专题六、因子分析和结构方程建模
Wei Wei
2022-10-20
教学目标
- 掌握探索性因子分析的方法（主成分分析法）
- 掌握验证性因子分析的方法（结构方程建模）
- 掌握常见的结构方程模型的建立方法（中介模型，潜变量关系模型，交叉滞后模型，潜变量增长模型）
探索性因子分析
- 数据准备
# 直接file-》import dataset，选择文件即可
library(haven)
# 导入数据默认为数据框格式（data.frame）
eee_data <- read_sav("https://gitee.com/vv_victorwei/r-language-data-analysis/raw/master/Untitled2.sav")
# head会呈现eee_data的前六行数据，以预览概况
head(eee_data)## # A tibble: 6 × 55
## V1 time_used IP id gender age birth…¹ roman…² hukou hair homet…³
## <chr> <dbl> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 175 594 223.66… 2225… 1 23 1 0 0 1 0
## 2 124 178 101.88… 2222… 1 23 1 0 0 1 0
## 3 138 207 211.16… 2225… 1 22 0 0 1 1 0
## 4 74 176 58.246… 2122… 1 23 1 0 1 1 0
## 5 149 232 120.25… 2222… 1 22 0 0 0 1 0
## 6 11 142 58.40.… 1601… 1 23 1 0 1 1 0
## # … with 44 more variables: family_income <dbl>, minzhu <dbl>, zhiyuan <dbl>,
## # height <dbl>, weight <dbl>, zhengzhi <dbl>, personality <dbl>, a1 <dbl>,
## # a2 <dbl>, a3 <dbl>, a4 <dbl>, a5 <dbl>, a6 <dbl>, a7 <dbl>, a8 <dbl>,
## # a9 <dbl>, a10 <dbl>, a11 <dbl>, a12 <dbl>, a13 <dbl>, b1 <dbl>, b2 <dbl>,
## # b3 <dbl>, b4 <dbl>, b5 <dbl>, b6 <dbl>, importance_ranking <chr>,
## # importance_ranking_d <dbl>, daode <dbl>, zhishi <dbl>, jineng <dbl>,
## # yishu <dbl>, food <chr>, chuan_food <dbl>, yue_food <dbl>, …#查看分布情况
summary(eee_data[19:31])## a1 a2 a3 a4 a5
## Min. :0.000 Min. :0.00 Min. :0.00 Min. :0.000 Min. :0.000
## 1st Qu.:3.000 1st Qu.:3.00 1st Qu.:2.00 1st Qu.:3.000 1st Qu.:3.000
## Median :3.000 Median :3.00 Median :3.00 Median :3.000 Median :3.000
## Mean :3.137 Mean :3.24 Mean :2.52 Mean :3.274 Mean :3.229
## 3rd Qu.:4.000 3rd Qu.:4.00 3rd Qu.:3.00 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :4.000 Max. :4.00 Max. :4.00 Max. :4.000 Max. :4.000
## a6 a7 a8 a9 a10
## Min. :0.000 Min. :0.000 Min. :0.000 Min. :0.000 Min. :0.00
## 1st Qu.:3.000 1st Qu.:3.000 1st Qu.:2.000 1st Qu.:3.000 1st Qu.:3.00
## Median :4.000 Median :4.000 Median :3.000 Median :3.000 Median :4.00
## Mean :3.503 Mean :3.463 Mean :2.583 Mean :3.343 Mean :3.56
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:4.000 3rd Qu.:4.00
