Chapter 6 测量恒等性检验
用来检验，研究者提出的测量模型所具有的因素结构，在不同样本（例如男/女，主管/下属）之间是否具有等同性，或者说，这套测量工具用于不同的对象，或同一对象不同时间点时，测量分数是否表述了相同的意义。
相比于前面几章的单样本CFA分析，需要增加一个平行样本，也就是另纳入一个类别变量来反映共变结构的跨样本变化。
library(tidyverse)
library(lavaan)
library(modelsummary)
library(semPlot)
library(corrplot)
# https://cran.r-project.org/web/packages/semptools/vignettes/semptools.html
library(semptools) # 给路径图标记显著性6.1 案例说明
在本册的第2章中，我们对《组织创新气氛量表》进行了CFA分析。本章将该CFA分析扩展到性别之间的恒等性比较，换句话说，本章旨在回答——组织创新气氛的知觉的因素结构，在不同性别之间是否具有恒等性？
回答该问题，主要操作分两个阶段：
- 阶段一：对全样本进行CFA分析，查看拟合情况，如果拟合情况良好，按性别分样本进行CFA分析，如每个样本的拟合状况都良好，则进入第二阶段。反之，则没有必要进行继续比较了。
- 阶段二：进行多样本结构方程模型分析，也就是在纳入性别变量的基础上，对全样本进行模型拟合。在这一过程中，会分别拟合多个嵌套模型，从而对不同性别因素结构中的系数恒等性进行检验。
第二阶段的模型逻辑相对比较复杂，请仔细阅读原书第九章。
6.2 第一阶段分析
首先读取数据
dat_female <- read_csv("data/Ch09f.csv") # 女性数据(n=140)
dat_male <- read_csv("data/Ch09m.csv") # 男性数据(n = 172)
# 数据组合起来
dat_all <- bind_rows(dat_female, dat_male, .id = "gender") |>
mutate(gender = ifelse(gender == 1, "female", "male"))分性别进行描述统计，结果见表6.1
datasummary(All(dat_all |> select(-gender)) ~ gender * (mean + sd + Histogram),
data = dat_all,
title = "描述统计表")## Warning in datasummary(All(select(dat_all, -gender)) ~ gender * (mean + :
## The `All()` function does not accept tibbles. Please convert `data` to a
## `data.frame` before calling `datasummary`. A bugfix patch has been submitted
## upstream to the `tables` package.| mean | sd | Histogram | mean | sd | Histogram | |
|---|---|---|---|---|---|---|
| A1 | 4.24 | 1.04 | ▁▃▆▇▁ | 4.55 | 0.91 | ▂▅▇▂ |
| A2 | 4.14 | 1.08 | ▂▃▇▇▁ | 4.44 | 0.96 | ▁▂▅▇▂ |
| A3 | 3.97 | 1.04 | ▁▃▇▃▁ | 4.15 | 0.90 | ▄▇▆▁ |
| B1 | 3.89 | 1.16 | ▂▇▄▇▁ | 4.11 | 1.15 | ▁▆▅▇▂ |
| B2 | 4.05 | 1.15 | ▁▁▅▇▇▂ | 4.41 | 1.14 | ▁▃▄▇▃ |
| B3 | 4.04 | 1.13 | ▁▅▇▇▂ | 4.39 | 1.02 | ▁▃▅▇▂ |
| C1 | 4.28 | 1.05 | ▁▃▆▇▂ | 4.44 | 0.91 | ▂▆▇▂ |
| C2 | 4.34 | 1.06 | ▁▃▆▇▂ | 4.33 | 1.00 | ▁▃▇▆▂ |
| C3 | 4.14 | 1.11 | ▁▁▃▇▅▂ | 4.44 | 0.97 | ▂▅▇▂ |
| D1 | 4.84 | 0.96 | ▁▄▇▄ | 4.83 | 0.92 | ▁▄▇▄ |
| D2 | 4.94 | 0.87 | ▁▄▇▅ | 4.95 | 0.81 | ▄▇▄ |
| D3 | 4.74 | 1.01 | ▂▆▇▅ | 4.91 | 0.82 | ▁▄▇▄ |
| E1 | 4.48 | 1.05 | ▂▇▇▃ | 4.75 | 0.89 | ▁▅▇▃ |
| E2 | 4.54 | 1.11 | ▁▂▆▇▄ | 4.88 | 0.89 | ▁▄▇▄ |
| E3 | 4.58 | 1.01 | ▃▅▇▃ | 4.80 | 0.94 | ▁▄▇▄ |
| F1 | 4.14 | 1.11 | ▁▄▇▄▂ | 4.30 | 1.20 | ▂▂▇▆▃ |
| F2 | 4.39 | 1.16 | ▁▂▇▅▃ | 4.81 | 1.00 | ▁▅▇▅ |
| F3 | 4.22 | 0.99 | ▂▇▄▁ | 4.70 | 0.84 | ▁▅▇▃ |
然后我们分别对总体，女性和男性拟合CFA模型，并将拟合度情况保存下来，如表6.2所示。
该表为原书289页的表9.3的上半部分。可以看到，拟合情形还算比较一致，而且各指数差异不大，因此进入第二阶段分析。
cfa_model<-'FA =~ NA*A1 + A2 + A3
FB =~ NA*B1 + B2 + B3
FC =~ NA*C1 + C2 + C3
FD =~ NA*D1 + D2 + D3
FE =~ NA*E1 + E2 + E3
FF =~ NA*F1 + F2 + F3
FA~~1*FA
FB~~1*FB
FC~~1*FC
FD~~1*FD
FE~~1*FE
FF~~1*FF'
# 将准备拟合cfa的各组数据放在一个list里面
dat_list <- list(全体样本 = dat_all, 男性样本 = dat_male, 女性样本 = dat_female)
# 写一个函数，执行cfa分析，并保存拟合结果
cal_fitness <- function(dat){
cfa_fit <- cfa(model = cfa_model,
data = dat)
fitness <- fitmeasures(cfa_fit)[c('chisq', 'df', 'rmsea','nnfi', 'nfi', 'cfi', 'gfi')]
split(fitness, names(fitness))
}
# 输出结果
map_dfr(dat_list, ~cal_fitness(.), .id = 'Sample') |>
select(Sample, chisq, df, rmsea, nnfi, cfi, gfi) |> # 给列排一下序
mutate(across(where(is.numeric), ~round(., 3))) |> # 四舍五入
knitr::kable(caption = "阶段一的拟合度评估") | Sample | chisq | df | rmsea | nnfi | cfi | gfi |
|---|---|---|---|---|---|---|
| 全体样本 | 242.856 | 120 | 0.057 | 0.941 | 0.954 | 0.924 |
| 男性样本 | 194.711 | 120 | 0.060 | 0.925 | 0.941 | 0.897 |
| 女性样本 | 218.681 | 120 | 0.077 | 0.911 | 0.930 | 0.857 |
6.3 第二阶段分析
这一阶段估计的，是纳入性别变量后的多样本CFA模型。
6.3.1 模型一：基准模型
跨样本的（1）因素结构相同。其他都不同。
此时，模型卡方值为两样本分别以同一因素结构进行估计的总和（即表6.2中男、女样本卡方值的加和）。
观测数据的数目（DP值）为两性别的加总，也就是\(18 * 19 / 2 + 18 * 19 / 2 = 342\)。
待估参数也是两性别模型的加和，也就是51 + 51 = 102，自由度\(df = 342 - 102 = 340\)。
注意，在
lavaan包中，如果使用多样本模型（即添加group选项），并且输入的数据是原始数据，summary的结果会自动带有intercept项，即截距。这些截距的估计不会影响模型的自由度和其他参数的估计，因为截距的大小与方差/协方差没有任何关系。
mod1_fit <- cfa(model=cfa_model,
data = dat_all,
group = 'gender') # 性别变量
mod1_res <- summary(mod1_fit, fit.measures=TRUE, standard=TRUE)
print(mod1_res)## lavaan 0.6.15 ended normally after 36 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 138
##
## Number of observations per group:
## female 140
## male 172
##
## Model Test User Model:
##
## Test statistic 413.391
## Degrees of freedom 240
## P-value (Chi-square) 0.000
## Test statistic for each group:
## female 218.681
## male 194.711
##
## Model Test Baseline Model:
##
## Test statistic 2997.338
## Degrees of freedom 306
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.936
## Tucker-Lewis Index (TLI) 0.918
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6654.498
## Loglikelihood unrestricted model (H1) -6447.803
##
## Akaike (AIC) 13584.997
## Bayesian (BIC) 14101.531
## Sample-size adjusted Bayesian (SABIC) 13663.841
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.068
