  The meaning of randomization tests has become obscure in statistics education
and practice over the last century. This article makes a fresh attempt at
rectifying this core concept of statistics. A new term -- "quasi-randomization
test" -- is introduced to define significance tests based on theoretical models
and distinguish these tests from the "randomization tests" based on the
physical act of randomization. The practical importance of this distinction is
illustrated through a real stepped-wedge cluster-randomized trial. Building on
the recent literature of randomization inference, a general framework of
conditional randomization tests is developed and some practical methods to
construct conditioning events are given. The proposed terminology and framework
are then applied to understand several widely used (quasi-)randomization tests,
including Fisher's exact test, permutation tests for treatment effect,
quasi-randomization tests for independence and conditional independence,
adaptive randomization, and conformal prediction.

Methodology