## Max. :4.000 Max. :4.000 Max. :4.000 Max. :4.000 Max. :4.00
## a11 a12 a13
## Min. :0.00 Min. :1.000 Min. :0.000
## 1st Qu.:2.00 1st Qu.:3.000 1st Qu.:3.000
## Median :2.00 Median :3.000 Median :4.000
## Mean :2.52 Mean :3.206 Mean :3.577
## 3rd Qu.:3.00 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :4.00 Max. :4.000 Max. :4.000#查看内部相关
library(corrgram)
corrgram(eee_data[19:31],upper.panel=panel.cor)## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
## Warning in par(usr): argument 1 does not name a graphical parameter
# 相关分析中，最后一道题与其他题目相关弱，建议删除
#13 16、为了教会孩子遵守规则，我会采用体罚（如打屁股、打手心）的方式
# 跟专业认同确实没有什么关系-前提条件
# KMO检验和Bartlett球形检验，反映内在相关情况，达到一定标准才可以继续分析
# KMO值应该达到0.7以上，越大越好
# Bartlett球形检验必须显著
library(psych)
KMO(eee_data[19:30])## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = eee_data[19:30])
## Overall MSA = 0.84
## MSA for each item =
## a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
## 0.92 0.84 0.82 0.88 0.85 0.83 0.89 0.80 0.76 0.68 0.81 0.83library(psych)
cortest.bartlett(eee_data[19:30])## R was not square, finding R from data## $chisq
## [1] 776.7912
##
## $p.value
## [1] 6.248433e-122
##
## $df
## [1] 66-正式开始
# 查看特征根与碎石图
ev <- eigen(cor(eee_data[19:30]))
ev$values## [1] 4.8294278 1.3838786 1.1066449 0.9029497 0.8174793 0.5861894 0.5552474
## [8] 0.4508588 0.4221654 0.3883831 0.2971355 0.2596400scree(eee_data[19:30], pc=FALSE)
# 析取因子,cutoff表示0.3以下的因子符合不限制
Nfacs <- 2
fit <- factanal(eee_data[19:30], Nfacs, rotation="promax")
print(fit, digits=2, cutoff=0.3, sort=TRUE)##
## Call:
## factanal(x = eee_data[19:30], factors = Nfacs, rotation = "promax")
##
## Uniquenesses:
## a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
## 0.50 0.55 0.53 0.48 0.49 0.42 0.51 0.77 0.78 0.85 0.26 0.75
##
## Loadings:
## Factor1 Factor2
## a1 0.68
## a2 0.74
## a4 0.69
## a5 0.81
## a6 0.85
## a7 0.61
## a8 0.56
## a11 0.98
## a3 0.31 0.44
## a9
## a10
## a12 0.41
##
## Factor1 Factor2
## SS loadings 3.48 1.86
## Proportion Var 0.29 0.15
## Cumulative Var 0.29 0.44
##
## Factor Correlations:
## Factor1 Factor2
## Factor1 1.00 -0.68
## Factor2 -0.68 1.00
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 136.54 on 43 degrees of freedom.
## The p-value is 1.15e-11Nfacs <- 3
fit <- factanal(eee_data[19:30], Nfacs, rotation="promax")