## 90 Percent confidence interval - lower 0.057
## 90 Percent confidence interval - upper 0.079
## P-value H_0: RMSEA <= 0.050 0.005
## P-value H_0: RMSEA >= 0.080 0.036
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.061
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [female]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 0.939 0.075 12.561 0.000 0.939 0.909
## A2 0.757 0.085 8.957 0.000 0.757 0.703
## A3 0.734 0.081 9.057 0.000 0.734 0.709
## FB =~
## B1 0.864 0.088 9.769 0.000 0.864 0.747
## B2 0.907 0.085 10.631 0.000 0.907 0.794
## B3 0.915 0.083 10.971 0.000 0.915 0.812
## FC =~
## C1 0.785 0.081 9.630 0.000 0.785 0.753
## C2 0.752 0.084 8.919 0.000 0.752 0.709
## C3 0.733 0.090 8.149 0.000 0.733 0.661
## FD =~
## D1 0.842 0.066 12.845 0.000 0.842 0.883
## D2 0.803 0.058 13.722 0.000 0.803 0.921
## D3 0.770 0.074 10.438 0.000 0.770 0.766
## FE =~
## E1 0.839 0.076 11.091 0.000 0.839 0.802
## E2 1.028 0.074 13.888 0.000 1.028 0.931
## E3 0.787 0.074 10.662 0.000 0.787 0.781
## FF =~
## F1 0.723 0.089 8.123 0.000 0.723 0.652
## F2 0.947 0.086 10.975 0.000 0.947 0.818
## F3 0.836 0.073 11.515 0.000 0.836 0.847
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.451 0.083 5.441 0.000 0.451 0.451
## FC 0.380 0.093 4.097 0.000 0.380 0.380
## FD 0.253 0.090 2.822 0.005 0.253 0.253
## FE 0.453 0.078 5.786 0.000 0.453 0.453
## FF 0.628 0.067 9.335 0.000 0.628 0.628
## FB ~~
## FC 0.807 0.056 14.296 0.000 0.807 0.807
## FD 0.521 0.075 6.984 0.000 0.521 0.521
## FE 0.639 0.064 9.976 0.000 0.639 0.639
## FF 0.327 0.092 3.547 0.000 0.327 0.327
## FC ~~
## FD 0.606 0.072 8.407 0.000 0.606 0.606
## FE 0.619 0.071 8.656 0.000 0.619 0.619
## FF 0.577 0.080 7.255 0.000 0.577 0.577
## FD ~~
## FE 0.626 0.060 10.359 0.000 0.626 0.626
## FF 0.403 0.083 4.858 0.000 0.403 0.403
## FE ~~
## FF 0.495 0.077 6.462 0.000 0.495 0.495
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.236 0.087 48.544 0.000 4.236 4.103
## .A2 4.136 0.091 45.431 0.000 4.136 3.840
## .A3 3.971 0.087 45.414 0.000 3.971 3.838
## .B1 3.893 0.098 39.813 0.000 3.893 3.365
## .B2 4.050 0.097 41.954 0.000 4.050 3.546
## .B3 4.043 0.095 42.454 0.000 4.043 3.588
## .C1 4.279 0.088 48.564 0.000 4.279 4.104
## .C2 4.343 0.090 48.440 0.000 4.343 4.094
## .C3 4.136 0.094 44.094 0.000 4.136 3.727
## .D1 4.836 0.081 60.022 0.000 4.836 5.073
## .D2 4.936 0.074 66.983 0.000 4.936 5.661
## .D3 4.736 0.085 55.791 0.000 4.736 4.715
## .E1 4.479 0.088 50.701 0.000 4.479 4.285
## .E2 4.543 0.093 48.673 0.000 4.543 4.114
## .E3 4.579 0.085 53.766 0.000 4.579 4.544
## .F1 4.136 0.094 44.094 0.000 4.136 3.727
## .F2 4.393 0.098 44.926 0.000 4.393 3.797
## .F3 4.221 0.083 50.653 0.000 4.221 4.281
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 0.185 0.067 2.780 0.005 0.185 0.173
## .A2 0.587 0.084 7.013 0.000 0.587 0.506
## .A3 0.532 0.077 6.953 0.000 0.532 0.497
## .B1 0.592 0.088 6.707 0.000 0.592 0.442
## .B2 0.483 0.079 6.100 0.000 0.483 0.370
## .B3 0.433 0.075 5.791 0.000 0.433 0.341
## .C1 0.471 0.076 6.172 0.000 0.471 0.433
## .C2 0.559 0.083 6.709 0.000 0.559 0.497
## .C3 0.694 0.097 7.120 0.000 0.694 0.564
## .D1 0.200 0.039 5.153 0.000 0.200 0.221
## .D2 0.116 0.031 3.771 0.000 0.116 0.153
## .D3 0.416 0.057 7.238 0.000 0.416 0.413
## .E1 0.389 0.058 6.703 0.000 0.389 0.356
## .E2 0.163 0.052 3.165 0.002 0.163 0.134
## .E3 0.397 0.057 6.971 0.000 0.397 0.391
## .F1 0.709 0.096 7.365 0.000 0.709 0.575
## .F2 0.442 0.082 5.415 0.000 0.442 0.330
## .F3 0.274 0.058 4.736 0.000 0.274 0.282
##
##
## Group 2 [male]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 0.673 0.067 10.013 0.000 0.673 0.739
## A2 0.615 0.072 8.537 0.000 0.615 0.644
## A3 0.482 0.070 6.921 0.000 0.482 0.540
## FB =~
## B1 0.701 0.086 8.131 0.000 0.701 0.610
## B2 0.979 0.079 12.395 0.000 0.979 0.863
## B3 0.751 0.074 10.206 0.000 0.751 0.736
## FC =~
## C1 0.612 0.068 8.930 0.000 0.612 0.677
## C2 0.754 0.074 10.157 0.000 0.754 0.755
## C3 0.609 0.075 8.146 0.000 0.609 0.627
## FD =~
## D1 0.797 0.061 13.129 0.000 0.797 0.865
## D2 0.682 0.054 12.745 0.000 0.682 0.847
## D3 0.548 0.059 9.352 0.000 0.548 0.668
## FE =~
## E1 0.759 0.056 13.592 0.000 0.759 0.860
## E2 0.768 0.056 13.702 0.000 0.768 0.864
## E3 0.807 0.059 13.563 0.000 0.807 0.858
## FF =~
## F1 0.559 0.098 5.707 0.000 0.559 0.465
## F2 0.668 0.077 8.633 0.000 0.668 0.670
## F3 0.659 0.064 10.271 0.000 0.659 0.789
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.604 0.076 7.980 0.000 0.604 0.604
## FC 0.565 0.085 6.673 0.000 0.565 0.565
## FD 0.627 0.071 8.851 0.000 0.627 0.627
## FE 0.525 0.077 6.815 0.000 0.525 0.525
## FF 0.717 0.074 9.681 0.000 0.717 0.717
## FB ~~
## FC 0.601 0.073 8.272 0.000 0.601 0.601
## FD 0.377 0.080 4.704 0.000 0.377 0.377
## FE 0.491 0.071 6.896 0.000 0.491 0.491
## FF 0.395 0.087 4.514 0.000 0.395 0.395
## FC ~~
## FD 0.426 0.082 5.169 0.000 0.426 0.426
## FE 0.555 0.072 7.750 0.000 0.555 0.555
## FF 0.596 0.080 7.438 0.000 0.596 0.596
## FD ~~
## FE 0.477 0.070 6.850 0.000 0.477 0.477
## FF 0.224 0.092 2.428 0.015 0.224 0.224
## FE ~~
## FF 0.314 0.087 3.603 0.000 0.314 0.314
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.552 0.069 65.586 0.000 4.552 5.001
