CHAPTER 4 CORRELATION

Cross product deviation 4.2 How to measure relationships 4.2.1 Covariance Adverts Watched and Packets bought Variance = ∑(xi – x)2 N - 1 Cov (x,y) = ∑(xi – x) (yi – y) N - 1 Cross product deviation = 4.25 for this example Figure 4.1 Graphical display of the difference between observed data and means of the two variables

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 Plot 4. 4 4.4 Graphing Relationships: The scatter Plot 4.4.1 Simple Scatter Plot : Load the file ExamAnxiety.sav SPSS Graphs-Interactive-Scatterplot Figure 4.5

4.4.2 3D Scatter plot Figure 4.6 – 3D Scatter Plot

4.4.3 Overlay Scatter plot SPSS Graphs-Scatter Figure 4.9

4.4.4 Matrix Scatter Plot Figure 4.11

4.5 Bivariate Correlation After 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 Part Correlation X2 X1 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. Y Part Correlation X2 The unique predictive effect of the due to a single independent variable among a set of independent variables. Y

X1 X2 c a = 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 a b d Y

4.6 Partial Correlation Figure 4.17

4.6.3 Semi Partial Correlation Figure 4.20