print(fit, digits=2, cutoff=0.3, sort=TRUE)##
## Call:
## factanal(x = eee_data[19:30], factors = Nfacs, rotation = "promax")
##
## Uniquenesses:
## a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
## 0.47 0.53 0.51 0.47 0.50 0.43 0.52 0.79 0.63 0.00 0.19 0.76
##
## Loadings:
## Factor1 Factor2 Factor3
## a1 0.72
## a2 0.76
## a4 0.69
## a5 0.79
## a6 0.82
## a7 0.61
## a11 0.99
## a10 1.03
## a3 0.39 0.44
## a8 0.48
## a9 0.45
## a12 0.36
##
## Factor1 Factor2 Factor3
## SS loadings 3.47 1.62 1.32
## Proportion Var 0.29 0.14 0.11
## Cumulative Var 0.29 0.42 0.53
##
## Factor Correlations:
## Factor1 Factor2 Factor3
## Factor1 1.00 0.39 -0.42
## Factor2 0.39 1.00 -0.63
## Factor3 -0.42 -0.63 1.00
##
## Test of the hypothesis that 3 factors are sufficient.
## The chi square statistic is 85.72 on 33 degrees of freedom.
## The p-value is 1.43e-06# 2-3个似乎是合适的，但第三个因子只有两道题目，采用2个，删除9题和10题
#9 16、我认为幼师行业是社会分工中最重要的职业之一
#10 16、当别人赞美幼儿教师时，我很欣慰
Nfacs <- 2
fit <- factanal(eee_data[c(19:26,29,30)], Nfacs, rotation="promax")
print(fit, digits=2, cutoff=0.3, sort=TRUE)##
## Call:
## factanal(x = eee_data[c(19:26, 29, 30)], factors = Nfacs, rotation = "promax")
##
## Uniquenesses:
## a1 a2 a3 a4 a5 a6 a7 a8 a11 a12
## 0.49 0.54 0.55 0.49 0.49 0.42 0.52 0.82 0.06 0.77
##
## Loadings:
## Factor1 Factor2
## a1 0.70
## a2 0.74
## a4 0.70
## a5 0.78
## a6 0.84
## a7 0.63
## a11 1.06
## a3 0.39 0.36
## a8 0.42
## a12 0.35
##
## Factor1 Factor2
## SS loadings 3.44 1.62
## Proportion Var 0.34 0.16
## Cumulative Var 0.34 0.51
##
## Factor Correlations:
## Factor1 Factor2
## Factor1 1.00 0.61
## Factor2 0.61 1.00
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 73.13 on 26 degrees of freedom.
## The p-value is 2.3e-06# 3题在两个因子上都有相同负荷，建议删除
#3 16、在社交方面，我常常因为自己将来是幼儿教师感到自豪
Nfacs <- 2
fit <- factanal(eee_data[c(19:20,22:26,29,30)], Nfacs, rotation="promax")
print(fit, digits=2, cutoff=0.3, sort=TRUE)##
## Call:
## factanal(x = eee_data[c(19:20, 22:26, 29, 30)], factors = Nfacs, rotation = "promax")
##
## Uniquenesses:
## a1 a2 a4 a5 a6 a7 a8 a11 a12
## 0.51 0.57 0.51 0.47 0.40 0.49 0.79 0.20 0.74
##
## Loadings:
## Factor1 Factor2
## a1 0.66
## a2 0.69
## a4 0.67
## a5 0.78
## a6 0.83
## a7 0.63
## a11 0.95
## a8 0.47
## a12 0.40
##
## Factor1 Factor2
## SS loadings 3.09 1.33
## Proportion Var 0.34 0.15
## Cumulative Var 0.34 0.49
##
## Factor Correlations:
## Factor1 Factor2
## Factor1 1.00 -0.57
## Factor2 -0.57 1.00
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 57.46 on 19 degrees of freedom.
## The p-value is 9.7e-06# 绘制因子归属示意图
loads <- fit$loadings
fa.diagram(loads)
#检查原始题目，给factor命名
#1 16、我喜欢和幼儿在一起
#2 16、我很认真地学习专业课知识
#4 16、我希望自己在幼教行业有所成就
#5 16、我相信自己能成为一名合格的幼儿教师
#6 16、我具备爱的能力，发自内心尊重每一位幼儿
#7 16、孩子健康快乐成长是我未来工作的目标
#8 16、当幼儿教师不能实现我的人生价值
#11 16、即使幼教工作再辛苦，我仍想从事幼儿教育事业
#12 16、我会留意观察幼儿教育的新动向
# 不是特别明显，但也看出前五道题偏向于幼教态度和情感，后面四道题偏向职业认同
# factor 1命名为幼教态度和情感，factor 2命名为幼教职业认同验证性因子分析
- 比较简单，借用lavaan的例子了
- lavaan的使用可以参考https://lavaan.ugent.be/tutorial/index.html
library(lavaan)## This is lavaan 0.6-12
## lavaan is FREE software! Please report any bugs.##
## Attaching package: 'lavaan'## The following object is masked from 'package:psych':
##
## cor2cov# 建立模型，=~表示由后面的题目测量
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
#进行分析
fit <- cfa(HS.model, data = HolzingerSwineford1939)
