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## Multivariate Analysis

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**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) Multivariate Analysis**Multivariate Analysis**• Correlation: Mathematical tool used to establish a dependency between two data sets. Covariance (shown below) Product of Standard deviations Deviations from average value X and Y represent the two sets of data**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. Multivariate Analysis**Correlation Continued**• Magnitude of association Correlation coefficient with a value between -1 and 1. Weak 0 - 0.3 Moderate 0.3 - 0.6 Strong 0.6 - 1.0 Discipline dependent**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**Multivariate Analysis**X Y • R close to zero**Multivariate Analysis**• R close to +0.5**Multivariate Analysis**• R close to +1**Excel Functions**COVAR(array1, array2) CORREL(array1, array2) Multivariate Analysis**Correlation Continued**• Correlation or association between two variable does not give causation • More experimentation must be done to determine causation.**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**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