Human Early Learning Partnership, The University of British Columbia, 2206 East Mall, Vancouver, British Columbia, V6T 1Z4, Canada.
BMC Med Res Methodol. 2019 May 9;19(1):97. doi: 10.1186/s12874-019-0742-8.
Despite its popularity, issues concerning the estimation of power in multilevel logistic regression models are prevalent because of the complexity involved in its calculation (i.e., computer-simulation-based approaches). These issues are further compounded by the fact that the distribution of the predictors can play a role in the power to estimate these effects. To address both matters, we present a sample of cases documenting the influence that predictor distribution have on statistical power as well as a user-friendly, web-based application to conduct power analysis for multilevel logistic regression.
Computer simulations are implemented to estimate statistical power in multilevel logistic regression with varying numbers of clusters, varying cluster sample sizes, and non-normal and non-symmetrical distributions of the Level 1/2 predictors. Power curves were simulated to see in what ways non-normal/unbalanced distributions of a binary predictor and a continuous predictor affect the detection of population effect sizes for main effects, a cross-level interaction and the variance of the random effects.
Skewed continuous predictors and unbalanced binary ones require larger sample sizes at both levels than balanced binary predictors and normally-distributed continuous ones. In the most extreme case of imbalance (10% incidence) and skewness of a chi-square distribution with 1 degree of freedom, even 110 Level 2 units and 100 Level 1 units were not sufficient for all predictors to reach power of 80%, mostly hovering at around 50% with the exception of the skewed, continuous Level 2 predictor.
Given the complex interactive influence among sample sizes, effect sizes and predictor distribution characteristics, it seems unwarranted to make generic rule-of-thumb sample size recommendations for multilevel logistic regression, aside from the fact that larger sample sizes are required when the distributions of the predictors are not symmetric or balanced. The more skewed or imbalanced the predictor is, the larger the sample size requirements. To assist researchers in planning research studies, a user-friendly web application that conducts power analysis via computer simulations in the R programming language is provided. With this web application, users can conduct simulations, tailored to their study design, to estimate statistical power for multilevel logistic regression models.
尽管多水平逻辑回归模型应用广泛，但由于其计算的复杂性（即基于计算机模拟的方法），在估计功效时仍存在诸多问题。此外，预测变量的分布也会影响这些效应的估计功效，这使得问题更加复杂。为了解决这两个问题，我们提供了一个案例样本，记录了预测变量分布对统计功效的影响，并提供了一个用户友好的、基于网络的应用程序，用于进行多水平逻辑回归的功效分析。
通过计算机模拟，我们在不同的聚类数量、不同的聚类样本量以及 1 级/2 级预测变量的非正态和非对称分布的情况下，估计了多水平逻辑回归的统计功效。模拟了功效曲线，以了解非正态/不平衡的二项预测变量和连续预测变量分布在何种程度上影响主要效应、跨水平交互作用和随机效应方差的总体效应大小的检测。
偏态连续预测变量和不平衡的二项预测变量比平衡的二项预测变量和正态分布的连续预测变量在两个水平上都需要更大的样本量。在最极端的不平衡（10%发生率）和自由度为 1 的卡方分布偏度的情况下，即使有 110 个 2 级单位和 100 个 1 级单位，所有预测变量的功效也无法达到 80%，除了偏态的连续 2 级预测变量外，大多数都在 50%左右徘徊。
鉴于样本量、效应大小和预测变量分布特征之间复杂的相互影响，除了预测变量分布不对称或不平衡时需要更大的样本量之外，针对多水平逻辑回归制定通用的经验法则样本量建议似乎是不合理的。预测变量越偏态或不平衡，所需的样本量就越大。为了帮助研究人员规划研究，我们提供了一个用户友好的网络应用程序，该程序通过计算机模拟在 R 编程语言中进行功效分析。使用这个网络应用程序，用户可以根据自己的研究设计进行模拟，以估计多水平逻辑回归模型的统计功效。
