Random effect models are popular statistical models for detecting and correcting spurious sample correlations due to hidden confounders in genome-wide gene expression data. In applications where some confounding factors are known, estimating simultaneously the contribution of known and latent variance components in random effect models is a challenge that has so far relied on numerical gradient-based optimizers to maximize the likelihood function. This is unsatisfactory because the resulting solution is poorly characterized and the efficiency of the method may be suboptimal. Here we prove analytically that maximum-likelihood latent variables can always be chosen orthogonal to the known confounding factors, in other words, that maximum-likelihood latent variables explain sample covariances not already explained by known factors. Based on this result we propose a restricted maximum-likelihood method which estimates the latent variables by maximizing the likelihood on the restricted subspace orthogonal to the known confounding factors, and show that this reduces to probabilistic PCA on that subspace. The method then estimates the variance-covariance parameters by maximizing the remaining terms in the likelihood function given the latent variables, using a newly derived analytic solution for this problem. Compared to gradient-based optimizers, our method attains greater or equal likelihood values, can be computed using standard matrix operations, results in latent factors that don't overlap with any known factors, and has a runtime reduced by several orders of magnitude. Hence the restricted maximum-likelihood method facilitates the application of random effect modelling strategies for learning latent variance components to much larger gene expression datasets than possible with current methods.
翻译:随机效应模型是常见的统计模型,用于检测和纠正基因组整体表达式数据中隐藏的混杂因素造成的虚假样本关联。在已知某些混杂因素的应用程序中,同时估计随机效应模型中已知和潜在差异元件的贡献是一个挑战,迄今为止,它依赖基于梯度的数值优化器来最大限度地发挥概率功能。这不尽如人意,因为由此产生的解决方案特征差,而且方法的效率可能不尽人意。在这里,我们从分析上证明,最大相似的潜在变量总是可以选择与已知的混杂因素(换言之,最大相似的潜在变量可以解释已知因素尚未解释的样本差异变异性。基于这个结果,我们提出一个有限的最大相似方法,通过在有限的子空间上尽可能增加可能性,或者通过已知的混杂因素,来估计潜在变量的变异异性变量。 使用新获得的最接近的变异异性数据, 使用最接近的变现的变异性方法, 使用更接近的变异性方法, 以最易变现的变异性方法来评估潜在变量的变异性值。