AVE(Average Variance Extracted)是平均萃取变异量，又叫平均提取方差值，是检验结构变量内部一致性的统计量。
CR(Composite Reliability)是组合信度，是指一个组合变量(composite score,由多于一个变量的总和做成的新变量)的信度。
如果你用过Amos运行SEM，你是无法在报表中找到AVE和CR值的。当然，SPSS也无法得到，而Mplus需要写冗长的语句。能直接在报表中给出AVE和CR值的目前有SmartPLS，其用户界面很友好，操作很容易上手。R中也有一些包可以计算信效度中的cronbach’s alpha以及AVE，CR。由于SmartPLS操作简单，不再介绍，本次教大家用R语言计算AVE和CR值。
semTools
首先需要在Lavaan包中做验证性因子分析，然后使用semTools包对CFA的结果计算我们需要的值，本次使用的数据集是HolzingerSwineford1939，该数据集在Lavaan包中。
library(lavaan) # 载入lavaan，没有安装的用户使用install.packages命令安装看一下数据集的基本情况，
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设置模型
cfamodel <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
进行CFA
cfafit <- cfa(cfamodel, HolzingerSwineford1939, std.lv = T)note: td.lv和std.all的区别
In the first column (labeled td.lv), only the latent variables are standardized. In the second column (labeled Std.all), both latent and observed variables are standardized. The latter is often called the ‘completely standardized solution‘.
显示CFA的总体结果，需要查看拟合指数
summary(cfafit, fit.measures = T)
lavaan 0.6-2 ended normally after 20 iterations
Optimization method NLMINB
Number of free parameters 21
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 11.128 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 7.779 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 1.000
textual 1.000
speed 1.000 显示所有拟合指数，能在文献中找到的拟合指数全都列了出来，这是其他软件做不到的
fitMeasures(cfafit, fit.measures = "all", baseline.model = NULL)
npar fmin chisq df pvalue baseline.chisq baseline.df
21.000 0.142 85.306 24.000 0.000 918.852 36.000
baseline.pvalue cfi tli nnfi rfi nfi pnfi
0.000 0.931 0.896 0.896 0.861 0.907 0.605
ifi rni logl unrestricted.logl aic bic ntotal
0.931 0.931 -3737.745 -3695.092 7517.490 7595.339 301.000
bic2 rmsea rmsea.ci.lower rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
7528.739 0.092 0.071 0.114 0.001 0.082 0.082
srmr srmr_bentler srmr_bentler_nomean srmr_bollen srmr_bollen_nomean srmr_mplus srmr_mplus_nomean
0.065 0.065 0.065 0.065 0.065 0.065 0.065
cn_05 cn_01 gfi agfi pgfi mfi ecvi
129.490 152.654 0.943 0.894 0.503 0.903 0.423 显示标准化结果
standardizedsolution(cfafit)
lhs op rhs est.std se z pvalue ci.lower ci.upper
1 visual =~ x1 0.772 0.055 14.041 0 0.664 0.880
2 visual =~ x2 0.424 0.060 7.105 0 0.307 0.540
3 visual =~ x3 0.581 0.055 10.539 0 0.473 0.689
4 textual =~ x4 0.852 0.023 37.776 0 0.807 0.896
5 textual =~ x5 0.855 0.022 38.273 0 0.811 0.899
6 textual =~ x6 0.838 0.023 35.881 0 0.792 0.884
7 speed =~ x7 0.570 0.053 10.714 0 0.465 0.674
8 speed =~ x8 0.723 0.051 14.309 0 0.624 0.822
9 speed =~ x9 0.665 0.051 13.015 0 0.565 0.765
10 x1 ~~ x1 0.404 0.085 4.763 0 0.238 0.571
11 x2 ~~ x2 0.821 0.051 16.246 0 0.722 0.920
12 x3 ~~ x3 0.662 0.064 10.334 0 0.537 0.788
13 x4 ~~ x4 0.275 0.038 7.157 0 0.200 0.350
14 x5 ~~ x5 0.269 0.038 7.037 0 0.194 0.344
15 x6 ~~ x6 0.298 0.039 7.606 0 0.221 0.374
16 x7 ~~ x7 0.676 0.061 11.159 0 0.557 0.794
17 x8 ~~ x8 0.477 0.073 6.531 0 0.334 0.620
18 x9 ~~ x9 0.558 0.068 8.208 0 0.425 0.691
19 visual ~~ visual 1.000 0.000 NA NA 1.000 1.000
20 textual ~~ textual 1.000 0.000 NA NA 1.000 1.000
21 speed ~~ speed 1.000 0.000 NA NA 1.000 1.000
22 visual ~~ textual 0.459 0.064 7.189 0 0.334 0.584
23 visual ~~ speed 0.471 0.073 6.461 0 0.328 0.613
24 textual ~~ speed 0.283 0.069 4.117 0 0.148 0.418调用semTools包
library(semTools)
###############################################################################
This is semTools 0.5-0
All users of R (or SEM) are invited to submit functions or ideas for functions.
###############################################################################计算AVE,CR等，结果中omega为CR，avevar为AVE，alpha为cronbach’s alpha
reliability(cfafit)
visual textual speed total
alpha 0.6261171 0.8827069 0.6884550 0.7604886
omega 0.6253185 0.8851756 0.6877607 0.8453355
omega2 0.6253185 0.8851756 0.6877607 0.8453355
omega3 0.6120058 0.8850625 0.6858435 0.8596225
avevar 0.3705594 0.7210168 0.4244891 0.5145880matrixpls
这个包是关于偏最小二乘法的，内部算法和一般的CB-SEM不一样，所以得出的结果也会不同。
来看下关于matrixpls包的基本介绍：
Estimates a weight matrix using Partial Least Squares or a related algorithm and then uses the weights to estimate the parameters of a statistical model.
matrixpls(S,
model,
W.model = NULL,
weightFun = weightFun.pls,
parameterEstim = parameterEstim.separate,
weightSign = NULL, ...,
validateInput = TRUE,
standardize = TRUE)参数介绍：
S
Covariance matrix of the data.
model
There are two options for this argument: 1. lavaan script or lavaan parameter table, or 2. a list containing three matrices inner, reflective, and formative defining the free regression paths in the model.
