Correlation Coefficients [AL]

Analysis and Interpretation of Correlation Including Spearman's Rho

Correlational analysis measures the strength and direction of the relationship between two variables without manipulating either. It is used to investigate naturally occurring associations between variables that cannot be ethically or practically manipulated experimentally. A correlation coefficient provides a numerical summary of both the direction and strength of the relationship, whilst a scatter diagram (covered in B1503) provides the visual display.

Direction and Strength

The direction of a correlation can be positive, negative, or zero. A positive correlation occurs when both variables increase together — high scores on one tend to accompany high scores on the other. A negative correlation occurs when one variable increases as the other decreases. A zero correlation indicates no systematic linear relationship.

The strength of a correlation is expressed as a coefficient ranging from −1.00 to +1.00. A coefficient of ±1.00 indicates a perfect relationship; 0 indicates no linear relationship. The closer the coefficient to ±1.00 (regardless of sign), the stronger the relationship. In psychology, correlations of ±0.30–0.70 are common.

Spearman's Rank Correlation (r_s)

Spearman's rho (r_s) is a non-parametric correlation coefficient that measures the strength and direction of the monotonic relationship between two variables. It works by converting raw scores to ranks before computing the correlation. This makes it appropriate when:

• Data are at the ordinal level (e.g. ranks, Likert-scale ratings)
• Interval/ratio data contain outliers that would distort a Pearson r
• The distribution is substantially non-normal
• The relationship may be monotonic but not linear

Spearman's r_s is interpreted identically to Pearson's r — the sign indicates direction and the absolute value indicates strength. A result is significant if the obtained |r_s| is equal to or greater than the critical value from the Spearman's rho table for the given N and significance level.

Limitations of Correlational Analysis

Correlation cannot establish cause and effect. Two problems prevent causal conclusions:

• Directionality problem: even a strong correlation cannot tell us which variable causes which — both causal directions (A→B or B→A) are equally consistent with the data.
• Third variable problem: an unmeasured third variable may independently cause both co-variables, producing a correlation with no direct causal link between them (e.g. ice cream sales and drowning deaths are positively correlated — both are caused by hot weather).

Additionally, Pearson's r and Spearman's r_s only detect linear and monotonic relationships respectively — a strong curvilinear relationship (e.g. the inverted-U of the Yerkes-Dodson law) can produce a near-zero correlation coefficient.