Mplus model47latent 模型讲解

复杂中介调节模型的Mplus实现

理论模型
数学模型
数学推导
Mplus代码解读

大家好，欢迎来到本期教程。今天我们将学习如何使用Mplus构建一个复杂的统计模型，它包含中介效应、调节效应以及两者兼而有之的调节中介效应。这个模型非常实用，可以帮助大家分析更复杂的社会科学数据。本教程会以一个具体的例子展开，这个例子包含一个或多个并行中介变量，以及四个调节变量，模型中包含了所有可能的二阶和三阶交互项，难度相当大。我们将会一步一步地讲解，首先介绍模型的理论框架，然后给出它的数学表达，再进行详细的数学推导，最后给出Mplus代码，并详细解读代码的含义和作用。本教程的目的是提供一个方便快捷的参考，帮助大家快速上手，解决类似问题的分析，所以并不会面面俱到地讲解Mplus的所有功能。我们会重点关注条件间接效应的计算、检验和可视化，这部分内容在实际研究中非常重要。希望通过本教程，大家能够掌握Mplus在处理复杂中介调节模型方面的应用技巧。感谢Data Sense提供本教程，Data Sense也提供统计咨询服务，如有需要欢迎联系我们。

理论模型

数学模型

这张图展示了一个复杂的调节中介模型。我们来看一下各个变量：X是自变量，Y是因变量，M是中介变量。W、Z、V、Q是四个调节变量，它们会影响自变量和中介变量之间的关系，以及中介变量和因变量之间的关系。图中展示了X如何通过不同的路径（a1到a5）影响M，其中a4和a5代表X与W和Z的交互作用。M又通过路径b1影响Y。V和Q分别通过b2和b3直接影响Y。此外，模型还包含了M与V的交互作用（b4），M与Q的交互作用（b5），V与Q的交互作用（b6），以及M、V、Q的三阶交互作用（b7）。最后，c’代表X对Y的直接效应。这个模型考虑了所有可能的二阶和三阶交互作用，让我们能够更全面地理解这些变量之间的复杂关系。

数学公式1

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X

数学公式2

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ

数学公式3

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3Q + b4(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + b5(a0 + a1X + a2W + a3Z + a4XW + a5XZ)Q + b6VQ + b7(a0 + a1X + a2W + a3Z + a4XW + a5XZ)VQ + c'X

数学公式4

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3Q + b4(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + b5(a0 + a1X + a2W + a3Z + a4XW + a5XZ)Q + b6VQ + b7(a0 + a1X + a2W + a3Z + a4XW + a5XZ)VQ + c'X
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + b3Q + a0b4V + a1b4XV + a2b4WV + a3b4ZV + a4b4XWV + a5b4XZV + a0b5Q + a1b5XQ + a2b5WQ + a3b5ZQ + a4b5XWQ + a5b5XZQ + b6VQ + a0b7VQ + a1b7XVQ + a2b7WVQ + a3b7ZVQ + a4b7XWVQ + a5b7XZVQ + c'X

数学公式5

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3Q + b4(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + b5(a0 + a1X + a2W + a3Z + a4XW + a5XZ)Q + b6VQ + b7(a0 + a1X + a2W + a3Z + a4XW + a5XZ)VQ + c'X
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + b3Q + a0b4V + a1b4XV + a2b4WV + a3b4ZV + a4b4XWV + a5b4XZV + a0b5Q + a1b5XQ + a2b5WQ + a3b5ZQ + a4b5XWQ + a5b5XZQ + b6VQ + a0b7VQ + a1b7XVQ + a2b7WVQ + a3b7ZVQ + a4b7XWVQ + a5b7XZVQ + c'X
Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + b3Q + a0b4V + a2b4WV + a3b4ZV + a0b5Q + a2b5WQ + a3b5ZQ + b6VQ + a0b7VQ + a2b7WVQ + a3b7ZVQ) + (a1b1 + a4b1W + a5b1Z + a1b4V + a4b4WV + a5b4ZV + a1b5Q + a4b5WQ + a5b5ZQ + a1b7VQ + a4b7WVQ + a5b7ZVQ + c')X

数学公式6

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3Q + b4(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + b5(a0 + a1X + a2W + a3Z + a4XW + a5XZ)Q + b6VQ + b7(a0 + a1X + a2W + a3Z + a4XW + a5XZ)VQ + c'X
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + b3Q + a0b4V + a1b4XV + a2b4WV + a3b4ZV + a4b4XWV + a5b4XZV + a0b5Q + a1b5XQ + a2b5WQ + a3b5ZQ + a4b5XWQ + a5b5XZQ + b6VQ + a0b7VQ + a1b7XVQ + a2b7WVQ + a3b7ZVQ + a4b7XWVQ + a5b7XZVQ + c'X
Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + b3Q + a0b4V + a2b4WV + a3b4ZV + a0b5Q + a2b5WQ + a3b5ZQ + b6VQ + a0b7VQ + a2b7WVQ + a3b7ZVQ) + (a1b1 + a4b1W + a5b1Z + a1b4V + a4b4WV + a5b4ZV + a1b5Q + a4b5WQ + a5b5ZQ + a1b7VQ + a4b7WVQ + a5b7ZVQ + c')X
X 对 Y 的间接效应 (在 W, Z, V, Q 条件下)：(a1 + a4W + a5Z)(b1 + b4V + b5Q + b7VQ)

