� _Gg_ =1<sup>_ng_.Weviewclustersizesasnon-random(seeBugnietal.,2023;SasakiandWang,2022,for</sup> an alternative sampling approach where cluster sizes are random). In a partial population experiment, clusters are randomly divided into categories or _saturations_ denoted by _Tg ∈{_ 0 _,_ 1 _,_ 2 _, . . . , M }_ , where by convention _Tg_ = 0 denotes a pure control cluster (i.e. a cluster where no unit is treated). Let P[ _Tg_ = _t_ ] = _qt ∈_ (0 _,_ 1) denote the probability that cluster _g_ is assigned to saturation _t_ . Within each cluster, a binary treatment _Dig_ is assigned to units with probability P[ _Dig_ = 1 _|Tg_ = _t_ ] where P[ _Dig_ = 0 _|Tg_ = 0] = 1.<sup>2</sup> We let **D** _g_ = ( _D_ 1 _g, D_ 2 _g, . . . , Dngg_ )<sup>_′_</sup> be the vector of unit-level treatment assignments in cluster _g_ , **D** = ( **D**<sup>_′_</sup> 1<sup>_, . . . ,_</sup><sup>**D**</sup><sup>_′_</sup> _G_<sup>)</sup><sup>_′_and</sup><sup>**T**=(</sup><sup>_T_1</sup><sup>_, . . . , TG_)</sup><sup>_′_.FigureA.3providesanexampleofapartialpopulationdesign</sup> with four saturations. Notice that both standard RCTs with independent observations and cluster RCTs are particular cases of partial population experiments, as we further illustrate in Section 3.5.

The observed outcome of interest for unit _i_ in cluster _g_ is denoted by _Yig_ and we let **Y** _g_ = ( _Y_ 1 _g, . . . , Yngg_ )<sup>_′_</sup> be the vector of observed outcomes in cluster _g_ . In partial population experiments, the estimands of interest are typically comparisons of average outcomes between treated or untreated units in treated clusters to pure control units, E[ _Yig|Dig_ = _d, Tg_ = _t_ ] _−_ E[ _Yig|Tg_ = 0], pooled across clusters. In the first part of the paper, we take these estimands as given since they are the most commonly analyzed estimands in the empirical literature. In Section 3.6, we set up a potential outcomes framework to rigorously justify the causal interpretation of these estimands. Let _μg_ ( _d, t_ ) = E[ _Yig|Dig_ = _d, Tg_ = _t_ ] be the conditional expectation of the outcome in cluster _g_ given assignment ( _d, t_ ). We consider the following sample means estimators:

where 1<sup>_t_</sup> _g_<sup>=1(</sup><sup>_Tg_=</sup><sup>_t_),</sup><sup>_N d_</sup> _g_<sup>=�</sup> _i_<sup>1(</sup><sup>_D_</sup> _ig_<sup>=</sup><sup>_d_)and</sup><sup>_Y_ ̄</sup> _g_<sup>_d_=�</sup> _i_<sup>_Y_</sup> _ig_<sup>1(</sup><sup>_D_</sup> _ig_<sup>=</sup><sup>_d_)</sup><sup>_/N_</sup> _g_<sup>_d_,definedwhenever</sup> _Ng_<sup>_d>_0.TheseestimatorsarecommonlycomputedbyrunninganOLSregressionoftheoutcomeon</sup> a full set of indicators ( 1 ( _Dig_ = _d, Tg_ = _t_ ))( _d,t_ ), without an intercept. Thus, in what follows, we refer to these estimators as OLS estimators. Our parameter of interest is the vector of cluster-size-weighted average of cluster-specific differences in means: We note that our framework can easily accommodate other parameters with different weighting schemes, such as the simple average across clusters<sup>�</sup><sup>_G_</sup> _g_ =1<sup>(</sup><sup>_μg_(</sup><sup>_d, t_)</sup><sup>_−μg_(0</sup><sup>_,_0))</sup><sup>_/G_.</sup>

> 2In practice, some desired saturations may not coincide with the observed proportion of treated units for some cluster sizes. For instance, if P[ _Dig_ = 1 _|Tg_ = _t_ ]=0.5 but _ng_ is odd, the observed proportion of treated cannot be exactly 0.5. Appendix A.4 proposes an assignment mechanism that ensures that the expected proportion of treated coincides with P[ _Dig_ = 1 _|Tg_ = _t_ ].

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