“中介分析”的版本间的差异

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中介效应分析
  
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[[File:Simple Mediation Model.png|thumb|Simple Mediation Model]]
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在统计学中，中介模型试图通过引入第三个假设变量，即中介变量(也称为中介变量、中介变量或中介变量)，来识别和解释自变量与因变量之间观察到的关系的基础机制或过程。与自变量和因变量之间的直接因果关系不同，中介模型所描绘的图景是自变量通过影响中介变量（不可观测）进而影响因变量。因此，中介变量的作用是澄清自变量和因变量之间关系的本质[2]。Baron and Kenny（1986）提出的中介效应(mediation)框架（简称BK框架）在社会心理和消费者行为等诸多社会科学研究中产生了十分深远的影响。基于回归的分析的传统 BK 框架存在一些局限性。例如，Zhao et al.（2010）指出了BK框架存在的三点问题：第一，直接效应的缺失不应成为评价中介效应强度的标准；第二，寻找中介效应无需以X对Y存在显著的净效应为前提；第三，Sobel z检验的效力并不强，存在改进方式。近年来，基于现代因果模型的因果中介分析框架缓解了部分问题，成为了中介分析研究热点。
  
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Simple Mediation Model
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BK 框架下的中介效应分析
  
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简单调解模式
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Baron and Kenny (1986) 提出了形成一个真正的中介关系必须满足的几个条件如下：
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1）让因变量对自变量进行回归，以确认自变量是因变量的显著预测因子，即
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<math>Y=\beta _{{10}}+\beta _{{11}}X+\varepsilon _{1}</math>
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的回归系数$$β_{11}$$ 是显著的。
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2）让中介变量对自变量进行回归，确认自变量是中介变量的显著预测因子，即
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$$Me=\beta _{{20}}+\beta _{{21}}X+\varepsilon _{2}$$
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的回归系数 $$\beta_{21}$$是显著的。如果中介变量与自变量没有关联，那么它就不可能中介任何事物。
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3）让因变量对中介和自变量同时进行回归，即
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$$Y=\beta _{{30}}+\beta _{{31}}X+\beta _{{32}}Me+\varepsilon _{3}$$
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的回归系数 $$\beta_{32}$$是显著的，并且 $$\beta_{31}$$的绝对值应该小于自变量的效应 $$\beta_{11}$$。从而确保了中介变量是因变量的重要预测因子，并且使得相对于第一步，自变量对结果的解释性降低。
  
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In [[statistics]], a '''mediation''' model seeks to identify and explain the mechanism or process that underlies an observed relationship between an [[independent variable]] and a [[dependent variable]] via the inclusion of a third hypothetical variable, known as a '''mediator variable''' (also a '''mediating variable''', '''intermediary variable''', or '''intervening variable''').<ref>{{Cite web|title=Types of Variables|website=[[University of Indiana]]|url=http://www.indiana.edu/~educy520/sec5982/week_2/variable_types.pdf}}</ref> Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the (non-observable) mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.<ref>MacKinnon, D. P. (2008). ''Introduction to Statistical Mediation Analysis''. New York: Erlbaum.</ref>
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中介变量可以解释两个变量之间观察到的全部或部分关系，如果中介变量的加入使自变量和因变量之间的相关性降为零，则中介的证据最大，也称为完全中介(full mediation)。而部分中介(partial mediation)是指不仅中介变量与因变量之间存在显著的关系，而且自变量与因变量之间也存在某种直接的关系。
  
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In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable (also a mediating variable, intermediary variable, or intervening variable). Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the (non-observable) mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.
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我们采用Sobel’s test[10]来检验中介变量加入后自变量与因变量之间的关系是否显著降低，从而评估中介效应是否显著。然而，这种方式的统计效力（Power）很低。因此，为了有足够的效力检测显著性影响，需要大的样本量。这是因为Sobel检验的关键假设是正态性假设。因为Sobel检验是根据正态分布来评估给定样本的，所以样本规模小和抽样分布的偏态可能会有问题(详见正态分布)。因此，MacKinnon et al .,(2002)[12]所建议的经验法是，检测较小的效应需要1000个样本，检测中等效应需要100个样本，检测较大效应需要50个样本。基于自助法的检验能减少对样本量的依赖，见 Preacher and Hayes(2004)。
  
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在[[统计学]]中，中介模型试图通过加入第三个假设变量，即中介变量(也是中介变量、中介变量或中间变量) ，来识别和解释所观察到的自变量与因变量之间关系的底层机制或过程。一个中介模型没有在自变量和因变量之间建立直接的因果关系，而是提出自变量影响(不可观察的)调解变量，这反过来又影响因变量。因此，中介变量的作用是阐明自变量和因变量之间关系的性质。与自变量和因变量之间存在直接的因果关系不同，中介模型主张：自变量通过影响中介变量（不可观测），进而影响因变量。
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因果中介分析
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固定（fixing）与条件化（conditioning）
  
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中介分析量化了变量参与从原因到其结果的变化传递的程度。它本质上是一个因果概念，因此不能用统计术语来定义。然而，传统上，大量的中介分析是在线性回归的范畴内进行的。统计术语掩盖了所涉及关系的因果特征，这导致了一些困难、偏差（biases）和局限性（limitations）。而基于因果图(causal diagrams)和反事实逻辑的现代因果分析方法缓解了这些困难、偏见和限制。
  
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这些困难的根源在于，在方法上，根据在回归方程中添加第三个变量所引起的变化来定义中介。虽然这种统计上的变化是伴随中介效应而来的附带现象。但这样的做法未能充分捕捉到中介分析的本质，即量化因果关系。
  
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Mediation analyses are employed to understand a known relationship by exploring the underlying mechanism or process by which one variable influences another variable through a mediator variable.<ref name=CCWA>Cohen, J.; Cohen, P.; West, S. G.; [[Leona S. Aiken|Aiken, L. S.]] (2003) ''Applied multiple regression/correlation analysis for the behavioral sciences'' (3rd ed.). Mahwah, NJ: Erlbaum.</ref> In particular, mediation analysis can contribute to better understanding the relationship between an independent variable and a dependent variable when these variables do not have an obvious direct connection.
 
  
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中介分析，通过探索一个变量通过中介影响其他变量背后的机制或程序，来理解一个已知关系。特别是当变量之间没有明显的直接连接时，中介分析可以更好的理解因变量与自变量之间的关系。
 
  
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因果方法的基本前提是，当我们试图估计自变量 X 对 因变量 Y 的直接影响时，并不总是适合对中介M进行“控制”(见上图)。对M进行“控制”的经典理论是，如果我们成功地阻止了M的变化，那么我们在Y中测量的任何变化都只能归因于X的变化，然后我们就有理由宣布观察到的效果是“X对Y的直接影响”。不幸的是，“控制M”并不能从物理上阻止M的改变；它只是把分析者的注意力集中在相等 M 值的情况下。而且，概率论的语言没有表示“阻止M改变”或“物理上保持M不变”的符号。唯一的运算是“以…为条件”（conditioning)，这是当我们“控制” M 时所做的。或者为 Y 的方程添加 M 作为其中的一个回归变量。 结果是，与在物理上保持 M 不变（例如 M = m ）并将 X = 1 下 Y 的单位 与 X = 0 下 Y 的单位进行比较的方法不同，我们允许 M 变化但忽略所有使得 M=m 的其他单位。这两个操作除了没有遗漏变量的情况，本质上是不同的，产生不同的结果[21][22]。
  
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==Baron and Kenny's (1986) steps for mediation==
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举例来说，假设 M 和 Y 的误差项是相关的。在这种情况下，通过对 Y 在 X 和 M 上进行回归，就无法对结构系数 B 和 A(在M和Y之间，在Y和X之间) 进行估计。事实上，即使当 C 等于 0 的时候，回归斜率也可能不等于 0 。这有两种后果。首先必须设计新的策略来估计结构系数 A、B 和 C。其次，直接和间接效应的基本定义必须超越回归分析，并且应该采用类似于“固定 M”的操作，而不是“在 M 的条件下”的操作。
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数学定义
  
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Baron and Kenny (1986) 的中介步骤
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Pearl(1994)[22]中定义了这样一个运算符 $$do(M = m)$$，它的作用是去除 M 的方程，代之以一个常数 m。例如，如果基本中介模型由以下方程组成:
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$$ {\displaystyle X=f(\varepsilon _{1}),M=g(X,\varepsilon _{2}),Y=h(X,M,\varepsilon _{3}),}$$
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那么应用了$$do(M = m)$$运算的模型将会变为：
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$${\displaystyle X=f(\varepsilon _{1}),M=m,Y=h(X,m,\varepsilon _{3})}$$
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同时，应用了$$do(X = x)$$ 运算的模型会变为：
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$${\displaystyle X=x,M=g(x,\varepsilon _{2}),Y=h(x,M,\varepsilon _{3})}$$
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其中函数 f 和 g 以及误差项 ε1 和 ε3 的分布保持不变。如果我们进一步将 $$do(X = x)$$ 得到的变量 $$M$$ 和 $$Y$$ 分别重新命名为 $$M(x)$$ 和 $$Y(x)$$ ，我们得到了所谓的“潜在结果(potential outcome)”[24]或“结构反事实(structural counterfactuals)”[25]这些新变量为定义直接和间接效应提供了便利的描述符号。具体来说，定义了从 $$X = 0$$ 到 $$X = 1$$ 变化的四种效应:
  
