In statistics, a monotonic relationship between two variables is one where, as one variable increases, the other consistently moves in a single direction — either always increasing (a monotonic increasing relationship) or always decreasing (a monotonic decreasing relationship) — without requiring that it does so at a constant rate. This is a broader, weaker condition than a linear relationship, and the distinction is the reason two of the most commonly reported statistics in research — Pearson’s correlation coefficient and Spearman’s rank correlation — can give very different answers about the same data.
This page covers the statistical sense of monotonicity: a property of the relationship between two variables in a dataset. It is closely related to, but distinct from, the pure-mathematics definition of a monotonic function (a function whose output never decreases, or never increases, as its input increases) — the statistical usage is the applied version of that same idea, evaluated on real, noisy, sampled data rather than an exact mathematical function.
Monotonic vs. linear: the distinction that matters
Every linear relationship is monotonic. If y = 3x + 2, y increases at a constant rate as x increases, and that constant increase is, trivially, also a consistent one-directional increase. But the reverse does not hold: most monotonic relationships are not linear.
Consider a relationship where y rises steeply for small values of x, then continues to rise but more slowly as x gets large — a decelerating curve, common in dose-response and learning-curve data. That relationship is perfectly monotonic (y never decreases as x increases) but it is not a straight line, and describing its strength with a statistic built for straight-line association will understate it.
This is precisely the gap between the two correlation coefficients researchers reach for most often:
- Pearson’s r measures the strength of a linear association. It assumes a constant rate of change and is calculated from the raw values.
- Spearman’s rho and Kendall’s tau measure the strength of a monotonic association. Both work by converting the raw values to ranks (or, for Kendall’s tau, comparing pairs of observations) before computing the statistic, which means they capture any consistent directional trend, curved or straight.
For a full treatment of how Pearson’s r, Spearman’s rho, and Kendall’s tau are calculated, interpreted, and reported, see the correlation coefficient guide. This page focuses on the underlying assumption — monotonicity itself — that determines when each is appropriate.
Illustrative example: a curve that Pearson misses
The following is an illustrative composite, not a reported dataset, constructed only to show the mechanism.
Suppose a study relates hours of practice to a skill score, and the true underlying pattern is a decelerating, concave curve: scores rise quickly at first and then level off as practice hours grow, but they never fall. Because the relationship curves rather than running in a straight line, Pearson’s r on this pattern typically comes out somewhere in the moderate range — clearly nonzero, but well short of reflecting how tight and consistent the relationship actually is. Spearman’s rho on the same ranked data comes out close to 1, because every additional hour of practice is associated with an equal or higher score, with no exceptions — exactly what a rank-based, monotonicity-sensitive statistic is built to detect. The two numbers describe the same relationship; they simply answer different questions (‘how linear is it’ versus ‘how consistently one-directional is it’), and only one of those questions matches what the data actually shows.
Strictly vs. weakly monotonic
A relationship is strictly monotonic if the direction never reverses and never flattens — each increase in x produces a genuine increase (or genuine decrease) in y, with no ties or plateaus. It is weakly (non-strictly) monotonic if the relationship is allowed to plateau — y can stay the same for a stretch as x increases, as long as it never reverses direction.
This distinction matters in practice because real data routinely contains ties: identical scores on a bounded scale, repeated measurements, or a variable that genuinely plateaus (a dose-response curve that reaches a ceiling effect, for example). Rank-based statistics handle ties through defined tie-correction procedures (Spearman’s rho typically uses average, or ‘mid’, ranks; Kendall’s tau has a tau-b variant designed specifically for tied data), so weak monotonicity with a reasonable number of ties does not break the method the way it would break a strict mathematical definition.
Non-monotonic relationships: where both Pearson and Spearman fail
Not every real relationship is monotonic in either direction, and this is the case that causes the most reporting errors. A U-shaped relationship (y is high at low x, falls, then rises again at high x) or an inverted-U relationship (y rises, peaks, then falls) is neither consistently increasing nor consistently decreasing — it is non-monotonic.
Classic examples of non-monotonic relationships in applied research include:
- Dose-response curves with a toxic ceiling — a compound that is beneficial at low doses and harmful at high doses (the basis of hormesis models in toxicology and pharmacology).
- The arousal-performance relationship (an inverted U commonly discussed under the Yerkes-Dodson framework in psychology) — performance improves with arousal or stress up to a point, then declines as arousal continues to rise.
- Age and many physiological or cognitive measures, which often rise through early life and decline later, producing a curve rather than a consistent direction across the full age range.
The critical, easy-to-miss point: both Pearson’s r and Spearman’s rho can return a value close to zero for a strong, real, U-shaped or inverted-U relationship. Neither statistic is built to detect a relationship that changes direction — Pearson because it is looking for a straight line, and Spearman because it is looking for a consistent rank ordering. A U-shaped relationship has no consistent direction to rank against, so the positive and negative portions of the curve cancel each other out in the summary statistic even though the underlying relationship is strong and systematic.
This is the strongest practical argument for never choosing a summary statistic before looking at the data. A histogram of each variable and, more importantly, a scatterplot of the two variables together will show a U-shape or inverted-U immediately, where a correlation coefficient alone will report ‘no relationship’ and mislead a reader who never sees the plot. Plot first, then choose (or decline to use) a summary statistic that matches the shape actually observed — never the reverse.
