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Covariate adjusted correlation analysis via varying coefficient models. (English) Zbl 1089.62068
Actual variables $$(Y,X)$$ are observed after being multiplied by smooth unknown functions $$\psi$$ and $$\varphi$$ of the confounder $$U$$, leading to the observations $$(U_i,\tilde Y_i=\psi(U_i)Y_i,\tilde X_i=\varphi(U_i)X_i)$$ with identifiability conditions $$E\varphi(U)=E\psi(U)=1$$. ($$U$$ is independent of $$X$$ and $$Y$$). The authors propose a cadcor estimator for $$\text{corr}\,(X,Y)$$ based on the relation $$\text{corr}\,(X,Y)=\text{sign}\,(\gamma)\sqrt{\gamma\eta}$$, where $$\gamma$$ and $$\eta$$ are linear regression coefficients for $$Y$$ by $$X$$ and $$X$$ by $$Y$$, respectively. For the observed data the estimation of $$\eta$$ and $$\gamma$$ leads to the multiple varying coefficient model. The asymptotic normality of the obtained estimator is demonstrated, and an estimate for the asymptotic variance is proposed. Application to Boston house price data and results of simulation studies are presented.

MSC:
 62H20 Measures of association (correlation, canonical correlation, etc.) 62J05 Linear regression; mixed models
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References:
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