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Baron and Kenny (1986) <ref>Baron, R. M. and Kenny, D. A. (1986) "The Moderator-Mediator Variable Distinction in Social Psychological Research &ndash; Conceptual, Strategic, and Statistical Considerations", [[Journal of Personality and Social Psychology]], Vol. 51(6), pp.&nbsp;1173&ndash;1182.</ref> laid out several requirements that must be met to form a true mediation relationship. They are outlined below using a real-world example. See the diagram above for a visual representation of the overall mediating relationship to be explained. Note: Hayes (2009)<ref name=Hayes/> critiqued Baron and Kenny's mediation steps approach, and as of 2019, [[David A. Kenny]] on his website stated that mediation can exist in the absence of a 'significant' total effect, and therefore step 1 below may not be needed. This situation is sometimes referred to as "inconsistent mediation". Later publications by Hayes also questioned the concepts of full or partial mediation and advocated for these terms, along with the classical mediation steps approach outlined below, to be abandoned.
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(a) 总体效应 –
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$$TE=E[Y(1)-Y(0)]$$
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(b) 受控直接效应 -
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$$CDE(m)=E[Y(1,m)-Y(0,m)]$$
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(c) 自然直接效应 -
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$$NDE=E[Y(1,M(0))-Y(0,M(0))]$$
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(d) 自然间接效应
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$$NIE = E [Y(0,M(1)) - Y(0,M(0))] $$
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其中 $$E[\cdot ]$$ 表示对误差项的期望，这些效应有如下一些解释：
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- $$TE$$ 表示的 $$X$$对 $$Y$$的总体因果效应。
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- $$CDE$$ 表示在某个条件 $$M=m$$下，$$X$$对 $$Y$$的因果效应。
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- $$NDE$$ 表示  $$X$$对 $$Y$$的直接产生的因果效应。
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- $$NIE$$ 表示  $$X$$对 $$Y$$的通过中介变量 $$M$$产生的因果效应。
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- 对于解释 $$X$$和 $$Y$$之间的效应，两个效应的差$$TE-NDE$$ 度量的是中介变量在何种程度上是必要的。而 $$NIE$$ 度量的是引入中介变量在充分性。
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间接效应的受控版本并不存在，因为没有办法通过将一个变量固定到一个常量来屏蔽直接效应。
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根据这些定义，总体效应可以如下分解
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$$TE=NDE-NIE_{r}$$
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其中 $$NIE_r$$ 表示在 $$NIE$$ 的定义中进行 $$X = 1$$ 到 $$X = 0$$ 的反向转换；线性系统中总体效应等于直接效应与间接效应之和，即负的反转间接效应等于间接效应 $$-NIE_r = NIE$$。这些定义的力量在于它们的普适性；它们适用于具有任意非线性相互作用，任意干扰之间的依赖关系，以及连续变量和离散变量的模型。
  
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Baron and Kenny (1986)  laid out several requirements that must be met to form a true mediation relationship. They are outlined below using a real-world example. See the diagram above for a visual representation of the overall mediating relationship to be explained. Note: Hayes (2009) explains each step of Baron and Kenny's requirements to understand further how a mediation effect is characterized. Step 1 and step 2 use simple regression analysis, whereas step 3 uses multiple regression analysis.
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中介效应公式
  
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Baron and Kenny (1986) 提出了要形成一个真正的中介关系必需满足的几个条件。下面使用一个真实的例子来概述它们。
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在线性分析中，所有的效应由结构系数的乘积决定，给出
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$${\displaystyle {\begin{aligned}TE&=C+AB\\CDE(m)&=NDE=C,{\text{ independent of }}m\\NIE&=AB.\end{aligned}}}$$
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因此，当模型被识别时，所有的效应都是可估计的。在非线性系统中，估计直接和间接效应需要更严格的条件，如不存在混杂因子(即 $$ε_1、ε_2、ε_3$$ 相互独立)，可推导出如下公式
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$${\displaystyle {\begin{aligned}TE&=E(Y\mid X=1)-E(Y\mid X=0)\\CDE(m)&=E(Y\mid X=1,M=m)-E(Y\mid X=0,M=m)\\NDE&=\sum _{m}[E(Y|X=1,M=m)-E(Y\mid X=0,M=m)]P(M=m\mid X=0)\\NIE&=\sum _{m}[P(M=m\mid X=1)-P(M=m\mid X=0)]E(Y\mid X=0,M=m).\end{aligned}}}$$
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后两个方程被称为中介公式[28][29][30]，已成为许多中介研究的估计对象。他们给出了直接和间接效应的无分布假设(distribution-free)表达式，并证明，尽管误差分布和函数 f, g, h 的性质难以确定，中介效应仍然可以通过使用回归方法利用数据来估计。调节中介和中介调节的分析属于因果中介分析的特例。中介公式确定了各种相互作用系数如何贡献于中介的必要和充分成分。
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简单案例
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假设模型采用这种形式
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$${\displaystyle {\begin{aligned}X&=\varepsilon _{1}\\M&=b_{0}+b_{1}X+\varepsilon _{2}\\Y&=c_{0}+c_{1}X+c_{2}M+c_{3}XM+\varepsilon _{3}\end{aligned}}}$$
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其中，参数 $$c_{3}$$ 量化了 M 对 X 对 Y 的影响的修正程度。即使所有参数都是从数据中估计出来的，仍然不清楚是哪些参数组合度量了 X 对 Y 的直接和间接影响，或者，更实际的是，如何评估由中介解释的总体效应 TE 的比例以及应归功于中介效应的 TE 的比例。在线性分析中，前者被 $$b_{1}c_{2}/TE$$ 所捕获，后者被差值 $$(TE-c_{1})/TE$$ 所捕获，并且这两个量重合。然而，在存在交互的情况下，每个部分都需要单独的分析。如中介公式所规定的那样，其结果是:
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$${\begin{aligned}NDE&=c_{1}+b_{0}c_{3}\\NIE&=b_{1}c_{2}\\TE&=c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})\\&=NDE+NIE+b_{1}c_{3}.\end{aligned}}$$
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因此，对于中介变量来说足够输出的部分是
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$${\displaystyle {\frac {NIE}{TE}}={\frac {b_{1}c_{2}}{c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})}},}$$
  
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请参阅上面的图表，它可视化了即将被解释的整体中介关系。注:Hayes(2009)[5]批评了Baron和Kenny的中介步骤方法，截至2019年，David a . Kenny在他的网站上表示，中介可以存在于没有“显著的”总体效应的情况下，因此下面的步骤1可能不需要。
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而需要中介的部分是
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$${\displaystyle 1-{\frac {NDE}{TE}}={\frac {b_{1}(c_{2}+c_{3})}{c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})}}.}$$
  
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这种情况有时被称为“不一致的中介”。后来Hayes在他的著作中也对完全或部分中介的概念提出了质疑，并主张放弃这些术语，以及下面概述的经典中介步骤方法。
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这些分数涉及模型参数的微妙的组合，并且可以在中介公式的帮助下机械地构造。值得注意的是，由于交互作用，即使参数 $$c_{1}$$ 为 0，直接效应也可以存在。而且，即使直接和间接效应都为 0，总效应也可以存在。 这说明孤立地估计参数几乎无法告诉我们中介的效果。更一般地说，中介和调节是交织在一起的，不能分开评估。
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'''Step 1:'''
 
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第一步:
 
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:Regress the dependent variable on the independent variable to confirm that the independent variable is a significant predictor of the dependent variable.
 
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: Independent variable <math> \to </math> dependent variable
 
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:: <math>Y=\beta_{10} +\beta_{11}X + \varepsilon_1</math>
 
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* ''β''<sub>11</sub> is significant
 
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'''Step 2:'''
 
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第二步:
 
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:Regress the mediator on the independent variable to confirm that the independent variable is a significant predictor of the mediator. If the mediator is not associated with the independent variable, then it couldn’t possibly mediate anything.
 
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: Independent variable <math> \to </math> mediator
 
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:: <math>Me=\beta_{20} +\beta_{21}X + \varepsilon_2</math>
 
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* ''β''<sub>21</sub> is significant
 
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'''Step 3:'''
 
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Step 3:
 
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第三步:
 
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:Regress the dependent variable on both the mediator and independent variable to confirm that a) the mediator is a significant predictor of the dependent variable, and b) the strength of the coefficient of the previously significant independent variable in Step #1 is now greatly reduced, if not rendered nonsignificant.
 
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:: <math>Y=\beta_{30} +\beta_{31}X +\beta_{32}Me + \varepsilon_3</math>
 
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* ''β''<sub>32</sub> is significant
 
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* ''β''<sub>31</sub> should be smaller in absolute value than the original effect for the independent variable (β<sub>11</sub> above)
 
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'''Example'''
 
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The following example, drawn from Howell (2009),<ref>Howell, D. C. (2009). Statistical methods for psychology (7th ed.). Belmot, CA: Cengage Learning.</ref> explains each step of Baron and Kenny's requirements to understand further how a mediation effect is characterized. Step 1 and step 2 use simple regression analysis, whereas step 3 uses [[multiple regression analysis]].
 
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Step 1:
 
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:How you were parented (i.e., independent variable) predicts how confident you feel about parenting your own children (i.e., dependent variable).
 
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:How you were parented <math> \to </math> confidence in own parenting abilities.
 
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Step 2:
 
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:How you were parented (i.e., independent variable) predicts your feelings of competence and self-esteem (i.e., mediator).
 
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:How you were parented <math> \to </math> Feelings of competence and self-esteem.
 
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Step 3:
 
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:Your feelings of competence and self-esteem (i.e., mediator) predict how confident you feel about parenting your own children (i.e., dependent variable), while controlling for how you were parented (i.e., independent variable).
 
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Such findings would lead to the conclusion implying that your feelings of competence and self-esteem mediate the relationship between how you were parented and how confident you feel about parenting your own children.
 
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Note: If step 1 does not yield a significant result, one may still have grounds to move to step 2. Sometimes there is actually a significant relationship between independent and dependent variables but because of small sample sizes, or other extraneous factors, there could not be enough power to predict the effect that actually exists (See Shrout & Bolger, 2002 <ref>{{cite journal | last1 = Shrout | first1 = P. E. | last2 = Bolger | first2 = N. | year = 2002 | title = Mediation in experimental and nonexperimental studies: New procedures and recommendations | journal = Psychological Methods | volume = 7 | issue = 4| pages = 422–445 | doi=10.1037/1082-989x.7.4.422}}</ref> for more info).
 
