Mplus model23 模型讲解

来自图书《MPlus中介调节模型》

Mplus模型23：包含平行中介、调节和调节中介效应的模型

  • 理论模型
  • 数学模型
  • 数学推导
  • Mplus代码解读

理论模型

数学模型

数学公式1

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ

数学公式2

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
将第二个方程代入第一个方程：

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + c'X

数学公式3

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
将第二个方程代入第一个方程：

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + c'X
展开括号：

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + a0b3V + a1b3XV + a2b3VW + a3b3ZV + a4b3XVW + a5b3XZV + c'X

数学公式4

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
将第二个方程代入第一个方程：

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + c'X
展开括号：

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + a0b3V + a1b3XV + a2b3VW + a3b3ZV + a4b3XVW + a5b3XZV + c'X
整理成 Y = a + bX 的形式，合并X的各项：

Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + a0b3V + a2b3VW + a3b3ZV) + (a1b1 + a4b1W + a5b1Z + a1b3V + a4b3VW + a5b3ZV + c')X

数学公式5

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
将第二个方程代入第一个方程：

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + c'X
展开括号：

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + a0b3V + a1b3XV + a2b3VW + a3b3ZV + a4b3XVW + a5b3XZV + c'X
整理成 Y = a + bX 的形式，合并X的各项：

Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + a0b3V + a2b3VW + a3b3ZV) + (a1b1 + a4b1W + a5b1Z + a1b3V + a4b3VW + a5b3ZV + c')X
X对Y的间接效应（在W, Z, V条件下）：

(a1 + a4W + a5Z)(b1 + b3V)

数学公式6

模型方程：

Y = b0 + b1M + b2V + b3MV + c'X

M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
将第二个方程代入第一个方程：

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + c'X
展开括号：

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + a0b3V + a1b3XV + a2b3VW + a3b3ZV + a4b3XVW + a5b3XZV + c'X
整理成 Y = a + bX 的形式，合并X的各项：

Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + a0b3V + a2b3VW + a3b3ZV) + (a1b1 + a4b1W + a5b1Z + a1b3V + a4b3VW + a5b3ZV + c')X
X对Y的间接效应（在W, Z, V条件下）：

(a1 + a4W + a5Z)(b1 + b3V)
X对Y的直接效应：

c'

代码解读1

USEVARIABLES = X M W Z V Y XW XZ MV;

代码解读2

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;

代码解读3

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;

代码解读4

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
MODEL:
 [Y] (b0);
 Y ON M (b1);
 Y ON V (b2);
 Y ON MV (b3);
 Y ON X (cdash);
 [M] (a0);
 M ON X (a1);
 M ON W (a2);
 M ON Z (a3);
 M ON XW (a4);
 M ON XZ (a5);

代码解读5

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
MODEL:
 [Y] (b0);
 Y ON M (b1);
 Y ON V (b2);
 Y ON MV (b3);
 Y ON X (cdash);
 [M] (a0);
 M ON X (a1);
 M ON W (a2);
 M ON Z (a3);
 M ON XW (a4);
 M ON XZ (a5);
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W LOW_Z MED_Z HIGH_Z LOW_V MED_V HIGH_V
 ILWLZLV IMWLZLV IHWLZLV ILWMZLV IMWMZLV IHWMZLV
 ILWHZLV IMWHZLV IHWHZLV
 ILWLZMV IMWLZMV IHWLZMV ILWMZMV IMWMZMV IHWMZMV
 ILWHZMV IMWHZMV IHWHZMV
 ILWLZHV IMWLZHV IHWLZHV ILWMZHV IMWMZHV IHWMZHV
 ILWHZHV IMWHZHV IHWHZHV
 TLWLZLV TMWLZLV THWLZLV TLWMZLV TMWMZLV THWMZLV
 TLWHZLV TMWHZLV THWHZLV
 TLWLZMV TMWLZMV THWLZMV TLWMZMV TMWMZMV THWMZMV
 TLWHZMV TMWHZMV THWHZMV
 TLWLZHV TMWLZHV THWLZHV TLWMZHV TMWMZHV THWMZHV
 TLWHZHV TMWHZHV THWHZHV);
 LOW_W = #LOWW;
 MED_W = #MEDW;
 HIGH_W = #HIGHW;
 LOW_Z = #LOWZ;
 MED_Z = #MEDZ;
 HIGH_Z = #HIGHZ;
 LOW_V = #LOWV;
 MED_V = #MEDV;
 HIGH_V = #HIGHV;
 ILWLZLV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*LOW_W + a5*b3*LOW_Z*LOW_V;
 IMWLZLV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*MED_W + a5*b3*LOW_Z*LOW_V;
 IHWLZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*HIGH_W + a5*b3*LOW_Z*LOW_V;
 ...
 TLWLZLV = ILWLZLV + cdash;
 TMWLZLV = IMWLZLV + cdash;
 THWLZLV = IHWLZLV + cdash;
 ...

