Bootstrap hypothesis testing in generalized additive models for comparing curves of treatments in longitudinal studies

The study of the effect of a treatment may involve the evaluation of a variable at a number of moments. When assuming a smooth curve for the mean response along time, estimation can be afforded by spline regression, in the context of generalized additive models. The novelty of our work lies in the construction of hypothesis tests to compare two curves of treatments in any interval of time for several types of response variables. The within-subject correlation is not modeled but is considered to obtain valid inferences by the use of bootstrap. We propose both semiparametric and nonparametric bootstrap approaches, based on resampling vectors of residuals or responses, respectively. Simulation studies revealed a good performance of the tests, considering, for the outcome, different distribution functions in the exponential family and varying the correlation between observations along time. We show that the sizes of bootstrap tests are close to the nominal value, with tests based on a standardized statistic having slightly better size properties. The power increases as the distance between curves increases and decreases when correlation gets higher. The usefulness of these statistical tools was confirmed using real data, thus allowing to detect changes in fish behavior when exposed to the toxin microcystin-RR.

针对干预效应的研究通常需要在多个时间节点对某一变量进行评估。若假设平均响应随时间呈平滑曲线，则可在广义加性模型（generalized additive models）的框架下，通过样条回归（spline regression）完成参数估计。本研究的创新之处在于构建了适用于多种响应变量类型的假设检验（hypothesis tests）方法，可在任意时间区间内对比两组干预的响应曲线。本研究未对受试者内相关性（within-subject correlation）进行建模，但通过自助法（bootstrap）开展统计推断以保证结果的有效性。我们分别提出了两种自助法方案：基于残差向量重采样的半参数自助法，以及基于响应向量重采样的非参数自助法。模拟实验结果表明，当结局变量服从指数族分布中的不同分布函数、且观测值的时间相关性存在差异时，所提检验方法均表现良好。研究显示，自助法检验的显著性水平趋近于名义水平，其中基于标准化统计量的检验在显著性水平控制方面表现略优。两条响应曲线的差异越大，检验功效越高；而观测的时间相关性越强，检验功效则越低。我们通过真实数据集验证了该统计工具的实用性：利用该方法可检测鱼类暴露于微囊藻毒素-RR（microcystin-RR）后的行为变化。