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注意: 如果步骤1没有产生显著的结果，一个人可能仍然有理由移动到步骤2。有时，独立变量和因变量之间确实存在显著的关系，但是由于样本量小，或者其他额外的因素，没有足够的能量来预测实际存在的影响(参见 Shrout & Bolger，2002)。
 
−
 
 
−
==Direct versus indirect effects==
 
−
[[File:Direct Effect in a Mediation Model.jpg|thumb|Direct Effect in a Mediation Model]]
 
−
In the diagram shown above, the indirect effect is the product of path coefficients "A" and "B". The direct effect is the coefficient "&nbsp;C'&nbsp;".
 
−
The direct effect measures the extent to which the dependent variable changes when the independent variable increases by one unit and the mediator variable remains unaltered. In contrast, the indirect effect measures the extent to which the dependent variable changes when the independent variable is held fixed and the mediator variable changes by the amount it would have changed had the independent variable increased by one unit.<ref name="Robins"/><ref name="Pearl-01"/>
 
−
[[File:Mediation Model.png|thumb|Indirect Effect in a Simple Mediation Model: The indirect effect constitutes the extent to which the X variable influences the Y variable through the mediator.]]
 
−
In linear systems, the total effect is equal to the sum of the direct and indirect  (''C' + AB'' in the model above). In nonlinear models, the total effect is not generally equal to the sum of the direct and indirect effects, but to a modified combination of the two.<ref name="Pearl-01"/>
 
−
 
 
−
==Full versus partial mediation==
 
−
 
 
−
A mediator variable can either account for all or some of the observed relationship between two variables.
 
−
 
 
−
'''Full mediation'''
 
−
 
 
−
Maximum evidence for mediation, also called full mediation, would occur if inclusion of the mediation variable drops the relationship between the independent variable and dependent variable (see pathway ''c'' in diagram above) to zero.
 
−
[[File:Full Mediation Model.png|thumb|Full Mediation Model]]
 
−
'''Partial mediation'''
 
−
[[File:Mediation.jpg|thumb|The Partial Mediation Model Includes a Direct Effect]]
 
−
Partial mediation maintains that the mediating variable accounts for some, but not all, of the relationship between the independent variable and dependent variable. Partial mediation implies that there is not only a significant relationship between the mediator and the dependent variable, but also some direct relationship between the independent and dependent variable.
 
−
 
 
−
In order for either full or partial mediation to be established, the reduction in variance explained by the independent variable must be significant as determined by one of several tests, such as the [[Sobel test]].<ref name="Sobel, M. E. 1982 pp. 290">{{cite journal | last1 = Sobel | first1 = M. E. | year = 1982 | title = Asymptotic confidence intervals for indirect effects in structural equation models |  journal = Sociological Methodology | volume = 13 | pages = 290–312 | doi = 10.2307/270723 | jstor = 270723 }}</ref> The effect of an independent variable on the dependent variable can become nonsignificant when the mediator is introduced simply because a trivial amount of variance is explained (i.e., not true mediation). Thus, it is imperative to show a significant reduction in variance explained by the independent variable before asserting either full or partial mediation.
 
−
It is possible to have statistically significant indirect effects in the absence of a total effect.<ref name=Hayes>{{cite journal | last1 = Hayes | first1 = A. F. | year = 2009 | title = Beyond Baron and Kenny: Statistical mediation analysis in the new millennium |  journal = Communication Monographs | volume = 76 | issue = 4| pages = 408–420 | doi = 10.1080/03637750903310360 }}</ref>  This can be explained by the presence of several mediating paths that cancel each other out, and become noticeable when one of the cancelling mediators is controlled for. This implies that the terms 'partial' and 'full' mediation should always be interpreted relative to the set of variables that are present in the model.
 
−
In all cases, the operation of "fixing a variable" must be distinguished from that of "controlling for a variable," which has been inappropriately used in the literature.<ref name="Robins"/><ref name="Kaufman"/> The former stands for physically fixing, while the latter stands for conditioning on, adjusting for, or adding to the regression model. The two notions coincide only when all error terms (not shown in the diagram) are statistically uncorrelated. When errors are correlated, adjustments must be made to neutralize those correlations before embarking on mediation analysis (see [[Bayesian Networks]]).
 
−
 
 
−
==Sobel's test==
 
−
{{main|Sobel test}}
 
−
 
 
−
As mentioned above, [[Sobel test|Sobel's test]]<ref name="Sobel, M. E. 1982 pp. 290"/> is performed to determine if the relationship between the independent variable and dependent variable has been significantly reduced after inclusion of the mediator variable. In other words, this test assesses whether a mediation effect is significant. It examines the relationship between the independent variable and the dependent variable compared to the relationship between the independent variable and dependent variable including the mediation factor.
 
−
 
 
−
The Sobel test is more accurate than the Baron and Kenny steps explained above; however, it does have low statistical power. As such, large sample sizes are required in order to have sufficient power to detect significant effects. This is because the key assumption of Sobel's test is the assumption of normality. Because Sobel's test evaluates a given sample on the normal distribution, small sample sizes and skewness of the sampling distribution can be problematic (see [[Normal distribution]] for more details). Thus, the rule of thumb as suggested by MacKinnon et al., (2002) <ref>{{cite journal | last1 = MacKinnon | first1 = D. P. | last2 = Lockwood | first2 = C. M. | last3 = Lockwood | first3 = J. M. | last4 = West | first4 = S. G. | last5 = Sheets | first5 = V. | year = 2002 | title = A comparison of methods to test mediation and other intervening variable effects |  journal =  Psychological Methods| volume = 7 | issue = 1| pages = 83–104 | doi=10.1037/1082-989x.7.1.83| pmid = 11928892 | pmc=2819363}}</ref> is that a sample size of 1000 is required to detect a small effect, a sample size of 100 is sufficient in detecting a medium effect, and a sample size of 50 is required to detect a large effect.
 
−
 
 
−
==Preacher and Hayes (2004) bootstrap method==
 
−
The bootstrapping method provides some advantages to the Sobel's test, primarily an increase in power. The Preacher and Hayes Bootstrapping method is a non-parametric test (See [[Non-parametric statistics]] for a discussion on non-parametric tests and their power). As such, the bootstrap method does not violate assumptions of normality and is therefore recommended for small sample sizes.
 
−
Bootstrapping involves repeatedly randomly sampling observations with replacement from the data set to compute the desired statistic in each resample. Computing over hundreds, or thousands, of bootstrap resamples provide an approximation of the sampling distribution of the statistic of interest. Hayes offers a macro <http://www.afhayes.com/> that calculates bootstrapping directly within [[SPSS]], a computer program used for statistical analyses. This method provides point estimates and confidence intervals by which one can assess the significance or nonsignificance of a mediation effect. Point estimates reveal the mean over the number of bootstrapped samples and if zero does not fall between the resulting confidence intervals of the bootstrapping method, one can confidently conclude that there is a significant mediation effect to report.-
 
−
 
 
−
==Significance of mediation==
 
−
 
 
−
As outlined above, there are a few different options one can choose from to evaluate a mediation model.
 
−
 
 
−
[[Bootstrapping (statistics)|Bootstrapping]]<ref>{{cite web |url=http://www.comm.ohio-state.edu/ahayes/sobel.htm |title=Testing of Mediation Models in SPSS and SAS |publisher=Comm.ohio-state.edu |access-date=2012-05-16 |archive-url=https://web.archive.org/web/20120518234943/http://www.comm.ohio-state.edu/ahayes/sobel.htm |archive-date=2012-05-18 |url-status=dead }}</ref><ref>{{cite web|url=http://www.comm.ohio-state.edu/ahayes/SPSS%20programs/indirect.htm |title=SPSS and SAS Macro for Bootstrapping Specific Indirect Effects in Multiple Mediation Models |publisher=Comm.ohio-state.edu |access-date=2012-05-16}}</ref> is becoming the most popular method of testing mediation because it does not require the normality assumption to be met, and because it can be effectively utilized with smaller sample sizes (''N''&nbsp;<&nbsp;25). However, mediation continues to be most frequently determined using the logic of Baron and Kenny <ref>[http://davidakenny.net/cm/mediate.htm "Mediation"]. ''davidakenny.net''. Retrieved April 25, 2012.</ref> or the [[Sobel test]]. It is becoming increasingly more difficult to publish tests of mediation based purely on the Baron and Kenny method or tests that make distributional assumptions such as the Sobel test. Thus, it is important to consider your options when choosing which test to conduct.<ref name=Hayes/>
 
−
 
 
−
==Approaches to mediation==
 
−
 
 
−
While the concept of mediation as defined within psychology is theoretically appealing, the methods used to study mediation empirically have been challenged by statisticians and epidemiologists<ref name="Robins">{{cite journal | last1 = Robins | first1 = J. M. | author-link = James Robins | author-link2 = Sander Greenland | last2 = Greenland | first2 = S. | year = 1992 | title = Identifiability and exchangeability for direct and indirect effects |  journal = Epidemiology | volume = 3 | issue = 2| pages = 143–55 | doi = 10.1097/00001648-199203000-00013 | pmid = 1576220 }}</ref><ref name="Kaufman">{{cite journal|pmc=526390|doi=10.1186/1742-5573-1-4|year=2004|last1=Kaufman|first1=J. S.|title=A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation|journal=Epidemiologic Perspectives & Innovations |volume=1|issue=1|pages=4|last2=MacLehose|first2=R. F.|last3=Kaufman|first3=S|pmid=15507130}}</ref><ref name="Bullock">{{cite journal|pmid=20307128|url=http://www2.psych.ubc.ca/~schaller/528Readings/BullockGreenHa2010.pdf|year=2010|last1=Bullock|first1=J. G.|title=Yes, but what's the mechanism? (don't expect an easy answer)|journal=Journal of Personality and Social Psychology|volume=98|issue=4|pages=550–8|last2=Green|first2=D. P.|last3=Ha|first3=S. E.|doi=10.1037/a0018933}}
 
−
</ref> and interpreted formally.<ref name="Pearl-01">[[Judea Pearl|Pearl, J.]] (2001) [http://ftp.cs.ucla.edu/pub/stat_ser/R273-U.pdf "Direct and indirect effects"]. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, [[Morgan Kaufmann]], 411&ndash;420.</ref>
 
−
 
 
−
(1) Experimental-causal-chain design
 
−
 
 
−
An experimental-causal-chain design is used when the proposed mediator is experimentally manipulated. Such a design implies that one manipulates some controlled third variable that they have reason to believe could be the underlying mechanism of a given relationship.
 
