Mplus model91 模型讲解

来自图书《MPlus中介调节模型》

复杂中介模型分析速查：Mplus实战

• 理论模型：深入理解包含两个串行中介和一个调节变量的模型架构
• 数学模型：精确表达模型中变量间的关系，构建可测量的方程组
• 数学推导：详细解析路径系数的计算方法，理解模型的内在逻辑
• 代码解读：逐行解读 Mplus 代码，掌握模型构建与结果分析技巧

大家好，欢迎来到 Data Sense 的教程。今天我们将学习如何使用 Mplus 分析一个复杂的串行中介模型，这个模型包含了两个串行中介变量和一个调节变量。首先，我们会从理论层面深入理解这个模型的基本架构，了解各个变量之间的关系。接着，我们将把这些关系用数学公式精确表达出来，构建出一套可测量的方程组。然后，我们会详细解析模型中路径系数的计算方法，让你理解模型背后隐藏的数学逻辑。最后，我们将逐行解读 Mplus 的代码，让你掌握如何用 Mplus 构建模型，并且解读分析结果。希望通过本教程，能帮助大家快速掌握复杂中介模型的分析方法。Data Sense 还会提供专业的统计咨询服务，如果你在数据分析过程中有任何疑问，欢迎随时联系我们。

理论模型

我们来看，这是一个复杂的路径模型。X 可以直接影响 Y，这就是一条直接路径。同时，X 也可以通过两个中间变量 M1 和 M2 来影响 Y，这就是一条间接路径。其中，X 先影响 M1，M1 再影响 M2。另外，变量 W 会影响 M1 到 M2 之间的路径。

数学模型

这张图展示的是一个复杂的路径模型，其中包含了五个关键变量，它们之间的关系用箭头来表示。首先，自变量X通过两条路径影响中间变量，分别是路径a1到中间变量M1，和路径a2到中间变量M2。中间变量M1本身也会影响M2，通过路径d1，同时M1也会通过路径b1影响最终的因变量Y。另外，调节变量W会通过路径d2和它与M1的交互项M1W通过路径d3来影响中间变量M2。中间变量M2最终也会通过路径b2影响因变量Y。最后，X也会通过路径c’直接影响Y。总而言之，这张图清晰地展示了各个变量之间相互影响的复杂关系，为我们后续的分析提供了基础。

数学公式1

模型方程:

Y = b_0 + b_1M_1 + b_2M_2 + c'X
M_1 = a_{01} + a_1X
M_2 = a_{02} + a_2X + d_1M_1 + d_2W + d_3M_1W

首先，我们定义了三个回归方程。第一个方程描述了因变量Y如何被中介变量M1、M2和自变量X影响。第二个方程描述了中介变量M1如何被自变量X影响。第三个方程描述了中介变量M2如何被自变量X、中介变量M1以及调节变量W和它们的交互项M1W影响。

数学公式2

模型方程:

Y = b_0 + b_1M_1 + b_2M_2 + c'X
M_1 = a_{01} + a_1X
M_2 = a_{02} + a_2X + d_1M_1 + d_2W + d_3M_1W
代入和展开:

Y = b_0 + b_1(a_{01} + a_1X) + b_2(a_{02} + a_2X + d_1(a_{01} + a_1X) + d_2W + d_3(a_{01} + a_1X)W) + c'X

Y = b_0 + a_{01}b_1 + a_1b_1X + a_{02}b_2 + a_2b_2X + a_{01}d_1b_2 + a_1d_1b_2X + d_2b_2W + a_{01}d_3b_2W + a_1d_3b_2XW + c'X

接下来，为了分析不同效应，我们将M1和M2的方程代入到Y的方程中。然后我们展开括号，得到一个更长的表达式，这个表达式包含了所有变量和它们之间的交互项。

数学公式3

模型方程:

Y = b_0 + b_1M_1 + b_2M_2 + c'X
M_1 = a_{01} + a_1X
M_2 = a_{02} + a_2X + d_1M_1 + d_2W + d_3M_1W
代入和展开:

Y = b_0 + b_1(a_{01} + a_1X) + b_2(a_{02} + a_2X + d_1(a_{01} + a_1X) + d_2W + d_3(a_{01} + a_1X)W) + c'X

