Mplus model84 模型讲解

来自图书《MPlus中介调节模型》

复杂中介调节模型 Mplus 速查

• 理论模型
• 数学模型
• 数学推导
• 代码解读

大家好，欢迎来到 Data Sense 统计咨询团队的 Mplus 教程。本节课我们将聚焦于复杂的串行中介调节模型。为了方便大家快速上手，本教程将采用速查手册的形式。每个模型都将从四个方面进行讲解：首先是模型的理论基础，帮助大家理解模型的含义；接着是模型的数学表达，包括公式的呈现；然后是数学推导，帮助大家深入理解模型；最后是Mplus代码的详细解读，让大家可以直接应用到自己的研究中。大家可以根据自己的需要，直接查阅相应的模型，无需逐一学习所有案例。希望本教程能够帮助大家轻松驾驭 Mplus。

理论模型

我们来看，这个模型里有五个变量，分别是X、W、M1、M2和Y。X和W同时影响M1，X和W也同时影响M2，M1会影响M2。M1和M2都影响Y。另外，X还会直接影响Y。

数学模型

这张图展示了一个复杂的统计模型，里面有六个变量，分别是X、W、XW、M1、M2和Y。X会通过a1影响M1，通过a2影响M2，同时通过c’直接影响Y。W会通过a3影响M1，通过a5影响M2。XW会通过a4影响M1，通过a6影响M2。M1会通过d1影响M2，同时通过b1影响Y。最后，M2会通过b2影响Y。

数学公式1

模型方程:

Y = b0 + b1M1 + b2M2 + c'X

M1 = a01 + a1X + a3W + a4XW

M2 = a02 + a2X + a5W + a6XW + d1M1

首先，我们定义了三个模型方程。第一个方程描述了因变量Y如何受中介变量M1和M2以及自变量X的影响。第二个方程描述了中介变量M1如何受自变量X和调节变量W的影响。第三个方程描述了中介变量M2如何受自变量X、调节变量W以及中介变量M1的影响。这里的XW代表X和W的交互项。

数学公式2

模型方程:

Y = b0 + b1M1 + b2M2 + c'X

M1 = a01 + a1X + a3W + a4XW

M2 = a02 + a2X + a5W + a6XW + d1M1
代入化简:

Y = b0 + b1(a01 + a1X + a3W + a4XW) + b2(a02 + a2X + a5W + a6XW + d1(a01 + a1X + a3W + a4XW)) + c'X

接下来，我们将M1和M2的表达式代入Y的表达式中，目的是为了得到一个Y关于X和W的直接关系式，这样我们可以更清楚地分析X如何通过M1和M2影响Y。

数学公式3

模型方程:

Y = b0 + b1M1 + b2M2 + c'X

M1 = a01 + a1X + a3W + a4XW

M2 = a02 + a2X + a5W + a6XW + d1M1
代入化简:

Y = b0 + b1(a01 + a1X + a3W + a4XW) + b2(a02 + a2X + a5W + a6XW + d1(a01 + a1X + a3W + a4XW)) + c'X
展开括号:

Y = b0 + a01b1 + a1b1X + a3b1W + a4b1XW + a02b2 + a2b2X + a5b2W + a6b2XW + a01d1b2 + a1d1b2X + a3d1b2W + a4d1b2XW + c'X

这里我们将上一步得到的表达式中的括号展开，以便下一步可以进行合并同类项。

数学公式4

模型方程:

Y = b0 + b1M1 + b2M2 + c'X

M1 = a01 + a1X + a3W + a4XW

M2 = a02 + a2X + a5W + a6XW + d1M1
代入化简:

Y = b0 + b1(a01 + a1X + a3W + a4XW) + b2(a02 + a2X + a5W + a6XW + d1(a01 + a1X + a3W + a4XW)) + c'X
展开括号:

Y = b0 + a01b1 + a1b1X + a3b1W + a4b1XW + a02b2 + a2b2X + a5b2W + a6b2XW + a01d1b2 + a1d1b2X + a3d1b2W + a4d1b2XW + c'X
整理:

Y = (b0 + a01b1 + a3b1W + a02b2 + a5b2W + a01d1b2 + a3d1b2W) + (a1b1 + a4b1W + a2b2 + a6b2W + a1d1b2 + a4d1b2W + c')X

现在，我们将含有X的项和不含有X的项分开整理，将其整理成Y = a + bX的形式。这样，我们可以清晰地看到X对Y的直接效应和通过M1、M2产生的间接效应。

数学公式5

模型方程:

Y = b0 + b1M1 + b2M2 + c'X

M1 = a01 + a1X + a3W + a4XW

M2 = a02 + a2X + a5W + a6XW + d1M1
代入化简:

Y = b0 + b1(a01 + a1X + a3W + a4XW) + b2(a02 + a2X + a5W + a6XW + d1(a01 + a1X + a3W + a4XW)) + c'X
展开括号:

Y = b0 + a01b1 + a1b1X + a3b1W + a4b1XW + a02b2 + a2b2X + a5b2W + a6b2XW + a01d1b2 + a1d1b2X + a3d1b2W + a4d1b2XW + c'X
整理:

Y = (b0 + a01b1 + a3b1W + a02b2 + a5b2W + a01d1b2 + a3d1b2W) + (a1b1 + a4b1W + a2b2 + a6b2W + a1d1b2 + a4d1b2W + c')X
效应分析:

间接效应:
(a1 + a4W)b1
(a2 + a6W)b2
(a1 + a4W)d1b2

直接效应:
c'

最后，我们可以从整理后的方程中分析出各种效应。这里我们有三个间接效应，它们分别表示X通过M1和M2对Y的影响，并且这些影响都受到调节变量W的调节。最后一个是直接效应，表示X对Y的直接影响，不受其他变量的影响。这个分析帮助我们理解X如何通过中介变量影响Y，以及调节变量W在其中的作用。

代码解读1

USEVARIABLES = X M1 M2 W Y XW;

第一步，我们使用USEVARIABLES命令，声明模型中用到的变量。包括预测变量X，两个中介变量M1和M2，调节变量W，结果变量Y，以及X和W的交互项XW。

代码解读2

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;

接下来，在DEFINE命令下，我们创建交互项XW，它是预测变量X和调节变量W的乘积。这一步是为了后续分析中考虑交互效应。

代码解读3

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;

然后，在ANALYSIS命令下，我们设置分析参数。TYPE指定分析类型为一般模型。ESTIMATOR指定使用最大似然法进行参数估计。BOOTSTRAP指定使用10000次自助抽样，来计算标准误和置信区间。

代码解读4

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);

现在，我们开始定义模型。这里我们使用ON语句定义回归路径，并给每个路径的回归系数命名。例如，Y ON M1 (b1) 表示M1对Y有回归路径，回归系数命名为b1。类似地，我们定义了其他路径，包括X对Y的直接效应，X、W和XW对M1和M2的影响，以及M1对M2的影响。

代码解读5

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);

接下来，我们使用MODEL CONSTRAINT命令来定义模型约束。首先，我们声明了一系列新的变量。这些变量用于存储调节变量W的低、中、高值，以及计算不同调节水平下的间接效应、调节中介指数和总效应。

代码解读6

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;

这里我们为调节变量W的低、中、高值赋值。#LOWW、#MEDW和#HIGHW需要在实际分析中替换为你选定的W的相应值，例如均值加减一个标准差。

代码解读7

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;

然后，我们计算在W的不同取值下，X通过M1、M2以及M1和M2对Y的条件间接效应。例如，LWa1b1计算了当W取低值时，X通过M1对Y的间接效应。这些公式结合了之前定义的回归系数和W的值。

代码解读8

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;
IMM_A = a4*b1;
IMM_B = a4*d1*b2;
IMM_C = a6*b2;

接着，我们计算调节中介指数。IMM_A表示W对X到M1路径的调节作用，IMM_B表示W对X通过M1到M2再到Y路径的调节作用，IMM_C表示W对X到M2路径的调节作用。