−
 
 
−
(2) Measurement-of-mediation design
 
−
 
 
−
A measurement-of-mediation design can be conceptualized as a statistical approach. Such a design implies that one measures the proposed intervening variable and then uses statistical analyses to establish mediation. This approach does not involve manipulation of the hypothesized mediating variable, but only involves measurement.<ref>{{cite journal|pmid=16393019|url=http://www2.psych.ubc.ca/~schaller/528Readings/SpencerZannaFong2005.pdf|year=2005|last1=Spencer|first1=S. J.|title=Establishing a causal chain: Why experiments are often more effective than mediational analyses in examining psychological processes|journal=Journal of Personality and Social Psychology|volume=89|issue=6|pages=845–51|last2=Zanna|first2=M. P.|last3=Fong|first3=G. T.|doi=10.1037/0022-3514.89.6.845}}
 
−
</ref>
 
−
 
 
−
==Criticisms of mediation measurement==
 
−
 
 
−
Experimental approaches to mediation must be carried out with caution. First, it is important to have strong theoretical support for the exploratory investigation of a potential mediating variable.
 
−
A criticism of a mediation approach rests on the ability to manipulate and measure a mediating variable. Thus, one must be able to manipulate the proposed mediator in an acceptable and ethical fashion. As such, one must be able to measure the intervening process without interfering with the outcome. The mediator must also be able to establish construct validity of manipulation.
 
−
One of the most common criticisms of the measurement-of-mediation approach is that it is ultimately a correlational design. Consequently, it is possible that some other third variable, independent from the proposed mediator, could be responsible for the proposed effect. However, researchers have worked hard to provide counter-evidence to this disparagement. Specifically, the following counter-arguments have been put forward:<ref name=CCWA/>
 
−
 
 
−
(1) Temporal precedence. For example, if the independent variable precedes the dependent variable in time, this would provide evidence suggesting a directional, and potentially causal, link from the independent variable to the dependent variable.
 
−
 
 
−
(2) Nonspuriousness and/or no confounds. For example, should one identify other third variables and prove that they do not alter the relationship between the independent variable and the dependent variable he/she would have a stronger argument for their mediation effect. See other 3rd variables below.
 
−
 
 
−
Mediation can be an extremely useful and powerful statistical test; however, it must be used properly. It is important that the measures used to assess the mediator and the dependent variable are theoretically distinct and that the independent variable and mediator cannot interact. Should there be an interaction between the independent variable and the mediator one would have grounds to investigate [[moderation (statistics)|moderation]].
 
−
 
 
−
==Other third variables==
 
−
 
 
−
(1) Confounding:
 
−
 
 
−
:Another model that is often tested is one in which competing variables in the model are alternative potential mediators or an unmeasured cause of the dependent variable. An additional variable in a [[causal model]] may obscure or confound the relationship between the independent and dependent variables. Potential confounders are variables that may have a causal impact on both the independent variable and dependent variable. They include common sources of measurement error (as discussed above) as well as other influences shared by both the independent and dependent variables.
 
−
 
 
−
[[File:Mediation model with two covariates.jpg|thumb|Mediation model with two covariates]]
 
−
In experimental studies, there is a special concern about aspects of the experimental manipulation or setting that may account for study effects, rather than the motivating theoretical factor. Any of these problems may produce spurious relationships between the independent and dependent variables as measured. Ignoring a confounding variable may bias empirical estimates of the causal effect of the independent variable.
 
−
 
 
−
(2) Suppression:
 
−
 
 
−
:A suppressor variable increases the predictive validity of another variable when included in a regression equation. Suppression can occur when a single causal variable is related to an outcome variable through two separate mediator variables, and when one of those mediated effects is positive and one is negative. In such a case, each mediator variable suppresses or conceals the effect that is carried through the other mediator variable. For example, higher intelligence scores (a causal variable, ''A'') may cause an increase in error detection (a mediator variable, ''B'') which in turn may cause a decrease in errors made at work on an assembly line (an outcome variable, ''X''); at the same time, intelligence could also cause an increase in boredom (''C''), which in turn may cause an ''increase'' in errors (''X''). Thus, in one causal path intelligence decreases errors, and in the other it increases them. When neither mediator is included in the analysis, intelligence appears to have no effect or a weak effect on errors. However, when boredom is controlled intelligence will appear to decrease errors, and when error detection is controlled intelligence will appear to increase errors. If intelligence could be increased while only boredom was held constant, errors would decrease; if intelligence could be increased while holding only error detection constant, errors would increase.
 
−
 
 
−
In general, the omission of suppressors or confounders will lead to either an underestimation or an overestimation of the effect of ''A'' on ''X'', thereby either reducing or artificially inflating the magnitude of a relationship between two variables.
 
−
 
 
−
(3) Moderators:
 
−
 
 
−
:Other important third variables are moderators. Moderators are variables that can make the relationship between two variables either stronger or weaker. Such variables further characterize interactions in regression by affecting the direction and/or strength of the relationship between ''X'' and ''Y''. A moderating relationship can be thought of as an [[interaction effect|interaction]]. It occurs when the relationship between variables A and B depends on the level of C. See [[moderation (statistics)|moderation]] for further discussion.
 
−
 
 
−
==Moderated mediation==
 
−
 
 
−
Mediation and [[moderation (statistics)|moderation]] can co-occur in statistical models.  It is possible to mediate moderation and moderate mediation.
 
−
 
 
−
[[Moderated mediation]] is when the effect of the treatment ''A'' on the mediator and/or the partial effect ''B'' on the dependent variable depend in turn on levels of another variable (moderator). Essentially, in moderated mediation, mediation is first established, and then one investigates if the mediation effect that describes the relationship between the independent variable and dependent variable is moderated by different levels of another variable (i.e., a moderator). This definition has been outlined by Muller, Judd, and Yzerbyt (2005)<ref name="Muller">{{cite journal | last1 = Muller | first1 = D. | last2 = Judd | first2 = C. M. | last3 = Yzerbyt | first3 = V. Y. | year = 2005 | title = When moderation is mediated and mediation is moderated |  journal = Journal of Personality and Social Psychology | volume = 89 | issue = 6| pages = 852–863 | doi = 10.1037/0022-3514.89.6.852 | pmid = 16393020 }}</ref> and Preacher, Rucker, and Hayes (2007).<ref name="Preacher">Preacher, K. J., Rucker, D. D. & Hayes, A. F. (2007).  Assessing moderated mediation hypotheses: Strategies, methods, and prescriptions.  Multivariate Behavioral Research, 42, 185&ndash;227.</ref>
 
−
 
 
−
===Models of moderated mediation===
 
−
There are five possible models of moderated mediation, as illustrated in the diagrams below.<ref name="Muller" />
 
−
 
 
−
# In the first model the independent variable also moderates the relationship between the mediator and the dependent variable.
 
−
# The second possible model of moderated mediation involves a new variable which moderates the relationship between the independent variable and the mediator (the ''A'' path).
 
−
# The third model of moderated mediation involves a new moderator variable which moderates the relationship between the mediator and the dependent variable (the ''B'' path).
 
−
# Moderated mediation can also occur when one moderating variable affects both the relationship between the independent variable and the mediator (the ''A'' path) and the relationship between the mediator and the dependent variable (the  ''B'' path).
 
−
# The fifth and final possible model of moderated mediation involves two new moderator variables, one moderating the ''A'' path and the other moderating the ''B'' path.
 
−
 
 
−
{|
 
−
 
 
−
| [[File:Mediated moderation model 1.png|centre|thumb|
 
−
First option: independent variable moderates the ''B'' path.]]
 
−
| [[File:Mediated moderation model 2.png|centre|thumb|
 
−
Second option: fourth variable moderates the ''A'' path.]]
 
−
| [[File:Mediated moderation model 3.png|centre|thumb|
 
−
Third option: fourth variable moderates the ''B'' path.]]
 
−
| [[File:Mediated moderation model 4.png|centre|thumb|
 
−
Fourth option: fourth variable moderates both the ''A'' path and the ''B'' path.]]
 
−
| [[File:Mediated moderation model 5.png|centre|thumb|
 
−
Fifth option: fourth variable moderates the ''A'' path and a fifth variable moderates the ''B'' path.]]
 
−
|}
 
−
 
 
−
==Mediated moderation==
 
−
Mediated moderation is a variant of both moderation and mediation. This is where there is initially overall moderation and the direct effect of the moderator variable on the outcome is mediated. The main difference between mediated moderation and moderated mediation is that for the former there is initial (overall) moderation and this effect is mediated and for the latter there is no moderation but the effect of either the treatment on the mediator (path ''A'') is moderated or the effect of the mediator on the outcome (path ''B'') is moderated.<ref name="Muller" />
 
−
 
 
−
In order to establish mediated moderation, one must first establish [[Moderation (statistics)|moderation]], meaning that the direction and/or the strength of the relationship between the independent and dependent variables (path ''C'') differs depending on the level of a third variable (the moderator variable). Researchers next look for the presence of mediated moderation when they have a theoretical reason to believe that there is a fourth variable that acts as the mechanism or process that causes the relationship between the independent variable and the moderator (path ''A'') or between the moderator and the dependent variable (path ''C'').
 
