When should I use Spearman instead of Pearson correlation?

Use Spearman correlation when: (1) one or both variables are not normally distributed (confirmed by Shapiro-Wilk p < 0.05 for n < 50, or histogram/Q-Q plot inspection for larger samples); (2) the relationship between variables appears monotonic but not linear on a scatter plot; (3) one or both variables are measured on an ordinal scale (e.g., Likert scores, VAS pain ratings, disease severity grading); (4) the data contain meaningful outliers that cannot be removed. Use Pearson when both variables are continuous, approximately normally distributed, and show a linear relationship on the scatter plot with no influential outliers.

How do I test for normality before choosing a correlation test?

The most common formal test is the Shapiro-Wilk test: if p 200), where trivial non-normality triggers a significant result. For large samples, rely more on visual inspection: a histogram should be approximately bell-shaped, a Q-Q plot should show points lying close to the diagonal line, and a box plot should show no extreme outliers. In practice: for n 200 use visual inspection primarily. When in doubt, report both Pearson and Spearman and note the similarity or difference.

Pearson vs Spearman Correlation in Medical Research: Complete Guide

Q: What is the difference between Pearson and Spearman correlation?

Pearson correlation (r) measures the strength and direction of the linear relationship between two continuous, normally distributed variables. It is a parametric test and is sensitive to outliers. Spearman correlation (ρ or rs) is a non-parametric test that converts raw values to ranks and then computes Pearson's formula on those ranks. It measures monotonic relationships (consistently increasing or decreasing, but not necessarily linear) and is robust to outliers and non-normal distributions. Pearson is more powerful when its assumptions are met; Spearman is more appropriate when normality is violated, data are ordinal, or the relationship is monotonic but not linear.

Q: How do I interpret Pearson r and Spearman ρ values?

Both r (Pearson) and ρ (Spearman) range from −1 to +1. The magnitude indicates strength and the sign indicates direction. Conventional thresholds (Cohen 1988; Evans 1996): |r| or |ρ| < 0.20 = negligible; 0.20–0.39 = weak; 0.40–0.59 = moderate; 0.60–0.79 = strong; ≥ 0.80 = very strong. For Pearson r only, squaring the coefficient gives the coefficient of determination (r²) — the proportion of variance in one variable explained by the other. For example, r = 0.58 means r² = 0.34, so 34% of the variance in the outcome is explained by the predictor. The p-value indicates whether the correlation is statistically distinguishable from zero, not whether it is clinically meaningful.

Q: Does a significant correlation prove causation?

No. Correlation measures association, not causation. A statistically significant correlation coefficient (p < 0.05) only indicates that the relationship is unlikely to be zero in the population — it does not establish that one variable causes the other. Both variables may be caused by an unmeasured third variable (confounding), the relationship may be coincidental, or reverse causation may apply. Establishing causation requires experimental designs (ideally RCTs), temporal ordering evidence, biological plausibility, and ruling out confounding. The statement 'A is correlated with B' should never be paraphrased as 'A causes B' in a thesis or manuscript.

Q: How do I test for normality before choosing a correlation test?

The most common formal test is the Shapiro-Wilk test: if p 200), where trivial non-normality triggers a significant result. For large samples, rely more on visual inspection: a histogram should be approximately bell-shaped, a Q-Q plot should show points lying close to the diagonal line, and a box plot should show no extreme outliers. In practice: for n 200 use visual inspection primarily. When in doubt, report both Pearson and Spearman and note the similarity or difference.

Q: Can I compare Pearson r values from different studies?

Yes, but cautiously. Raw r values can be compared across studies using Fisher's z-transformation, which converts r to a normally distributed z-score that is suitable for meta-analytic pooling and significance testing of the difference between two independent correlations. The formula is: z = 0.5 × ln[(1+r)/(1−r)]. This transformed z is used to construct confidence intervals around r and to formally test whether two correlations differ significantly. When comparing correlations across studies, also consider differences in sample characteristics (age range, disease severity), variable measurement methods (different pain scales, different glucose assays), and restriction of range (a study restricted to severe cases will yield a lower r than one spanning the full severity spectrum).

Q: What is the coefficient of determination and why does it matter?

The coefficient of determination (r²) is Pearson r squared and represents the proportion of variance in one variable explained by the other. It transforms r from an index of direction and strength into an effect-size measure with direct interpretive meaning. Example: r = 0.44 → r² = 0.19, meaning 19% of the variance in HbA1c is explained by BMI, and 81% is explained by other factors. r² applies only to Pearson r; squaring Spearman ρ does not have the same variance-explained interpretation. Reporting r² alongside r is required by APA Publication Manual and recommended by most statistical reporting guidelines to prevent overinterpretation of moderate correlation coefficients.

Q: Is Spearman correlation less powerful than Pearson?

When both variables are perfectly normally distributed and the relationship is exactly linear, Pearson r has slightly higher statistical power than Spearman ρ — the asymptotic relative efficiency (ARE) of Spearman vs Pearson under ideal normal conditions is approximately 91%, meaning Spearman needs about 10% more subjects to achieve equivalent power. However, when the normality assumption is violated (which is common in real clinical data), Spearman ρ can be more powerful than Pearson r, because Pearson's standard error is inflated by non-normality. In practice, the power difference is rarely the deciding factor; assumption compliance and data scale (continuous vs ordinal) are the primary criteria for test selection.

