S3.5 - Regression and correlation

Learning objectives

S3.5.1Spearman’s rank correlationSpearman’s rank correlation Numerical questions involving ties will not be set.; Some coefficient, its use, interpretation understanding of how to deal with ties will be expected. and limitations.S3.5.2Hypothesis tests for zero correlationTesting the hypothesis that a Use of tables for Spearman’s and product moment correlation is zero. correlation coefficients.

Spearman's rank correlation coefficient $r_s$ measures the strength and direction of a monotonic association between two variables by comparing their ranks. It is useful for ordinal data or when the relationship is monotonic but a linear model for the original measurements is unsuitable.

Rank both variables in a consistent direction, calculate $d_i$ as one rank minus the other for each pair, square the differences, and sum them. Reversing both ranking directions changes no differences; reversing only one changes the sign of the coefficient.

r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}\qquad\text{when there are no tied ranks}

For five pairs with ranks $(1,2,3,4,5)$ and $(1,3,2,5,4)$ , $\sum d^2=4$ . Hence $r_s=1-6(4)/[5(25-1)]=0.8$ , showing a fairly strong positive monotonic association.

When values are tied, assign average ranks: each tied value receives the average of the rank positions it occupies. Numerical questions involving ties will not be set, but the correct approach is to calculate the product moment correlation coefficient of the two average-rank lists rather than use the no-ties shortcut unchanged.

$r_s$ lies between $-1$ and $1$ : its sign gives direction and its magnitude gives strength of monotonic association. It uses order, not the sizes of gaps, so it loses quantitative information; it does not prove causation and need not describe a non-monotonic relationship well.

A correlation test asks whether a sample coefficient is extreme enough to provide evidence that the corresponding population correlation is not zero. Select the table by coefficient, sample size $n$ , significance level and test direction.

For Spearman use $H_0:\rho_s=0$ ; for product moment correlation use $H_0:\rho=0$ . Choose a $>0$ or $<0$ alternative for a predicted direction, or $\ne0$ when either direction counts. A two-sided claim requires the two-tailed table value.

Calculate $r_s$ or $r$ and compare it with the matching critical value. For an upper-tail test reject $H_0$ above the positive critical value; for a two-tailed test reject when the coefficient's absolute value exceeds the critical magnitude.

Suppose $n=7$ , $r_s=0.679$ and the 5% upper-tail Spearman critical value is 0.7143. Since $0.679<0.7143$ , do not reject $H_0$ ; there is insufficient evidence of a positive population rank correlation at the 5% level.

Product moment correlation targets linear association in the original values; Spearman targets monotonic association through ranks. Do not interchange their tables.

Choose the direction before inspecting the coefficient. Non-significance does not establish zero correlation, and significance does not establish causation. Conclude using the named variables and tested direction.

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—Pearson Edexcel IAL MathematicsRead

S3.5 - Regression and correlation

Learning objectives

Measure monotonic association with Spearman's rank

Test whether a population correlation is zero

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