代码解读9

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;
IMM_A = a4*b1;
IMM_B = a4*d1*b2;
IMM_C = a6*b2;
TOT_LOWW = LWa1d1b2 + LWa1b1 + LWa2b2 + cdash;
TOT_MEDW = MWa1d1b2 + MWa1b1 + MWa2b2 + cdash ;
TOT_HIW = HWa1d1b2 + HWa1b1 + HWa2b2 + cdash;

接下来，我们计算在W的不同取值下，X对Y的总效应。总效应包括了X对Y的直接效应以及通过M1和M2的间接效应。

代码解读10

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;
IMM_A = a4*b1;
IMM_B = a4*d1*b2;
IMM_C = a6*b2;
TOT_LOWW = LWa1d1b2 + LWa1b1 + LWa2b2 + cdash;
TOT_MEDW = MWa1d1b2 + MWa1b1 + MWa2b2 + cdash ;
TOT_HIW = HWa1d1b2 + HWa1b1 + HWa2b2 + cdash;
PLOT(LOMOD MEDMOD HIMOD);
LOOP(XVAL,1,5,0.1);
LOMOD = TOT_LOWW*XVAL;
MEDMOD = TOT_MEDW*XVAL;
HIMOD = TOT_HIW*XVAL;

然后，我们使用PLOT和LOOP命令来绘制不同W值下X对Y总效应的图表。LOOP命令设定X值的范围和步长，LOMOD、MEDMOD和HIMOD则计算了不同X值和W值下对应的Y值，用于绘制图表。

代码解读11

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;
IMM_A = a4*b1;
IMM_B = a4*d1*b2;
IMM_C = a6*b2;
TOT_LOWW = LWa1d1b2 + LWa1b1 + LWa2b2 + cdash;
TOT_MEDW = MWa1d1b2 + MWa1b1 + MWa2b2 + cdash ;
TOT_HIW = HWa1d1b2 + HWa1b1 + HWa2b2 + cdash;
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;

指定图表的类型为plot2，用于绘制双变量的图表。

代码解读12

USEVARIABLES = X M1 M2 W Y XW;
DEFINE:
XW = X*W;
ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;
MODEL:
Y ON M1 (b1);
Y ON M2 (b2);
Y ON X (cdash);
M1 ON X (a1);
M1 ON W (a3);
M1 ON XW (a4);
M2 ON X (a2);
M2 ON W (a5);
M2 ON XW (a6);
M2 ON M1 (d1);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
LWa1b1 MWa1b1 HWa1b1
LWa2b2 MWa2b2 HWa2b2
LWa1d1b2 MWa1d1b2 HWa1d1b2
IMM_A IMM_B IMM_C
TOT_LOWW TOT_MEDW TOT_HIW);
LOW_W = #LOWW;
MED_W = #MEDW;
HIGH_W = #HIGHW;
LWa1b1 = a1*b1 + a4*b1*LOW_W;
MWa1b1 = a1*b1 + a4*b1*MED_W;
HWa1b1 = a1*b1 + a4*b1*HIGH_W;
LWa2b2 = a2*b2 + a6*b2*LOW_W;
MWa2b2 = a2*b2 + a6*b2*MED_W;
HWa2b2 = a2*b2 + a6*b2*HIGH_W;
LWa1d1b2 = a1*d1*b2 + a4*d1*b2*LOW_W;
MWa1d1b2 = a1*d1*b2 + a4*d1*b2*MED_W;
HWa1d1b2 = a1*d1*b2 + a4*d1*b2*HIGH_W;
IMM_A = a4*b1;
IMM_B = a4*d1*b2;
IMM_C = a6*b2;
TOT_LOWW = LWa1d1b2 + LWa1b1 + LWa2b2 + cdash;
TOT_MEDW = MWa1d1b2 + MWa1b1 + MWa2b2 + cdash ;
TOT_HIW = HWa1d1b2 + HWa1b1 + HWa2b2 + cdash;
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，我们指定输出选项。STAND表示输出标准化系数，CINT(bcbootstrap)表示使用偏差校正自助法计算置信区间。至此，整个代码解读完毕。

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