Q: Can I use Pearson correlation with Likert scale data?

This is actively debated. Strict statistical convention holds that Likert scale items (e.g., 1–5 or 1–7 ordered categories) are ordinal, not interval, and therefore Spearman correlation is technically more appropriate for individual Likert items. However, many researchers and methodologists argue that summed Likert scales (composite scores across multiple items) can be treated as approximately continuous and analysed with Pearson correlation when: the composite has many items (≥ 5–7), the distribution of composite scores is approximately normal, and the scale has demonstrated metric properties. In practice: use Spearman for individual Likert items; consider Pearson acceptable for normally distributed composite scores with ≥ 7 items; and always check and report the distribution of both variables.

Q: How should I report correlation results in a thesis or paper?

APA 7th Edition format for Pearson: r(df) = [value], p = [value], 95% CI [lower, upper]. For Spearman: rs(df) = [value], p = [value]. Always report: the specific test used (Pearson or Spearman) and its justification; the test statistic (r or ρ); degrees of freedom (n − 2); p-value; and for Pearson, the 95% confidence interval (calculated via Fisher's z-transformation) and r². For Spearman, report the 95% CI for ρ using bootstrapping if software supports it. Example: 'Age was significantly positively correlated with systolic blood pressure (r(282) = 0.58, p < 0.001, 95% CI [0.49, 0.66], r² = 0.34), indicating that age accounted for 34% of the variance in blood pressure in this cohort.'

A PhD candidate submits a thesis reporting r = 0.68 (p < 0.001) between pain scores and analgesic dosing, using Pearson correlation. The examiner flags it immediately: pain was measured on a 0–10 visual analogue scale with a right-skewed distribution and three extreme outliers. Pearson’s result is biased — the outliers inflated both the correlation coefficient and its significance. The correct method, Spearman’s rank correlation, yields ρ = 0.51 (p < 0.001) — still significant, but meaningfully weaker, and more defensible. This kind of mistake is extraordinarily common in published medical literature. The decision between Pearson and Spearman is not a detail — it is one of the most frequent methodological choices in clinical research, and getting it wrong distorts effect size estimates and undermines the validity of reported associations.

Pearson Correlation: The Parametric Standard

The Pearson product-moment correlation coefficient (r) quantifies the strength and direction of the linear relationship between two continuous variables. It was developed by Karl Pearson in 1895 and remains the most widely used measure of bivariate association in medical research. The coefficient ranges from −1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive linear relationship).

Pearson Correlation Formula

r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √[Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²]

Where x̄ and ȳ are the means of variables X and Y, and n is the sample size.
Degrees of freedom: df = n − 2 | Significance tested with t = r√(n−2) / √(1−r²)

Pearson’s Assumptions

Pearson r is a parametric test and its validity rests on five assumptions that must all be checked before use:

Continuous data: Both variables must be measured on a continuous (interval or ratio) scale. Discrete ordinal data (Likert items, pain grades) violate this assumption.
Bivariate normality: Both variables should be approximately normally distributed. The test is moderately robust to mild non-normality for large samples (n > 100) but sensitive to severe skewness and outliers.
Linearity: The relationship between the two variables should follow a straight line, not a curve. A scatter plot is the primary diagnostic tool — a curvilinear relationship produces a Pearson r that underestimates the true association strength.
Homoscedasticity: The variance of Y should be approximately constant across all levels of X (equal spread of points around the fitted line). Funnel-shaped scatter plots indicate heteroscedasticity.
No influential outliers: Pearson r is highly sensitive to single extreme values. One outlier pair can dramatically inflate or deflate the correlation coefficient, producing a coefficient that does not represent the general trend in the data.

The outlier problem: Because Pearson r is calculated from raw values, a single pair of extreme observations (e.g., one patient with extremely high BMI and very high HbA1c) can shift the correlation coefficient by 0.10–0.20 in either direction. Always inspect a scatter plot before reporting r. If you see outliers that cannot be clinically justified for exclusion, Spearman’s ρ is more appropriate because rank transformation dramatically reduces outlier influence.

Spearman Rank Correlation: The Non-Parametric Alternative

The Spearman rank-order correlation coefficient (ρ, “rho”, or sometimes written r ₛ) is a non-parametric measure of the monotonic relationship between two variables. Proposed by Charles Spearman in 1904, it works by converting each variable’s raw values to ranks and then computing Pearson’s formula on those ranks. Because it operates on ranks rather than raw values, it is robust to outliers, non-normal distributions, and ordinal data.

A monotonic relationship is one where the variables consistently move in the same direction (or consistently opposite directions) — but not necessarily at a constant rate. A linear relationship is always monotonic, but a monotonic relationship is not always linear. This means Spearman detects a wider class of associations than Pearson, at the cost of slightly reduced power when the relationship is truly linear and normality holds.