Y = b_0 + a_{01}b_1 + a_1b_1X + a_{02}b_2 + a_2b_2X + a_{01}d_1b_2 + a_1d_1b_2X + d_2b_2W + a_{01}d_3b_2W + a_1d_3b_2XW + c'X
整理成 Y = a + bX 的形式:

Y = (b_0 + a_{01}b_1 + a_{02}b_2 + a_{01}d_1b_2 + d_2b_2W + a_{01}d_3b_2W) + (a_1b_1 + a_1d_1b_2 + a_2b_2 + a_1d_3b_2W + c')X

现在，我们将方程整理成标准的线性形式Y = a + bX。其中a是截距项，它包含了所有不含X的项，而b是斜率项，它包含了所有X的系数，同时注意，这里的b其实是受W影响的。

数学公式4

模型方程:

Y = b_0 + b_1M_1 + b_2M_2 + c'X
M_1 = a_{01} + a_1X
M_2 = a_{02} + a_2X + d_1M_1 + d_2W + d_3M_1W
代入和展开:

Y = b_0 + b_1(a_{01} + a_1X) + b_2(a_{02} + a_2X + d_1(a_{01} + a_1X) + d_2W + d_3(a_{01} + a_1X)W) + c'X

Y = b_0 + a_{01}b_1 + a_1b_1X + a_{02}b_2 + a_2b_2X + a_{01}d_1b_2 + a_1d_1b_2X + d_2b_2W + a_{01}d_3b_2W + a_1d_3b_2XW + c'X
整理成 Y = a + bX 的形式:

Y = (b_0 + a_{01}b_1 + a_{02}b_2 + a_{01}d_1b_2 + d_2b_2W + a_{01}d_3b_2W) + (a_1b_1 + a_1d_1b_2 + a_2b_2 + a_1d_3b_2W + c')X
提取效应:

X对Y的三个间接效应:
   a_1b_1  (X -> M_1 -> Y)
   a_2b_2  (X -> M_2 -> Y)
   a_1b_2(d_1 + d_3W) (X -> M_1 -> M_2 -> Y, 受W调节)

X对Y的直接效应: c'

最后，我们从整理后的方程中提取出我们关心的效应。包括X通过M1和M2对Y的三个间接效应，以及X对Y的直接效应c’。注意，第三个间接效应受调节变量W的影响，体现了有调节的中介效应。这个过程可以帮助我们在Mplus代码中进行效应计算。

代码解读1

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;

首先，我们定义变量。X是预测变量，也就是自变量。M1和M2是中介变量，它们在X和Y之间起作用。W是调节变量，它会影响X和Y之间的关系。Y是结果变量，也就是因变量。最后，我们声明了一个交互项M1W，它是M1和W的乘积。USEVARIABLES 这一行声明了模型中用到的所有变量。

代码解读2

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;

接下来，我们创建交互项。DEFINE命令用来定义新变量，这里我们把M1和W相乘，得到交互项M1W。注意，这个定义必须放在USEVARIABLES之后。

代码解读3

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;

然后，我们设置分析选项。ANALYSIS 命令开始定义分析的细节。TYPE = GENERAL 表示使用一般模型。ESTIMATOR = ML 表示使用最大似然估计。BOOTSTRAP = 10000 表示使用1万次自助抽样来计算标准误差和置信区间。

代码解读4

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);

现在，我们开始定义模型结构。MODEL命令描述变量之间的关系。例如，Y ON M1 (b1) 表示Y对M1的回归，路径系数命名为b1。我们用括号给每个路径命名，方便后续引用。这里定义了X对Y的直接效应(cdash)，以及X对M1和M2的效应(a1, a2)，还有M1, W 和 交互项 M1W 对 M2 的效应（d1, d2, d3）。

代码解读5

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);

接下来，我们使用 MODEL CONSTRAINT 命令进行参数约束和计算。首先，我们用 NEW 命令声明一些新的变量，包括调节变量W的低，中，高值，以及一些中间变量和效应值。注意，这里需要选择调节变量W的低、中、高值，比如均值加减一个标准差。

代码解读6

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);
 LOW_W = #LOWW;
! replace #LOWW in the code with your chosen low value of W