## .A2 4.442 0.073 61.087 0.000 4.442 4.658
## .A3 4.145 0.068 60.834 0.000 4.145 4.639
## .B1 4.110 0.088 46.935 0.000 4.110 3.579
## .B2 4.407 0.087 50.930 0.000 4.407 3.883
## .B3 4.390 0.078 56.447 0.000 4.390 4.304
## .C1 4.442 0.069 64.473 0.000 4.442 4.916
## .C2 4.326 0.076 56.768 0.000 4.326 4.329
## .C3 4.442 0.074 59.948 0.000 4.442 4.571
## .D1 4.831 0.070 68.748 0.000 4.831 5.242
## .D2 4.953 0.061 80.641 0.000 4.953 6.149
## .D3 4.913 0.063 78.561 0.000 4.913 5.990
## .E1 4.750 0.067 70.512 0.000 4.750 5.377
## .E2 4.884 0.068 72.116 0.000 4.884 5.499
## .E3 4.797 0.072 66.943 0.000 4.797 5.104
## .F1 4.297 0.092 46.938 0.000 4.297 3.579
## .F2 4.808 0.076 63.304 0.000 4.808 4.827
## .F3 4.703 0.064 73.889 0.000 4.703 5.634
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 0.376 0.059 6.380 0.000 0.376 0.454
## .A2 0.532 0.069 7.684 0.000 0.532 0.585
## .A3 0.566 0.067 8.386 0.000 0.566 0.709
## .B1 0.828 0.102 8.155 0.000 0.828 0.628
## .B2 0.329 0.085 3.896 0.000 0.329 0.256
## .B3 0.476 0.070 6.801 0.000 0.476 0.458
## .C1 0.442 0.062 7.156 0.000 0.442 0.542
## .C2 0.430 0.072 5.931 0.000 0.430 0.430
## .C3 0.573 0.075 7.670 0.000 0.573 0.607
## .D1 0.214 0.045 4.711 0.000 0.214 0.251
## .D2 0.184 0.035 5.263 0.000 0.184 0.283
## .D3 0.372 0.046 8.141 0.000 0.372 0.553
## .E1 0.204 0.033 6.182 0.000 0.204 0.261
## .E2 0.200 0.033 6.048 0.000 0.200 0.253
## .E3 0.232 0.037 6.216 0.000 0.232 0.263
## .F1 1.129 0.132 8.544 0.000 1.129 0.783
## .F2 0.547 0.078 6.967 0.000 0.547 0.551
## .F3 0.263 0.056 4.732 0.000 0.263 0.3786.3.2 模型二：因素载荷恒等模型
跨样本的（1）因素结构相同，（2）因素载荷相同，其他都不相同。
和上一模型相比，待估参数减去少估计的一半载荷参数，也就是102-18 = 84，自由度\(df = 342 - 84 = 258\)。
mod2_fit <- cfa(model=cfa_model,
data = dat_all,
group.equal=c("loadings"), # 跨样本因素载荷相同
group = 'gender') # 性别变量
mod2_res <- summary(mod2_fit, fit.measures=TRUE, standard=TRUE)
print(mod2_res)## lavaan 0.6.15 ended normally after 34 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 138
## Number of equality constraints 18
##
## Number of observations per group:
## female 140
## male 172
##
## Model Test User Model:
##
## Test statistic 448.511
## Degrees of freedom 258
## P-value (Chi-square) 0.000
## Test statistic for each group:
## female 235.170
## male 213.341
##
## Model Test Baseline Model:
##
## Test statistic 2997.338
## Degrees of freedom 306
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.929
## Tucker-Lewis Index (TLI) 0.916
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6672.058
## Loglikelihood unrestricted model (H1) -6447.803
##
## Akaike (AIC) 13584.116
## Bayesian (BIC) 14033.277
## Sample-size adjusted Bayesian (SABIC) 13652.677
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.069
## 90 Percent confidence interval - lower 0.058
## 90 Percent confidence interval - upper 0.079
## P-value H_0: RMSEA <= 0.050 0.003
## P-value H_0: RMSEA >= 0.080 0.040
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.112
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [female]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.818 0.050 16.372 0.000 0.818 0.868
## A2 (.p2.) 0.697 0.054 12.808 0.000 0.697 0.680
## A3 (.p3.) 0.619 0.053 11.770 0.000 0.619 0.642
## FB =~
## B1 (.p4.) 0.801 0.061 13.053 0.000 0.801 0.719
## B2 (.p5.) 0.939 0.058 16.247 0.000 0.939 0.812
## B3 (.p6.) 0.846 0.055 15.429 0.000 0.846 0.784
## FC =~
## C1 (.p7.) 0.700 0.052 13.368 0.000 0.700 0.707
## C2 (.p8.) 0.762 0.055 13.734 0.000 0.762 0.719
## C3 (.p9.) 0.675 0.057 11.763 0.000 0.675 0.629
## FD =~
## D1 (.10.) 0.819 0.044 18.576 0.000 0.819 0.884
## D2 (.11.) 0.750 0.039 19.184 0.000 0.750 0.907
## D3 (.12.) 0.657 0.046 14.307 0.000 0.657 0.706
## FE =~
## E1 (.13.) 0.804 0.045 17.802 0.000 0.804 0.797
## E2 (.14.) 0.882 0.045 19.446 0.000 0.882 0.882
## E3 (.15.) 0.806 0.047 17.298 0.000 0.806 0.791
## FF =~
## F1 (.16.) 0.645 0.065 9.891 0.000 0.645 0.607
## F2 (.17.) 0.812 0.057 14.212 0.000 0.812 0.765
## F3 (.18.) 0.751 0.047 15.906 0.000 0.751 0.822
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.420 0.087 4.819 0.000 0.420 0.420
## FC 0.339 0.098 3.467 0.001 0.339 0.339
## FD 0.216 0.092 2.348 0.019 0.216 0.216
## FE 0.428 0.083 5.150 0.000 0.428 0.428
## FF 0.593 0.075 7.882 0.000 0.593 0.593
## FB ~~
## FC 0.781 0.062 12.666 0.000 0.781 0.781
## FD 0.486 0.076 6.415 0.000 0.486 0.486
## FE 0.617 0.066 9.300 0.000 0.617 0.617
## FF 0.273 0.095 2.867 0.004 0.273 0.273
## FC ~~
## FD 0.572 0.076 7.559 0.000 0.572 0.572
## FE 0.582 0.076 7.627 0.000 0.582 0.582
## FF 0.536 0.086 6.203 0.000 0.536 0.536
## FD ~~
## FE 0.610 0.062 9.910 0.000 0.610 0.610
## FF 0.360 0.087 4.155 0.000 0.360 0.360
## FE ~~
## FF 0.446 0.083 5.387 0.000 0.446 0.446
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.236 0.080 53.125 0.000 4.236 4.490
