今天教大家如何在R中做验证性因子分析，R中能做CFA的包不止一个，但lavaan是最常用的SEM包。lavaan是latent variable analysis的组合，从名称中我们就可以看出它是一个关于潜变量分析的工具。
lavaan的相关资料网站有：
lavaan: An R Package for Structural Equation Modeling
install.packages("lavaan")
library(lavaan) # 安装并载入lavaan包Lavaan包中有个数据集叫HolzingerSwineford1939，本次验证性因子分析的数据就在这个数据集中，以下是关于这个数据集的描述：
The classic Holzinger and Swineford (1939) dataset consists of mental ability test scores of seventh- and eighth-grade children from two different schools (Pasteur and Grant-White). In the original dataset (available in the MBESS package), there are scores for 26 tests. However, a smaller subset with 9 variables is more widely used in the literature (for example in Joreskog’s 1969 paper, which also uses the 145 subjects from the Grant-White school only).
在安装lavaan包后，简要查看下数据集中的数据大致是个什么样子
head(HolzingerSwineford1939)
id sex ageyr agemo school grade x1 x2 x3 x4 x5 x6 x7 x8 x9
1 1 1 13 1 Pasteur 7 3.333333 7.75 0.375 2.333333 5.75 1.2857143 3.391304 5.75 6.361111
2 2 2 13 7 Pasteur 7 5.333333 5.25 2.125 1.666667 3.00 1.2857143 3.782609 6.25 7.916667
3 3 2 13 1 Pasteur 7 4.500000 5.25 1.875 1.000000 1.75 0.4285714 3.260870 3.90 4.416667
4 4 1 13 2 Pasteur 7 5.333333 7.75 3.000 2.666667 4.50 2.4285714 3.000000 5.30 4.861111
5 5 2 12 2 Pasteur 7 4.833333 4.75 0.875 2.666667 4.00 2.5714286 3.695652 6.30 5.916667
6 6 2 14 1 Pasteur 7 5.333333 5.00 2.250 1.000000 3.00 0.8571429 4.347826 6.65 7.500000查看数据集的结构，数据集中包含15个变量variables，共301个观察值obs，5个整数型变量int，1个因子型变量factor，9个数值型变量num。
str(HolzingerSwineford1939)
'data.frame': 301 obs. of 15 variables:
$ id : int 1 2 3 4 5 6 7 8 9 11 ...
$ sex : int 1 2 2 1 2 2 1 2 2 2 ...
$ ageyr : int 13 13 13 13 12 14 12 12 13 12 ...
$ agemo : int 1 7 1 2 2 1 1 2 0 5 ...
$ school: Factor w/ 2 levels "Grant-White",..: 2 2 2 2 2 2 2 2 2 2 ...
$ grade : int 7 7 7 7 7 7 7 7 7 7 ...
$ x1 : num 3.33 5.33 4.5 5.33 4.83 ...
$ x2 : num 7.75 5.25 5.25 7.75 4.75 5 6 6.25 5.75 5.25 ...
$ x3 : num 0.375 2.125 1.875 3 0.875 ...
$ x4 : num 2.33 1.67 1 2.67 2.67 ...
$ x5 : num 5.75 3 1.75 4.5 4 3 6 4.25 5.75 5 ...
$ x6 : num 1.286 1.286 0.429 2.429 2.571 ...
$ x7 : num 3.39 3.78 3.26 3 3.7 ...
$ x8 : num 5.75 6.25 3.9 5.3 6.3 6.65 6.2 5.15 4.65 4.55 ...
$ x9 : num 6.36 7.92 4.42 4.86 5.92 ...