−
 
 
−
'''Example'''
 
−
 
 
−
The following is a published example of mediated moderation in psychological research.<ref>{{cite journal | last1 = Smeesters | first1 = D. | last2 = Warlop | first2 = L. | last3 = Avermaet | first3 = E. V. | last4 = Corneille | first4 = O. | last5 = Yzerbyt | first5 = V. | year = 2003 | title = Do not prime hawks with doves: The interplay of construct activation and consistency of social value orientation on cooperative behavior |  journal = Journal of Personality and Social Psychology | volume = 84 | issue = 5| pages = 972–987 | doi = 10.1037/0022-3514.84.5.972 | pmid = 12757142 }}</ref>
 
−
Participants were presented with an initial stimulus (a prime) that made them think of morality or made them think of might. They then participated in the [[Prisoner's dilemma|Prisoner's Dilemma Game]] (PDG), in which participants pretend that they and their partner in crime have been arrested, and they must decide whether to remain loyal to their partner or to compete with their partner and cooperate with the authorities. The researchers found that prosocial individuals were affected by the morality and might primes, whereas proself individuals were not. Thus, [[Social value orientations|social value orientation]] (proself vs. prosocial) moderated the relationship between the prime (independent variable: morality vs. might) and the behaviour chosen in the PDG (dependent variable: competitive vs. cooperative).
 
−
 
 
−
The researchers next looked for the presence of a mediated moderation effect. Regression analyses revealed that the type of prime (morality vs. might) mediated the moderating relationship of participants’ [[Social value orientations|social value orientation]] on PDG behaviour. Prosocial participants who experienced the morality prime expected their partner to cooperate with them, so they chose to cooperate themselves. Prosocial participants who experienced the might prime expected their partner to compete with them, which made them more likely to compete with their partner and cooperate with the authorities. In contrast, participants with a pro-self social value orientation always acted competitively.
 
−
 
 
−
==Regression equations for moderated mediation and mediated moderation==
 
−
 
 
−
Muller, Judd, and Yzerbyt (2005)<ref name="Muller"/> outline three fundamental models that underlie both moderated mediation and mediated moderation. ''Mo'' represents the moderator variable(s), ''Me'' represents the mediator variable(s), and ''ε<sub>i</sub>'' represents the measurement error of each regression equation.
 
−
 
 
−
[[File:Mediation.jpg|434px|right|A simple statistical mediation model.]]
 
−
 
 
−
'''Step 1''': Moderation of the relationship between the independent variable (X) and the dependent variable (Y), also called the overall treatment effect (path ''C'' in the diagram).
 
−
: <math>Y=\beta_{40} +\beta_{41}X +\beta_{42}Mo +\beta_{43}XMo + \varepsilon_4</math>
 
−
* To establish overall moderation, the ''β''<sub>43</sub> regression weight must be significant (first step for establishing mediated moderation).
 
−
* Establishing moderated mediation requires that there be no moderation effect, so the ''β''<sub>43</sub> regression weight must not be significant.
 
−
 
 
−
'''Step 2''': Moderation of the relationship between the independent variable and the mediator (path ''A'').
 
−
: <math>Me=\beta_{50} +\beta_{51}X +\beta_{52}Mo +\beta_{53}XMo + \varepsilon_5</math>
 
−
* If the ''β''<sub>53</sub> regression weight is significant, the moderator affects the relationship between the independent variable and the mediator.
 
−
 
 
−
'''Step 3''': Moderation of both the relationship between the independent and dependent variables (path ''A'') and the relationship between the mediator and the dependent variable (path ''B'').
 
−
: <math>Y=\beta_{60} +\beta_{61}X +\beta_{62}Mo +\beta_{63}XMo +\beta_{64}Me +\beta_{65}MeMo  + \varepsilon_6</math>
 
−
* If both ''β''<sub>53</sub> in step 2 and ''β''<sub>63</sub> in step 3 are significant, the moderator affects the relationship between the independent variable and the mediator (path ''A'').
 
−
* If both ''β''<sub>53</sub> in step 2 and ''β<sub>65</sub>'' in step 3 are significant, the moderator affects the relationship between the mediator and the dependent variable (path ''B'').
 
−
* Either or both of the conditions above may be true.
 
−
 
 
−
==Causal mediation analysis==
 
−
 
 
−
===Fixing versus conditioning===
 
−
 
 
−
Mediation analysis quantifies the
 
−
extent to which a variable participates in the transmittance
 
−
of change from a cause to its effect. It is inherently a causal
 
−
notion, hence it cannot be defined in statistical terms.  Traditionally,
 
−
however, the bulk of mediation analysis has been conducted
 
−
within the confines of linear regression, with statistical
 
−
terminology masking the causal character of the
 
−
relationships involved. This led to difficulties,
 
−
biases, and limitations that have been alleviated by
 
−
modern methods of causal analysis, based on causal diagrams
 
−
and counterfactual logic.
 
−
 
 
−
The source of these difficulties lies in defining mediation
 
−
in terms of changes induced by adding a third variables into
 
−
a regression equation. Such statistical changes are
 
−
epiphenomena which sometimes accompany mediation but,
 
−
in general, fail to capture the causal relationships that
 
−
mediation analysis aims to quantify.
 
−
 
 
−
The basic premise of the causal approach is that it is
 
−
not always appropriate to "control" for the mediator ''M''
 
−
when we seek to estimate the direct effect of ''X'' on ''Y''
 
−
(see the Figure above).
 
−
The classical rationale for "controlling" for ''M''"
 
−
is that, if we succeed in preventing ''M'' from changing, then
 
−
whatever changes we measure in Y are attributable solely
 
−
to variations in ''X'' and we are justified then in proclaiming the
 
−
effect observed as "direct effect of ''X'' on ''Y''." Unfortunately,
 
−
"controlling for ''M''" does not physically prevent ''M'' from changing;
 
−
it merely narrows the analyst's attention to cases
 
−
of equal ''M'' values.  Moreover, the language of probability
 
−
theory does not possess the notation to express the idea
 
−
of "preventing ''M'' from changing" or "physically holding ''M'' constant".
 
−
The only operator probability provides is "Conditioning"
 
−
which is what we do when we "control" for ''M'',
 
−
or add ''M'' as a regressor in the equation for ''Y''.
 
−
The result is that, instead of physically holding ''M" constant
 
−
(say at ''M'' = ''m'') and comparing ''Y'' for units under ''X''&nbsp;=&nbsp;1' to those under
 
−
''X'' = 0, we allow ''M'' to vary but ignore all units except those in
 
−
which ''M'' achieves the value ''M''&nbsp;=&nbsp;''m''. These two operations are
 
−
fundamentally different, and yield different results,<ref>{{cite journal|last1=Robins|first1=J.M.|last2=Greenland|first2=S.|title=Identifiability and exchangeability for direct and indirect effects|journal=Epidemiology|date=1992|volume=3|issue=2|pages=143–155|doi=10.1097/00001648-199203000-00013|pmid=1576220}}</ref><ref name="pearl1994" >{{cite journal|last1=Pearl|first1=Judea|editor1-last=Lopez de Mantaras|editor1-first=R.|editor2-last=Poole|editor2-first=D.|title=A probabilistic calculus of actions|journal=Uncertainty in Artificial Intelligence 10|volume=1302|date=1994|pages=454–462|publisher=[[Morgan Kaufmann]]|location=San Mateo, CA|bibcode=2013arXiv1302.6835P|arxiv=1302.6835}}</ref> except in the case of no omitted variables.
 
−
 
 
−
To illustrate, assume that the error terms of ''M'' and ''Y''
 
−
are correlated. Under such conditions, the
 
−
structural coefficient ''B'' and ''A'' (between ''M'' and ''Y'' and between ''Y'' and ''X'')
 
−
can no longer be estimated by regressing ''Y'' on ''X'' and ''M''.
 
−
In fact, the regression slopes may both be nonzero
 
−
even when ''C'' is zero.<ref>{{cite journal|pmid=24885338|year=2014|last1=Pearl|first1=J|title=Interpretation and identification of causal mediation|journal=Psychological Methods|volume=19|issue=4|pages=459–81|doi=10.1037/a0036434|url=ftp://ftp.cs.ucla.edu/pub/stat_ser/r389-imai-etal-commentary-r421-reprint.pdf}}</ref>  This has two
 
−
consequences. First, new strategies must be devised for
 
−
estimating the structural coefficients ''A,B'' and ''C''. Second,
 
−
the basic definitions of direct and indirect effects
 
−
must go beyond regression analysis, and should
 
−
invoke an operation that mimics "fixing ''M''",
 
−
rather than "conditioning on ''M''."
 