## .A2 4.136 0.087 47.722 0.000 4.136 4.033
## .A3 3.971 0.081 48.772 0.000 3.971 4.122
## .B1 3.893 0.094 41.359 0.000 3.893 3.495
## .B2 4.050 0.098 41.461 0.000 4.050 3.504
## .B3 4.043 0.091 44.338 0.000 4.043 3.747
## .C1 4.279 0.084 51.138 0.000 4.279 4.322
## .C2 4.343 0.090 48.521 0.000 4.343 4.101
## .C3 4.136 0.091 45.552 0.000 4.136 3.850
## .D1 4.836 0.078 61.728 0.000 4.836 5.217
## .D2 4.936 0.070 70.653 0.000 4.936 5.971
## .D3 4.736 0.079 60.189 0.000 4.736 5.087
## .E1 4.479 0.085 52.534 0.000 4.479 4.440
## .E2 4.543 0.085 53.739 0.000 4.543 4.542
## .E3 4.579 0.086 53.140 0.000 4.579 4.491
## .F1 4.136 0.090 46.081 0.000 4.136 3.895
## .F2 4.393 0.090 48.961 0.000 4.393 4.138
## .F3 4.221 0.077 54.702 0.000 4.221 4.623
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 0.220 0.060 3.650 0.000 0.220 0.247
## .A2 0.566 0.081 6.953 0.000 0.566 0.538
## .A3 0.545 0.075 7.235 0.000 0.545 0.587
## .B1 0.599 0.086 6.934 0.000 0.599 0.483
## .B2 0.455 0.079 5.739 0.000 0.455 0.341
## .B3 0.448 0.072 6.203 0.000 0.448 0.385
## .C1 0.490 0.074 6.613 0.000 0.490 0.500
## .C2 0.541 0.084 6.480 0.000 0.541 0.483
## .C3 0.698 0.096 7.262 0.000 0.698 0.605
## .D1 0.188 0.038 4.895 0.000 0.188 0.219
## .D2 0.121 0.030 4.097 0.000 0.121 0.178
## .D3 0.434 0.058 7.535 0.000 0.434 0.501
## .E1 0.372 0.057 6.558 0.000 0.372 0.365
## .E2 0.223 0.047 4.752 0.000 0.223 0.223
## .E3 0.389 0.059 6.626 0.000 0.389 0.374
## .F1 0.712 0.096 7.417 0.000 0.712 0.631
## .F2 0.468 0.078 5.975 0.000 0.468 0.415
## .F3 0.270 0.055 4.891 0.000 0.270 0.324
##
##
## Group 2 [male]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.818 0.050 16.372 0.000 0.818 0.807
## A2 (.p2.) 0.697 0.054 12.808 0.000 0.697 0.689
## A3 (.p3.) 0.619 0.053 11.770 0.000 0.619 0.640
## FB =~
## B1 (.p4.) 0.801 0.061 13.053 0.000 0.801 0.669
## B2 (.p5.) 0.939 0.058 16.247 0.000 0.939 0.824
## B3 (.p6.) 0.846 0.055 15.429 0.000 0.846 0.790
## FC =~
## C1 (.p7.) 0.700 0.052 13.368 0.000 0.700 0.730
## C2 (.p8.) 0.762 0.055 13.734 0.000 0.762 0.748
## C3 (.p9.) 0.675 0.057 11.763 0.000 0.675 0.667
## FD =~
## D1 (.10.) 0.819 0.044 18.576 0.000 0.819 0.859
## D2 (.11.) 0.750 0.039 19.184 0.000 0.750 0.873
## D3 (.12.) 0.657 0.046 14.307 0.000 0.657 0.739
## FE =~
## E1 (.13.) 0.804 0.045 17.802 0.000 0.804 0.869
## E2 (.14.) 0.882 0.045 19.446 0.000 0.882 0.900
## E3 (.15.) 0.806 0.047 17.298 0.000 0.806 0.848
## FF =~
## F1 (.16.) 0.645 0.065 9.891 0.000 0.645 0.519
## F2 (.17.) 0.812 0.057 14.212 0.000 0.812 0.745
## F3 (.18.) 0.751 0.047 15.906 0.000 0.751 0.828
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.640 0.065 9.846 0.000 0.640 0.640
## FC 0.626 0.070 8.937 0.000 0.626 0.626
## FD 0.661 0.059 11.148 0.000 0.661 0.661
## FE 0.585 0.064 9.150 0.000 0.585 0.585
## FF 0.760 0.057 13.301 0.000 0.760 0.760
## FB ~~
## FC 0.634 0.066 9.550 0.000 0.634 0.634
## FD 0.436 0.073 5.934 0.000 0.436 0.436
## FE 0.529 0.065 8.093 0.000 0.529 0.529
## FF 0.457 0.077 5.912 0.000 0.457 0.457
## FC ~~
## FD 0.492 0.074 6.691 0.000 0.492 0.492
## FE 0.609 0.063 9.682 0.000 0.609 0.609
## FF 0.652 0.067 9.682 0.000 0.652 0.652
## FD ~~
## FE 0.533 0.062 8.632 0.000 0.533 0.533
## FF 0.316 0.081 3.895 0.000 0.316 0.316
## FE ~~
## FF 0.389 0.076 5.098 0.000 0.389 0.389
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.552 0.077 58.890 0.000 4.552 4.490
## .A2 4.442 0.077 57.548 0.000 4.442 4.388
## .A3 4.145 0.074 56.174 0.000 4.145 4.283
## .B1 4.110 0.091 45.047 0.000 4.110 3.435
## .B2 4.407 0.087 50.750 0.000 4.407 3.870
## .B3 4.390 0.082 53.752 0.000 4.390 4.099
## .C1 4.442 0.073 60.707 0.000 4.442 4.629
## .C2 4.326 0.078 55.725 0.000 4.326 4.249
## .C3 4.442 0.077 57.554 0.000 4.442 4.388
## .D1 4.831 0.073 66.461 0.000 4.831 5.068
## .D2 4.953 0.065 75.687 0.000 4.953 5.771
## .D3 4.913 0.068 72.397 0.000 4.913 5.520
## .E1 4.750 0.070 67.379 0.000 4.750 5.138
## .E2 4.884 0.075 65.395 0.000 4.884 4.986
## .E3 4.797 0.072 66.171 0.000 4.797 5.046
## .F1 4.297 0.095 45.350 0.000 4.297 3.458
## .F2 4.808 0.083 57.855 0.000 4.808 4.411
## .F3 4.703 0.069 68.017 0.000 4.703 5.186
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 0.358 0.057 6.237 0.000 0.358 0.348
