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The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), [5] and some value in the open interval (,) in all other cases, indicating the degree of linear dependence between the variables. As it ...
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables. [ a ] The variables may be two columns of a given data set of observations, often called a sample , or two components of a multivariate random variable with a known distribution .
The term significance does not imply importance here, and the term statistical significance is not the same as research significance, theoretical significance, or practical significance. [1] [2] [18] [19] For example, the term clinical significance refers to the practical importance of a treatment effect. [20]
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
In broad usage, the "practical clinical significance" answers the question, how effective is the intervention or treatment, or how much change does the treatment cause. In terms of testing clinical treatments, practical significance optimally yields quantified information about the importance of a finding, using metrics such as effect size, number needed to treat (NNT), and preventive fraction ...
However, without knowing the correlates of immunity, scientists cannot know exactly what sort of immune response a vaccine would need to stimulate, and the only method of assessing vaccine effectiveness will be through large phase III trials with clinical outcomes (i.e. infection and/or disease, not just laboratory markers). [citation needed]
"Correlation does not imply causation" is a common theme for much of the epidemiological literature. For epidemiologists, the key is in the term inference . Correlation, or at least association between two variables, is a necessary but not sufficient criterion for the inference that one variable causes the other.
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.