Spearman Correlation Formula (No Tied Ranks)

ρ = 1 − (6 · Σdᵢ²) / (n(n² − 1))

Where dᵢ = difference in ranks between each observation pair, and n = sample size.
When tied ranks exist, apply the standard Pearson formula to the rank-transformed data.
Significance tested with t = ρ√(n−2) / √(1−ρ²) for n ≥ 10

When Spearman Is the Right Choice

One or both variables fail normality testing (Shapiro-Wilk p < 0.05 for small samples, or clear skew on histogram/Q-Q plot)
Data are ordinal (Likert scales, pain scores, disease severity grades, NYHA class)
The scatter plot shows a monotonic but non-linear relationship (e.g., logarithmic, exponential curve)
The data contain meaningful outliers that are clinically real and cannot be excluded
Small sample sizes (n < 30) where normality cannot be reliably assessed or assumed
Variables with natural floor or ceiling effects (e.g., a questionnaire where most patients score near the maximum)

Pearson vs Spearman: Side-by-Side Comparison

Feature	Pearson (r)	Spearman (ρ)
Type	Parametric	Non-parametric
Data scale	Continuous (interval/ratio)	Continuous or ordinal
Normality required	Yes (bivariate normality)	No
Relationship type	Linear only	Monotonic (linear or curved)
Outlier sensitivity	High — outliers distort r	Low — ranks dampen outlier influence
Operates on	Raw values	Ranked values
Statistical power	Higher when assumptions met	~91% ARE vs Pearson under normality; superior when non-normal
Coefficient of determination	r² = % variance explained	ρ² has no variance-explained interpretation
Confidence interval	Via Fisher’s z-transformation	Via bootstrapping
Likert scale items	Not appropriate (ordinal)	Appropriate
Small samples (n < 30)	Caution — normality unverifiable	Preferred
APA reporting symbol	r	r ₛ or ρ

Checking Assumptions Before You Analyse

The assumption-checking workflow should be completed and documented before running any correlation analysis. Skipping this step and choosing a test by habit or convention is the most common source of correlation errors in published medical research.

Step 1 — Inspect the Scatter Plot

Always start here, before any formal test. A scatter plot of X vs Y reveals: the direction of the relationship (positive or negative); whether the relationship is linear or curved (use Pearson for linear, Spearman for curved monotonic); whether extreme outliers are present; whether the spread of Y values changes systematically across X (heteroscedasticity). A scatter plot takes 30 seconds to produce and prevents a large category of analytical errors.

Step 2 — Test Normality

The Shapiro-Wilk test is the gold standard for samples up to n ≈ 2000. Apply it separately to each variable:

Sample Size	Recommended Approach	Decision Rule
n < 30	Shapiro-Wilk + histogram	If S-W p < 0.05 → Spearman
n 30–50	Shapiro-Wilk + Q-Q plot	If S-W p < 0.05 or clear Q-Q deviation → Spearman
n 50–200	Shapiro-Wilk + histogram + Q-Q	S-W plus visual — both must support normality for Pearson
n > 200	Histogram + Q-Q plot primarily	S-W nearly always significant; rely on visual inspection
Any n, ordinal data	—	Always Spearman regardless of distribution

Step 3 — Identify Outliers

Compute standardised scores (z-scores) for both variables. Any observation with |z| > 3.3 (p < 0.001 under normality) is a potential outlier. Inspect these cases: are they data entry errors (correct or exclude), genuine extreme clinical cases (retain and use Spearman), or measurement artefacts (exclude with documentation)? Never silently remove outliers without documentation and sensitivity analysis.

Decision Flowchart: Which Correlation Test?

Inspect a scatter plot of X vs Y

Does the relationship appear linear (roughly straight)? Does it appear curved (logarithmic, exponential)?

Curved monotonic relationship → Spearman Linear-looking → proceed to Step 2

Check the measurement scale of both variables

Are both variables measured on a continuous (interval or ratio) scale? Or is one measured on an ordinal scale (Likert, ranked grades, VAS treated as ordinal)?

Any ordinal variable → Spearman Both continuous → proceed to Step 3

Test normality of both variables

Run Shapiro-Wilk (for n < 200) and inspect histograms and Q-Q plots. Both variables must be approximately normally distributed.

Either variable non-normal → Spearman Both approximately normal → proceed to Step 4

Check for influential outliers

Are there pairs of extreme observations (|z| > 3.3) that are clinically real and cannot be excluded?

Clinically real outliers present → Spearman No influential outliers → proceed to Step 5

All assumptions satisfied

Both variables continuous, normally distributed, linearly related, no influential outliers.

Use Pearson r — report r, df, p, 95% CI, and r²

Linear vs Monotonic: What the Scatter Plot Reveals

Scatter Plot Patterns: Pearson vs Spearman Applicability

The middle panel illustrates why Spearman is more powerful for curved monotonic relationships: the true association is stronger (ρ = 0.96) than Pearson suggests (r = 0.82) because Pearson penalises the deviation from linearity. The right panel shows how a single outlier inflates Pearson r from the true value of ~0.31 to a misleading 0.72, while Spearman ρ correctly reflects the association in the main data cluster.

Interpreting Correlation Coefficients

Strength Thresholds

Both r (Pearson) and ρ (Spearman) use the same conventional interpretation thresholds. The most widely cited system in medical research combines Cohen (1988) and Evans (1996):

Correlation Strength: Conventional Thresholds

Very Strong

|r| ≥ 0.80

≥ 0.80

Strong

0.60 – 0.79

0.60–0.79

Moderate

0.40 – 0.59

0.40–0.59

Weak

0.20 – 0.39

0.20–0.39

Negligible

< 0.20

Domain matters: Interpretation thresholds are context-dependent. In clinical physiology, r = 0.50 between two indirect measures of the same construct (e.g., estimated vs measured GFR) may be disappointingly weak. In epidemiology, r = 0.30 between a dietary exposure and a distal health outcome may represent a substantial and clinically important effect. Always interpret correlation coefficients in the context of the research question, the measurement precision of both variables, and what prior literature reports for similar associations.