## .A2 0.539 0.068 7.922 0.000 0.539 0.526
## .A3 0.554 0.067 8.254 0.000 0.554 0.591
## .B1 0.791 0.100 7.943 0.000 0.791 0.552
## .B2 0.416 0.074 5.648 0.000 0.416 0.321
## .B3 0.431 0.067 6.394 0.000 0.431 0.376
## .C1 0.430 0.061 7.036 0.000 0.430 0.467
## .C2 0.456 0.067 6.761 0.000 0.456 0.440
## .C3 0.569 0.074 7.720 0.000 0.569 0.555
## .D1 0.238 0.041 5.833 0.000 0.238 0.262
## .D2 0.175 0.032 5.405 0.000 0.175 0.237
## .D3 0.360 0.046 7.885 0.000 0.360 0.454
## .E1 0.209 0.033 6.431 0.000 0.209 0.245
## .E2 0.182 0.034 5.394 0.000 0.182 0.189
## .E3 0.253 0.036 6.953 0.000 0.253 0.280
## .F1 1.128 0.131 8.639 0.000 1.128 0.731
## .F2 0.529 0.076 6.942 0.000 0.529 0.445
## .F3 0.259 0.050 5.212 0.000 0.259 0.3156.3.3 模型三：因素载荷与测量残差恒等模型
跨样本的（1）因素结构相同，（2）因素载荷相同，（3）测量变量残差相同。
和上一模型相比，待估参数再减去一半的测量变量残差方差，也就是84-18=66，自由度\(df = 342 - 66 = 276\)。
mod3_fit <- cfa(model=cfa_model,
data = dat_all,
group.equal=c("loadings", "residuals"),
group = 'gender') # 性别变量
mod3_res <- summary(mod3_fit, fit.measures=TRUE, standard=TRUE)
print(mod3_res)## lavaan 0.6.15 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 138
## Number of equality constraints 36
##
## Number of observations per group:
## female 140
## male 172
##
## Model Test User Model:
##
## Test statistic 481.176
## Degrees of freedom 276
## P-value (Chi-square) 0.000
## Test statistic for each group:
## female 250.957
## male 230.219
##
## Model Test Baseline Model:
##
## Test statistic 2997.338
## Degrees of freedom 306
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.924
## Tucker-Lewis Index (TLI) 0.915
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6688.391
## Loglikelihood unrestricted model (H1) -6447.803
##
## Akaike (AIC) 13580.782
## Bayesian (BIC) 13962.568
## Sample-size adjusted Bayesian (SABIC) 13639.058
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.069
## 90 Percent confidence interval - lower 0.059
## 90 Percent confidence interval - upper 0.079
## P-value H_0: RMSEA <= 0.050 0.002
## P-value H_0: RMSEA >= 0.080 0.038
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.110
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [female]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.797 0.050 15.976 0.000 0.797 0.816
## A2 (.p2.) 0.707 0.054 12.985 0.000 0.707 0.695
## A3 (.p3.) 0.614 0.053 11.602 0.000 0.614 0.636
## FB =~
## B1 (.p4.) 0.794 0.062 12.869 0.000 0.794 0.687
## B2 (.p5.) 0.929 0.058 16.072 0.000 0.929 0.814
## B3 (.p6.) 0.848 0.055 15.485 0.000 0.848 0.791
## FC =~
## C1 (.p7.) 0.703 0.053 13.384 0.000 0.703 0.722
## C2 (.p8.) 0.759 0.056 13.663 0.000 0.759 0.733
## C3 (.p9.) 0.676 0.058 11.716 0.000 0.676 0.649
## FD =~
## D1 (.10.) 0.818 0.044 18.529 0.000 0.818 0.870
## D2 (.11.) 0.744 0.039 19.039 0.000 0.744 0.886
## D3 (.12.) 0.660 0.046 14.294 0.000 0.660 0.725
## FE =~
## E1 (.13.) 0.795 0.046 17.294 0.000 0.795 0.827
## E2 (.14.) 0.897 0.045 19.748 0.000 0.897 0.903
## E3 (.15.) 0.788 0.047 16.775 0.000 0.788 0.810
## FF =~
## F1 (.16.) 0.637 0.066 9.593 0.000 0.637 0.548
## F2 (.17.) 0.806 0.057 14.066 0.000 0.806 0.750
## F3 (.18.) 0.750 0.047 15.820 0.000 0.750 0.825
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.439 0.088 4.971 0.000 0.439 0.439
## FC 0.360 0.098 3.685 0.000 0.360 0.360
## FD 0.223 0.095 2.354 0.019 0.223 0.223
## FE 0.457 0.082 5.592 0.000 0.457 0.457
## FF 0.617 0.075 8.194 0.000 0.617 0.617
## FB ~~
## FC 0.762 0.061 12.471 0.000 0.762 0.762
## FD 0.494 0.076 6.512 0.000 0.494 0.494
## FE 0.607 0.066 9.227 0.000 0.607 0.607
## FF 0.270 0.096 2.817 0.005 0.270 0.270
## FC ~~
## FD 0.573 0.075 7.679 0.000 0.573 0.573
## FE 0.562 0.075 7.514 0.000 0.562 0.562
## FF 0.526 0.086 6.110 0.000 0.526 0.526
## FD ~~
## FE 0.609 0.061 9.999 0.000 0.609 0.609
## FF 0.353 0.088 4.002 0.000 0.353 0.353
## FE ~~
## FF 0.439 0.082 5.334 0.000 0.439 0.439
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.236 0.082 51.354 0.000 4.236 4.340
## .A2 4.136 0.086 48.086 0.000 4.136 4.064
## .A3 3.971 0.082 48.701 0.000 3.971 4.116
## .B1 3.893 0.098 39.881 0.000 3.893 3.371
## .B2 4.050 0.096 41.970 0.000 4.050 3.547
## .B3 4.043 0.091 44.607 0.000 4.043 3.770
## .C1 4.279 0.082 51.936 0.000 4.279 4.389
## .C2 4.343 0.087 49.676 0.000 4.343 4.198
## .C3 4.136 0.088 46.984 0.000 4.136 3.971
## .D1 4.836 0.079 60.849 0.000 4.836 5.143
## .D2 4.936 0.071 69.525 0.000 4.936 5.876
## .D3 4.736 0.077 61.553 0.000 4.736 5.202
## .E1 4.479 0.081 55.144 0.000 4.479 4.661
## .E2 4.543 0.084 54.135 0.000 4.543 4.575
## .E3 4.579 0.082 55.720 0.000 4.579 4.709
## .F1 4.136 0.098 42.102 0.000 4.136 3.558
## .F2 4.393 0.091 48.391 0.000 4.393 4.090
## .F3 4.221 0.077 54.965 0.000 4.221 4.645
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 (.25.) 0.318 0.043 7.308 0.000 0.318 0.333
