HIV模型的统计诊断

描述HIV的数学模型是一组非线性常微分方程,其中包括CD4+T细胞再感染率和死亡率,HIV病毒死亡率等多个重要未知参数,准确估计这些参数有助于正确了解患者病情发展,以采用个人化治疗方案.针对此模型已提出多个参数估计方法,包括基于数值解的非线性最小二乘,截面似然及两步估计.另一方面,由于客观因素影响,各观测数据对于参数估计的影响大小不同,找出对参数估计影响大的观测值,进一步分析其原因具有重要意义.基于HIV模型的两步估计,构造了用于影响分析的非参数Cook型统计量,并给出其大样本分布.通过模拟数据和临床数据分析发现:(1)所构造统计量可以检测中度以上偏移.(2)相对于内点,边界点对HIV模型的参数估计影响更大.基于以上结论,建议对HIV模型进行参数估计时,对边界点应加以格外关注.提出的方法也可以推广到其他线性常微分方程模型的统计诊断问题中.

Statistical diagnosis of HIV model

The dynamics of the human immunodeficiency virus (HIV) can be described by ordinary differential equations (ODEs), in which the unknown parameters include the infected rate and death rate of CD4+ T cell, the death rate of virus etc. Accurate estimation of these parameters plays a major role in the personalized treatment. Presently the main parameter estimation approaches include numerical solution based nonlinear OLS, generalized profiling likelihood and two stage estimation. On the other hand, influential analysis for dynamics of HIV, though important for the reliable parameter estimation, has received little attention in the literatures. Based on the two stage estimation of HIV dynamics, a nonparametric Cook type statistic is constructed in this paper to detect the influential measurements. The limit distribution of such Cook type statistic is established. By applying the proposed Cook statistic to the simulated and clinical trial data we find (1) the middle bias can be detected by the proposed Cook statistic, (2) compared to the interior measurements, the boundary measurements have more impact on the estimation of the parameters. Based on these observations, we suggest that the boundary measurements should be paid more attentions than the interior measurements for the parameters estimation of HIV dynamics. The proposed influential measurements detection procedure can be generalized to other linear ODEs with measurement error.