−
 
 
−
===Definitions===
 
−
Such an operator, denoted do(''M''&nbsp;=&nbsp;''m''), was defined in Pearl (1994)<ref name="pearl1994"/> and it operates by removing the equation of ''M'' and replacing it by a constant ''m''. For example, if the basic mediation model consists of the equations:
 
−
 
 
−
: <math> X=f(\varepsilon_1),~~M=g(X,\varepsilon_2),~~Y=h(X,M,\varepsilon_3) , </math>
 
−
 
 
−
then after applying the operator do(''M''&nbsp;=&nbsp;''m'') the model becomes:
 
−
 
 
−
: <math> X=f(\varepsilon_1),~~M=m,~~Y=h(X,m,\varepsilon_3) </math>
 
−
 
 
−
and after applying the operator do(''X''&nbsp;=&nbsp;''x'') the model becomes:
 
−
 
 
−
: <math>X=x, M=g(x, \varepsilon_2), Y=h(x,M,\varepsilon_3) </math>
 
−
 
 
−
where the functions ''f'' and ''g'', as well as the
 
−
distributions of the error terms ε<sub>1</sub> and ε<sub>3</sub> remain
 
−
unaltered. If we further rename the variables ''M'' and ''Y'' resulting from do(''X''&nbsp;=&nbsp;''x'')
 
−
as ''M''(''x'') and ''Y''(''x''), respectively, we obtain what
 
−
came to be known as "potential
 
−
outcomes"<ref>{{cite journal|last1=Rubin|first1=D.B.|title=Estimating causal effects of treatments in randomized and nonrandomized studies|journal=Journal of Educational Psychology|date=1974|volume=66|issue=5|pages=688–701|doi=10.1037/h0037350|url=https://semanticscholar.org/paper/545122e2990590524459ec9b59ccac6ce71e3b6a}}</ref> or "structural counterfactuals".<ref>{{cite journal|last1=Balke|first1=A.|last2=Pearl|first2=J.|editor1-last=Besnard|editor1-first=P.|editor2-last=Hanks|editor2-first=S.|title=Counterfactuals and Policy Analysis in Structural Models|journal=Uncertainty in Artificial Intelligence 11|volume=1302|date=1995|pages=11–18|publisher=[[Morgan Kaufmann]]|location=San Francisco, CA|bibcode=2013arXiv1302.4929B|arxiv=1302.4929}}</ref>
 
−
These new variables provide convenient notation
 
−
for defining direct and indirect effects. In particular,
 
−
four types of effects have been defined for the
 
−
transition from ''X''&nbsp;=&nbsp;0 to ''X''&nbsp;=&nbsp;1:
 
−
 
 
−
(a) Total effect – 
 
−
: <math>TE = E [Y(1) - Y(0)] </math>         
 
−
(b) Controlled direct effect -
 
−
: <math> CDE(m) = E [Y(1,m) - Y(0,m) ]  </math>
 
−
(c) Natural direct effect -
 
−
: <math>NDE = E [Y(1,M(0))  - Y(0,M(0))] </math>
 
−
(d) Natural indirect effect
 
−
: <math> NIE = E [Y(0,M(1)) - Y(0,M(0))] </math>
 
−
Where ''E''[ ] stands for expectation taken over the error terms.
 
−
 
 
−
These effects have the following interpretations:
 
−
* ''TE'' measures the expected increase in the outcome ''Y'' as ''X'' changes from ''X=0'' to ''X''&nbsp;=1'', while the mediator is allowed to track the change in ''X'' as dictated by the function ''M = g(X, ε<sub>2</sub>)''.
 
−
* CDE measures the expected increase in the outcome ''Y'' as ''X'' changes from ''X'' = 0 to ''X'' = 1, while the mediator is fixed at a pre-specified level ''M = m'' uniformly over the entire population
 
−
* ''NDE'' measures the expected increase in ''Y'' as ''X'' changes from ''X'' = 0 to ''X'' = 1, while setting the mediator variable to whatever value it ''would have obtained''  under ''X'' = 0, i.e., before the change.
 
−
* ''NIE'' measures the expected increase in ''Y'' when the ''X'' is held constant, at ''X'' = 1, and ''M'' changes to whatever value it would have attained (for each individual) under ''X'' = 1.
 
−
* The difference ''TE-NDE'' measures the extent to which mediation is ''necessary'' for explaining the effect, while the ''NIE'' measures the extent to which mediation is ''sufficient'' for sustaining it.
 
−
 
 
−
A controlled version of the indirect effect does not
 
−
exist because there is no way of disabling the
 
−
direct effect by fixing a variable to a constant.
 
−
 
 
−
According to these definitions the total effect can be decomposed as a sum
 
−
: <math>TE = NDE - NIE_r </math>
 
−
where ''NIE<sub>r</sub>'' stands for the reverse transition, from
 
−
''X''&nbsp;=&nbsp;1 to ''X'' = 0; it becomes additive in linear systems,
 
−
where reversal of transitions entails sign reversal.
 
−
 
 
−
The power of these definitions lies in their generality; they are applicable to models with arbitrary nonlinear interactions, arbitrary dependencies among the disturbances, and both continuous and categorical variables.
 
−
 
 
−
===The mediation formula===
 
−
[[File:Formulation of the indirect effect.png|thumb|Formulation of the indirect effect]]
 
−
 
 
−
In linear analysis, all effects are determined by sums
 
−
of products of structural coefficients, giving
 
−
: <math>
 
−
\begin{align}
 
−
TE        & = C + AB \\
 
−
CDE(m) & = NDE = C, \text{ independent of } m\\
 
−
NIE        & = AB.
 
−
\end{align}
 
−
</math>
 
−
Therefore, all effects are estimable whenever the model
 
−
is identified.  In non-linear systems, more stringent
 
−
conditions are needed for estimating the
 
−
direct and indirect effects <ref name="Pearl-01"/><ref name="imai-etal-2010">{{cite journal|last1=Imai|first1=K.|last2=Keele|first2=L.|last3=Yamamoto|first3=T.|title=Identification, inference, and sensitivity analysis for causal mediation effects|journal=Statistical Science|date=2010|volume=25|issue=1|pages=51–71|doi=10.1214/10-sts321|arxiv=1011.1079|bibcode=2010arXiv1011.1079I}}</ref>
 
−
.<ref name="vanderweele-2009">{{cite journal|last1=VanderWeele|first1=T.J.|title=Marginal structural models for the estimation of direct and indirect effects|journal=Epidemiology|date=2009|volume=20|issue=1|pages=18–26|doi=10.1097/ede.0b013e31818f69ce|pmid=19234398}}</ref>
 
−
For example, if no confounding exists,
 
−
(i.e.,  ε<sub >1</sub>, ε<sub>2</sub>, and ε<sub>3</sub> are mutually independent) the
 
−
following formulas can be derived:<ref name="Pearl-01"/>
 
−
 
 
−
<!-- : <math> TE = \sum_m P(M=m) [E(Y\mid X=1, M=m) - E(Y\mid X=0, M=m) ]</math> -->
 
−
: <math>
 
−
\begin{align}
 
−
TE        & = E(Y\mid X=1)- E(Y\mid X=0)\\
 
−
CDE(m) & = E(Y\mid X=1, M=m) - E(Y\mid X=0, M=m) \\
 
−
NDE    & = \sum_m [E(Y|X=1, M=m)  - E(Y\mid X=0, M=m) ] P(M=m\mid X=0) \\
 
−
NIE      & = \sum_m [P(M=m\mid X=1) - P(M=m\mid X=0)] E(Y\mid X=0, M=m).
 
−
\end{align}
 
−
</math>
 
−
 
 
−
The last two equations are called ''Mediation Formulas'' <ref name="pearl-2009-r350">{{cite journal|last1=Pearl|first1=Judea|title=Causal inference in statistics: An overview|journal=Statistics Surveys|date=2009|volume=3|pages=96–146|url=http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf|doi=10.1214/09-ss057|doi-access=free}}</ref><ref name="vansteelandt-2012">{{cite journal|last1=Vansteelandt|first1=Stijn|last2=Bekaert|first2=Maarten|last3=Lange|first3=Theis|title=Imputation strategies for the estimation of natural direct and indirect effects|journal=Epidemiologic Methods|date=2012|volume=1|issue=1, Article 7|doi=10.1515/2161-962X.1014}}</ref><ref name="albert-2012">{{cite journal|last1=Albert|first1=Jeffrey|title=Distribution-Free Mediation Analysis for Nonlinear Models with Confounding|journal=Epidemiology|date=2012|volume=23|issue=6|pages=879–888|doi=10.1097/ede.0b013e31826c2bb9|pmid=23007042|pmc=3773310}}</ref>
 
−
and have become the target of estimation in many studies of mediation.<ref name="imai-etal-2010"/><ref name="vanderweele-2009"/><ref name="vansteelandt-2012"/><ref name="albert-2012"/> They give
 
−
distribution-free expressions for direct and indirect
 
−
effects and demonstrate that, despite the arbitrary nature of
 
−
the error distributions and the functions ''f'', ''g'', and ''h'',
 
−
mediated effects can nevertheless be estimated from data using
 
−
regression.
 
−
The analyses of ''moderated mediation''
 
−
and ''mediating moderators'' fall as special cases of the causal mediation
 
−
analysis, and the mediation formulas identify how various interactions coefficients contribute to the necessary and sufficient components of mediation.<ref name="vanderweele-2009"/><ref name="pearl-2009-r350"/>
 
−
 
 
−
[[File:Serial Mediation Model.png|thumb|A serial mediation model with two mediator variables.]]
 
−
 
 
−
===Example===
 
−
[[File:Parallel Mediation Model.pdf|thumb|A conceptual diagram that depicts a parallel mediation model with two mediator variables.]]
 
−
Assume the model takes the form
 
−
: <math>
 
−
\begin{align}
 
−
X & = \varepsilon_1 \\
 
−
M & = b_0 + b_1X + \varepsilon_2 \\
 
−
Y & = c_0 + c_1X + c_2M + c_3XM + \varepsilon_3
 
−
\end{align}
 
−
</math>
 
−
where the parameter <math>c_3</math> quantifies the degree to which ''M'' modifies the effect of ''X'' on ''Y''. Even when all parameters are estimated from data, it is still not obvious what combinations of parameters measure the direct and indirect effect of ''X'' on ''Y'', or, more practically, how to assess the fraction of the total effect <math>TE</math> that is ''explained'' by mediation and the fraction of <math>TE</math> that is ''owed'' to mediation.  In linear analysis, the former fraction is captured by the product <math>b_1 c_2 / TE</math>, the latter by the difference <math>(TE - c_1)/TE</math>, and the two quantities coincide.  In the presence of interaction, however, each fraction demands a separate analysis, as dictated by the Mediation Formula, which yields:
 
−
: <math>
 
−
\begin{align}
 
−
NDE & = c_1 + b_0 c_3 \\
 
−
NIE & = b_1 c_2 \\
 
−
TE  & = c_1 + b_0 c_3 + b_1(c_2 + c_3) \\
 
−
    & = NDE + NIE + b_1 c_3.
 