## .A2 (.26.) 0.535 0.053 10.115 0.000 0.535 0.517
## .A3 (.27.) 0.554 0.051 10.788 0.000 0.554 0.595
## .B1 (.28.) 0.703 0.068 10.393 0.000 0.703 0.527
## .B2 (.29.) 0.441 0.057 7.791 0.000 0.441 0.338
## .B3 (.30.) 0.430 0.051 8.444 0.000 0.430 0.374
## .C1 (.31.) 0.456 0.049 9.347 0.000 0.456 0.479
## .C2 (.32.) 0.494 0.054 9.108 0.000 0.494 0.462
## .C3 (.33.) 0.628 0.060 10.423 0.000 0.628 0.579
## .D1 (.34.) 0.214 0.030 7.182 0.000 0.214 0.242
## .D2 (.35.) 0.151 0.023 6.458 0.000 0.151 0.215
## .D3 (.36.) 0.394 0.036 10.807 0.000 0.394 0.475
## .E1 (.37.) 0.291 0.032 9.246 0.000 0.291 0.315
## .E2 (.38.) 0.182 0.029 6.195 0.000 0.182 0.185
## .E3 (.39.) 0.325 0.034 9.679 0.000 0.325 0.343
## .F1 (.40.) 0.945 0.083 11.353 0.000 0.945 0.699
## .F2 (.41.) 0.505 0.057 8.816 0.000 0.505 0.438
## .F3 (.42.) 0.264 0.040 6.602 0.000 0.264 0.319
##
##
## Group 2 [male]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.797 0.050 15.976 0.000 0.797 0.816
## A2 (.p2.) 0.707 0.054 12.985 0.000 0.707 0.695
## A3 (.p3.) 0.614 0.053 11.602 0.000 0.614 0.636
## FB =~
## B1 (.p4.) 0.794 0.062 12.869 0.000 0.794 0.687
## B2 (.p5.) 0.929 0.058 16.072 0.000 0.929 0.814
## B3 (.p6.) 0.848 0.055 15.485 0.000 0.848 0.791
## FC =~
## C1 (.p7.) 0.703 0.053 13.384 0.000 0.703 0.722
## C2 (.p8.) 0.759 0.056 13.663 0.000 0.759 0.733
## C3 (.p9.) 0.676 0.058 11.716 0.000 0.676 0.649
## FD =~
## D1 (.10.) 0.818 0.044 18.529 0.000 0.818 0.870
## D2 (.11.) 0.744 0.039 19.039 0.000 0.744 0.886
## D3 (.12.) 0.660 0.046 14.294 0.000 0.660 0.725
## FE =~
## E1 (.13.) 0.795 0.046 17.294 0.000 0.795 0.827
## E2 (.14.) 0.897 0.045 19.748 0.000 0.897 0.903
## E3 (.15.) 0.788 0.047 16.775 0.000 0.788 0.810
## FF =~
## F1 (.16.) 0.637 0.066 9.593 0.000 0.637 0.548
## F2 (.17.) 0.806 0.057 14.066 0.000 0.806 0.750
## F3 (.18.) 0.750 0.047 15.820 0.000 0.750 0.825
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB 0.627 0.065 9.604 0.000 0.627 0.627
## FC 0.625 0.070 8.917 0.000 0.625 0.625
## FD 0.643 0.059 10.808 0.000 0.643 0.643
## FE 0.585 0.064 9.159 0.000 0.585 0.585
## FF 0.750 0.056 13.345 0.000 0.750 0.750
## FB ~~
## FC 0.636 0.067 9.492 0.000 0.636 0.636
## FD 0.425 0.074 5.770 0.000 0.425 0.425
## FE 0.525 0.066 7.895 0.000 0.525 0.525
## FF 0.448 0.078 5.767 0.000 0.448 0.448
## FC ~~
## FD 0.482 0.075 6.466 0.000 0.482 0.482
## FE 0.626 0.063 9.946 0.000 0.626 0.626
## FF 0.663 0.067 9.948 0.000 0.663 0.663
## FD ~~
## FE 0.523 0.063 8.349 0.000 0.523 0.523
## FF 0.306 0.081 3.779 0.000 0.306 0.306
## FE ~~
## FF 0.397 0.076 5.196 0.000 0.397 0.397
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.552 0.074 61.176 0.000 4.552 4.665
## .A2 4.442 0.078 57.244 0.000 4.442 4.365
## .A3 4.145 0.074 56.345 0.000 4.145 4.296
## .B1 4.110 0.088 46.676 0.000 4.110 3.559
## .B2 4.407 0.087 50.620 0.000 4.407 3.860
## .B3 4.390 0.082 53.683 0.000 4.390 4.093
## .C1 4.442 0.074 59.763 0.000 4.442 4.557
## .C2 4.326 0.079 54.842 0.000 4.326 4.182
## .C3 4.442 0.079 55.933 0.000 4.442 4.265
## .D1 4.831 0.072 67.386 0.000 4.831 5.138
## .D2 4.953 0.064 77.339 0.000 4.953 5.897
## .D3 4.913 0.069 70.777 0.000 4.913 5.397
## .E1 4.750 0.073 64.826 0.000 4.750 4.943
## .E2 4.884 0.076 64.506 0.000 4.884 4.919
## .E3 4.797 0.074 64.701 0.000 4.797 4.933
## .F1 4.297 0.089 48.481 0.000 4.297 3.697
## .F2 4.808 0.082 58.708 0.000 4.808 4.476
## .F3 4.703 0.069 67.881 0.000 4.703 5.176
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 (.25.) 0.318 0.043 7.308 0.000 0.318 0.333
## .A2 (.26.) 0.535 0.053 10.115 0.000 0.535 0.517
## .A3 (.27.) 0.554 0.051 10.788 0.000 0.554 0.595
## .B1 (.28.) 0.703 0.068 10.393 0.000 0.703 0.527
## .B2 (.29.) 0.441 0.057 7.791 0.000 0.441 0.338
## .B3 (.30.) 0.430 0.051 8.444 0.000 0.430 0.374
## .C1 (.31.) 0.456 0.049 9.347 0.000 0.456 0.479
## .C2 (.32.) 0.494 0.054 9.108 0.000 0.494 0.462
## .C3 (.33.) 0.628 0.060 10.423 0.000 0.628 0.579
## .D1 (.34.) 0.214 0.030 7.182 0.000 0.214 0.242
## .D2 (.35.) 0.151 0.023 6.458 0.000 0.151 0.215
## .D3 (.36.) 0.394 0.036 10.807 0.000 0.394 0.475
## .E1 (.37.) 0.291 0.032 9.246 0.000 0.291 0.315
## .E2 (.38.) 0.182 0.029 6.195 0.000 0.182 0.185
## .E3 (.39.) 0.325 0.034 9.679 0.000 0.325 0.343
## .F1 (.40.) 0.945 0.083 11.353 0.000 0.945 0.699
## .F2 (.41.) 0.505 0.057 8.816 0.000 0.505 0.438
## .F3 (.42.) 0.264 0.040 6.602 0.000 0.264 0.3196.3.4 模型四：协方差恒等模型
最严格的CFA恒等检测，跨样本的（1）因素结构相同，（2）因素载荷相同，（3）测量变量残差相同，（4）新增方差和协方差恒等
和上一模型相比，待估参数再减去一半的潜变量的协方差，也就是66-15=51，自由度\(df = 342 - 51 = 291\)。
mod4_fit <- cfa(model=cfa_model,
data = dat_all,
group.equal=c("loadings", "residuals", "lv.covariances"),
group = 'gender') # 性别变量
mod4_res <- summary(mod4_fit, fit.measures=TRUE, standard=TRUE)
print(mod4_res)## lavaan 0.6.15 ended normally after 25 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 138
## Number of equality constraints 51
##
## Number of observations per group:
## female 140
## male 172
##
## Model Test User Model:
##
## Test statistic 522.269
## Degrees of freedom 291
## P-value (Chi-square) 0.000
## Test statistic for each group:
## female 270.584
## male 251.685
##
## Model Test Baseline Model:
##
## Test statistic 2997.338
## Degrees of freedom 306
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.914
