  Ever since the seminal work of R. A. Fisher and F. Yates, factorial designs
have been an important experimental tool to simultaneously estimate the effects
of multiple treatment factors. In factorial designs, the number of treatment
combinations grows exponentially with the number of treatment factors, which
motivates the forward selection strategy based on the sparsity, hierarchy, and
heredity principles for factorial effects. Although this strategy is intuitive
and has been widely used in practice, its rigorous statistical theory has not
been formally established. To fill this gap, we establish design-based theory
for forward factor selection in factorial designs based on the potential
outcome framework. We not only prove a consistency property for the factor
selection procedure but also discuss statistical inference after factor
selection. In particular, with selection consistency, we quantify the
advantages of forward selection based on asymptotic efficiency gain in
estimating factorial effects. With inconsistent selection in higher-order
interactions, we propose two strategies and investigate their impact on
subsequent inference. Our formulation differs from the existing literature on
variable selection and post-selection inference because our theory is based
solely on the physical randomization of the factorial design and does not rely
on a correctly specified outcome model.
