Correlation and Regression Modelling for CMA Inter

Understand Correlation and Regression Modelling

Key rules to remember

Karl Pearson's correlation coefficient

r = [nΣXY − ΣX·ΣY] ÷ √{[nΣX² − (ΣX)²] × [nΣY² − (ΣY)²]}

Use when you have raw paired data. r is always between −1 and +1.

Correlation using covariance

r = Cov(X, Y) ÷ (σx × σy)

Cov(X, Y) = ΣXY ÷ n − X̄·Ȳ. Use when the question gives covariance and standard deviations.

Regression line of Y on X

Y = a + bX, where b = [nΣXY − ΣX·ΣY] ÷ [nΣX² − (ΣX)²] and a = Ȳ − b·X̄

Use this to predict Y from X. The line always passes through (X̄, Ȳ).

Regression coefficients using r

b(yx) = r × σy ÷ σx ; b(xy) = r × σx ÷ σy

b(yx) is for Y on X. b(xy) is for X on Y. Both have the same sign as r.

Link between r and the two coefficients

r² = b(yx) × b(xy)

r = √(b(yx) × b(xy)), with the sign of the coefficients. Both coefficients must have the same sign.

Coefficient of determination

R² = 1 − SSE ÷ SST

SSE is the sum of squared errors. SST is the total sum of squares. In simple regression, R² = r².

Adjusted R²

Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1)

k is the number of independent variables. It penalises adding variables that add little.

Spearman's rank correlation (no tied ranks)

ρ = 1 − 6Σd² ÷ [n(n² − 1)]

d is the difference between the two ranks of each item. Use when data are ranks.

How to solve Correlation and Regression Modelling questions

1. 1Identify which variable is dependent (Y) and which is independent (X). The one to be predicted is Y.
2. 2Check what is given. If you have raw pairs, build a table. If you have means, standard deviations and r, use the b = r × σ ratio formulas.
3. 3For raw data, make columns for X, Y, XY, X² and Y². Total each column and write n.
4. 4Compute b first, then a = Ȳ − b·X̄. Write the equation Y = a + bX.
5. 5Compute r if asked, and check that its sign matches the sign of b.
6. 6Predict by putting the given X into the equation. Say whether the X value is within the range of the data.
7. 7Interpret in words: the direction and strength of the relationship, what b means in business terms, and what R² explains. State that correlation does not prove causation.

Quickest way: Deviation-free shortcut with totals

Common mistakes in Correlation and Regression Modelling

• Writing that a high correlation proves that X causes Y.

  Students read a strong r as a cause-and-effect link.

  Fix: Say that r shows association only. Two variables may move together because of a third factor or by chance.

• Regressing in the wrong direction, using X on Y when Y on X is asked.

  Students use the same b for both lines.

  Fix: Decide which variable is to be predicted. The two lines differ, with b(yx) = r·σy ÷ σx and b(xy) = r·σx ÷ σy.

• Confusing ΣX² with (ΣX)² in the denominator.

  Both look similar and are rushed in the table.

  Fix: Square each X first and then add for ΣX². Add all X first and then square for (ΣX)². Write both clearly.

• Giving r a positive sign when b is negative, or finding r > 1.

  Students take the square root and forget the sign, or make an arithmetic slip.

  Fix: The sign of r is the sign of b. r must lie between −1 and +1. If not, recheck the sums.

• Predicting far outside the data range.

  Students trust the equation for any X.

  Fix: Predict only within or close to the range of observed X. State that extrapolation is unreliable.

• Treating a higher R² as always better in multiple regression.

  R² never falls when you add a variable.

  Fix: Compare adjusted R², and check that each added variable makes business sense.

Worked examples