代码解读6

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
MODEL:
 [Y] (b0);
 Y ON M (b1);
 Y ON V (b2);
 Y ON MV (b3);
 Y ON X (cdash);
 [M] (a0);
 M ON X (a1);
 M ON W (a2);
 M ON Z (a3);
 M ON XW (a4);
 M ON XZ (a5);
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W LOW_Z MED_Z HIGH_Z LOW_V MED_V HIGH_V
 ILWLZLV IMWLZLV IHWLZLV ILWMZLV IMWMZLV IHWMZLV
 ILWHZLV IMWHZLV IHWHZLV
 ILWLZMV IMWLZMV IHWLZMV ILWMZMV IMWMZMV IHWMZMV
 ILWHZMV IMWHZMV IHWHZMV
 ILWLZHV IMWLZHV IHWLZHV ILWMZHV IMWMZHV IHWMZHV
 ILWHZHV IMWHZHV IHWHZHV
 TLWLZLV TMWLZLV THWLZLV TLWMZLV TMWMZLV THWMZLV
 TLWHZLV TMWHZLV THWHZLV
 TLWLZMV TMWLZMV THWLZMV TLWMZMV TMWMZMV THWMZMV
 TLWHZMV TMWHZMV THWHZMV
 TLWLZHV TMWLZHV THWLZHV TLWMZHV TMWMZHV THWMZHV
 TLWHZHV TMWHZHV THWHZHV);
 LOW_W = #LOWW;
 MED_W = #MEDW;
 HIGH_W = #HIGHW;
 LOW_Z = #LOWZ;
 MED_Z = #MEDZ;
 HIGH_Z = #HIGHZ;
 LOW_V = #LOWV;
 MED_V = #MEDV;
 HIGH_V = #HIGHV;
 ILWLZLV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*LOW_W + a5*b3*LOW_Z*LOW_V;
 IMWLZLV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*MED_W + a5*b3*LOW_Z*LOW_V;
 IHWLZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*HIGH_W + a5*b3*LOW_Z*LOW_V;
 ...
 TLWLZLV = ILWLZLV + cdash;
 TMWLZLV = IMWLZLV + cdash;
 THWLZLV = IHWLZLV + cdash;
 ...
PLOT(PLWLZLV PMWLZLV PHWLZLV PLWMZLV PMWMZLV PHWMZLV
 PLWHZLV PMWHZLV PHWHZLV
 PLWLZMV PMWLZMV PHWLZMV PLWMZMV PMWMZMV PHWMZMV
 PLWHZMV PMWHZMV PHWHZMV
 PLWLZHV PMWLZHV PHWLZHV PLWMZHV PMWMZHV PHWMZHV
 PLWHZHV PMWHZHV PHWHZHV);
 LOOP(XVAL,1,5,0.1);
 PLWLZLV = ILWLZLV*XVAL;
 PMWLZLV = IMWLZLV*XVAL;
 PHWLZLV = IHWLZLV*XVAL;
 ...

代码解读7

USEVARIABLES = X M W Z V Y XW XZ MV;
DEFINE:
 MV = M*V;
 XW = X*W;
 XZ = X*Z;
ANALYSIS:
 TYPE = GENERAL;
 ESTIMATOR = ML;
 BOOTSTRAP = 10000;
MODEL:
 [Y] (b0);
 Y ON M (b1);
 Y ON V (b2);
 Y ON MV (b3);
 Y ON X (cdash);
 [M] (a0);
 M ON X (a1);
 M ON W (a2);
 M ON Z (a3);
 M ON XW (a4);
 M ON XZ (a5);
MODEL CONSTRAINT:
 NEW(LOW_W MED_W HIGH_W LOW_Z MED_Z HIGH_Z LOW_V MED_V HIGH_V
 ILWLZLV IMWLZLV IHWLZLV ILWMZLV IMWMZLV IHWMZLV
 ILWHZLV IMWHZLV IHWHZLV
 ILWLZMV IMWLZMV IHWLZMV ILWMZMV IMWMZMV IHWMZMV
 ILWHZMV IMWHZMV IHWHZMV
 ILWLZHV IMWLZHV IHWLZHV ILWMZHV IMWMZHV IHWMZHV
 ILWHZHV IMWHZHV IHWHZHV
 TLWLZLV TMWLZLV THWLZLV TLWMZLV TMWMZLV THWMZLV
 TLWHZLV TMWHZLV THWHZLV
 TLWLZMV TMWLZMV THWLZMV TLWMZMV TMWMZMV THWMZMV
 TLWHZMV TMWHZMV THWHZMV
 TLWLZHV TMWLZHV THWLZHV TLWMZHV TMWMZHV THWMZHV
 TLWHZHV TMWHZHV THWHZHV);
 LOW_W = #LOWW;
 MED_W = #MEDW;
 HIGH_W = #HIGHW;
 LOW_Z = #LOWZ;
 MED_Z = #MEDZ;
 HIGH_Z = #HIGHZ;
 LOW_V = #LOWV;
 MED_V = #MEDV;
 HIGH_V = #HIGHV;
 ILWLZLV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*LOW_W + a5*b3*LOW_Z*LOW_V;
 IMWLZLV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*MED_W + a5*b3*LOW_Z*LOW_V;
 IHWLZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a1*b3*LOW_V +
 a4*b3*LOW_V*HIGH_W + a5*b3*LOW_Z*LOW_V;
 ...
 TLWLZLV = ILWLZLV + cdash;
 TMWLZLV = IMWLZLV + cdash;
 THWLZLV = IHWLZLV + cdash;
 ...
PLOT(PLWLZLV PMWLZLV PHWLZLV PLWMZLV PMWMZLV PHWMZLV
 PLWHZLV PMWHZLV PHWHZLV
 PLWLZMV PMWLZMV PHWLZMV PLWMZMV PMWMZMV PHWMZMV
 PLWHZMV PMWHZMV PHWHZMV
 PLWLZHV PMWLZHV PHWLZHV PLWMZHV PMWMZHV PHWMZHV
 PLWHZHV PMWHZHV PHWHZHV);
 LOOP(XVAL,1,5,0.1);
 PLWLZLV = ILWLZLV*XVAL;
 PMWLZLV = IMWLZLV*XVAL;
 PHWLZLV = IHWLZLV*XVAL;
 ...
PLOT:
 TYPE = plot2;
OUTPUT:
 STAND CINT(bcbootstrap);

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