Limits...
On measures of association among genetic variables.

Gianola D, Manfredi E, Simianer H - Anim. Genet. (2012)

Bottom Line: These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution.Our measures are based on logarithmic (Kullback-Leibler) and on relative 'distances' between distributions.Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.

View Article: PubMed Central - PubMed

Affiliation: Department of Animal Sciences, University of Wisconsin-Madison, Madison, WI, 53706, USA. gianola@ansci.wisc.edu

Show MeSH

Related in: MedlinePlus

Scatter diagrams of 5000 samples from each of four Olkin-Liu bivariate beta distributions. Plot 1 illustrates a strong association with essentially no correlation. Plot 2 (‘meteorite’) depicts a limitation of the correlation as a parameter for describing association. Plot 3 suggests association clearly. Plot 4 shows a bivariate distribution that is not trivial: the true correlation (0.46) arises primarily due to weaker association in the ‘middle’ of the bivariate sampling space.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3569618&req=5

fig04: Scatter diagrams of 5000 samples from each of four Olkin-Liu bivariate beta distributions. Plot 1 illustrates a strong association with essentially no correlation. Plot 2 (‘meteorite’) depicts a limitation of the correlation as a parameter for describing association. Plot 3 suggests association clearly. Plot 4 shows a bivariate distribution that is not trivial: the true correlation (0.46) arises primarily due to weaker association in the ‘middle’ of the bivariate sampling space.

Mentions: Where 0 < X < 1 and 0 < Y < 1. Moments E(XkYl) cannot be written in closed form but can be approximated numerically or using sampling methods. Large c and small a, b produce correlations close to 0, whereas large a, b or small c produce correlations close to 1 (Olkin & Liu 2003). For example, a correlation equal to 0.002 is obtained for a = b = 0.01 and c = 5, whereas the correlation is 0.91 for a = 2.5, b = 4 and c = 0.1. Figure 4 displays scatter plots of 5000 samples obtained from each of four bivariate beta distributions. Plot 1 represents a distribution in which the correlation is very low, and yet, there is considerable association between pairs of values near 0, illustrating inadequacy of correlation to reveal association. In plot 2 (resembling a ‘meteorite’) clustering takes place primarily at large values of X and Y. The two bottom plots depict bivariate beta distributions with similar correlations but with a completely different pattern of association. Clearly, correlation often fails as measure of statistical association.


On measures of association among genetic variables.

Gianola D, Manfredi E, Simianer H - Anim. Genet. (2012)

Scatter diagrams of 5000 samples from each of four Olkin-Liu bivariate beta distributions. Plot 1 illustrates a strong association with essentially no correlation. Plot 2 (‘meteorite’) depicts a limitation of the correlation as a parameter for describing association. Plot 3 suggests association clearly. Plot 4 shows a bivariate distribution that is not trivial: the true correlation (0.46) arises primarily due to weaker association in the ‘middle’ of the bivariate sampling space.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3569618&req=5

fig04: Scatter diagrams of 5000 samples from each of four Olkin-Liu bivariate beta distributions. Plot 1 illustrates a strong association with essentially no correlation. Plot 2 (‘meteorite’) depicts a limitation of the correlation as a parameter for describing association. Plot 3 suggests association clearly. Plot 4 shows a bivariate distribution that is not trivial: the true correlation (0.46) arises primarily due to weaker association in the ‘middle’ of the bivariate sampling space.
Mentions: Where 0 < X < 1 and 0 < Y < 1. Moments E(XkYl) cannot be written in closed form but can be approximated numerically or using sampling methods. Large c and small a, b produce correlations close to 0, whereas large a, b or small c produce correlations close to 1 (Olkin & Liu 2003). For example, a correlation equal to 0.002 is obtained for a = b = 0.01 and c = 5, whereas the correlation is 0.91 for a = 2.5, b = 4 and c = 0.1. Figure 4 displays scatter plots of 5000 samples obtained from each of four bivariate beta distributions. Plot 1 represents a distribution in which the correlation is very low, and yet, there is considerable association between pairs of values near 0, illustrating inadequacy of correlation to reveal association. In plot 2 (resembling a ‘meteorite’) clustering takes place primarily at large values of X and Y. The two bottom plots depict bivariate beta distributions with similar correlations but with a completely different pattern of association. Clearly, correlation often fails as measure of statistical association.

Bottom Line: These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution.Our measures are based on logarithmic (Kullback-Leibler) and on relative 'distances' between distributions.Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.

View Article: PubMed Central - PubMed

Affiliation: Department of Animal Sciences, University of Wisconsin-Madison, Madison, WI, 53706, USA. gianola@ansci.wisc.edu

Show MeSH
Related in: MedlinePlus