数学公式7

Y = b0 + b1M + b2V + b3Q + b4MV + b5MQ + b6VQ + b7MVQ + c'X
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ
Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ) + b2V + b3Q + b4(a0 + a1X + a2W + a3Z + a4XW + a5XZ)V + b5(a0 + a1X + a2W + a3Z + a4XW + a5XZ)Q + b6VQ + b7(a0 + a1X + a2W + a3Z + a4XW + a5XZ)VQ + c'X
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + b2V + b3Q + a0b4V + a1b4XV + a2b4WV + a3b4ZV + a4b4XWV + a5b4XZV + a0b5Q + a1b5XQ + a2b5WQ + a3b5ZQ + a4b5XWQ + a5b5XZQ + b6VQ + a0b7VQ + a1b7XVQ + a2b7WVQ + a3b7ZVQ + a4b7XWVQ + a5b7XZVQ + c'X
Y = (b0 + a0b1 + a2b1W + a3b1Z + b2V + b3Q + a0b4V + a2b4WV + a3b4ZV + a0b5Q + a2b5WQ + a3b5ZQ + b6VQ + a0b7VQ + a2b7WVQ + a3b7ZVQ) + (a1b1 + a4b1W + a5b1Z + a1b4V + a4b4WV + a5b4ZV + a1b5Q + a4b5WQ + a5b5ZQ + a1b7VQ + a4b7WVQ + a5b7ZVQ + c')X
X 对 Y 的间接效应 (在 W, Z, V, Q 条件下)：(a1 + a4W + a5Z)(b1 + b4V + b5Q + b7VQ)
X 对 Y 的直接效应：c'

代码解读1

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;

代码解读2

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;

代码解读3

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;

代码解读4

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;

代码解读5

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;

代码解读6

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
Y ON M (b1);
Y ON V (b2);
Y ON Q (b3);
Y ON MV (b4);
Y ON MQ (b5);
Y ON VQ (b6);
Y ON MVQ (b7);
Y ON X (cdash);

代码解读7

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
Y ON M (b1);
Y ON V (b2);
Y ON Q (b3);
Y ON MV (b4);
Y ON MQ (b5);
Y ON VQ (b6);
Y ON MVQ (b7);
Y ON X (cdash);
M ON X (a1);
M ON W (a2);
M ON Z (a3);
M ON XW (a4);
M ON XZ (a5);

代码解读8

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
Y ON M (b1);
Y ON V (b2);
Y ON Q (b3);
Y ON MV (b4);
Y ON MQ (b5);
Y ON VQ (b6);
Y ON MVQ (b7);
Y ON X (cdash);
M ON X (a1);
M ON W (a2);
M ON Z (a3);
M ON XW (a4);
M ON XZ (a5);
MODEL CONSTRAINT:
...
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILLLL = a1*b1 + a4*b1*LOW_W + ...; ...

代码解读9

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
Y ON M (b1);
Y ON V (b2);
Y ON Q (b3);
Y ON MV (b4);
Y ON MQ (b5);
Y ON VQ (b6);
Y ON MVQ (b7);
Y ON X (cdash);
M ON X (a1);
M ON W (a2);
M ON Z (a3);
M ON XW (a4);
M ON XZ (a5);
MODEL CONSTRAINT:
...
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILLLL = a1*b1 + a4*b1*LOW_W + ...; ...
PLOT(...);
LOOP(XVAL,-3,3,0.1);
PLLLL = ILLLL*XVAL; ...

代码解读10

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4 W1 W2 W3 W4 Z1 Z2 Z3 Z4 V1 V2 V3 V4 Q1 Q2 Q3 Q4 Y1 Y2 Y3 Y4;
ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;
MODEL:
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
Z BY Z1* Z2 Z3 Z4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;
W@1;  Z@1;  V@1;  Q@1;
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XZ | X XWITH Z;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
Y ON M (b1);
Y ON V (b2);
Y ON Q (b3);
Y ON MV (b4);
Y ON MQ (b5);
Y ON VQ (b6);
Y ON MVQ (b7);
Y ON X (cdash);
M ON X (a1);
M ON W (a2);
M ON Z (a3);
M ON XW (a4);
M ON XZ (a5);
MODEL CONSTRAINT:
...
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILLLL = a1*b1 + a4*b1*LOW_W + ...; ...
PLOT(...);
LOOP(XVAL,-3,3,0.1);
PLLLL = ILLLL*XVAL; ...
OUTPUT: CINT;

资源汇总

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