## Tucker-Lewis Index (TLI) 0.910
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6708.937
## Loglikelihood unrestricted model (H1) -6447.803
##
## Akaike (AIC) 13591.874
## Bayesian (BIC) 13917.516
## Sample-size adjusted Bayesian (SABIC) 13641.581
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.071
## 90 Percent confidence interval - lower 0.061
## 90 Percent confidence interval - upper 0.081
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 0.075
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.105
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [female]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.795 0.051 15.562 0.000 0.795 0.822
## A2 (.p2.) 0.691 0.056 12.433 0.000 0.691 0.684
## A3 (.p3.) 0.603 0.054 11.206 0.000 0.603 0.628
## FB =~
## B1 (.p4.) 0.784 0.062 12.641 0.000 0.784 0.680
## B2 (.p5.) 0.941 0.058 16.272 0.000 0.941 0.826
## B3 (.p6.) 0.833 0.055 15.069 0.000 0.833 0.779
## FC =~
## C1 (.p7.) 0.690 0.053 13.014 0.000 0.690 0.712
## C2 (.p8.) 0.749 0.056 13.387 0.000 0.749 0.729
## C3 (.p9.) 0.673 0.058 11.613 0.000 0.673 0.649
## FD =~
## D1 (.10.) 0.815 0.045 18.275 0.000 0.815 0.870
## D2 (.11.) 0.740 0.040 18.722 0.000 0.740 0.885
## D3 (.12.) 0.654 0.046 14.096 0.000 0.654 0.721
## FE =~
## E1 (.13.) 0.792 0.046 17.168 0.000 0.792 0.825
## E2 (.14.) 0.894 0.046 19.639 0.000 0.894 0.902
## E3 (.15.) 0.788 0.047 16.763 0.000 0.788 0.812
## FF =~
## F1 (.16.) 0.639 0.067 9.583 0.000 0.639 0.550
## F2 (.17.) 0.801 0.058 13.844 0.000 0.801 0.747
## F3 (.18.) 0.745 0.048 15.528 0.000 0.745 0.822
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB (.43.) 0.520 0.056 9.249 0.000 0.520 0.520
## FC (.44.) 0.475 0.062 7.613 0.000 0.475 0.475
## FD (.45.) 0.419 0.058 7.163 0.000 0.419 0.419
## FE (.46.) 0.507 0.054 9.423 0.000 0.507 0.507
## FF (.47.) 0.677 0.048 13.978 0.000 0.677 0.677
## FB ~~
## FC (.48.) 0.687 0.048 14.331 0.000 0.687 0.687
## FD (.49.) 0.449 0.055 8.126 0.000 0.449 0.449
## FE (.50.) 0.558 0.049 11.381 0.000 0.558 0.558
## FF (.51.) 0.353 0.064 5.561 0.000 0.353 0.353
## FC ~~
## FD (.52.) 0.520 0.055 9.411 0.000 0.520 0.520
## FE (.53.) 0.591 0.051 11.658 0.000 0.591 0.591
## FF (.54.) 0.589 0.056 10.560 0.000 0.589 0.589
## FD ~~
## FE (.55.) 0.560 0.046 12.127 0.000 0.560 0.560
## FF (.56.) 0.321 0.062 5.170 0.000 0.321 0.321
## FE ~~
## FF (.57.) 0.416 0.058 7.176 0.000 0.416 0.416
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.236 0.082 51.828 0.000 4.236 4.380
## .A2 4.136 0.085 48.407 0.000 4.136 4.091
## .A3 3.971 0.081 48.973 0.000 3.971 4.139
## .B1 3.893 0.097 39.972 0.000 3.893 3.378
## .B2 4.050 0.096 42.104 0.000 4.050 3.558
## .B3 4.043 0.090 44.742 0.000 4.043 3.781
## .C1 4.279 0.082 52.280 0.000 4.279 4.418
## .C2 4.343 0.087 50.017 0.000 4.343 4.227
## .C3 4.136 0.088 47.236 0.000 4.136 3.992
## .D1 4.836 0.079 61.130 0.000 4.836 5.166
## .D2 4.936 0.071 69.858 0.000 4.936 5.904
## .D3 4.736 0.077 61.750 0.000 4.736 5.219
## .E1 4.479 0.081 55.233 0.000 4.479 4.668
## .E2 4.543 0.084 54.239 0.000 4.543 4.584
## .E3 4.579 0.082 55.807 0.000 4.579 4.717
## .F1 4.136 0.098 42.161 0.000 4.136 3.563
## .F2 4.393 0.091 48.519 0.000 4.393 4.101
## .F3 4.221 0.077 55.140 0.000 4.221 4.660
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 (.25.) 0.303 0.047 6.499 0.000 0.303 0.324
## .A2 (.26.) 0.544 0.055 9.920 0.000 0.544 0.532
## .A3 (.27.) 0.558 0.052 10.627 0.000 0.558 0.606
## .B1 (.28.) 0.713 0.068 10.429 0.000 0.713 0.537
## .B2 (.29.) 0.411 0.057 7.225 0.000 0.411 0.317
## .B3 (.30.) 0.449 0.052 8.631 0.000 0.449 0.392
## .C1 (.31.) 0.462 0.049 9.348 0.000 0.462 0.493
## .C2 (.32.) 0.495 0.055 9.019 0.000 0.495 0.469
## .C3 (.33.) 0.621 0.060 10.297 0.000 0.621 0.578
## .D1 (.34.) 0.212 0.031 6.952 0.000 0.212 0.242
## .D2 (.35.) 0.152 0.024 6.287 0.000 0.152 0.217
## .D3 (.36.) 0.396 0.037 10.793 0.000 0.396 0.481
## .E1 (.37.) 0.294 0.032 9.261 0.000 0.294 0.319
## .E2 (.38.) 0.182 0.030 6.160 0.000 0.182 0.186
## .E3 (.39.) 0.321 0.033 9.607 0.000 0.321 0.341
## .F1 (.40.) 0.939 0.083 11.282 0.000 0.939 0.697
## .F2 (.41.) 0.507 0.058 8.662 0.000 0.507 0.442
## .F3 (.42.) 0.266 0.041 6.448 0.000 0.266 0.324
##
##
## Group 2 [male]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA =~
## A1 (.p1.) 0.795 0.051 15.562 0.000 0.795 0.822
## A2 (.p2.) 0.691 0.056 12.433 0.000 0.691 0.684
## A3 (.p3.) 0.603 0.054 11.206 0.000 0.603 0.628
## FB =~
## B1 (.p4.) 0.784 0.062 12.641 0.000 0.784 0.680
## B2 (.p5.) 0.941 0.058 16.272 0.000 0.941 0.826
## B3 (.p6.) 0.833 0.055 15.069 0.000 0.833 0.779
## FC =~
## C1 (.p7.) 0.690 0.053 13.014 0.000 0.690 0.712
## C2 (.p8.) 0.749 0.056 13.387 0.000 0.749 0.729
## C3 (.p9.) 0.673 0.058 11.613 0.000 0.673 0.649
## FD =~
## D1 (.10.) 0.815 0.045 18.275 0.000 0.815 0.870
## D2 (.11.) 0.740 0.040 18.722 0.000 0.740 0.885
## D3 (.12.) 0.654 0.046 14.096 0.000 0.654 0.721
## FE =~
## E1 (.13.) 0.792 0.046 17.168 0.000 0.792 0.825
## E2 (.14.) 0.894 0.046 19.639 0.000 0.894 0.902
## E3 (.15.) 0.788 0.047 16.763 0.000 0.788 0.812
## FF =~
## F1 (.16.) 0.639 0.067 9.583 0.000 0.639 0.550
## F2 (.17.) 0.801 0.058 13.844 0.000 0.801 0.747
## F3 (.18.) 0.745 0.048 15.528 0.000 0.745 0.822
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA ~~