一共3个潜变量，分别为visual, textual, speed
visual包含3个观察变量，x1,x2,x3
textual包含3个观察变量，x4,x5,x6
speed包含3个观察变量，x7,x8,x9
划重点了，=~符号左边是潜变量，右边是指标
latent variable =~ indicator1 + indicator2 + indicator3我们按照上述格式，列出本例中3个潜变量的组成情况
visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9然后组建模型，我们将这个模型命名为HS.model
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '下一步用cfa命令做验证性因子分析，cfa即为验证性因子分析的英文缩写，所用数据为HolzingerSwineford1939，并将cfa的结果赋值给fit
fit <- cfa(HS.model, data=HolzingerSwineford1939)用summary命令查看fit结果，fit.measures = TRUE表示我们需要拟合度指标，默认不给出。
summary(fit, fit.measures=TRUE)
lavaan (0.6-1) converged normally after 35 iterations
Number of observations 301
Estimator ML
Model Fit Test Statistic 85.306
Degrees of freedom 24
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 918.852
Degrees of freedom 36
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.931
Tucker-Lewis Index (TLI) 0.896
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -3737.745
Loglikelihood unrestricted model (H1) -3695.092
Number of free parameters 21
Akaike (AIC) 7517.490
Bayesian (BIC) 7595.339
Sample-size adjusted Bayesian (BIC) 7528.739
Root Mean Square Error of Approximation:
RMSEA 0.092
90 Percent Confidence Interval 0.071 0.114
P-value RMSEA <= 0.05 0.001
Standardized Root Mean Square Residual:
SRMR 0.065
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
visual =~
x1 1.000
x2 0.554 0.100 5.554 0.000
x3 0.729 0.109 6.685 0.000
textual =~
x4 1.000
x5 1.113 0.065 17.014 0.000
x6 0.926 0.055 16.703 0.000
speed =~
x7 1.000
x8 1.180 0.165 7.152 0.000
x9 1.082 0.151 7.155 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
visual ~~
textual 0.408 0.074 5.552 0.000
speed 0.262 0.056 4.660 0.000
textual ~~
speed 0.173 0.049 3.518 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.x1 0.549 0.114 4.833 0.000
.x2 1.134 0.102 11.146 0.000
.x3 0.844 0.091 9.317 0.000
.x4 0.371 0.048 7.779 0.000
.x5 0.446 0.058 7.642 0.000
.x6 0.356 0.043 8.277 0.000
.x7 0.799 0.081 9.823 0.000
.x8 0.488 0.074 6.573 0.000
.x9 0.566 0.071 8.003 0.000
visual 0.809 0.145 5.564 0.000
textual 0.979 0.112 8.737 0.000
speed 0.384 0.086 4.451 0.000结果中包含了拟合指标，指标载荷，协方差和方差等信息，但我们注意到题项载荷那一块有超过1的数值，说明这是未标准化的。鉴于文章大多要报告标准化的数据，那么如何才能得到标准化的题项载荷呢？
我们需要在cfa命令中加入http://std.lv = TRUE，然后再查看fit，此时便得到了标准化后的题项载荷。
fit <- cfa(HS.model, data=HolzingerSwineford1939, std.lv=T)
summary(fit, fit.measures=TRUE)
lavaan (0.6-1) converged normally after 22 iterations
Number of observations 301
Estimator ML
Model Fit Test Statistic 85.306
Degrees of freedom 24
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 918.852
Degrees of freedom 36
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.931
Tucker-Lewis Index (TLI) 0.896
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -3737.745
Loglikelihood unrestricted model (H1) -3695.092
Number of free parameters 21
Akaike (AIC) 7517.490
Bayesian (BIC) 7595.339
Sample-size adjusted Bayesian (BIC) 7528.739
Root Mean Square Error of Approximation:
RMSEA 0.092
90 Percent Confidence Interval 0.071 0.114
P-value RMSEA <= 0.05 0.001
Standardized Root Mean Square Residual:
SRMR 0.065
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
visual =~
x1 0.900 0.081 11.127 0.000
x2 0.498 0.077 6.429 0.000
x3 0.656 0.074 8.817 0.000
textual =~
x4 0.990 0.057 17.474 0.000
x5 1.102 0.063 17.576 0.000
x6 0.917 0.054 17.082 0.000
speed =~
x7 0.619 0.070 8.903 0.000
x8 0.731 0.066 11.090 0.000
x9 0.670 0.065 10.305 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
visual ~~
textual 0.459 0.064 7.189 0.000
speed 0.471 0.073 6.461 0.000
textual ~~
speed 0.283 0.069 4.117 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.x1 0.549 0.114 4.833 0.000
.x2 1.134 0.102 11.146 0.000
.x3 0.844 0.091 9.317 0.000
.x4 0.371 0.048 7.778 0.000
.x5 0.446 0.058 7.642 0.000
.x6 0.356 0.043 8.277 0.000
.x7 0.799 0.081 9.823 0.000
.x8 0.488 0.074 6.573 0.000
.x9 0.566 0.071 8.003 0.000
visual 1.000
textual 1.000
speed 1.000 参考文献：
Yves Rosseel (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1-36. URL http://www.jstatsoft.org/v48/i02/