The Coefficient of Determination (r²)

For Pearson r only, squaring the coefficient gives the coefficient of determination — the proportion of variance in one variable explained by the other. This is a far more informative metric than r alone, and APA guidelines require its reporting:

r = 0.44 → r² = 0.19 → BMI explains 19% of HbA1c variance
r = 0.70 → r² = 0.49 → 49% of variance explained (51% from other sources)

The unexplained variance (1 − r²) is typically more important for clinical understanding: even a “strong” correlation of 0.70 leaves 51% of the outcome variance unexplained. Reporting r without r² invites overinterpretation of moderate correlations.

Confidence Intervals via Fisher’s z-Transformation

A p-value tells you whether the correlation differs from zero; a 95% confidence interval tells you the plausible range of the true population correlation. For Pearson r, the CI is calculated using Fisher’s z-transformation, because r is not normally distributed and CIs cannot be computed directly:

Fisher’s z-Transformation for Pearson r 95% CI

z = 0.5 × ln[(1 + r) / (1 − r)]
SEᵢ = 1 / √(n − 3)
95% CI for z: z ± 1.96 × SEᵢ
Back-transform: r = (e²ᵜ − 1) / (e²ᵜ + 1)

Example: r = 0.58, n = 284 → z = 0.662, SE = 0.0596 → z CI [0.545, 0.779] → r CI [0.499, 0.652]

Clinical Examples

Age and Systolic Blood Pressure in a Hypertension Cohort

A cross-sectional study in a cardiology outpatient clinic examines the association between age and resting systolic blood pressure (SBP) in 284 patients with essential hypertension, all off antihypertensive medication for 48 hours. The research question: does age independently predict blood pressure level in this cohort?

284

Patients (n)

58.4

Mean age (years, SD 11.2)

152.3

Mean SBP (mmHg, SD 18.6)

Pearson

Correct test

Step 1 — Normality Testing Age: Shapiro-Wilk W = 0.982, p = 0.312 → approximately normal ✓ SBP: Shapiro-Wilk W = 0.976, p = 0.184 → approximately normal ✓ Histograms: both approximately bell-shaped; Q-Q plots: points close to diagonal ✓ Step 2 — Scatter Plot Inspection Relationship: linear, positive gradient ✓ No apparent curvature; no extreme outlier pairs (all |z| < 2.8) ✓ Homoscedasticity: roughly constant spread across age range ✓ Step 3 — Pearson Correlation r(282) = 0.58 t = 0.58 × √(282) / √(1 − 0.58²) = 11.90 p < 0.001 Step 4 — Coefficient of Determination r² = 0.58² = 0.336 → age explains 33.6% of SBP variance Step 5 — 95% CI via Fisher's z z = 0.5 × ln(1.58/0.42) = 0.662 SE_z = 1/√(284−3) = 0.0596 z CI: [0.545, 0.779] → r CI: [0.499, 0.652]

r(282) = 0.58, p < 0.001, 95% CI [0.50, 0.65], r² = 0.34 — moderate-to-strong positive correlation. Age accounts for 34% of the variance in systolic blood pressure in this hypertensive cohort.

Clinical interpretation: A moderate-to-strong correlation confirms that age is an important predictor of blood pressure elevation in hypertensive patients, consistent with known arterial stiffening mechanisms. However, r² = 0.34 also means 66% of SBP variance is not explained by age — underscoring that other factors (sodium intake, kidney function, obesity, medication adherence) must be addressed in comprehensive cardiovascular risk management. The narrow 95% CI (0.50 to 0.65) indicates a precise estimate from this adequately powered sample.

Pain VAS Score and 24-Hour Opioid Dose — Non-Normal Data

A prospective study in a post-surgical ward investigates the relationship between pain intensity (VAS, 0–10) and total opioid analgesic requirement (mg morphine equivalent, MEQ) in the first 24 hours post-operatively, in 68 patients undergoing elective abdominal surgery. The ward pharmacist suspects patients reporting higher pain receive higher doses, but wants to quantify the association.

Patients (n)

6.1

Median VAS (IQR 4–8)

Median 24h MEQ mg (IQR 22–78)

Spearman

Correct test

Step 1 — Normality Testing VAS (0–10): Shapiro-Wilk W = 0.934, p = 0.041 → significant departure ✗ 24h MEQ: Shapiro-Wilk W = 0.881, p = 0.003 → highly non-normal (right-skewed) ✗ Histogram of MEQ: pronounced right skew with 5 extreme high-dose outliers (range 180–320 mg) Step 2 — Decision Both variables fail normality → Spearman correlation required MEQ is also right-skewed (log-normal) with clinically real outliers → Pearson inappropriate Step 3 — Spearman Correlation ρ(66) = 0.61, p < 0.001 (via rank transformation then Pearson formula on ranks) Comparison: Pearson r on Same Data (Incorrect) r(66) = 0.72, p < 0.001 Difference: outliers pulled r up by 0.11 — a 18% inflation in r, biasing toward stronger association Step 4 — 95% CI for ρ (Bootstrap) ρ 95% CI [0.44, 0.74] (10,000 bootstrap iterations, BCa method)

ρ(66) = 0.61, p < 0.001, 95% CI [0.44, 0.74] — strong positive association between post-operative pain intensity and opioid consumption. Using Pearson r on the same data (r = 0.72) overestimated the association by 18% due to outlier influence.

