Mplus model67latent 模型讲解

来自图书《MPlus中介调节模型》

使用Mplus进行复杂调节中介分析：完整代码示例

• 理论模型
• 数学模型
• 数学推导
• 代码解读

大家好，欢迎来到本教程。本教程将手把手教大家如何使用Mplus软件进行复杂的调节中介分析。 我们会从理论模型开始，逐步讲解数学模型，然后进行关键步骤的数学推导，最后通过Mplus代码详解，一步步地构建和运行模型。本教程内容翔实，但结构清晰，方便大家按需查阅。不需要从头到尾学习，遇到问题可以直接翻到对应的部分。我们将会覆盖潜变量自变量、中介变量、调节变量和因变量，并计算在不同调节变量水平下的条件间接效应、条件直接效应和条件总效应。 相信通过本教程的学习，大家能够熟练掌握使用Mplus进行高级统计分析的技巧。

理论模型

我们来看这个模型图，它展示了一个比较复杂的调节中介模型。 X是我们的自变量，Y是因变量，M是中介变量，W和V是调节变量。 箭头表示变量之间的影响关系，比如X影响M，X也直接影响Y，M又影响Y。 W和V不光影响M和Y，还和X、M发生交互作用，共同调节X对Y的直接效应和M对Y的效应。W还单独调节X对M的效应。 接下来，我们会一步步地用数学模型和Mplus代码来分析这个模型。

数学模型

这张图展示了我们研究的数学模型。 它包含几个关键变量：X是自变量，M是中介变量，Y是因变量。W和V是两个调节变量，它们会影响X对M和Y的影响。 X通过不同的路径影响M，这些路径分别用a1、a2、a3表示。 同时，X与W和V的交互作用，产生了新的变量XW和XV，它们也会影响M。M直接影响Y，这个路径用b1表示。 此外，M与W和V的交互作用，也产生了新的变量MW和MV，它们同样影响Y。 X也直接影响Y，路径用c1’表示。 W和V也直接影响Y，路径分别用c2’和c4’表示。XW和XV也分别直接影响Y，路径分别用c3’和c5’表示。 总而言之，X对Y的影响包含直接效应和间接效应（通过M），而W和V则调节了X对M和Y的影响。

数学公式1

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV

这是我们的初始模型方程式，其中Y是因变量，X是自变量，M，W，V是中介变量。

数学公式2

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW

这是中介变量M的模型方程式，它与自变量X以及调节变量W有关。

数学公式3

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW
Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + c1'X + c2'W + c3'XW + c4'V + c5'XV

将第二个方程式代入第一个方程式。

数学公式4

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW
Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2W + a1b2XW + a2b2WW + a3b2XWW + a0b3V + a1b3XV + a2b3WV + a3b3XWV + c1'X + c2'W + c3'XW + c4'V + c5'XV

展开括号后的方程式。

数学公式5

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW
Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2W + a1b2XW + a2b2WW + a3b2XWW + a0b3V + a1b3XV + a2b3WV + a3b3XWV + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = (b0 + a0b1 + a2b1W + a0b2W + a2b2WW + a0b3V + a2b3WV + c2'W + c4'V) + (a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV + c1' + c3'W + c5'V)X

整理合并同类项，使方程式符合Y = a + bX的形式。

数学公式6

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW
Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2W + a1b2XW + a2b2WW + a3b2XWW + a0b3V + a1b3XV + a2b3WV + a3b3XWV + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = (b0 + a0b1 + a2b1W + a0b2W + a2b2WW + a0b3V + a2b3WV + c2'W + c4'V) + (a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV + c1' + c3'W + c5'V)X
条件间接效应 (X对Y的间接效应，在W, V条件下)：a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV = (a1 + a3W)(b1 + b2W + b3V)

这是X对Y的间接效应，考虑了W和V的调节作用。 注意这个简化后的形式。

数学公式7

Y = b0 + b1M + b2MW + b3MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW
Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2W + a1b2XW + a2b2WW + a3b2XWW + a0b3V + a1b3XV + a2b3WV + a3b3XWV + c1'X + c2'W + c3'XW + c4'V + c5'XV
Y = (b0 + a0b1 + a2b1W + a0b2W + a2b2WW + a0b3V + a2b3WV + c2'W + c4'V) + (a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV + c1' + c3'W + c5'V)X
条件间接效应 (X对Y的间接效应，在W, V条件下)：a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV = (a1 + a3W)(b1 + b2W + b3V)
条件直接效应 (X对Y的直接效应，在W, V条件下): c1' + c3'W + c5'V

这是X对Y的直接效应，同样考虑了W和V的调节作用。

代码解读1

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
Y1 Y2 Y3 Y4;

