Correlation coefficients
- Helpful page on Cross and autocorrelation
- Key (from Matlab help files): if X is a matrix of data (say, time series in each column),
then corrcoef(X) is the zeroth lag of the covariance function, that is, the zeroth lag of xcov(x,'coeff') packed into a square array.
Pearson's Correlation Coefficient
Definition: Measures the strength of the linear relationship between two variables.
Assumptions: Both variables (often called X and Y) are interval/ratio and approximately normally distributed, and their joint distribution is bivariate normal.
Characteristics: Pearson's Correlation Coefficient is usually signified by r (rho), and can take on the values from -1.0 to 1.0. Where -1.0 is a perfect negative (inverse)
correlation, 0.0 is no correlation, and 1.0 is a perfect positive correlation.
Related statistics: R2 (called the coefficient of determination or r squared) can be interpreted as the proportion of variance in Y that is contained in X.
Tests: The statistical significance of r is tested using a t-test. The hypotheses for this test are:
H0: rho = 0
Ha: rho <> 0
A low p-value for this test (less than 0.05 for example) means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis, or that there is a
statistically significant relationship between the two variables.
Note: This test is equivalent to the test of no slope in the simple linear regression procedure.