Why this matters: Had the researcher used Pearson r without checking assumptions, they would have reported r = 0.72 (“strong” correlation). Spearman ρ = 0.61 is still clinically meaningful and statistically significant — but it accurately represents the association in the full sample, including the clinically real high-opioid patients (likely those with higher body weight, comorbid chronic pain, or prior opioid tolerance) rather than being artificially inflated by them. Always report the test used and state why, e.g., “Spearman’s rank correlation was used because the 24-hour opioid requirement was right-skewed (Shapiro-Wilk p = 0.003) with extreme values consistent with known high-opioid-requirement subgroups.”

BMI and HbA1c in Type 2 Diabetes: Moderate Association

A retrospective audit of 412 patients with established type 2 diabetes attending a diabetes outpatient clinic examines the cross-sectional association between body mass index (BMI, kg/m²) and glycaemic control measured by HbA1c (mmol/mol). The aim is to quantify the strength of the BMI–glycaemia relationship to support a hospital policy on weight management referrals.

412

Patients (n)

31.8

Mean BMI kg/m² (SD 5.4)

58.4

Mean HbA1c mmol/mol (SD 14.2)

Pearson

Correct test (both normal)

Normality (large n — visual inspection primary) BMI: Histogram approximately normal; Q-Q plot close to diagonal ✓ Shapiro-Wilk p = 0.023 (n = 412: S-W underpowered; visual inspection overrides) ✓ HbA1c: Histogram approximately normal; Q-Q plot shows mild right tail ✓ Scatter plot: linear trend, mild heteroscedasticity at high BMI, no extreme outliers ✓ Decision: Pearson appropriate for this large continuous dataset with visually normal distributions Pearson Correlation Result r(410) = 0.44, p < 0.001 Coefficient of Determination r² = 0.44² = 0.194 → BMI explains 19.4% of HbA1c variance → 80.6% of HbA1c variance explained by OTHER factors (diet quality, medication adherence, disease duration, beta-cell function decline, comorbidities) 95% CI (Fisher's z) z = 0.5 × ln(1.44/0.56) = 0.472 SE_z = 1/√(409) = 0.0495 z CI: [0.375, 0.569] → r CI: [0.360, 0.516] Spearman Sensitivity Check ρ(410) = 0.42 (p < 0.001) — consistent with Pearson; results are robust

r(410) = 0.44, p < 0.001, 95% CI [0.36, 0.52], r² = 0.19 — moderate positive correlation. BMI accounts for approximately 19% of the variance in HbA1c in this type 2 diabetes cohort.

Clinical and policy implication: A moderate correlation of r = 0.44 confirms a statistically robust association between obesity and glycaemic control, supporting the policy rationale for weight management referrals. However, r² = 0.19 is the crucial number for the policy document: BMI alone explains only 19% of glycaemic variance. The remaining 81% — dominated by medication type, adherence, diet quality, disease duration, and genetic factors — means that weight reduction alone should be expected to partially, not fully, control HbA1c. The Spearman sensitivity check (ρ = 0.42, very close to r = 0.44) confirms that results are not driven by outliers or distributional assumptions, strengthening the robustness of the reported association.

Thesis Writing Recommendations

Correlation analysis is one of the most frequently tested skills in postgraduate medical research vivas. Examiners expect candidates to demonstrate that they checked assumptions before choosing a test, understand the difference between statistical and clinical significance, and can distinguish correlation from causation.

In Your Methods Section

State: (1) which correlation coefficient was used (Pearson r or Spearman ρ); (2) the justification for that choice (normality test results and scatter plot findings); (3) how normality was assessed (Shapiro-Wilk test, histograms, Q-Q plots); (4) the significance threshold used (conventionally α = 0.05); (5) for Pearson, that CIs were calculated via Fisher’s z-transformation; (6) that scatter plots were inspected before analysis.

Model Methods Paragraph — Pearson Correlation

“The association between age and systolic blood pressure was assessed using Pearson product-moment correlation. Normality of both variables was confirmed by the Shapiro-Wilk test (age: p = 0.312; systolic blood pressure: p = 0.184) and visual inspection of histograms and Q-Q plots. Scatter plot inspection confirmed a linear relationship with no extreme outliers. The Pearson correlation coefficient (r), degrees of freedom, p-value, 95% confidence interval (calculated using Fisher’s z-transformation), and coefficient of determination (r²) were reported. A p-value < 0.05 was considered statistically significant.”

Model Methods Paragraph — Spearman Correlation

“The association between post-operative pain score (VAS, 0–10) and 24-hour opioid analgesic requirement was assessed using Spearman’s rank correlation coefficient (ρ). Spearman’s ρ was selected because the 24-hour opioid requirement was significantly right-skewed (Shapiro-Wilk p = 0.003) with extreme high values and because VAS, being an ordinal measure, does not meet the continuous-variable assumption required for Pearson correlation. The Spearman correlation coefficient, degrees of freedom, and p-value are reported. Ninety-five percent confidence intervals for ρ were calculated using 10,000 bootstrap iterations with bias-corrected and accelerated (BCa) correction.”