 MED_W = #MEDW;
! replace #MEDW in the code with your chosen medium value of W

 HIGH_W = #HIGHW;
! replace #HIGHW in the code with your chosen high value of W
! Now calc indirect and total effects for each value of W

 a1b1 = a1*b1;
! Specific indirect effect of X on Y via M1 only

 a2b2 = a2*b2;
! Specific indirect effect of X on Y via M2 only

然后，我们给调节变量W的低、中、高值赋值。这里的 #LOWW, #MEDW 和 #HIGHW 需要替换成具体的数值。之后我们计算X通过M1和M2分别对Y的间接效应，分别为a1乘以b1和a2乘以b2。

代码解读7

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);
 LOW_W = #LOWW;
! replace #LOWW in the code with your chosen low value of W

 MED_W = #MEDW;
! replace #MEDW in the code with your chosen medium value of W

 HIGH_W = #HIGHW;
! replace #HIGHW in the code with your chosen high value of W
! Now calc indirect and total effects for each value of W

 a1b1 = a1*b1;
! Specific indirect effect of X on Y via M1 only

 a2b2 = a2*b2;
! Specific indirect effect of X on Y via M2 only
! Conditional indirect effects of X on Y via M1 and M2 given values of W
 LWa1d1b2 = a1*d1*b2 + a1*d3*b2*LOW_W;
 MWa1d1b2 = a1*d1*b2 + a1*d3*b2*MED_W;
 HWa1d1b2 = a1*d1*b2 + a1*d3*b2*HIGH_W;
! Index of Moderated Mediation
 IMM = a1*d3*b2;

接着，我们计算在W不同取值下，X通过M1和M2对Y的条件间接效应。公式中包含了W的低、中、高值。IMM 表示调节中介作用的指标，等于a1乘以d3乘以b2。

代码解读8

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);
 LOW_W = #LOWW;
! replace #LOWW in the code with your chosen low value of W

 MED_W = #MEDW;
! replace #MEDW in the code with your chosen medium value of W

 HIGH_W = #HIGHW;
! replace #HIGHW in the code with your chosen high value of W
! Now calc indirect and total effects for each value of W

 a1b1 = a1*b1;
! Specific indirect effect of X on Y via M1 only

 a2b2 = a2*b2;
! Specific indirect effect of X on Y via M2 only
! Conditional indirect effects of X on Y via M1 and M2 given values of W
 LWa1d1b2 = a1*d1*b2 + a1*d3*b2*LOW_W;
 MWa1d1b2 = a1*d1*b2 + a1*d3*b2*MED_W;
 HWa1d1b2 = a1*d1*b2 + a1*d3*b2*HIGH_W;
! Index of Moderated Mediation
 IMM = a1*d3*b2;
! Conditional total effects of X on Y given values of W
 TOT_LOWW = LWa1d1b2 + a1b1 + a2b2 + cdash;
 TOT_MEDW = MWa1d1b2 + a1b1 + a2b2 + cdash ;
 TOT_HIW = HWa1d1b2 + a1b1 + a2b2 + cdash;

然后，我们计算在W不同取值下，X对Y的条件总效应。总效应包括了间接效应和直接效应。注意，这里把之前计算的条件间接效应加在一起，再加上a1b1, a2b2 和 cdash。

代码解读9

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);
 LOW_W = #LOWW;
! replace #LOWW in the code with your chosen low value of W

 MED_W = #MEDW;
! replace #MEDW in the code with your chosen medium value of W

 HIGH_W = #HIGHW;
! replace #HIGHW in the code with your chosen high value of W
! Now calc indirect and total effects for each value of W

