Correlation vs Causation Notation

A scatter plot with a regression line is one of the most visually persuasive figures in science —and one of the most frequently misinterpreted. The regression line implies a directional relationship; readers infer causation. Licklider requires a disclosure statement in the figure legend whenever a fitted line is displayed over observational data.

STEP 1 —The Pitfall: The Regression Line as Causal Signal

A regression line is a mathematical description of the linear relationship between two variables in the observed sample. It does not indicate that X causes Y. Yet the visual convention of displaying a fitted line —particularly with an upward or downward slope — reliably produces causal inferences in readers, even when the figure legend explicitly states "correlation."

The practical consequences are significant:

• Observational data presented as evidence of mechanism. A scatter plot showing a positive correlation between biomarker A and tumor size, with a regression line, is routinely cited as evidence that A "drives" tumor growth —when the data support only an association, and A may be a downstream consequence rather than a cause.
• Confounding variables not visible in the figure. The regression line summarizes the marginal relationship between X and Y. If both are driven by a third variable Z, the apparent relationship is spurious —and the figure does not show Z.
• Extrapolation beyond the data range. A regression line extends beyond the range of observed data by default in most software. The extrapolated portions suggest a predictable relationship in a domain where no data exist. (See also: Regression Diagnostics Guard, 4.7.1.)

STEP 2 —Journal Requirement

Nature's editorial standards require that "correlational analyses should not be described using causal language in the abstract, results, or figure legends unless the study design supports causal inference." Science similarly requires that "the distinction between correlation and causation must be made explicit when regression or correlation analyses are reported."

The EQUATOR network guidelines (STROBE for observational studies, CONSORT for trials) require explicit statements about the limitations of observational inference. STROBE checklist item 21 asks: "Discuss limitations of the study, taking into account sources of potential bias or imprecision —in particular, discuss any potential for reverse causality."

STEP 3 —Licklider's Solution

Input

• Any scatter plot with a regression line (linear or non-linear), Loess curve, or other fitted line.
• The study design declared in the Data Contract (observational / experimental / randomized). Randomized experiments may qualify for weaker disclosure language; observational studies require the full disclosure.

Output

• Auto-inserted disclosure statement. The figure legend automatically includes a disclosure sentence: "The fitted line describes the observed linear association between [X] and [Y] in this sample and does not imply a causal relationship."
• Extrapolation zone marking. The portion of the regression line that extends beyond the observed data range is automatically rendered as a dashed line, with a note in the figure legend: "Dashed line: extrapolation beyond the range of observed data."
• Adjusted disclosure for experimental designs. When the study design supports causal inference (e.g., randomized controlled experiment with manipulation of X), the disclosure language is softened: "The fitted line describes the observed dose-response relationship from a randomized experiment in which [X] was experimentally manipulated."

Guard

STEP 4 —Draft Output (Draft / Needs review)

Figure legend —Observational study

Figure 3A. Scatter plot of biomarker_A expression versus tumor volume at day 21.
The solid line represents the ordinary least squares regression fit
(slope = 2.31, 95% CI [1.08, 3.54]; rﾂｲ = 0.41; n = 47).
The dashed line indicates extrapolation beyond the range of observed data.
The fitted line describes the observed association between biomarker_A and
tumor volume in this sample and does not imply a causal relationship.
Both variables may be influenced by unmeasured confounders.
Licklider regression disclosure ID: reg-5b3c.

Figure legend —Randomized experiment

Figure 2B. Dose-response scatter plot (drug concentration vs. cell viability, %).
The curve represents the 4-parameter logistic (4PL) regression fit
(IC50 = 2.4 ﾎｼM, 95% CI [1.9, 3.1]; Hill slope = 1.8; n = 6 per concentration).
Drug concentration was experimentally manipulated across the range shown.
The fitted curve describes the observed dose-response relationship
in this randomized experiment.

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