FS2 3.1 - Product moment correlation coefficient

FS2 3.1 - Product moment correlation coefficient

The product moment correlation coefficient measures linear association on a scale that does not depend on ordinary measurement units. You will calculate it from summaries, recognise when it is informative and track the effect of coding either variable.

A zero coefficient can hide a perfect curve

Can two variables follow an exact relationship and still have product moment correlation zero?

Take x=−2,−1,0,1,2x=-2,-1,0,1,2 and y=4,1,0,1,4y=4,1,0,1,4. Every point obeys y=x2y=x^2, yet the product moment correlation coefficient is zero. The upward and downward parts of the curve cancel in its measure of linear association.

You already know

From regression, SxxS_{xx} measures centred variation in the explanatory values and SxyS_{xy} measures their centred cross-product with the response. Inspecting a scatter diagram helps judge whether a line is reasonable. Association alone does not establish causation.

Correlation treats the variables symmetrically; exchanging their roles does not change the coefficient. A regression slope does depend on which variable is the response.

Product moment correlation coefficient

The sample product moment correlation coefficient, or PMCC, is denoted by rr. For two non-constant quantitative variables it measures the direction and strength of linear association, with

−1≤r≤1-1\le r\le1

Values 11 and −1-1 describe exact straight-line relationships with positive and negative slopes respectively. Zero means no linear correlation, not necessarily no relationship. The population coefficient is denoted by ρ\rho.

Use correctly paired quantitative measurements on the same individuals or cases. A nearly linear scatter without a dominating unusual point makes a single linear summary more informative. Ordinal ratings or a curved relationship may call for a different description; the coefficient can still be calculated where its denominator is non-zero, but it need not capture the pattern well.

[DIAGRAM: asset_name: An exact curve with zero linear correlation; asset_slug: 9fm0-fs2-124-zero-pmcc; caption: PMCC measures linear association rather than every possible relationship; description: Five exact points at x minus two,minus one,zero,one,two and y four,one,zero,one,four lie on the parabola y equals x squared. The curve is shown faintly behind the points. A side panel states that Sxy equals zero and r equals zero despite the exact curved relationship. A note says a zero coefficient does not mean there is no association]
PMCC measures linear association rather than every possible relationship

The population assumption for a formal PMCC critical-table test is stronger: the paired variables should come from a bivariate normal population. This is a joint distribution assumption, not merely a claim that each variable separately looks normal. Formal verification is not required. Normality is not needed just to evaluate the descriptive coefficient.

Key idea

Read PMCC with the scatter diagram: a linear coefficient can miss curvature or be strongly affected by an unusual point.

Use the opening curve to test the interpretation.

Calculating PMCC from paired summaries

Use the same observations in all summaries. Centre both variables before combining their variation; a raw cross-product alone depends on the locations of the measurement scales.

Product moment correlation coefficient

For nn paired observations with Sxx>0S_{xx}>0 and Syy>0S_{yy}>0,

Syy=∑yi2−(∑yi)2nS_{yy}=\sum y_i^2-\frac{(\sum y_i)^2}{n} Sxy=∑xiyi−(∑xi)(∑yi)n,Sxx=∑xi2−(∑xi)2nS_{xy}=\sum x_iy_i-\frac{(\sum x_i)(\sum y_i)}n,\qquad S_{xx}=\sum x_i^2-\frac{(\sum x_i)^2}n r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

The denominator is positive, so rr has the sign of SxyS_{xy}. The coefficient is undefined if either variable is constant. In the formula booklet.

Unlike regression slope, this expression divides by a square root involving both variables. The units cancel, leaving a dimensionless coefficient.

Worked example 1: a positive coefficient

Question. Five paired observations have

∑x=15,∑x2=55,∑y=20,∑y2=86,∑xy=66\sum x=15,\quad\sum x^2=55,\quad \sum y=20,\quad\sum y^2=86,\quad\sum xy=66

Calculate PMCC to three significant figures.

Plan. Find the centred sums and use the supplied coefficient formula.

Sxx=55−1525=10Syy=86−2025=6Sxy=66−15(20)5=6r=610(6)=0.774596…\begin{aligned} S_{xx}&=55-\frac{15^2}{5}=10\\ S_{yy}&=86-\frac{20^2}{5}=6\\ S_{xy}&=66-\frac{15(20)}5=6\\ r&=\frac6{\sqrt{10(6)}}=0.774596\ldots \end{aligned}

Why this step? Dividing by SxxS_{xx} alone would give a regression slope, not correlation.

Answer. r=0.775r=0.775, indicating positive linear association in this sample.

Check. The sign matches Sxy>0S_{xy}>0, and the magnitude lies below 11.

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