Tangent exponential models at a data point lead to third order significance for component parameters (Fraser and Reid, 1989,1993a, 1993b, 1995). With nuisance parameters, adjustment factors to a tangent exponential model can arise; for example, with conditioning for exponential models and with marginalization for transformation and general asymptotic models. Methods of calculating these adjustments are discussed and different versions of the resulting significance formulas are derived. Examples and numerical comparisons are given.