In the previous chapter the emphasis has been on the equivalence to first order of the likelihood ratio statistic, the directed likelihood, the score statistic and the statistic based on the maximum likelihood estimate, the last two statistics having a number of variants. These statistics often, but not always, give virtually identical answers in applications and it is therefore necessary to consider three rather different higher-order aspects: arguments for choosing among the different procedures; approximations to the distribution theory; and modifications to the procedures, in particular to achieve better behaviour and especially when the number of nuisance parameters is appreciable.