Monotonic transformations
A monotonic transformation is a function applied to a variable that preserves its rank order — if a < b before the transformation, the transformed values still satisfy transformed(a) < transformed(b) afterward (for a monotonic increasing transformation), even though the actual numeric values change. Common examples include the logarithm, the square root, and converting raw scores to ranks.
Monotonic transformations preserve monotonic relationships and rank-based statistics, but they change linear ones. This is precisely why Spearman’s rho is invariant under any monotonic transformation of either variable (it will return the same value on the raw data and on the log-transformed data, for instance, because the rank order is unchanged) while Pearson’s r is not (log-transforming a variable changes the spacing between values and therefore can change r, sometimes substantially). Researchers who log-transform a skewed variable specifically to make a relationship ‘look more linear’ before running Pearson’s r are, in effect, relying on the fact that the underlying relationship was already monotonic — the transformation cannot create a monotonic relationship that was not there to begin with.
Where the monotonicity assumption shows up in analysis
Monotonicity is not only a lens for choosing a correlation statistic — it is an explicit assumption or constraint in several other common methods:
- Proportional-odds (ordinal logistic) models. Ordinal logistic regression, most commonly fit as a proportional-odds model, assumes the effect of each predictor is consistent (monotonic and, more specifically, of constant direction) across the ordered outcome categories. The proportional-odds assumption itself is a stronger, constant-effect version of monotonicity, and it is routinely tested (e.g., via a Brant test) before the model’s output is trusted. See the related guide on logistic regression and the logit model for the standard (binary-outcome) case this extends from.
- Isotonic regression. This is a regression method that fits the best-fitting curve to a set of points under the explicit constraint that the fitted curve must be monotonic (non-decreasing or non-increasing throughout). It is used when a researcher has strong prior reason to believe the true relationship is monotonic — for example, a calibration curve that should never decrease — but does not want to assume a specific linear or parametric shape.
- Monotonicity constraints in dose-response modelling. Pharmacological and toxicological dose-response models frequently constrain the fitted curve to be monotonic within a defined dose range where the underlying biology is expected to behave that way (e.g., increasing efficacy up to a plateau), even when the broader dose-response relationship (including toxicity at very high doses) is not monotonic across its full range.
Practical guidance: plot first, then choose the statistic
- Plot the two variables against each other before calculating anything. A scatterplot will show you whether the relationship looks like a straight line, a curve that consistently rises or falls, or a shape that changes direction.
- If the relationship looks like a straight line, and the other assumptions behind it hold (see the correlation coefficient guide for the full assumption checklist, and normality of distribution for the relevant background), Pearson’s r is appropriate and interpretable as a linear-association strength.
- If the relationship curves but never reverses direction — monotonic but not linear — report Spearman’s rho or Kendall’s tau instead of, or alongside, Pearson’s r. Say explicitly in the write-up that the relationship is monotonic rather than linear; this is a real, reportable finding about the shape of the data, not just a technicality of statistic choice.
- If the relationship changes direction (U-shaped or inverted-U), neither Pearson’s r nor Spearman’s rho will represent it faithfully, and reporting either alone as ‘the correlation’ will understate or entirely miss a real effect. Describe the shape directly, consider a quadratic or piecewise model, or split the range at the turning point and report the (now monotonic) relationship within each segment.
- Never choose the statistic first and let it dictate how the relationship gets described. The purpose of plotting first is to let the shape of the actual data determine which number, if any, is worth reporting.
Frequently asked questions
Is every monotonic relationship also linear?
No. The reverse is true: every linear relationship is monotonic, but most monotonic relationships are not linear. A relationship can curve, accelerate, or decelerate and still be monotonic, as long as it never reverses direction.
What is the difference between a monotonic function and a monotonic relationship?
A monotonic function, in the pure-mathematics sense, is an exact property of a defined mathematical function: its output never decreases (or never increases) as its input increases, with no exceptions. A monotonic relationship, in the statistical sense used on this page, is the applied version of that idea evaluated on real, sampled, noisy data — it describes the general tendency of one variable to move consistently with another, assessed using statistics like Spearman’s rho or Kendall’s tau rather than proven algebraically.
Can a strong relationship have a correlation coefficient near zero?
Yes, if the relationship is non-monotonic. A U-shaped or inverted-U relationship can produce a Pearson’s r and a Spearman’s rho both close to zero even when the two variables are strongly and systematically related, because neither statistic is designed to detect a relationship that reverses direction. This is why plotting the data before interpreting a correlation coefficient of zero as ‘no relationship’ is essential.
Does a monotonic transformation change Spearman’s rho?
No. Spearman’s rho is invariant under any monotonic transformation of either variable, because the transformation cannot change the rank order of the observations, and rank order is all the statistic uses. Pearson’s r, by contrast, is generally changed by a monotonic transformation such as a log or square-root transform, because the transformation changes the spacing between values.
Why does the proportional-odds assumption in ordinal logistic regression relate to monotonicity?
The proportional-odds assumption requires that each predictor’s effect be consistent in direction and magnitude across all thresholds of the ordered outcome. That consistent, single-direction effect is a stronger, constant-effect version of a monotonic relationship between the predictor and the ordinal outcome, which is why violations of the assumption are typically detected as effects that reverse direction or change magnitude sharply across categories.
