  Many model selection algorithms produce a path of fits specifying a sequence
of increasingly complex models. Given such a sequence and the data used to
produce them, we consider the problem of choosing the least complex model that
is not falsified by the data. Extending the selected-model tests of Fithian et
al. (2014), we construct p-values for each step in the path which account for
the adaptive selection of the model path using the data. In the case of linear
regression, we propose two specific tests, the max-t test for forward stepwise
regression (generalizing a proposal of Buja and Brown (2014)), and the
next-entry test for the lasso. These tests improve on the power of the
saturated-model test of Tibshirani et al. (2014), sometimes dramatically. In
addition, our framework extends beyond linear regression to a much more general
class of parametric and nonparametric model selection problems.
  To select a model, we can feed our single-step p-values as inputs into
sequential stopping rules such as those proposed by G'Sell et al. (2013) and Li
and Barber (2015), achieving control of the familywise error rate or false
discovery rate (FDR) as desired. The FDR-controlling rules require the null
p-values to be independent of each other and of the non-null p-values, a
condition not satisfied by the saturated-model p-values of Tibshirani et al.
(2014). We derive intuitive and general sufficient conditions for independence,
and show that our proposed constructions yield independent p-values.
