CHAPTER 4

CORRELATION

4.2 How to measure relationships4.2.1 Covariance Adverts Watched and Packets bought

Figure 4.1 Graphical display of the difference between observed data and means of the two variables

Variance = ∑(xi – x)2

N - 1

Cov (x,y) = ∑(xi – x) (yi – y)

N - 1

Cross product deviation

= 4.25 for this example

4.2.2. Standardisation and the correlation coefficient

• To overcome the problem of dependence on measurement scale, we convert covariance in to

standard set of units.

• By standardising we end up with a value that lies between -1 and +1.

• +1 means variables are perfectly positively related.

• This correlation coefficient can be called as the

• Pearson product moment correlation coeffiecient or Pearson correlation coefficient

r = Cov (x,y) = ∑(xi – x) (yi – y)

sx sy (N – 1) sx sy

4.4 Graphing Relationships: The scatter Plot4.4.1 Simple Scatter Plot : Load the file ExamAnxiety.sav

Figure 4.5

SPSS

Graphs-Interactive-Scatterplot

4.4.2 3D Scatter plot

Figure 4.6 – 3D Scatter Plot

4.4.3 Overlay Scatter plot

Figure 4.9

SPSS

Graphs-Scatter

4.4.4 Matrix Scatter Plot

Figure 4.11

4.5 Bivariate CorrelationAfter a prelimnary glance at the data, we can conduct the correlation

analysis.

• Access the File Advert.sav• SPSS Analyze –Correlate-Bivariate

• 4.5.1 Pearsons Correlation Coefficient– Reload the file ExamAnxiety.sav

• 4.5.2 A word of warning about interpretation: Causality– The third variable problem (There may be other measured or

unmeasured variables affecting the results)– Direction of causality (The correlation coefficients say nothing

about which variable causes the other to change)

4.5.3 Using R2 for Interpretation• We can go a step further by squaring r

• The correlation coefficient squared (coefficient of determination) is a measure of the amount of variability in one variable that is explained by the other.

• R2 = 19.4

• Exam anxiety accounts for 19.4% of the variability in exam performance.

• 4.5.4 Spearman´s correlation coefficient

• 4.5.5 Kendall´ tau (non parametric)

Partial

Correlation

Y

X2X1

The shaded area of X1 as a portion of the area Y represents the partial correlation of X1 with Y given X2. This shaded area as a proportion of Y, denotes the incremental variance explained by X1, given that X2 is already in the equation.

Part Correlation The unique predictive effect of the due to a single

independent variable among a set of independent variables.

X2

Y

Y

X2X1

bcaa = Variance of Y uniquely explained by X1

b = Variance of Y uniquely explained by X2

c = Variance of Y explained jointly by X1and X2

d = Variance of Y not explained by X1 or X2 d

4.6 Partial Correlation

Figure 4.17

4.6.3 Semi Partial Correlation

Figure 4.20