Understanding and Applying Basic Statistical Methods Using R by Wilcox Rand R.;

Author:Wilcox, Rand R.; [Wilcox, Rand R.]
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2016-06-07T00:00:00+00:00

So if , the least squares regression line has a positive slope, and if , the reverse is true. Perhaps this interpretation is usually correct, but this should not be taken for granted. In Figure 8.1, for example, is negative, butgenerally, as increases, increases too. Many methods have been derived to get a more exact understanding of how and are related. One simple recommendation is to always plot the data, as well as the least squares regression line, as opposed to relying completely on the value of to interpret how two variables are related. At a minimum, the least squares regression line should look reasonable when viewed within a plot of the data. But even if it gives a reasonable summary of the data, there can be a considerable practical advantage to using an alternative regression estimator.

A Summary of How to Use

To summarize, Pearson's correlation, , has two useful functions. First, it can be used to establish dependence between two variables by testing and rejecting the hypothesis that is equal to zero. Second, (the coefficient of determination), reflects the extent to which the least squares regression estimate of , namely , improves upon the sample mean, , in terms of predicting .

However, even if is close to 1, this does not necessarily mean that the least squares estimate of is performing well. It might be, for example, that both and perform poorly. Even when Pearson's correlation is very close to 1, this does not necessarily mean that the least squares regression line provides a highly accurate estimate of , given . Finally, compared to more modern methods, Pearson's correlation can be relatively ineffective in terms of describing and detecting dependence. This is not always the case, but it is imprudent to assume that it always provides an adequate method for detecting and describing an association. Put another way, rejecting the hypothesis that provides empirical evidence that there is dependence, but failing to reject is not a compelling reason to conclude that two variables are independent. More modern methods can be sensitive to types of dependence that are difficult to discover when using Pearson's correlation only.

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