Tests of linearity in regression functions are frequently applied in practice. If a researcher hasapriori information about the shape of the regression function, then incorporating this information into the test typically increases the power at alternatives that satisfy the hypothesized shape restriction. We study likelihood ratio tests of linearity with the alternative constrained to be concave (or convex) as well as tests of concavity as the null hypothesis. To complete the development of the test statistics and their null distributions, we only need to study the level probabilities. A numerical example and a discussion of the use of modifications of these tests as alternatives to the likelihood ratio test for homogeneity versus a unimodal (umbrella) ordering are included