−
\end{align}
 
−
</math>
 
−
 
 
−
Thus, the fraction of output response for which mediation would be ''sufficient'' is
 
−
 
 
−
: <math> \frac{NIE}{TE} = \frac{b_1 c_2}{c_1 + b_0 c_3 + b_1 (c_2 + c_3)}, </math>
 
−
 
 
−
while the fraction for which mediation would be ''necessary'' is
 
−
 
 
−
: <math> 1- \frac{NDE}{TE} = \frac{b_1 (c_2 +c_3)}{c_1 + b_0c_3 + b_1 (c_2 + c_3)}. </math>
 
−
 
 
−
These fractions involve non-obvious combinations
 
−
of the model's parameters, and can be constructed
 
−
mechanically with the help of the Mediation Formula. Significantly, due to interaction, a direct effect can be sustained even when the parameter <math>c_1</math> vanishes and, moreover, a total effect can be sustained even when both the direct and indirect effects vanish.  This illustrates that estimating parameters in isolation tells us little about the effect of mediation and, more generally, mediation and moderation are intertwined and cannot be assessed separately.
 
−
 
 
−
 
 
−
 
 
−
==References==
 
−
{{Dual|source=Causal Analysis in Theory and Practice|sourcepath=http://www.mii.ucla.edu/causality/?p=713|date=19 June 2014}}{{Dead link|date=February 2020}}
 
−
 
 
−
;Notes
 
−
{{reflist|30em}}
 
−
;Bibliography
 
−
*{{Cite journal
 
−
  | last = Preacher
 
−
  | first = Kristopher J.
 
−
  | last2 = Hayes
 
−
  | first2 = Andrew F.
 
−
  | title = SPSS and SAS procedures for estimating indirect effects in simple mediation models
 
−
  | journal = Behavior Research Methods, Instruments, and Computers
 
−
  | volume = 36
 
−
  | issue = 4
 
−
  | pages = 717–731
 
−
  | url = http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html
 
−
  | year = 2004
 
−
  | doi = 10.3758/BF03206553
 
−
      | pmid = 15641418
 
−
| doi-access = free
 
−
  }}
 
−
*{{Cite journal
 
−
  | last = Preacher
 
−
  | first = Kristopher J.
 
−
  | last2 = Hayes
 
−
  | first2 = Andrew F.
 
−
  | title = Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models
 
−
  | journal = Behavior Research Methods
 
−
  | volume = 40
 
−
  | issue = 3
 
−
  | pages = 879–891
 
−
  | url = http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html
 
−
  | year = 2008
 
−
  | doi = 10.3758/BRM.40.3.879
 
−
  | pmid = 18697684
 
−
      | doi-access = free
 
−
  }}
 
−
*{{Cite journal
 
−
  | last = Preacher
 
−
  | first = K. J.
 
−
  | last2 = Zyphur
 
−
  | first2 = M. J.
 
−
  | last3 = Zhang
 
−
  | first3 = Z.
 
−
  | title = A general multilevel SEM framework for assessing multilevel mediation
 
−
  | journal = Psychological Methods
 
−
  | volume = 15
 
−
  | issue = 3
 
−
  | pages = 209–233
 
−
  | year = 2010
 
−
  | doi = 10.1037/a0020141
 
−
  | pmid = 20822249
 
−
  | citeseerx = 10.1.1.570.7747
 
−
}}
 
−
* Baron, R. M. and Kenny, D. A. (1986) "The Moderator-Mediator Variable Distinction in Social Psychological Research &ndash; Conceptual, Strategic, and Statistical Considerations", [[Journal of Personality and Social Psychology]], Vol. 51(6), pp.&nbsp;1173&ndash;1182.
 
−
*Cohen, J. (1988). ''Statistical power analysis for the behavioral sciences'' (2nd ed.). New York, NY: Academic Press.
 
−
* {{cite journal | last1 = Hayes | first1 = A. F. | year = 2009 | title = Beyond Baron and Kenny: Statistical mediation analysis in the new millennium | journal = Communication Monographs | volume = 76 | issue = 4| pages = 408–420 | doi = 10.1080/03637750903310360 }}
 
−
*Howell, D. C. (2009). ''Statistical methods for psychology'' (7th ed.). Belmot, CA: Cengage Learning.
 
−
*{{cite journal | last1 = MacKinnon | first1 = D. P. | last2 = Lockwood | first2 = C. M. | year = 2003 | title = Advances in statistical methods for substance abuse prevention research |  journal = Prevention Science | volume = 4 | issue = 3| pages = 155–171 | doi = 10.1023/A:1024649822872 | pmid = 12940467 | pmc = 2843515 }}
 
−
*{{cite journal | last1 = Preacher | first1 = K. J. | last2 = Kelley | first2 = K. | year = 2011 | title = Effect sizes measures for mediation models: Quantitative strategies for communicating indirect effects |  journal = Psychological Methods | volume = 16 | issue = 2| pages = 93–115 | doi = 10.1037/a0022658 | pmid = 21500915 }}
 
−
*Rucker, D.D., Preacher, K.J., Tormala, Z.L. & Petty, R.E. (2011). "Mediation analysis in social psychology: Current practices and new recommendations". ''Social and Personality Psychology Compass'', 5/6, 359–371.
 
−
*{{cite journal | last1 = Sobel | first1 = M. E. | year = 1982 | title = Asymptotic confidence intervals for indirect effects in structural equation models |  journal = Sociological Methodology | volume = 13 | pages = 290–312 | doi = 10.2307/270723 | jstor = 270723 }}
 
−
*{{cite journal | last1 = Spencer | first1 = S. J. | last2 = Zanna | first2 = M. P. | last3 = Fong | first3 = G. T. | year = 2005 | title = Establishing a causal chain: why experiments are often more effective than mediational analyses in examining psychological processes |  journal =  Journal of Personality and Social Psychology| volume = 89 | issue = 6| pages = 845–851 | doi=10.1037/0022-3514.89.6.845 | pmid=16393019}}
 
−
*{{cite book|last1=Pearl|first1=Judea|editor1-last=Berzuini|editor1-first=C.|editor2-last=Dawid|editor2-first=P.|editor3-last=Bernardinelli|editor3-first=L.|title=Causality: Statistical Perspectives and Applications|date=2012|publisher=John Wiley and Sons, Ltd.|location=Chichester, UK|pages=151–179|chapter=The Mediation Formula: A guide to the assessment of causal pathways in nonlinear models}}
 
−
*Shaughnessy J.J., Zechmeister E. & Zechmeister J. (2006). ''Research Methods in Psychology'' (7th ed., pp.&nbsp;51–52). New York: McGraw Hill.
 
−
*{{cite journal | last1 = Tolman | first1 = E. C. | year = 1938 | title = The Determiners of Behavior at a Choice Point | journal = Psychological Review | volume = 45 | pages = 1–41 | doi=10.1037/h0062733}}
 
−
*{{cite journal | last1 = Tolman | first1 = E. C. | last2 = Honzik | first2 = C. H. | year = 1930 | title = Degrees of hunger, reward and nonreward, and maze learning in rats | journal = University of California Publications in Psychology | volume = 4 | pages = 241–275 }}
 
−
*{{cite book| last1 = Vanderweele| first1 = Tyler J. | year = 2015| title = Explanation in Causal Inference }}
 
−
 
 
−
 
 
−
 
 
−
Category:Statistical models
 
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类别: 统计模型
 
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Category:Independence (probability theory)
 
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类别: 独立概率论
 
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Category:Psychometrics
 
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类别: 心理测量学
 
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−
<noinclude>
 
−
 
 
−
<small>This page was moved from [[wikipedia:en:Mediation (statistics)]]. Its edit history can be viewed at [[中介变量/edithistory]]</small></noinclude>
 
−
 
 
−
[[Category:待整理页面]]
 

2021年6月10日 (四) 11:55的版本

中介效应分析

在统计学中，中介模型试图通过引入第三个假设变量，即中介变量(也称为中介变量、中介变量或中介变量)，来识别和解释自变量与因变量之间观察到的关系的基础机制或过程。与自变量和因变量之间的直接因果关系不同，中介模型所描绘的图景是自变量通过影响中介变量（不可观测）进而影响因变量。因此，中介变量的作用是澄清自变量和因变量之间关系的本质[2]。Baron and Kenny（1986）提出的中介效应(mediation)框架（简称BK框架）在社会心理和消费者行为等诸多社会科学研究中产生了十分深远的影响。基于回归的分析的传统 BK 框架存在一些局限性。例如，Zhao et al.（2010）指出了BK框架存在的三点问题：第一，直接效应的缺失不应成为评价中介效应强度的标准；第二，寻找中介效应无需以X对Y存在显著的净效应为前提；第三，Sobel z检验的效力并不强，存在改进方式。近年来，基于现代因果模型的因果中介分析框架缓解了部分问题，成为了中介分析研究热点。

BK 框架下的中介效应分析

Baron and Kenny (1986) 提出了形成一个真正的中介关系必须满足的几个条件如下： 1）让因变量对自变量进行回归，以确认自变量是因变量的显著预测因子，即 [math]\displaystyle{ Y=\beta _{{10}}+\beta _{{11}}X+\varepsilon _{1} }[/math] 的回归系数$$β_{11}$$ 是显著的。 2）让中介变量对自变量进行回归，确认自变量是中介变量的显著预测因子，即 $$Me=\beta _模板:20+\beta _模板:21X+\varepsilon _{2}$$ 的回归系数 $$\beta_{21}$$是显著的。如果中介变量与自变量没有关联，那么它就不可能中介任何事物。 3）让因变量对中介和自变量同时进行回归，即 $$Y=\beta _模板:30+\beta _模板:31X+\beta _模板:32Me+\varepsilon _{3}$$ 的回归系数 $$\beta_{32}$$是显著的，并且 $$\beta_{31}$$的绝对值应该小于自变量的效应 $$\beta_{11}$$。从而确保了中介变量是因变量的重要预测因子，并且使得相对于第一步，自变量对结果的解释性降低。