第一步，我们先定义要使用的变量。X1到X4是自变量X的指标，M1到M4是中介变量M的指标，W1到W4和V1到V4分别是调节变量W和V的指标，Y1到Y4是因变量Y的指标。 这些变量的名字清晰地表明了它们在模型中的作用。

代码解读2

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

接下来是分析设置。我们选择一般随机效应模型（TYPE = GENERAL RANDOM），使用最大似然估计法（ESTIMATOR = ML），并使用积分算法（ALGORITHM = INTEGRATION）来处理潜在变量。这些设置对于处理模型中的潜在变量非常重要。

代码解读3

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;

第三步，我们定义测量模型。 这里我们指定了每个潜在变量（X, M, W, V, Y）是如何由其对应的观测变量测量的。星号(*)表示W和V的第一个指标作为参考指标，它们的方差被固定为1，进行了标准化。

代码解读4

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;

然后，我们定义了潜在变量的交互项。例如，‘MW | M XWITH W’ 表示中介变量M与调节变量W的交互作用。类似地，我们也定义了M与V、X与W以及X与V的交互作用。

代码解读5

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);

这一步定义了结构模型中因变量Y与其他变量的关系，也就是路径系数。例如，‘Y ON M (b1)’ 表示Y受到M的影响，路径系数为b1。 我们定义了所有可能的路径。

代码解读6

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);

同样，我们也定义了中介变量M与其他变量的关系，即M如何受到X, W和XW的影响。

代码解读7

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...

现在进入模型约束部分。我们定义了新的参数，来代表调节变量W和V的不同水平（低、中、高），分别用-1, 0, 1表示，这代表标准差。

代码解读8

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILOW_LOV = a1*b1 + a3*b1*LOW_W + ...;
...其他间接效应计算...

接下来是关键部分，我们计算不同调节变量水平下的间接效应。这些公式是根据模型推导出的。例如，‘ILOW_LOV’ 表示在W处于低水平，V处于低水平时，X对Y的间接效应。

代码解读9

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILOW_LOV = a1*b1 + a3*b1*LOW_W + ...;
...其他间接效应计算...
DLOW_LOV = cdash1 + cdash3*LOW_W + cdash5*LOW_V;
...其他直接效应计算...

然后我们计算不同调节变量水平下的直接效应。这些公式也是根据模型推导出来的。

代码解读10

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILOW_LOV = a1*b1 + a3*b1*LOW_W + ...;
...其他间接效应计算...
DLOW_LOV = cdash1 + cdash3*LOW_W + cdash5*LOW_V;
...其他直接效应计算...
TLOW_LOV = ILOW_LOV + DLOW_LOV;
...其他总效应计算...

总效应是间接效应和直接效应的总和，我们也计算了不同条件下的总效应。

代码解读11

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILOW_LOV = a1*b1 + a3*b1*LOW_W + ...;
...其他间接效应计算...
DLOW_LOV = cdash1 + cdash3*LOW_W + cdash5*LOW_V;
...其他直接效应计算...
TLOW_LOV = ILOW_LOV + DLOW_LOV;
...其他总效应计算...
PLOT(PLOW_LOV PMEW_LOV PHIW_LOV ...);
LOOP(XVAL,-3,3,0.1); ...

最后，我们定义要绘制的图。使用循环语句，我们让X值从-3到3变化，计算不同X值下的间接效应，并绘制曲线图。

代码解读12

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4
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;
V BY V1* V2 V3 V4;
Y BY Y1 Y2 Y3 Y4;
W@1; V@1;
MW | M XWITH W;
MV | M XWITH V;
XW | X XWITH W;
XV | X XWITH V;
Y ON M (b1);
Y ON MW (b2);
Y ON MV (b3);
Y ON X(cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (cdash5);
M ON X (a1);
M ON W (a2);
M ON XW (a3);
MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V ...);
LOW_W = -1; MED_W = 0; HIGH_W = 1; ...
ILOW_LOV = a1*b1 + a3*b1*LOW_W + ...;
...其他间接效应计算...
DLOW_LOV = cdash1 + cdash3*LOW_W + cdash5*LOW_V;
...其他直接效应计算...
TLOW_LOV = ILOW_LOV + DLOW_LOV;
...其他总效应计算...
PLOT(PLOW_LOV PMEW_LOV PHIW_LOV ...);
LOOP(XVAL,-3,3,0.1); ...
PLOT: TYPE = plot2;
OUTPUT: CINT;

指定绘图类型为plot2，并输出置信区间。

资源汇总

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• Mplus 模型模板教程列表: https://mlln.cn/mplus-model-templates
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