Checking linearity of non-parametric component in partially linear models with an application in systemic inflammatory response syndrome study

Two tests are proposed for checking the linearity of nonparametric function in partially linear models. The first one is based on a Crámer-von Mises statistic. This test can detect the local alternative converging to the null at the parametric rate 1/square root n. A bootstrap resample technique is provided to calculate the critical values. The second one is constructed in a penalized spline framework along with linear mixed-effects (LME) modeling. This is an extension likelihood ratio test for testing zero variance of random effects in LME models. Simulation experiments are conducted to explore the numerical performance of two tests. It is observed that two tests have good level properties, and the first test has a substantially superior power property over the second test in a variety of cases. A real data set is analysed with the proposed tests.