Summary: An adjustment to the profile likelihood proposed by D. R. Cox and N. Reid [J. R. Stat. Soc., Ser. B 49, 1-39 (1987; Zbl 0616.62006)] when the parameters are orthogonal is shown to agree with modified profile likelihood in a number of instances in which the parameters are not orthogonal. Several examples of the phenomenon are given. In the case of composite transformation models with invariant parameter as the parameter of interest, it is shown that the Cox-Reid adjustment agrees with modified profile likelihood in a large-deviation sense if Lebesgue measure is right-invariant. Necessary and sufficient conditions for equivalence in the moderate-deviation sense of modified profile likelihood and the Cox-Reid adjusted profile likelihood are derived.