## FB (.43.) 0.520 0.056 9.249 0.000 0.520 0.520
## FC (.44.) 0.475 0.062 7.613 0.000 0.475 0.475
## FD (.45.) 0.419 0.058 7.163 0.000 0.419 0.419
## FE (.46.) 0.507 0.054 9.423 0.000 0.507 0.507
## FF (.47.) 0.677 0.048 13.978 0.000 0.677 0.677
## FB ~~
## FC (.48.) 0.687 0.048 14.331 0.000 0.687 0.687
## FD (.49.) 0.449 0.055 8.126 0.000 0.449 0.449
## FE (.50.) 0.558 0.049 11.381 0.000 0.558 0.558
## FF (.51.) 0.353 0.064 5.561 0.000 0.353 0.353
## FC ~~
## FD (.52.) 0.520 0.055 9.411 0.000 0.520 0.520
## FE (.53.) 0.591 0.051 11.658 0.000 0.591 0.591
## FF (.54.) 0.589 0.056 10.560 0.000 0.589 0.589
## FD ~~
## FE (.55.) 0.560 0.046 12.127 0.000 0.560 0.560
## FF (.56.) 0.321 0.062 5.170 0.000 0.321 0.321
## FE ~~
## FF (.57.) 0.416 0.058 7.176 0.000 0.416 0.416
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .A1 4.552 0.074 61.740 0.000 4.552 4.708
## .A2 4.442 0.077 57.626 0.000 4.442 4.394
## .A3 4.145 0.073 56.659 0.000 4.145 4.320
## .B1 4.110 0.088 46.782 0.000 4.110 3.567
## .B2 4.407 0.087 50.781 0.000 4.407 3.872
## .B3 4.390 0.082 53.845 0.000 4.390 4.106
## .C1 4.442 0.074 60.159 0.000 4.442 4.587
## .C2 4.326 0.078 55.218 0.000 4.326 4.210
## .C3 4.442 0.079 56.233 0.000 4.442 4.288
## .D1 4.831 0.071 67.697 0.000 4.831 5.162
## .D2 4.953 0.064 77.710 0.000 4.953 5.925
## .D3 4.913 0.069 71.003 0.000 4.913 5.414
## .E1 4.750 0.073 64.931 0.000 4.750 4.951
## .E2 4.884 0.076 64.630 0.000 4.884 4.928
## .E3 4.797 0.074 64.801 0.000 4.797 4.941
## .F1 4.297 0.088 48.549 0.000 4.297 3.702
## .F2 4.808 0.082 58.863 0.000 4.808 4.488
## .F3 4.703 0.069 68.097 0.000 4.703 5.192
## FA 0.000 0.000 0.000
## FB 0.000 0.000 0.000
## FC 0.000 0.000 0.000
## FD 0.000 0.000 0.000
## FE 0.000 0.000 0.000
## FF 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## FA 1.000 1.000 1.000
## FB 1.000 1.000 1.000
## FC 1.000 1.000 1.000
## FD 1.000 1.000 1.000
## FE 1.000 1.000 1.000
## FF 1.000 1.000 1.000
## .A1 (.25.) 0.303 0.047 6.499 0.000 0.303 0.324
## .A2 (.26.) 0.544 0.055 9.920 0.000 0.544 0.532
## .A3 (.27.) 0.558 0.052 10.627 0.000 0.558 0.606
## .B1 (.28.) 0.713 0.068 10.429 0.000 0.713 0.537
## .B2 (.29.) 0.411 0.057 7.225 0.000 0.411 0.317
## .B3 (.30.) 0.449 0.052 8.631 0.000 0.449 0.392
## .C1 (.31.) 0.462 0.049 9.348 0.000 0.462 0.493
## .C2 (.32.) 0.495 0.055 9.019 0.000 0.495 0.469
## .C3 (.33.) 0.621 0.060 10.297 0.000 0.621 0.578
## .D1 (.34.) 0.212 0.031 6.952 0.000 0.212 0.242
## .D2 (.35.) 0.152 0.024 6.287 0.000 0.152 0.217
## .D3 (.36.) 0.396 0.037 10.793 0.000 0.396 0.481
## .E1 (.37.) 0.294 0.032 9.261 0.000 0.294 0.319
## .E2 (.38.) 0.182 0.030 6.160 0.000 0.182 0.186
## .E3 (.39.) 0.321 0.033 9.607 0.000 0.321 0.341
## .F1 (.40.) 0.939 0.083 11.282 0.000 0.939 0.697
## .F2 (.41.) 0.507 0.058 8.662 0.000 0.507 0.442
## .F3 (.42.) 0.266 0.041 6.448 0.000 0.266 0.3246.3.5 结果
首先，我们复现原书289页表9.3的下半部分，结果见表6.3。
原书进行数据拟合时，输入的是协方差矩阵；而我们输入的是原始数据，这造成部分拟合指数不一样。不过卡方值和自由度一致就可以了。
get_fit <- function(my_fit){
fitness <- fitmeasures(my_fit)[c('chisq', 'df', 'rmsea','nnfi', 'nfi', 'cfi', 'gfi', 'pvalue')]
split(fitness, names(fitness))
}
nest_models <- list(
模型一 = mod1_fit,
模型二 = mod2_fit,
模型三 = mod3_fit,
模型四 = mod4_fit
)
map_dfr(nest_models, .f = get_fit, .id = "Model") |>
mutate(Delta_chisq = chisq - lag(chisq),
Delta_df = df - lag(df),
across(where(is.numeric), ~round(., 3))) |>
select(Model, chisq, Delta_chisq, df, Delta_df, pvalue, rmsea, nnfi, cfi, gfi) |>
knitr::kable(caption = "阶段二的拟合度评估") | Model | chisq | Delta_chisq | df | Delta_df | pvalue | rmsea | nnfi | cfi | gfi |
|---|---|---|---|---|---|---|---|---|---|
| 模型一 | 413.391 | NA | 240 | NA | 0 | 0.068 | 0.918 | 0.936 | 0.983 |
| 模型二 | 448.511 | 35.120 | 258 | 18 | 0 | 0.069 | 0.916 | 0.929 | 0.981 |
| 模型三 | 481.176 | 32.665 | 276 | 18 | 0 | 0.069 | 0.915 | 0.924 | 0.980 |
| 模型四 | 522.269 | 41.093 | 291 | 15 | 0 | 0.071 | 0.910 | 0.914 | 0.978 |
我们对模型二和模型一间的差异进行检验。可以看到，二者的拟合程度存在显著差异，也就是增加了因素载荷的限制之后，模型的拟合度降低了。研究者应该停止进一步的恒等性检验，就两个样本的因素载荷的数值内容进行讨论。
anova(mod1_fit, mod2_fit)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## mod1_fit 240 13585 14102 413.39
## mod2_fit 258 13584 14033 448.51 35.12 0.078081 18 0.00913 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1进一步说，研究者应该找到两套参数数值，分别计算男性和女性工作者的得分，从而更有效的反应两个性别的测量结构。