  Neyman (1923/1990) introduced the randomization model, which contains the
notation of potential outcomes to define causal effects and a framework for
large-sample inference based on the design of the experiment. However, the
existing theory for this framework is far from complete especially when the
number of treatment levels diverges and the treatment group sizes vary. We
provide a unified discussion of statistical inference under the randomization
model with general treatment group sizes. We formulate the estimator in terms
of a linear permutational statistic and use results based on Stein's method to
derive various Berry--Esseen bounds on the linear and quadratic functions of
the estimator. These new Berry--Esseen bounds serve as basis for design-based
causal inference with possibly diverging treatment levels and a diverging
number of causal parameters of interest. We also fill an important gap by
proposing novel variance estimators for experiments with possibly many
treatment levels without replications. Equipped with the newly developed
results, design-based causal inference in general settings becomes more
convenient with stronger theoretical guarantees.
