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The basic case<ref name="Rosenbaum 1983 41–55" /> is of two treatments (numbered 1 and 0), with ''N'' [Independent and identically distributed random variables|i.i.d] subjects. Each subject ''i'' would respond to the treatment with <math>r_{1i}</math> and to the control with <math>r_{0i}</math>. The quantity to be estimated is the [[average treatment effect]]: <math>E[r_1]-E[r_0]</math>. The variable <math>Z_i</math> indicates if subject ''i'' got treatment (''Z'' = 1) or control (''Z'' = 0). Let <math>X_i</math> be a vector of observed pretreatment measurement (or covariate) for the ''i''th subject. The observations of <math>X_i</math> are made prior to treatment assignment, but the features in <math>X_i</math> may not include all (or any) of the ones used to decide on the treatment assignment. The numbering of the units (i.e.: ''i'' = 1, ..., ''i'' = ''N'') are assumed to not contain any information beyond what is contained in <math>X_i</math>. The following sections will omit the ''i'' index while still discussing about the stochastic behavior of some subject.
The basic case<ref name="Rosenbaum 1983 41–55" /> is of two treatments (numbered 1 and 0), with ''N'' [Independent and identically distributed random variables|i.i.d] subjects. Each subject ''i'' would respond to the treatment with <math>r_{1i}</math> and to the control with <math>r_{0i}</math>. The quantity to be estimated is the [[average treatment effect]]: <math>E[r_1]-E[r_0]</math>. The variable <math>Z_i</math> indicates if subject ''i'' got treatment (''Z'' = 1) or control (''Z'' = 0). Let <math>X_i</math> be a vector of observed pretreatment measurement (or covariate) for the ''i''th subject. The observations of <math>X_i</math> are made prior to treatment assignment, but the features in <math>X_i</math> may not include all (or any) of the ones used to decide on the treatment assignment. The numbering of the units (i.e.: ''i'' = 1, ..., ''i'' = ''N'') are assumed to not contain any information beyond what is contained in <math>X_i</math>. The following sections will omit the ''i'' index while still discussing about the stochastic behavior of some subject.
基本场景<ref name="Rosenbaum 1983 41–55" />是，有两种处理方式（分别记为1和0），''N''个[[Independent and identically distributed random variables|独立同分布]]物体。每个物体''i''如果接受了处理则响应为<math>r_{1i}</math>，接受控制则响应为<math>r_{0i}</math>。被估计量是[[average treatment effect|平均处理效应]]：<math>E[r_1]-E[r_0]</math>。变量<math>Z_i</math>指示物体''i''接受处理(''Z'' = 1)还是接受控制(''Z'' = 0)。让<math>X_i</math>代表第''i''个物体处理前观测值（或者协变量）的向量。对<math>X_i</math>的测量发生于处理前，但是<math>X_i</math>中也可以不包括那些决定是否接受处理的特征。单元编号（即：''i'' = 1, ..., ''i'' = ''N''）不包含任何<math>X_i</math>所包含信息之外的的信息。以下部分在讨论某些物体的随机行为的时候将省略索引''i''。
基本场景<ref name="Rosenbaum 1983 41–55" />是，有两种处理方式（分别记为1和0），''N''个[[Independent and identically distributed random variables|独立同分布]]个体。每个个体''i''如果接受了处理则响应为<math>r_{1i}</math>，接受控制则响应为<math>r_{0i}</math>。被估计量是[[average treatment effect|平均处理效应]]：<math>E[r_1]-E[r_0]</math>。变量<math>Z_i</math>指示个体''i''接受处理(''Z'' = 1)还是接受控制(''Z'' = 0)。让<math>X_i</math>代表第''i''个个体处理前观测值（或者协变量）的向量。对<math>X_i</math>的测量发生于处理前，但是<math>X_i</math>中也可以不包括那些决定是否接受处理的特征。单元编号（即：''i'' = 1, ..., ''i'' = ''N''）不包含任何<math>X_i</math>所包含信息之外的的信息。以下部分在讨论某些物体的随机行为的时候将省略索引''i''。