多水平Logistic回归模型的层内相关系数（ICC）计算

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收藏 2016-03-17

虽然多水平模型的层内相关系数和量表信度评价的ICC (详见：http://user.qzone.qq.com/569473320/blog/1457505780) 都是ICC，但是意义完全不一样，很容易混淆。多水平模型的层内相关系数是在研究设计是处于高水平单位的内部个体之间存在的同质性。如从社区为基础的整群随机抽样得到的样本，每一个社区内部的个体之间存在一定的同质性，也即ICC。

ICC计算的最基本的思路是方差分析，也即定义为高水平变量的均方占总体误差的比例。对于多水平线性模型而言，stata和SAS的MIXED model可以分别得到高水平变量的均方（MSB）及残差的均方(MSW)，ICC=MSB/(MSB+MSW)。而对于多水平的logistic回归而言，STATA中的melogit model能够计算MSB，但是模型并不能显示MSW。从《Appliedmultilevel analysis: a practical guide for medical researchers》查阅得到多水平logistic模型的ICC计算公式为：ICC =MSB / [MSB+(π^2/3)] = MSB / [MSB+3.29]. 虽然通过以下STATA相对官方的多水平模型ICC计算实例中验证后，结果与该公式计算的结果基本接近，但是为什么多水平logistic模型的MSW是一个固定值，很难理解。

以下示例来源：http://www.stata.com/features/overview/intraclass-correlations-for-multilevel-models/

Stata’s estat icc command is a postestimation command that can be used after linear, logistic, or probit random-effects models. It estimates intraclass correlations for multilevel models.

We fit a three-level mixed model for gross state product using mixed. Fixed-effects covariates include the state unemployment rate and different categories of public capital stock: hwy, water, and other. Random intercepts are present at both the region and state levels. Seventeen years of annual data are used. We use estat icc to estimate the intraclass correlations for this model.

estat icc reports two intraclass correlations for this three-level nested model. The first is the level-3 intraclass correlation at the region level, the correlation between productivity years in the same region. The second is the level-2 intraclass correlation at the state-within-region level, the correlation between productivity years in the same state and region.

Conditional on the fixed-effects covariates, we find that annual productivity is only slightly correlated within the same region, but it is highly correlated within the same state and region. We estimate that state and region random effects compose approximately 85% of the total residual variance.

Now we fit a three-level logistic model for successful completion of the Tower of London computerized task. The variable group is used to classify individuals as controls (1), relatives of a schizophrenic (2), or schizophrenic (3). The difficulty level of the task and separate indicators for the different values of group are fixed-effect covariates. Random intercepts are present at both the family and subject levels.

We use estat icc to estimate the intraclass correlations for this model.

estat icc reports two intraclass correlations for this three-level nested model. The first is the level-3 intraclass correlation at the family level, the correlation between latent measurements of the cognitive ability in the same family. The second is the level-2 intraclass correlation at the subject-within-family level, the correlation between the latent measurements of cognitive ability in the same subject and family.

There is not a strong correlation between individual realizations of the latent response, even within the same subject.