CORRELATION CURVES AS LOCAL MEASURES OF VARIANCE EXPLAINED BY REGRESSION
CORRELATION CURVES AS LOCAL MEASURES OF VARIANCE EXPLAINED BY REGRESSION
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DOI：
10.2307/2290860
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发表时间：
1994-06-01
影响因子：
3.7
通讯作者：
ZHAO, HY
中科院分区：
文献类型：
--
作者：
DOKSUM, K;BLYTH, S;ZHAO, HY
如果响应变量Y和协变量X之间的关系强度在协变量空间的不同区域中不同，则我们称实验（的模型）为异相关。对于这样的实验，我们引入了一个相关曲线，测量异相关性的方差解释回归本地在每个协变量值。更准确地说，平方相关曲线是通过首先用残差方差和回归斜率来表示通常的线性模型“方差解释为总方差”公式，然后用依赖于x的条件残差方差和Y的条件平均值的斜率（给定X = x）来代替它们来获得的。相关性曲线rho（x）满足相关性的不变性，它在线性模型中简化为Galton-Pearson相关性rho，它在-1和1之间，当X和Y独立时为0，当Y是X的函数时为+/-1。我们介绍的相关曲线的估计的基础上最近邻估计的（条件）残差方差函数和（条件）回归斜率函数，以及对这些功能的Gasser-Muller内核估计。我们得到的一致性和渐近正态性的结果，并给出简单的渐近同时置信区间的相关曲线。真实的数据和模拟数据的例子来说明的局部相关程序。
We call (a model for) an experiment heterocorrelations if the strength of the relationship between a response variable Y and a covariate X is different in different regions of the covariate space. For such experiments we introduce a correlation curve that measures heterocorrelaticity in terms of the variance explained by regression locally at each covariate value. More precisely, the squared correlation curve is obtained by first expressing the usual linear model ''variance explained to total variance'' formula in terms of the residual variance and the regression slope and then replacing these by the conditional residual variance depending on x and the slope of the conditional mean of Y given X = x. The correlation curve rho(x) satisfies the invariance properties of correlation, it reduces to the Galton-Pearson correlation rho in linear models, it is between -1 and 1, it is 0 when X and Y are independent, and it is +/-1 when Y is a function of X. We introduce estimates of the correlation curve based on nearest-neighbor estimates of the (conditional) residual variance function and the (conditional) regression slope function, as well as on Gasser-Muller kernel estimates of these functions. We obtain consistency and asymptotic normality results and give simple asymptotic simultaneous confidence intervals for the correlation curve. Real data and simulated data examples are used to illustrate the local correlation procedures.