As metabolite concentration in culture medium increased, cell viability also increased. If the correlation coefficient between the two variables is r=.78, can we conclude that increasing the metabolite will improve viability? Correlation is a starting point, not a causal conclusion.

A correlation coefficient compresses linear co-movement into −1 to +1

Pearson's correlation coefficient r is a standardized summary of whether two variables tend to depart from their own means in the same direction.

• r>0: observations with larger X also tend to have larger Y in a linear pattern
• r < 0: observations with larger X tend to have smaller Y in a linear pattern
• |r| close to 1: points lie close to a straight line
• r≈0: linear co-movement is weak

r=0 does not mean that no relationship exists. In a perfect U-shaped relationship, the slopes on the two sides can cancel and leave r near zero. Draw a scatterplot before calculating one number.

Correlation removes the units of X and Y, which makes comparisons easier, but it does not state an actual amount of change. “How much does Y change when X increases by one?” is a question about a regression slope and is covered in the next unit.

The same r can conceal very different data

Check the following features in a scatterplot.

1. Linearity: do points lie around a line, or are there curves, thresholds, or saturation?
2. Outliers: do one or two points control the direction and size of the result?
3. Range restriction: was only a narrow part of X observed?
4. Clusters: are there groups by plate, donor, or date?
5. Heteroscedasticity: does the spread of Y increase as X increases?
6. Measurement error: does repeat precision in X and Y attenuate the relationship?

Restricting a sample to healthy adults can weaken an age–biomarker relationship that is strong in the full population. Conversely, combining two batch clusters can make overall r large even when no relationship exists within either cluster.

Correlation has the same value when direction is reversed and does not assign a cause

cor(X,Y)=cor(Y,X). Correlation does not determine which variable is explanatory and which is the response. A high r has several possible explanations.

• X affects Y.
• Y affects X.
• A third variable Z affects both.
• Selection, measurement, or a time trend created an artificial relationship.
• Chance and multiple exploration selected a large r.

Causal judgment requires temporal ordering, an intervenable design, randomization, control of confounding, mechanism, and replication. A small correlation p-value does not automatically satisfy those conditions.

p-values and CIs depend on n

A test of H0: population linear correlation ρ=0 uses r and n. With a large n, even a small r can be significant; with a small n, the CI for a large r can be wide. Report r, its CI, the independent n, and the scatterplot together.

If thousands of correlations are tested in a matrix of genes and endpoints, many small p-values arise by chance. Restrict candidates before analysis, or plan multiplicity control such as FDR and independent validation.

In-Silico Lab: create the fragility of r yourself

1. Change the slope to −2, 0, and +2 to see the direction of r.
2. Increase curvature to create a situation where the relationship is strong but r is weak.
3. Add one outlier and see how much r moves.
4. Check whether the sample r changes under the same generating conditions with a new seed.

The line in the Lab is a simple regression line used to support explanation. Seeing a line does not establish causation or justify extrapolation.

In JMP, inspect each scatterplot before the matrix

A Correlation Matrix compresses many pairs but hides nonlinearity and outliers. Inspect the shape of every pair in a Scatterplot Matrix, and read a density ellipse as an auxiliary summary of a linear-normal structure. The slope of an ellipse does not indicate causal direction.

Example result statement

Across 42 independent batches, the Pearson correlation between metabolite X and viability was .61 (95% CI .38–.77). The scatterplot showed a positive linear trend, but two high-concentration points were influential, and coloring by batch date suggested that clustering explained part of the pattern. This observational correlation was not interpreted as a causal effect of increasing X.

At the end of this unit

• r gives the direction and standardized size of linear co-movement.
• A nonlinear relationship can exist even when r≈0.
• Outliers, range restriction, clusters, and confounding can change r substantially.
• Correlation does not establish a causal direction between X and Y.
• Consider r, its CI, independent n, and the scatterplot together rather than a p-value alone.
• Large-scale correlation searches require multiplicity control and validation.

In the next unit, we express a relationship as Y=β₀+β₁X+ε and use residuals to find what the straight line missed.

Official supplementary resources

• NIST/SEMATECH · Measures of association
• JMP Statistics Knowledge Portal · Correlation

The examples and Lab in this article use synthetic educational data.