In Your Results Section

APA 7th edition formats: Pearson: r(df) = [value], p = [value], 95% CI [lower, upper]. Spearman: r ₛ(df) = [value], p = [value]. Always include r² for Pearson (not Spearman). Include a scatter plot in your Results if the visual pattern is meaningful — especially if the relationship is curvilinear or outliers are informative.

Common Mistakes Researchers Make

Mistake 1: Choosing Pearson Without Checking Normality

The single most common correlation error in published medical research. Pearson r is chosen by default — often because it is the software default — without running Shapiro-Wilk or inspecting the distributions. When one or both variables are skewed, ordinal, or contain outliers, Pearson r is invalid and the reported coefficient is biased.

Fix: Make assumption checking the first step of every correlation analysis. Run Shapiro-Wilk and inspect histograms and a scatter plot before choosing Pearson or Spearman. If in doubt, report Spearman as the primary result and Pearson as a sensitivity check — if they agree, this strengthens your conclusion.

Mistake 2: Interpreting Correlation as Causation

A significant r or ρ with a large coefficient does not establish that X causes Y. Both variables may be driven by an unmeasured confounder; the relationship may be coincidental at the population level; reverse causation may apply. Despite being a foundational statistical principle, this error appears in peer-reviewed papers with alarming frequency: phrases like “higher BMI leads to higher HbA1c (r = 0.44, p < 0.001)” confuse association with causation.

Fix: Use causal language only when your design permits causal inference (RCT, well-designed cohort with confounding adjustment). For cross-sectional correlational studies, use “is associated with,” “is correlated with,” or “is positively related to.” Never write that one variable “causes,” “leads to,” “produces,” or “results in” the other based on correlation data alone.

Mistake 3: Treating Statistical Significance as Clinical Significance

With large samples (n > 300), even trivially small correlations reach statistical significance. r = 0.11 with n = 400 yields p = 0.028 — statistically significant but explaining only 1.2% of variance, which is clinically meaningless. Reporting only “a significant correlation was found” without quantifying the effect size invites misleading interpretation.

Fix: Always report r² (for Pearson) alongside the p-value and interpret its magnitude explicitly. State what percentage of variance is explained. For large samples, apply the conventional strength thresholds and acknowledge when a statistically significant finding represents a clinically negligible association. Consider using the 95% CI width to also convey precision.

Mistake 4: Ignoring the Scatter Plot and Missing Non-Linear Relationships

Both Pearson r and Spearman ρ only detect monotonic relationships. A U-shaped or inverted-U relationship (e.g., the relationship between cortisol and cognitive performance, or the J-shaped alcohol–cardiovascular risk curve) produces a correlation near zero despite a strong, clinically important relationship. Reporting r = 0.04, p = 0.72 as “no association” when the scatter plot shows a clear J-curve is a major analytical error.

Fix: Always inspect the scatter plot before concluding that no association exists. If the scatter plot shows a non-monotonic pattern, report that a non-linear relationship is present and consider polynomial regression, restricted cubic splines, or segmented regression rather than correlation analysis.

Mistake 5: Not Reporting Which Test Was Used or Why

Many published papers report simply “correlation analysis was performed” without specifying whether Pearson or Spearman was used. This is a serious reporting deficiency that prevents readers from evaluating methodological appropriateness and makes results impossible to compare with other studies or include in meta-analyses.

Fix: Always specify the test name (“Pearson product-moment correlation” or “Spearman’s rank correlation”), the test statistic symbol used in reporting (r vs r ₛ or ρ), the justification for the choice (normality results), and the degrees of freedom. Use APA format: r(282) = 0.58, p < 0.001. Vague statements like “a correlation was run” are not acceptable in modern research reporting.

Mistake 6: Applying Pearson to Likert Scale Items

Individual Likert items (e.g., “Rate your pain 1–5”) are ordinal categorical variables — the intervals between categories are not guaranteed to be equal, and the distribution is typically far from normal. Applying Pearson correlation to individual Likert items violates both the continuous-data and normality assumptions. Despite this, the practice is common in published medical literature, particularly in nursing, psychology, and patient satisfaction research.

Fix: Use Spearman correlation for individual Likert items. Pearson may be acceptable for summed composite scales with ≥ 7 items that demonstrate a close-to-normal distribution — but this should be verified and explicitly justified, not assumed. Report Shapiro-Wilk p-values for any composite score treated as continuous.

Scientific Reporting Standards

Several major reporting guidelines address correlation analysis in medical research:

APA 7th Edition: Requires reporting of r (or r ₛ), degrees of freedom, exact p-value, and 95% confidence interval. For Pearson, also requires r². Uses the format r(df) = [value], p = [value], 95% CI [lower, upper].
STROBE Statement (Item 12e): For observational studies, requires specification of “statistical methods, including those used to control for confounding” and description of sensitivity analyses. Correlation methods should be explicitly named and justified.
EQUATOR ARRIVE (for animal studies) and CONSORT (for RCTs): Both require full specification of all statistical methods including correlation tests and their assumptions.
ICMJE guidelines: Require reporting of actual p-values rather than “p < 0.05” and recommend inclusion of measures of uncertainty (95% CIs) for all primary correlation findings.
Preferred journal standards: Most high-impact clinical journals (BMJ, JAMA, Lancet) now require scatter plots to accompany reported correlations as a supplement or figure, particularly when the correlation is a primary study outcome.

