["baseline"],
    data_function["month_1"],
    data_function["month_3"],
    data_function["month_6"]
)

print(data_function.head())
print(f"Friedman statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

import numpy as np
import pandas as pd
from scipy.stats import friedmanchisquare

np.random.seed(42)

n = 35

baseline = np.random.choice([3, 4, 5], size=n, p=[0.25, 0.45, 0.30])
month_1 = np.clip(baseline - np.random.choice([0, 1], size=n, p=[0.45, 0.55]), 1, 5)
month_3 = np.clip(month_1 - np.random.choice([0, 1], size=n, p=[0.40, 0.60]), 1, 5)
month_6 = np.clip(month_3 - np.random.choice([0, 1], size=n, p=[0.55, 0.45]), 1, 5)

data_function = pd.DataFrame({
    "patient_id": range(1, n + 1),
    "baseline": baseline,
    "month_1": month_1,
    "month_3": month_3,
    "month_6": month_6
})

stat, p_value = friedmanchisquare(
    data_function["baseline"],
    data_function["month_1"],
    data_function["month_3"],
    data_function["month_6"]
)

print(data_function.head())
print(f"Friedman statistic:{stat:.3f}")
print(f"p-value:{p_value:.4f}")

Interpretation

A significant Friedman test suggests that the repeated measurements are not all drawn from the same distribution. In our example, it would support the presence of a change in functional limitation over time.

As with Kruskal-Wallis, the global test does not identify which time points differ. When needed, post-hoc paired comparisons can be performed with appropriate correction.

A reasonable reporting sentence:

Functional limitation improved over time. The overall change across follow-up visits was significant according to the Friedman test.

Practical Decision Table

Common Mistakes in Medical Papers

1. Using non-parametric tests automatically after a significant normality test

Normality tests can be overly sensitive in large samples and underpowered in small samples. The decision should also rest on histograms, Q-Q plots, clinical plausibility, outliers, and the actual scale of measurement. Mechanical reliance on a single test is rarely a good idea.

2. Saying that Mann-Whitney always compares medians

This is a common oversimplification. When the two distributions have a similar shape, the Mann-Whitney U test can be interpreted as a test of location shift. When the distributions differ in shape or spread, the interpretation is broader: one group tends to have larger or smaller values than the other.

3. Reporting mean and standard deviation for markedly skewed variables

For strongly skewed variables, median and interquartile range are usually more clinically informative.

4. Reporting only p-values

A p-value does not quantify clinical relevance. Whenever possible, the report should include effect sizes, median differences, confidence intervals, and clinically interpretable summaries.

5. Ignoring multiple comparisons

After a significant Kruskal-Wallis or Friedman test, post-hoc comparisons must account for multiplicity. Otherwise, the probability of false-positive findings rises quickly.

6. Confusing statistical significance with clinical importance

A statistically significant reduction in pain score may still be clinically irrelevant when the magnitude is small. Conversely, a clinically relevant difference can easily fail to reach statistical significance in a small pilot study.

How to Report Non-Parametric Analyses

A concise reporting style could be:

Continuous skewed variables were summarized as median and interquartile range. Between-group comparisons were performed using the Mann-Whitney U test for two independent groups and the Kruskal-Wallis test for more than two independent groups. Paired before-after comparisons were performed using the Wilcoxon signed-rank test. Repeated ordinal measurements were analyzed using the Friedman test. A two-sided p-value < 0.05 was considered statistically significant.

For a more complete report, effect sizes and confidence intervals should be added where possible. Practical options include:

• Wilcoxon signed-rank: rank-biserial correlation or an r-type effect size (when available).
• Mann-Whitney: rank-biserial correlation or Cliff’s delta.
• Kruskal-Wallis: epsilon-squared (or a similar rank-based η²) to quantify the global effect.
• Friedman: Kendall’s W as an overall effect size.

Conclusion

Non-parametric statistics are particularly useful in medicine, where clinical data are routinely skewed, ordinal, bounded, or affected by outliers. These methods are not a fallback for weak data analysis. In many cases they represent the most appropriate way to analyze real-world clinical variables.

The practical message is straightforward:

Choose the test according to the clinical question, the study design, and the structure of the variable, not only according to a mechanical normality test.

In clinical research the goal is not to deploy the most sophisticated method available. The goal is to use a method that respects the data and produces an interpretation that is clinically meaningful.

Suggested References

Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods. Wiley.

Altman DG. Practical Statistics for Medical Research. Chapman & Hall/CRC.

Bland M. An Introduction to Medical Statistics. Oxford University Press.

Conover WJ. Practical Nonparametric Statistics. Wiley.

See also on this site: Nonparametric statistics