. Author manuscript; available in PMC: 2023 Jan 19.

Published in final edited form as: Comput Stat Data Anal. 2020 Apr 23;150:106987. doi: 10.1016/j.csda.2020.106987

Table 1:

Summary of the existing methods

Author(s), Methods	Same x(s)	Correlation	>2 Groups	Curve/Surface	Additional Comments
Bowman (2006)[17]:	Y	N	N	Curve/Surface	(+) Simple to implement and understand as a derivation from ANOVA test; (−) Assume equal variance across groups.
Dette & Neumeyer (2001)[19], Pardo-Fernandez & Van Keilegom (2007)[21], Wang & Ye (2010)[23]:	N	N	Y	Curve/Surface	(+) Demonstrated asymptotic normality of all three kernel-based statistics under H₀; Recommended wild bootstrap when studying finite samples.
Zhang & Lin (1998)[24]:	Y	Y	N	Curve	(+) Spline-based semiparametric additive model; (−) χ² approximation can be biased with different covariate values.
Wang & Ye (2010)[23]:	N	Y	Y	Curve/Surface	(+) Able to adjust for spatial correlation; (−) Larger bias in estimating regression surface hence decreased power.
Kulasekera (1995)[13]:	N	N	N	Curve	(+) Low computational demand; (−) Low power when curve functions are similar.
Park, Hannig,& Kang (2014)[18]:	N	N	Y	Curve	(+) A visualization tool to present differences between curves across multiple locations and scales; (−) Type 1 error rate below nominal level; relatively low power