Practical Guidance

When in Doubt, Report Both

If normality results are borderline or the sample is small, run both Pearson and Spearman. If they agree closely (within ~0.05), report Spearman as the primary result and note that Pearson gave a similar value, strengthening confidence. If they diverge substantially, Spearman is the more trustworthy estimate and you should investigate why they differ (usually outliers or non-linearity).

Report Scatter Plots for Primary Correlations

A scatter plot is irreplaceable — it reveals non-linearity, outliers, heteroscedasticity, clusters, and floor/ceiling effects that no single statistic captures. For any correlation that is a primary study outcome, include the scatter plot as a figure with a fitted regression line and 95% confidence band. A plot immediately makes the report more transparent and more convincing to reviewers.

Always Compute and Report r²

r² converts the correlation coefficient from a directional index into an effect-size measure. r = 0.44 sounds impressive; r² = 0.19 is sobering. Report both in every Pearson analysis. Remember r² does not apply to Spearman ρ — do not square ρ and report it as variance explained; it has no such interpretation without additional assumptions.

Beware Restriction of Range

If your sample covers only part of the variable's natural range (e.g., you study only patients with BMI 25–32 in a clinic that excludes very obese patients), the observed correlation will be artificially attenuated compared to the full-range population. When comparing your r with literature values, note whether your sample is restricted in range — and consider correction formulas (Thorndike correction) if this is important for interpretation.

Use Bootstrapped CIs for Spearman

Unlike Pearson r, there is no exact closed-form confidence interval for Spearman ρ. Some software applies Fisher's z-transformation to ρ, which produces approximate CIs but is known to be inaccurate under non-normality. The preferred method is bootstrapping (BCa method, ≥ 5,000 iterations). In R, the boot package or cor.test() with bootstrapping handles this; SPSS does not offer bootstrap CIs for Spearman natively (use R or the SPSS bootstrap module).

Multiple Correlations: Correct for Multiple Testing

Running a correlation matrix of 10 variables produces 45 pairwise correlations — at α = 0.05, you expect ~2 false positives by chance alone. Apply Bonferroni correction (threshold α/m where m is the number of tests) or the less conservative Benjamini-Hochberg procedure to control the false discovery rate. Report the number of comparisons made and the correction applied. A table of 45 “significant” correlations without multiple comparison correction is not publishable in most clinical journals.

Frequently Asked Questions

What is the difference between Pearson and Spearman correlation? +

Pearson r is a parametric test measuring the linear association between two continuous, normally distributed variables, operating on raw values. Spearman ρ is a non-parametric test measuring monotonic association between two variables, operating on their ranks rather than raw values. Pearson assumes normality, linearity, and no influential outliers; Spearman requires none of these. Use Pearson when all assumptions are met and the relationship is linear; use Spearman for ordinal data, non-normal distributions, monotonic curved relationships, or data with meaningful outliers.

When should I use Spearman instead of Pearson? +

Use Spearman when: (1) either variable fails the Shapiro-Wilk test for normality (p < 0.05 for n < 200); (2) either variable is measured on an ordinal scale (Likert items, pain grades, disease severity scores); (3) the scatter plot shows a curved monotonic rather than linear relationship; (4) meaningful outliers are present that cannot be clinically justified for exclusion; (5) the sample is small (n < 30) and normality cannot be reliably assessed. When in doubt between the two, Spearman is the safer choice because it loses only ~10% efficiency under ideal normal conditions but is substantially more robust under real-world departures from normality.

How do I interpret Pearson r and Spearman ρ values? +

Both range from −1 to +1. Sign indicates direction (+ = both variables increase together; − = as one increases, the other decreases). Magnitude indicates strength by convention: < 0.20 = negligible; 0.20–0.39 = weak; 0.40–0.59 = moderate; 0.60–0.79 = strong; ≥ 0.80 = very strong. For Pearson r, always compute r² to quantify the proportion of variance explained (e.g., r = 0.58 → r² = 0.34 → 34% of variance explained). Interpret in clinical context — the same r value may be impressive in epidemiology but weak in a method comparison study.

Does a significant correlation prove causation? +

No — never. Correlation, however strong and significant, establishes only association. Both variables may be caused by an unmeasured confounder (spurious correlation), the relationship may be coincidental, or causation may be reversed. In cross-sectional correlational studies, always use associative language: “is associated with,” “is correlated with,” “is related to.” Never write “causes,” “leads to,” or “results in” based on correlation data. Causal inference requires experimental designs (RCT), temporal sequence evidence, and confounding adjustment.

How do I test normality before choosing a correlation test? +

Run the Shapiro-Wilk test on each variable separately (not as a bivariate test): p < 0.05 indicates significant departure from normality → use Spearman. For n < 50, Shapiro-Wilk is the primary tool. For n 50–200, use Shapiro-Wilk combined with histogram and Q-Q plot inspection. For n > 200, Shapiro-Wilk is overly sensitive (trivial non-normality becomes significant) — rely primarily on visual inspection. Also check the scatter plot for the bivariate relationship: a markedly non-normal bivariate distribution (banana-shaped cloud, fan spread, clustered outliers) is an additional indication for Spearman regardless of univariate tests.

