1. Multivariate Analysis Trying to establish a mathematical relationship between multiple data sets. (e.g. smoking/cancer, salary/productivity, pressure/volume, etc.) (We will limit our examples to two variables)

2. Multivariate Analysis Correlation: Mathematical tool used to establish a dependency between two data sets. X and Y represent the two sets of data Covariance (shown below) Product of Standard deviations Deviations from average value

3. Multivariate Analysis R: Value ranges between -1 and +1. Applies only to linear systems Interpretation is subjective but in general the closer to -1 or +1 the more highly correlated is the data.

4. Correlation Continued Magnitude of association Correlation coefficient with a value between -1 and 1. Weak 0 - 0.3 Moderate 0.3 - 0.6 Strong0.6 - 1.0 Discipline dependent

5. Correlation Continued Direction Sign of the correlation coefficient Positive correlation (+) - an increase in one variable is accompanied by an increase in the other Negative correlation (-) - an increase in one variable is accompanied by an decrease in the other

6. Multivariate Analysis R close to zero 15.4 26.8 36.2 4-27.4 5-32 6-65.6 776.8 874.2 9-87.4 10142 11102.4 1248.8 13-78.8 14-134.4 15-91 166.4 1734.8 18162.2 19-94.4 2027 X Y

7. Multivariate Analysis R close to +0.5

8. Multivariate Analysis R close to +1

9. Multivariate Analysis Excel Functions –COVAR(array1, array2) –CORREL(array1, array2)

10. Correlation Continued Correlation or association between two variable does not give causation More experimentation must be done to determine causation.

11. Causation Consistency - measurements of the same two properties yield nearly the same results Responsiveness - Change in one variable elicits corresponding change in the other variable Mechanism - finding a valid and reasonable explanation for the cause and effect between the pair of variables

12. Establishing Causation Continued A valid and reasonable explanation establishes the sequential ordering of events and, thus, provides the rationale for the existence of such a relationship. Cause must precede effect, otherwise there is no causality