Testing for distinct marginal DFEs in a joint DFE context

[Emma Howell]

Hi dadi developers,

I was wondering if you have any suggestions about how best to test for distinct marginal DFEs in a joint DFE context. I'd like to do so in a manner that still allows for fitness effects to be highly correlated between the two populations.

I suppose I could conduct likelihood ratio tests to determine whether the bivariate lognormal distribution with distinct parameters (mu1, mu2, sigma1, sigma2, and rho) provides a significantly better fit than a bivariate lognormal with shared parameters (mu, sigma, and rho). But, my understanding from the Huang et al. (2021) paper is that a mixture model may be preferable for modeling the joint DFE when fitness effects are highly correlated.

However, as the mixture models are currently defined, they require the parameters of the bivariate and univariate probability distributions to be shared. For the lognormal case, is it possible to to define a mixture of two bivariate probability distributions (rather than a univariate and bivariate probability distribution)? Similarly to how it is defined in the paper, the uncorrelated selective effects would correspond to a bivariate distribution (but this time with distinct marginal parameters: mu1, mu2, sigma1, sigma2, and rho = 0), but the correlated selective effects would also be modeled with a bivariate distribution (with the same set of parameters as above, but fixing rho = 1). 

I would appreciate any thoughts you might have about this. Thanks!

Best,

Emma

[Ryan Gutenkunst]

Hello Emma,

Sorry for the slow reply. This is a subtle issue, and I don’t think we have a great solution to it.

The most direct route is to test the bivariate lognormal distribution with and without distinct mus and sigmas. There is a challenge that if the correlation is very high, then the distribution is narrow, in that most of the gamma1, gamma2 space has almost zero probability, and only a very small set of values of large probability. If the gamma_pts grid is not fine enough, those values with high probability might not be captured well, and the integration will be inaccurate.

The mixture model was designed to avoid this problem with narrow distributions. The perfectly correlated component is infinitely thin, but since it consists only of points where s1=s2, the density is always at points we actually sample with the gamma_pts grid. Unfortunately, that doesn’t work if mu1 != mu2 or sigma1 != sigma2. So I don’t think it’s possible to get the benefits of the mixture model and not have that perfectly correlated component be symmetric.

I’m curious about the circumstances you’re testing. I’d be happy to talk more, on list or off, about the details of the species you’re studying.

Best,

Ryan

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