 a1b1 = a1*b1;
! Specific indirect effect of X on Y via M1 only

 a2b2 = a2*b2;
! Specific indirect effect of X on Y via M2 only
! Conditional indirect effects of X on Y via M1 and M2 given values of W
 LWa1d1b2 = a1*d1*b2 + a1*d3*b2*LOW_W;
 MWa1d1b2 = a1*d1*b2 + a1*d3*b2*MED_W;
 HWa1d1b2 = a1*d1*b2 + a1*d3*b2*HIGH_W;
! Index of Moderated Mediation
 IMM = a1*d3*b2;
! Conditional total effects of X on Y given values of W
 TOT_LOWW = LWa1d1b2 + a1b1 + a2b2 + cdash;
 TOT_MEDW = MWa1d1b2 + a1b1 + a2b2 + cdash ;
 TOT_HIW = HWa1d1b2 + a1b1 + a2b2 + cdash;
! Use loop plot to plot total effect of X on Y for low, med, high values of W
! NOTE - values of 1,5 in LOOP() statement need to be replaced by
! logical min and max limits of predictor X used in analysis
 PLOT(LOMOD MEDMOD HIMOD);
 LOOP(XVAL,1,5,0.1);
 LOMOD = TOT_LOWW*XVAL;
 MEDMOD = TOT_MEDW*XVAL;
 HIMOD = TOT_HIW*XVAL;

接下来，我们使用LOOP循环和PLOT命令来绘制X对Y的总效应图。 LOOP 命令定义了一个循环，XVAL从1变化到5，步长是0.1。请务必将这里的1和5替换为X变量的实际最小值和最大值。我们计算了在W不同取值下，模型预测的Y值，方便我们画图。

代码解读10

! Predictor variable - X
! Mediator variable(s) – M1, M2
! Moderator variable(s) - W
! Outcome variable - Y
USEVARIABLES = X M1 M2 W Y M1W;
! Create interaction term
! Note that it has to be placed at end of USEVARIABLES subcommand above
DEFINE:
 M1W = M1*W;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
! In model statement name each path using parentheses
MODEL:
 Y ON M1 (b1);
 Y ON M2 (b2);
 Y ON X (cdash);
! direct effect of X on Y
 M1 ON X (a1);
 M2 ON X (a2);
 M2 ON M1 (d1);
 M2 ON W (d2);
 M2 ON M1W (d3);
! Use model constraint subcommand to test simple slopes
! You need to pick low, medium and high moderator values,
! for example, of 1 SD below mean, mean, 1 SD above mean
! Also calc total effects at lo, med, hi values of moderator
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W
 a1b1 a2b2
 LWa1d1b2 MWa1d1b2 HWa1d1b2
 IMM
 TOT_LOWW TOT_MEDW TOT_HIW);
 LOW_W = #LOWW;
! replace #LOWW in the code with your chosen low value of W

 MED_W = #MEDW;
! replace #MEDW in the code with your chosen medium value of W

 HIGH_W = #HIGHW;
! replace #HIGHW in the code with your chosen high value of W
! Now calc indirect and total effects for each value of W

 a1b1 = a1*b1;
! Specific indirect effect of X on Y via M1 only

 a2b2 = a2*b2;
! Specific indirect effect of X on Y via M2 only
! Conditional indirect effects of X on Y via M1 and M2 given values of W
 LWa1d1b2 = a1*d1*b2 + a1*d3*b2*LOW_W;
 MWa1d1b2 = a1*d1*b2 + a1*d3*b2*MED_W;
 HWa1d1b2 = a1*d1*b2 + a1*d3*b2*HIGH_W;
! Index of Moderated Mediation
 IMM = a1*d3*b2;
! Conditional total effects of X on Y given values of W
 TOT_LOWW = LWa1d1b2 + a1b1 + a2b2 + cdash;
 TOT_MEDW = MWa1d1b2 + a1b1 + a2b2 + cdash ;
 TOT_HIW = HWa1d1b2 + a1b1 + a2b2 + cdash;
! Use loop plot to plot total effect of X on Y for low, med, high values of W
! NOTE - values of 1,5 in LOOP() statement need to be replaced by
! logical min and max limits of predictor X used in analysis
 PLOT(LOMOD MEDMOD HIMOD);
 LOOP(XVAL,1,5,0.1);
 LOMOD = TOT_LOWW*XVAL;
 MEDMOD = TOT_MEDW*XVAL;
 HIMOD = TOT_HIW*XVAL;
PLOT:
 TYPE = plot2;
OUTPUT:
 STAND CINT(bcbootstrap);

最后，我们指定PLOT类型为plot2。 在OUTPUT命令中，我们要求输出标准化系数(STAND)和基于bootstrap的置信区间 (CINT(bcbootstrap))。 这就是整个代码的解释。

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