TABLE III.

Simulation results at 95% significance level of testing additive genetic effect H0: α₁ = 0 in model y_i(t) = μ(t) + sex_iβ_sex + x_i1α₁ + U_i(t) + E_i + ε_i by nonparametric linear penalized spline model, a correctly specified nonlinear function, and misspecified parametric functions of μ(t)

μ(t): Mean function	Type I error or power (bias)	α1	No. of obsc	Empirical results: type I error or power (bias ${\mathrm{\alpha }}_1-{\widehat{\mathrm{\alpha }}}_1$)
				Nonparametric modeld	Correctly specifiede	Misspecified: Linearf	Misspecified: Cubicg
Logarithma	Type I error (bias)	0	926	0.0535 (0.011)	0.0555 (0.005)	0.0615 (0.0161)	0.0540 (0.009)
		0.5	896	0.1620 (−0.068)	0.1595 (−0.054)	0.2430 (−0.254)	0.1605 (0.509)
	Power (bias)	1	926	0.4010 (0.005)	0.4055 (0.003)	0.5695 (−0.291)	0.4015 (1.009)
		2	889	0.9465 (−0.008)	0.9470 (−0.001)	0.9000 (0.137)	0.9475 (2.009)
		3	902	0.9980 (−0.007)	0.9980 (0.001)	0.9970 (0.124)	0.9980 (3.010)
Exponentialb	Type I error (bias)	0	916	0.0525 (−0.015)	0.0525 (−0.018)	0.009 (−0.242)	0.0525 (−0.032)
		0.5	908	0.1215 (0.003)	0.1240 (0.003)	0.0785 (−0.343)	0.1205 (0.467)
	Power (bias)	1	900	0.3540 (−0.002)	0.3505 (0.001)	0.3120 (−0.522)	0.3430 (0.967)
		2	906	0.8545 (−0.005)	0.8535 (−0.008)	0.7175 (−0.358)	0.8430 (1.966)
		3	916	0.9990 (−0.015)	0.9990 (−0.018)	0.9910 (−0.242)	0.9990 (2.968)

^aTrue μ(t) = −34.2 + 81.7log(0.3(t + 21.7)).

^bTrue μ(t) = 110exp(0.0002×(t − 25)²).

^cThe total number of observations.

^dμ(t) is estimated by nonparametric linear penalized spline model.

^eμ(t) is correctly specified.

^fμ(t) is misspecified as μ(t) = μ₀ +μ₁t.

^gμ(t) is misspecified as μ(t) = μ₀ + μ₁t + μ₂t² + μ₃t³.