中介变量可以解释两个变量之间观察到的全部或部分关系，如果中介变量的加入使自变量和因变量之间的相关性降为零，则中介的证据最大，也称为完全中介(full mediation)。而部分中介(partial mediation)是指不仅中介变量与因变量之间存在显著的关系，而且自变量与因变量之间也存在某种直接的关系。

我们采用Sobel’s test[10]来检验中介变量加入后自变量与因变量之间的关系是否显著降低，从而评估中介效应是否显著。然而，这种方式的统计效力（Power）很低。因此，为了有足够的效力检测显著性影响，需要大的样本量。这是因为Sobel检验的关键假设是正态性假设。因为Sobel检验是根据正态分布来评估给定样本的，所以样本规模小和抽样分布的偏态可能会有问题(详见正态分布)。因此，MacKinnon et al .,(2002)[12]所建议的经验法是，检测较小的效应需要1000个样本，检测中等效应需要100个样本，检测较大效应需要50个样本。基于自助法的检验能减少对样本量的依赖，见 Preacher and Hayes(2004)。

因果中介分析 固定（fixing）与条件化（conditioning）

中介分析量化了变量参与从原因到其结果的变化传递的程度。它本质上是一个因果概念，因此不能用统计术语来定义。然而，传统上，大量的中介分析是在线性回归的范畴内进行的。统计术语掩盖了所涉及关系的因果特征，这导致了一些困难、偏差（biases）和局限性（limitations）。而基于因果图(causal diagrams)和反事实逻辑的现代因果分析方法缓解了这些困难、偏见和限制。

这些困难的根源在于，在方法上，根据在回归方程中添加第三个变量所引起的变化来定义中介。虽然这种统计上的变化是伴随中介效应而来的附带现象。但这样的做法未能充分捕捉到中介分析的本质，即量化因果关系。


因果方法的基本前提是，当我们试图估计自变量 X 对 因变量 Y 的直接影响时，并不总是适合对中介M进行“控制”(见上图)。对M进行“控制”的经典理论是，如果我们成功地阻止了M的变化，那么我们在Y中测量的任何变化都只能归因于X的变化，然后我们就有理由宣布观察到的效果是“X对Y的直接影响”。不幸的是，“控制M”并不能从物理上阻止M的改变；它只是把分析者的注意力集中在相等 M 值的情况下。而且，概率论的语言没有表示“阻止M改变”或“物理上保持M不变”的符号。唯一的运算是“以…为条件”（conditioning)，这是当我们“控制” M 时所做的。或者为 Y 的方程添加 M 作为其中的一个回归变量。 结果是，与在物理上保持 M 不变（例如 M = m ）并将 X = 1 下 Y 的单位 与 X = 0 下 Y 的单位进行比较的方法不同，我们允许 M 变化但忽略所有使得 M=m 的其他单位。这两个操作除了没有遗漏变量的情况，本质上是不同的，产生不同的结果[21][22]。

举例来说，假设 M 和 Y 的误差项是相关的。在这种情况下，通过对 Y 在 X 和 M 上进行回归，就无法对结构系数 B 和 A(在M和Y之间，在Y和X之间) 进行估计。事实上，即使当 C 等于 0 的时候，回归斜率也可能不等于 0 。这有两种后果。首先必须设计新的策略来估计结构系数 A、B 和 C。其次，直接和间接效应的基本定义必须超越回归分析，并且应该采用类似于“固定 M”的操作，而不是“在 M 的条件下”的操作。 数学定义

Pearl(1994)[22]中定义了这样一个运算符 $$do(M = m)$$，它的作用是去除 M 的方程，代之以一个常数 m。例如，如果基本中介模型由以下方程组成: $$ {\displaystyle X=f(\varepsilon _{1}),M=g(X,\varepsilon _{2}),Y=h(X,M,\varepsilon _{3}),}$$ 那么应用了$$do(M = m)$$运算的模型将会变为： $${\displaystyle X=f(\varepsilon _{1}),M=m,Y=h(X,m,\varepsilon _{3})}$$ 同时，应用了$$do(X = x)$$ 运算的模型会变为： $${\displaystyle X=x,M=g(x,\varepsilon _{2}),Y=h(x,M,\varepsilon _{3})}$$ 其中函数 f 和 g 以及误差项 ε1 和 ε3 的分布保持不变。如果我们进一步将 $$do(X = x)$$ 得到的变量 $$M$$ 和 $$Y$$ 分别重新命名为 $$M(x)$$ 和 $$Y(x)$$ ，我们得到了所谓的“潜在结果(potential outcome)”[24]或“结构反事实(structural counterfactuals)”[25]这些新变量为定义直接和间接效应提供了便利的描述符号。具体来说，定义了从 $$X = 0$$ 到 $$X = 1$$ 变化的四种效应:

(a) 总体效应 – $$TE=E[Y(1)-Y(0)]$$ (b) 受控直接效应 - $$CDE(m)=E[Y(1,m)-Y(0,m)]$$ (c) 自然直接效应 - $$NDE=E[Y(1,M(0))-Y(0,M(0))]$$ (d) 自然间接效应 $$NIE = E [Y(0,M(1)) - Y(0,M(0))] $$ 其中 $$E[\cdot ]$$ 表示对误差项的期望，这些效应有如下一些解释： - $$TE$$ 表示的 $$X$$对 $$Y$$的总体因果效应。 - $$CDE$$ 表示在某个条件 $$M=m$$下，$$X$$对 $$Y$$的因果效应。 - $$NDE$$ 表示 $$X$$对 $$Y$$的直接产生的因果效应。 - $$NIE$$ 表示 $$X$$对 $$Y$$的通过中介变量 $$M$$产生的因果效应。 - 对于解释 $$X$$和 $$Y$$之间的效应，两个效应的差$$TE-NDE$$ 度量的是中介变量在何种程度上是必要的。而 $$NIE$$ 度量的是引入中介变量在充分性。 间接效应的受控版本并不存在，因为没有办法通过将一个变量固定到一个常量来屏蔽直接效应。 根据这些定义，总体效应可以如下分解 $$TE=NDE-NIE_{r}$$ 其中 $$NIE_r$$ 表示在 $$NIE$$ 的定义中进行 $$X = 1$$ 到 $$X = 0$$ 的反向转换；线性系统中总体效应等于直接效应与间接效应之和，即负的反转间接效应等于间接效应 $$-NIE_r = NIE$$。这些定义的力量在于它们的普适性；它们适用于具有任意非线性相互作用，任意干扰之间的依赖关系，以及连续变量和离散变量的模型。

中介效应公式

在线性分析中，所有的效应由结构系数的乘积决定，给出 $${\displaystyle {\begin{aligned}TE&=C+AB\\CDE(m)&=NDE=C,{\text{ independent of }}m\\NIE&=AB.\end{aligned}}}$$ 因此，当模型被识别时，所有的效应都是可估计的。在非线性系统中，估计直接和间接效应需要更严格的条件，如不存在混杂因子(即 $$ε_1、ε_2、ε_3$$ 相互独立)，可推导出如下公式 $${\displaystyle {\begin{aligned}TE&=E(Y\mid X=1)-E(Y\mid X=0)\\CDE(m)&=E(Y\mid X=1,M=m)-E(Y\mid X=0,M=m)\\NDE&=\sum _{m}[E(Y|X=1,M=m)-E(Y\mid X=0,M=m)]P(M=m\mid X=0)\\NIE&=\sum _{m}[P(M=m\mid X=1)-P(M=m\mid X=0)]E(Y\mid X=0,M=m).\end{aligned}}}$$ 后两个方程被称为中介公式[28][29][30]，已成为许多中介研究的估计对象。他们给出了直接和间接效应的无分布假设(distribution-free)表达式，并证明，尽管误差分布和函数 f, g, h 的性质难以确定，中介效应仍然可以通过使用回归方法利用数据来估计。调节中介和中介调节的分析属于因果中介分析的特例。中介公式确定了各种相互作用系数如何贡献于中介的必要和充分成分。 简单案例 假设模型采用这种形式 $${\displaystyle {\begin{aligned}X&=\varepsilon _{1}\\M&=b_{0}+b_{1}X+\varepsilon _{2}\\Y&=c_{0}+c_{1}X+c_{2}M+c_{3}XM+\varepsilon _{3}\end{aligned}}}$$ 其中，参数 $$c_{3}$$ 量化了 M 对 X 对 Y 的影响的修正程度。即使所有参数都是从数据中估计出来的，仍然不清楚是哪些参数组合度量了 X 对 Y 的直接和间接影响，或者，更实际的是，如何评估由中介解释的总体效应 TE 的比例以及应归功于中介效应的 TE 的比例。在线性分析中，前者被 $$b_{1}c_{2}/TE$$ 所捕获，后者被差值 $$(TE-c_{1})/TE$$ 所捕获，并且这两个量重合。然而，在存在交互的情况下，每个部分都需要单独的分析。如中介公式所规定的那样，其结果是: $${\begin{aligned}NDE&=c_{1}+b_{0}c_{3}\\NIE&=b_{1}c_{2}\\TE&=c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})\\&=NDE+NIE+b_{1}c_{3}.\end{aligned}}$$ 因此，对于中介变量来说足够输出的部分是 $${\displaystyle {\frac {NIE}{TE}}={\frac {b_{1}c_{2}}{c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})}},}$$

而需要中介的部分是 $${\displaystyle 1-{\frac {NDE}{TE}}={\frac {b_{1}(c_{2}+c_{3})}{c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})}}.}$$

这些分数涉及模型参数的微妙的组合，并且可以在中介公式的帮助下机械地构造。值得注意的是，由于交互作用，即使参数 $$c_{1}$$ 为 0，直接效应也可以存在。而且，即使直接和间接效应都为 0，总效应也可以存在。 这说明孤立地估计参数几乎无法告诉我们中介的效果。更一般地说，中介和调节是交织在一起的，不能分开评估。

取自“https://wiki.swarma.org/index.php?title=中介分析&oldid=23461”