A Lower Bound for the Correlation of Exponentiated Bivariate Normal Pairs
Greenland, Sander, The American Statistician
The conceptual limitations of correlation coefficients and related statistics have been extensively discussed, especially in regression problems and in contexts involving binary outcomes (Tukey 1954; Rosenthal and Rubin 1979; Fleiss 1983; Greenland, Schlesselman, and Criqui 1986; Greenland, Maclure, Schlesselman, Poole, and Morgenstern 1991; Willett and Singer 1988; Scott and Wild 1991; Cox and Wermuth 1992; and Bring 1994 are some examples.) In my experience students often confront these discussions without awareness of the basic mathematical limits of correlation coefficients. I have found the following example useful in emphasizing that Pearson correlation coefficients need not be preserved by nonlinear transforms, and that such transforms can impose sharp bounds on such coefficients even when the variates are continuous. The example requires only elementary algebra applied to the formula for the mean of the lognormal distribution. It seems to have particular force for students in biomedical fields, perhaps because measurements (such as blood pressure and serum cholesterol) are often modeled as lognormal variates. It also serves as an introduction to more sophisticated regression examples concerning [R.sup.2] (e.g., Scott and Wild 1991).
2. A LOGNORMAL CORRELATION FORMULA
Suppose X is normal with mean [Mu] and variance [[Sigma].sup.2]. Then T = [e.sup.X] is lognormal with mean exp([Mu] + [[Sigma].sup.2]/2) (Mood, Graybill, and Boes 1974), and the variance of T is
E([T.sup.2]) - E[(T).sup.2] = E([e.sup.2X]) - E[(e.sup.X]).sup.2]
= exp(2[Mu] + 2[[Sigma].sup.2]) - exp(2[Mu] + [[Sigma].sup.2]). (1)
Now suppose X and Y are bivariate normal with zero means, equal variances of [[Sigma].sup.2], and a correlation coefficient [Rho]. Then X + Y has mean zero and variance [[Sigma].sup.2] + [[Sigma].sup.2] + 2[Rho][[Sigma].sup.2] = 2(1 + [Rho])[[Sigma].sup.2], and the covariance of T = [e.sup.X] and U = [e.sup.Y] is
E(TU) - E(T)E(U) = E([e.sup.X+Y]) - E([e.sup.X])E([e.sup.Y])
= exp([[Sigma].sup.2] + [Rho][[Sigma].sup.2])
= exp([[Sigma].sup.2])(exp([Rho][[Sigma].sup.2]) - 1). (2)
Because T and U both have variance exp([[Sigma].sup.2])(exp([[Sigma].sup.2]) - 1), the correlation of T and U is
r = (exp([Rho][[Sigma].sup.2]) - 1)/(exp([[Sigma].sup.2]) - 1). (3)
It can be easily verified that the T-U correlation r attains a maximum of 1 when [Rho] = 1, but has a minimum of -exp(-[[Sigma].sup.2]) when [Rho] = -1. These are profoundly asymmetric bounds: for any [[Sigma].sup.2] the correlation r of T and U can always be as high as 1, but can be no less than -.61 when [[Sigma].sup.2] = 1/2 and no less than -.05 when [[Sigma].sup.2] = 3.
When r = -exp(-[[Sigma].sup.2]) in the above example T can be perfectly predicted from U and vice versa because in this case [Rho] = -1, X = -Y, and hence T = 1/U. That is, r = -exp(-[[Sigma].sup.2]) corresponds to a perfect inverse relationship of T and U. Yet a perfect direct relation (T = U) corresponds to r = 1. Thus the example shows that the correlation coefficient r is not a symmetrically bounded index of predictive ability or association for T and U, even when the regressions of T on U and U on T are monotone and continuous. More generally, a correlation coefficient of 1 or -1 can arise only from perfectly linear relationships (T = a + bU with b [not equal to] 0), and so cannot occur whenever the ranges of T and U logically exclude such a relationship. In particular, r must exceed -1 if both T and U are bounded below but unbounded above (as are lognormal variates), because r = -1 requires T = a + bU with b [less than] 0.
The example also illustrates how even linear transformations can have profound effects on a correlation-based analysis if those transformations are applied before a nonlinear transformation to the correlational scale. …