a Department of Psychology , Arizona State University.
Multivariate Behav Res. 2019 May-Jun;54(3):444-455. doi: 10.1080/00273171.2018.1545630. Epub 2019 Jan 20.
Recent methodological studies have investigated the properties of multilevel models with small samples. Previous work has primarily focused on continuous outcomes and little attention has been paid to count outcomes. The estimation of count outcome models can be difficult because the likelihood has no closed-form solution, meaning that approximation methods are required. Although adaptive Gaussian quadrature (AGQ) is generally seen as the gold standard, its comparative performance has been investigated with larger samples. AGQ approximates the full likelihood, a function that is known to produce biased estimates with small samples with continuous outcomes. Conversely, penalized quasi-likelihood (PQL) is considered to be a less desirable approximation; however, it can approximate the restricted likelihood function, a function that is known to perform well with smaller samples with continuous outcomes. The goal of this paper is to compare the small sample bias of full likelihood methods to the linearization bias of PQL with restricted likelihood. Simulation results indicate that the linearization bias of PQL is preferable to the finite sample bias of AGQ with smaller samples.
最近的方法学研究调查了小样本下多层次模型的特性。之前的工作主要集中在连续结果上，很少关注计数结果。计数结果模型的估计可能很困难，因为似然函数没有闭式解，这意味着需要近似方法。虽然自适应高斯求积（AGQ）通常被视为黄金标准，但它在更大的样本中已经被研究了其比较性能。AGQ 近似完全似然，对于连续结果的小样本，这个函数已知会产生有偏估计。相反，惩罚拟似然（PQL）被认为是一种不太理想的近似；然而，它可以近似受限似然函数，对于连续结果的小样本，这个函数已知表现良好。本文的目的是比较完全似然方法的小样本偏差与受限似然的 PQL 的线性化偏差。模拟结果表明，对于小样本，PQL 的线性化偏差优于 AGQ 的有限样本偏差。