Can I compare Pearson r values from different studies? +

Yes, using Fisher’s z-transformation: convert both r values to z-scores (z = 0.5 × ln[(1+r)/(1−r)]), test whether they differ using a z-test, and back-transform. This is the basis for meta-analytic pooling of correlation coefficients. When comparing r values across studies, account for differences in: sample range (restriction of range attenuates r), variable measurement methods (different pain scales yield different r values for the same true relationship), and sample characteristics (age range, disease severity). Studies in restricted samples will yield systematically lower r values than studies spanning the full clinical range.

What is the coefficient of determination and why does it matter? +

r² is Pearson r squared: the proportion of variance in Y explained by X. It converts correlation from a directional index into an effect-size measure with direct practical meaning. r = 0.44 sounds substantial; r² = 0.19 makes it concrete — only 19% of Y’s variance is explained. The remaining 81% comes from other sources. Always report r² alongside r to prevent overinterpretation of moderate correlations. Note: r² applies only to Pearson r. Squaring Spearman ρ produces a number, but it does NOT have the variance-explained interpretation. For Spearman, report only ρ and its CI, not ρ².

Is Spearman correlation less powerful than Pearson? +

Only marginally, and only when all parametric assumptions are perfectly met. The asymptotic relative efficiency (ARE) of Spearman vs Pearson under ideal bivariate normality is ~91%, meaning Spearman needs roughly 10% more subjects to achieve equivalent power. In practice, real clinical data rarely meet the strict normality and linearity assumptions perfectly, so Spearman frequently provides equal or greater power than Pearson in real-world datasets. The small power disadvantage of Spearman is almost never the deciding factor — assumption compliance, data scale, and presence of outliers should drive the choice.

Can I use Pearson correlation with Likert scale data? +

For individual Likert items (e.g., a single 1–5 response): No — Spearman is the appropriate choice because Likert items are ordinal and their distributions are not normal. For summed composite scales (e.g., a 20-item anxiety questionnaire summed to a total score): Pearson may be acceptable if the composite score has ≥ 7 items, is approximately normally distributed (verify with Shapiro-Wilk or histogram), and the scale has established metric properties. In this case, always check and report the distribution of the composite and justify the Pearson choice explicitly in the methods section. When uncertain, Spearman is the safe default.

How should I report correlation results in a thesis or paper? +

APA 7th edition format: Pearson: r(df) = [value], p = [value], 95% CI [lower, upper]. Spearman: r ₛ(df) = [value], p = [value]. Always include: the test name and justification; df (= n − 2); exact p-value; 95% CI; and r² for Pearson. Example: “Age was significantly positively correlated with systolic blood pressure (r(282) = 0.58, p < 0.001, 95% CI [0.50, 0.65], r² = 0.34).” For Spearman: “Post-operative pain score was significantly positively correlated with opioid consumption (r ₛ(66) = 0.61, p < 0.001, 95% CI [0.44, 0.74]).” Include a scatter plot for any correlation that is a primary outcome.

Calculate Pearson and Spearman Correlation Online

Run assumption checks, compute r and ρ with exact p-values, Fisher’s z confidence intervals, and r² automatically for your dataset.

Open StatClinic →

Pearson vs Spearman Correlation in Medical Research:The Complete Researcher's Guide

Pearson Correlation: The Parametric Standard

Pearson’s Assumptions

Spearman Rank Correlation: The Non-Parametric Alternative

When Spearman Is the Right Choice

Pearson vs Spearman: Side-by-Side Comparison

Checking Assumptions Before You Analyse

Step 1 — Inspect the Scatter Plot

Step 2 — Test Normality

Step 3 — Identify Outliers

Decision Flowchart: Which Correlation Test?

Inspect a scatter plot of X vs Y

Check the measurement scale of both variables

Test normality of both variables

Check for influential outliers

All assumptions satisfied

Linear vs Monotonic: What the Scatter Plot Reveals

Scatter Plot Patterns: Pearson vs Spearman Applicability

Interpreting Correlation Coefficients

Strength Thresholds

Correlation Strength: Conventional Thresholds

The Coefficient of Determination (r²)

Confidence Intervals via Fisher’s z-Transformation

Clinical Examples

Thesis Writing Recommendations

In Your Methods Section

In Your Results Section

Common Mistakes Researchers Make

Mistake 1: Choosing Pearson Without Checking Normality

Mistake 2: Interpreting Correlation as Causation

Mistake 3: Treating Statistical Significance as Clinical Significance

Mistake 4: Ignoring the Scatter Plot and Missing Non-Linear Relationships

Mistake 5: Not Reporting Which Test Was Used or Why

Mistake 6: Applying Pearson to Likert Scale Items

Scientific Reporting Standards

Practical Guidance

When in Doubt, Report Both

Report Scatter Plots for Primary Correlations

Always Compute and Report r²

Beware Restriction of Range

Use Bootstrapped CIs for Spearman

Multiple Correlations: Correct for Multiple Testing

Frequently Asked Questions

Calculate Pearson and Spearman Correlation Online

Pearson vs Spearman Correlation in Medical Research:
The Complete Researcher's Guide