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Background The CKD-EPI equation reduces bias and improves accuracy for GFR estimation compared to the MDRD Study equation. Creatinine generation differs among racial-ethnic groups but both equations only consider Blacks vs other. We developed and validated a GFR-estimating equation that includes a 4-level race variable. Methods Equations were developed in pooled data from 10 studies (N=8254) and validated in 17 additional studies from the US and Europe [CKD-EPI validation database (N=4014)], and in studies from China (N=675), Japan (N=248) and South Africa (N=99). Race was defined as a 2-level variable (Black vs other) and a 4-level variable (Black, Asian, Native American and Hispanic vs other). Results Coefficients for Black, Asian and Native American and Hispanic resulted in 15%, 5% and 1% higherlevels of estimated GFR, respectively, compared to others. The 2-level race equation had minimal bias in Blacks, Native Americans, Hispanics and others [−0.8 (−2.0,0.6), 2.3 (−2.1,5.1), and 2.8 (2.4,3.2) ml/min/1.73 m2, respectively) in the CKD-EPI validation database. The 4-level race equation improved bias in CKD-EPI Asians (0.8 (−2.2,2.6) vs 2.1 (0.3,4.4) ml/min/1.73 m2) and in Chinese (1.3 (0.6,2.2) vs 2.7 (1.9,3.7) ml/min/1.73 m2). Both equations had a large bias in Japanese [−17.8 (−0.1,−14.7) and −21.4 (−23.2,−18.2) ml/min/1.73 m2)] and South Africans [−12.4 (−18.3,−7.6) and −12.5 (−18.3,−7.6) ml/min/1.73 m2. Conclusions A multilevel variable for race developed in one geographic region may not be applicable in other regions. The 2-level race variable in the CKD-EPI equation can be used for all racial-ethnic groups in the US and Europe.

Abstract. We study the existence of positive solutions of the equation u" + a{t)f{u) = 0 with linear boundary conditions. We show the existence of at least one positive solution if / is either superlinear or sublinear by a simple application of a Fixed Point Theorem in cones.

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