CORRELATION CURVES AS LOCAL MEASURES OF VARIANCE EXPLAINED BY REGRESSION

• 成果类型: Article
• 署名作者: DOKSUM, K; BLYTH, S; BRADLOW, E; MENG, XL; ZHAO, HY
• 署名单位: Imperial College London; Harvard University; University of Chicago
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.2307/2290860
• 发表日期: 1994
• 页码: 571-582
• 关键词: derivatives

摘要

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.