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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

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Related in: MedlinePlus

Plots of the densities of four Lee-Sarmanov bivariate beta distributions having the same Beta (2,2) marginal distributions but differing in the strength of association: as the correlation increases multi-modality emerges.
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fig05: Plots of the densities of four Lee-Sarmanov bivariate beta distributions having the same Beta (2,2) marginal distributions but differing in the strength of association: as the correlation increases multi-modality emerges.

Mentions: Note that D′AI is solely a function of the correlation, so it was the same for the three pairs of beta distributions examined. Using (25), D′IA was estimated by averaging over draws. Because is a ratio between density functions, negative realizations were not used when forming the Monte Carlo estimate (this produces an upward bias but a more precise estimate). As shown in Table 4, the KL distance between the bivariate beta distribution and the independence process increased monotonically with the strength of the correlation. The indexes of association θ′ and 1 − θ′ also drifted away monotonically from only in setting 2. For instance, in setting 1 θ′ increased from 0.50 to 0.66 as the correlation increased from 0 to but decreased to 0.60 when the correlation was . The shape of the bivariate beta distribution was examined for this setting by plotting the density at 20 000 random points drawn from each of the marginal Beta(2,2) distributions. This is shown in Fig. 5: the Lee-Sarmanov density ‘evolves’ towards a more complex topography as correlation increases. When correlation grows from 0 to , the shape of the joint distribution is not too different from that under independence, as also suggested by KL and θ′. However, when the correlation increases by a factor of 3 from to , the KL distance grows only about two times, while θ′ and 1 − θ′ move away from at an even slower rate; this is expected because θ has an upper bound at 1, whereas KL takes any positive value in the real line. The bimodal shape of the density also suggests that more intensive sampling is needed to obtain reliable estimates of the indexes of association.


On measures of association among genetic variables.

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

Plots of the densities of four Lee-Sarmanov bivariate beta distributions having the same Beta (2,2) marginal distributions but differing in the strength of association: as the correlation increases multi-modality emerges.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig05: Plots of the densities of four Lee-Sarmanov bivariate beta distributions having the same Beta (2,2) marginal distributions but differing in the strength of association: as the correlation increases multi-modality emerges.
Mentions: Note that D′AI is solely a function of the correlation, so it was the same for the three pairs of beta distributions examined. Using (25), D′IA was estimated by averaging over draws. Because is a ratio between density functions, negative realizations were not used when forming the Monte Carlo estimate (this produces an upward bias but a more precise estimate). As shown in Table 4, the KL distance between the bivariate beta distribution and the independence process increased monotonically with the strength of the correlation. The indexes of association θ′ and 1 − θ′ also drifted away monotonically from only in setting 2. For instance, in setting 1 θ′ increased from 0.50 to 0.66 as the correlation increased from 0 to but decreased to 0.60 when the correlation was . The shape of the bivariate beta distribution was examined for this setting by plotting the density at 20 000 random points drawn from each of the marginal Beta(2,2) distributions. This is shown in Fig. 5: the Lee-Sarmanov density ‘evolves’ towards a more complex topography as correlation increases. When correlation grows from 0 to , the shape of the joint distribution is not too different from that under independence, as also suggested by KL and θ′. However, when the correlation increases by a factor of 3 from to , the KL distance grows only about two times, while θ′ and 1 − θ′ move away from at an even slower rate; this is expected because θ has an upper bound at 1, whereas KL takes any positive value in the real line. The bimodal shape of the density also suggests that more intensive sampling is needed to obtain reliable estimates of the indexes of 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