Simple Penalties on Maximum-Likelihood Estimates of Genetic Parameters to Reduce Sampling Variation

Abstract

Multivariate estimates of genetic parameters are subject to substantial sampling variation, especially for smaller data sets and more than a few traits. A simple modification of standard, maximum-likelihood procedures for multivariate analyses to estimate genetic covariances is described, which can improve estimates by substantially reducing their sampling variances. This is achieved by maximizing the likelihood subject to a penalty. Borrowing from Bayesian principles, we propose a mild, default penalty-derived assuming a Beta distribution of scale-free functions of the covariance components to be estimated-rather than laboriously attempting to determine the stringency of penalization from the data. An extensive simulation study is presented, demonstrating that such penalties can yield very worthwhile reductions in loss, i.e., the difference from population values, for a wide range of scenarios and without distorting estimates of phenotypic covariances. Moreover, mild default penalties tend not to increase loss in difficult cases and, on average, achieve reductions in loss of similar magnitude to computationally demanding schemes to optimize the degree of penalization. Pertinent details required for the adaptation of standard algorithms to locate the maximum of the likelihood function are outlined.KEYWORDS genetic parameters; improved estimates; regularization; maximum likelihood; penalty E STIMATION of genetic parameters, i.e., partitioning of phenotypic variation into its causal components, is one of the fundamental tasks in quantitative genetics. For multiple characteristics of interest, this involves estimation of covariance matrices due to genetic, residual, and possibly other random effects. It is well known that such estimates can be subject to substantial sampling variation. This holds especially for analyses comprising more than a few traits, as the number of parameters to be estimated increases quadratically with the number of traits considered, unless the covariance matrices of interest have a special structure and can be modeled more parsimoniously. Indeed, a sobering but realistic view is that "Few datasets, whether from livestock, laboratory or natural populations, are of sufficient size to obtain useful estimates of many genetic parameters" (Hill 2010, p. 75). This not only emphasizes the importance of appropriate data, but also implies that a judicious choice of methodology for estimation-which makes the most of limited and precious records available-is paramount.A measure of the quality of an estimator is its "loss," i.e., the deviation of the estimate from the true value. This is an aggregate of bias and sampling variation. We speak of improving an estimator if we can modify it so that the expected loss is lessened. In most cases, this involves reducing sampling variance at the expense of some bias-if the additional bias is small and the reduction in variance sufficiently large, the loss is reduced. In statistical parlance "regularization" refers to the use of some kind of additional ...

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“…Genome-wide significance was then declared at 10% FDR after Bonferroni correction for multiple tests based on the effective number of markers ( M eff ), the number of eigenvalues of the marker correlation matrix R that explain at least 99% of the variance in linkage disequilibrium (as in Davis et al (2016)). Covariance matrices sampled from an unobserved population are often poorly behaved and biased in the distribution of eigen-values when the number of markers (or phenotypes) p is much larger than the number of observations n , as is common using genomic data (Meyer 2016). Following Davis et al (2016), we applied the non-parametric Ledoit-Wolf shrinkage estimator to R before eigenvalue decomposition.…”

“Bending” and beyond: Better estimates of quantitative genetic parameters?

“…More recently, Meyer () suggested “simple,” scale‐free penalty functions based on Beta distributions, considering either the canonical eigenvalues or genetic correlations. For the latter, this allowed for shrinkage of estimates towards simple structures, such as on identity matrix, or their phenotypic counterparts, and was simplified by the transformation to partial correlations, π ij , that is, for the variables i and j (with i < j) the correlation given the intervening variables i + 1 to j − 1, which vary independently over the interval [−1, 1] (Daniels & Pourahmadi, ; Joe, ).…”

“…Similarly, assuming independent shifted Beta distributions on [−1, 1] for partial correlations π ij and shrinking all values towards zero, the penalty is as follows: with C 0 = log[B(1 + ( κ − 2)/2, 1 + ( κ − 2)/2)]. A generalization of (Equation ) to allow for different shrinkage targets for individual π ij is given in Meyer ().…”

“…It thus provides an intuitive parameter to specify the strength of penalization. Based on simulation results, Meyer () suggested to simply employ default values for κ (or corresponding hyperparameters) which are expected to yield worthwhile reductions in average loss whilst not being likely to have a detrimental effect for cases where the penalty does not match the distribution of population values. For penalties based on a Beta distribution, default values for κ in the range of 4 to 8–10 were recommended.…”

“…Extensive simulation studies demonstrated that penalized estimation could substantially reduce the average loss in estimates compared to standard, unpenalized values (Meyer, , ). These considered a wide range of population values, deliberately including some cases for which the distributions of the targeted variables differed markedly from those implied when deriving the penalties.…”