TESTING FOR ADDITIVITY OF A REGRESSION FUNCTION

Observations y(ij) are made at points (x1i, x2j) according to the model y(ij) = F(x1i, x2j) + e(ij), where the e(ij) are independent normals with constant variance. In order to test that F(x1, x2) is an additive function of x1 and x2, a likelihood ratio test is constructed comparing F(x1, X2) = mu + Z1(x1) + Z2(x2) with F(x1, x2) = mu + Z1(x1) + Z2(x2) + Z(x1, x2), where Z1, Z2 are Brownian motions and Z is a Brownian sheet. The ratio of Brownian sheet variance to error variance alpha is chosen by maximum likelihood and the likelihood ratio test statistic W of H-0: alpha = 0 used to test for departures from additivity. The asymptotic null distribution of W is derived, and its finite sample size behaviour is compared with two standard tests in a simulation study. The W test performs well on the five alternatives considered.