W.model
An optional numeric matrix representing the weight pattern and starting weights (i.e. the how the indicators are combined to form the composite variables). If this argument is not specified, the weight patter is defined based on the relationships in the reflective and formative elements of model.
weightFun
A function for calculating indicator weights using the data covariance matrix S, a model specification model, and a weight pattern W.model. Returns a weigth matrix W. The default is weightFun.pls
parameterEstim
A function for estimating the model parameters using the data covariance matrix S, model specification model, and weight matrix W. Returns a named vector of parameter estimates. The default is parameterEstim.separate
weightSign
A function for resolving weight sign ambiquity based on the data covariance matrix S and a weight matrix W. Returns a weigth matrix W. See weightSign for details.
...
All other arguments are passed through to weightFun and parameterEstim.
validateInput
If TRUE, the arguments are validated.
standardize
If TRUE, S is converted to a correlation matrix before analysis.上面参数众多，本次只用到S，model及standardize，其他参数大家有兴趣可以自行研究。
# 载入matrixpls包
library(matrixpls)
# 建立模型，并赋值给cfamodel
cfamodel <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
# 我们只用到数据集中的x1-x9，前几行不是数值型数据，没无法计算协方差，所以这里取子集
mydata <- subset(HolzingerSwineford1939, select = c(x1, x2, x3, x4, x5, x6, x7, x8, x9))
# 计算数据的协方差矩阵，并赋值给cov
cov <- cov(mydata)
# 用matrixpls计算模型，并赋值给fit
fit <- matrixpls(cov, cfamodel, standardize = T)
# 查看fit的总体结果
summary(fit)
matrixpls parameter estimates # 载荷值
Est.
visual=~x1 0.7742312
visual=~x2 0.6951567
visual=~x3 0.7996425
textual=~x4 0.9012916
textual=~x5 0.9076744
textual=~x6 0.8956943
speed=~x7 0.7738933
speed=~x8 0.8343229
speed=~x9 0.7475307
matrixpls weights # weights
x1 x2 x3 x4 x5 x6 x7 x8 x9
visual 0.4495824 0.4036703 0.464339 0.0000000 0.0000000 0.0000000 0.0000000 0.000000 0.0000000
textual 0.0000000 0.0000000 0.000000 0.3696139 0.3722315 0.3673185 0.0000000 0.000000 0.0000000
speed 0.0000000 0.0000000 0.000000 0.0000000 0.0000000 0.0000000 0.4174613 0.450059 0.4032409
Weight algorithm converged in 10 iterations.
Total Effects (column on row)
<0 x 0 matrix>
Direct Effects
<0 x 0 matrix>
Indirect Effects
<0 x 0 matrix>
Inner model squared multiple correlations (R2)
visual textual speed
0 0 0
Inner model (composite) residual covariance matrix
<0 x 0 matrix>
Outer model (indicator) residual covariance matrix
x1 x2 x3 x4 x5 x6 x7 x8 x9
x1 0.59943401 0.53821206 0.61910818 0.11473134 0.05278178 0.13633379 -0.027837523 0.1161719 0.2669108
x2 0.53821206 0.48324288 0.55587684 0.24748588 0.19288654 0.26598479 0.101806987 0.2410096 0.3621323
x3 0.61910818 0.55587684 0.63942808 0.32621467 0.26370287 0.34723616 0.009335276 0.1610452 0.3124602
x4 0.32705111 0.14917604 0.13006763 0.81232650 0.81807926 0.80728176 0.130107846 0.1293709 0.2143993
x5 0.39212931 0.20658448 0.19581075 0.81807926 0.82387277 0.81299880 0.138153278 0.1379674 0.2227990
x6 0.30270509 0.12821376 0.10621252 0.80728176 0.81299880 0.80226835 0.128504702 0.1277103 0.2122997
x7 0.36400676 0.20002830 0.33786745 0.18214614 0.17631204 0.18181012 0.598910899 0.6456770 0.5785091
x8 0.24624719 0.08439455 0.21326888 0.20726547 0.20105298 0.20683553 0.645676971 0.6960948 0.6236820
x9 0.05780685 -0.07057904 0.02291509 0.08721782 0.08095414 0.08744424 0.578509057 0.6236820 0.5588022
Residual-based fit indices
Value
Communality 0.6682645
Redundancy 0.0000000
SMC NaN
RMS outer residual covariance 0.3821996
RMS inner residual covariance NaN
SRMR 0.3418497
SRMR (Henseler) 0.2067663
Absolute goodness of fit: NaN
Composite Reliability indices # CR值
visual textual speed
0.8011490 0.9287099 0.8288176
Average Variance Extracted indices # AVE值
visual textual speed
0.5740350 0.8128225 0.6179360
AVE - largest squared correlation
visual textual speed
0.4738310 0.7126186 0.5392400
Heterotrait-monotrait matrix
visual textual speed
visual 0.0000000 0.0000000 0.0000000
textual 0.4243230 0.0000000 0.0000000
speed 0.4235419 0.2895679 0.0000000本文使用了R语言中的semTools和matrixpls包计算了AVE和CR值，大家一般使用semTools已经足够，操作相对简洁。
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