Theme 7 Correlation

Overview Researchers are often interested in exploring whether or not two variables are associated This lecture will consider Scatter plots Pearson correlation coefficient Spearman correlation coefficient

Scatter Plots To get a pictorial view of the relationship between two variables you can create a scatter plot Each data point is marked on a graph If there is a positive relationship between your variables then there will be a general trend from the bottom left to the top right If there is a negative relationship between your variables then there will be a general trend from the top left to the bottom right

Line of best fit Imagine drawing a line on your scatter plot which best fits the data The closer your data points are in relation to this line, the stronger the relationship between your two variables Similarly, the further your data points are in relation to this line, the weaker the relationship between your two variables Remember the direction of the line indicates the direction of the relationship

Example of a Scatter Plot and Line of Best Fit Figure 10.1 A scattergram showing the relationship between maths ability and musical ability

Correlation Coefficients A correlation coefficient indicates the amount of variance shared by two variables A correlation of 1.0 shows that all of the variance can be explained A correlation of zero shows that none of the variance can be explained The closer to 1 (or -1) the stronger the relationship; the closer to 0 the weaker the relationship Positive values indicate a positive relationship Negative values indicate a negative relationship

Proportion of Variance Explained One can calculate this by squaring the correlation coefficient and multiplying by 100 A coefficient of 1.0 indicates that 100% of the variance is explained (i.e. 1.0 ×1.0) × 100 A coefficient of 0.6 indicates that 36% of the variance is explained (i.e. 0.6 × 0.6) × 100 A coefficient of -0.6 also indicates that 36% of the variance is explained (i.e. -0.6 × -0.6) × 100

Types of Correlation There are two key tests of correlation Pearson Product Moment Correlation Coefficient – This can be used to explore the linear relationship between two variables Spearman Rho Correlation Coefficient – This is similar to the PPMCC but uses ranked scores rather than raw data

Raw Data Case Music score Maths score 1 2 8 6 3 4 9 7 5 10

Scatter Plot

Interpretation You should never report a correlation coefficient without examining the scatter plot for problems such as curved relationships or outliers We would write of the scatter plot: ‘A scatter plot of the relationship between mathematical ability and musical ability was examined. There was no evidence of a curvilinear relationship or the undue influence of outliers.’

Steps on SPSS Data Analysis Output In Variable View, ‘name’ the variables and enter the data Analysis For the correlation, select ‘Analyze’, ‘Correlate’ and ‘Bivariate...’. Move appropriate variables to the ‘Variables:’ box, Select the appropriate correlation and then ‘OK’. For the scatter plot, select ‘Graphs’. Then select ‘Chart Builder’ ‘OK’, ‘Scatter/Dot’ and move the ‘Simple Scatter’ figure to the box above. Move appropriate variable names to the vertical and horizontal axes. Output The correlation table shows the correlation, its significance level and the sample size.

Pearson Output

Interpretation The correlation coefficient is −.90 The relationship is negative and statistically significant In a report, we would write ‘There is a significant negative relationship between musical ability and mathematical ability, r(8) = −.90, p < 0.001. Children with more musical ability have lower mathematical ability.’

Spearman Output

Interpretation The correlation coefficient is −.89 Again the relationship is negative and statistically significant We would report this in the following way: ‘There is a statistically significant negative correlation between musical ability and mathematical ability, ρ(8) = −.89, p < 0.001. Those with the highest musical ability tend to be those with the lowest mathematical ability and vice versa.’

Conclusion When exploring the association between two variables it is important to first inspect the scatter plot The Pearson or Spearman correlation coefficient is a statistical indicator of the relationship They inform you about the strength and direction of the relationship as well as the amount of variance explained