summary(fit, fit.measures = TRUE)## lavaan 0.6-12 ended normally after 35 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 21
##
## Number of observations 301
##
## Model Test User Model:
##
## Test statistic 85.306
## Degrees of freedom 24
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## 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
##
## 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 - lower 0.071
## 90 Percent confidence interval - upper 0.114
## P-value RMSEA <= 0.05 0.001
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.065
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## 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# 输入完整拟合指数结果
fitMeasures(fit)## npar fmin chisq df
## 21.000 0.142 85.306 24.000
## pvalue baseline.chisq baseline.df baseline.pvalue
## 0.000 918.852 36.000 0.000
## cfi tli nnfi rfi
## 0.931 0.896 0.896 0.861
## nfi pnfi ifi rni
## 0.907 0.605 0.931 0.931
## logl unrestricted.logl aic bic
## -3737.745 -3695.092 7517.490 7595.339
## ntotal bic2 rmsea rmsea.ci.lower
## 301.000 7528.739 0.092 0.071
## rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
## 0.114 0.001 0.082 0.082
## srmr srmr_bentler srmr_bentler_nomean crmr
## 0.065 0.065 0.065 0.073
## crmr_nomean srmr_mplus srmr_mplus_nomean cn_05
## 0.073 0.065 0.065 129.490
## cn_01 gfi agfi pgfi
## 152.654 0.943 0.894 0.503
## mfi ecvi
## 0.903 0.423#输出修正指数
modificationindices(fit, sort = TRUE, maximum.number = 5)## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 30 visual =~ x9 36.411 0.577 0.519 0.515 0.515
## 76 x7 ~~ x8 34.145 0.536 0.536 0.859 0.859
## 28 visual =~ x7 18.631 -0.422 -0.380 -0.349 -0.349
## 78 x8 ~~ x9 14.946 -0.423 -0.423 -0.805 -0.805
## 33 textual =~ x3 9.151 -0.272 -0.269 -0.238 -0.238#输出结构图（仅供参考，文章中使用时需要重新在ppt里绘制）
library(lavaanPlot)
lavaanPlot(model = fit,coefs = TRUE,stand = TRUE, sig = 0.05,stars = c("regress",'latent'))模型的多组比较
三种水平的模型等值 结构相同configural invariance,负荷相同metric invariance, 截距（或残差）相同scale invariance 分别建立三个模型：其中第一个模型负荷和截距都自由估计，第二个模型负荷限定组间相等，截距自由估计，第三个模型限定负荷和截距都组间相等
library(lavaan)
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
## 对男女样本进行结构不变性检验
# configural invariance
fit1 <- cfa(HS.model, data = HolzingerSwineford1939, group = "sex")
# weak invariance
fit2 <- cfa(HS.model, data = HolzingerSwineford1939, group = "sex",
group.equal = "loadings")
# strong invariance
fit3 <- cfa(HS.model, data = HolzingerSwineford1939, group = "sex",
group.equal = c("intercepts", "loadings"))
# model comparison tests
lavTestLRT(fit1, fit2, fit3)## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit1 48 7517.6 7740.1 105.80
## fit2 54 7526.1 7726.2 126.23 20.431 6 0.002320 **
## fit3 60 7532.8 7710.8 145.00 18.771 6 0.004567 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## 对不同学校中结构不变形进行检验
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
# configural invariance
fit1 <- cfa(HS.model, data = HolzingerSwineford1939, group = "school")
# weak invariance
fit2 <- cfa(HS.model, data = HolzingerSwineford1939, group = "school",
group.equal = "loadings")
# strong invariance
fit3 <- cfa(HS.model, data = HolzingerSwineford1939, group = "school",
group.equal = c("intercepts", "loadings"))
# model comparison tests
lavTestLRT(fit1, fit2, fit3)## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit1 48 7484.4 7706.8 115.85
## fit2 54 7480.6 7680.8 124.04 8.192 6 0.2244
## fit3 60 7508.6 7686.6 164.10 40.059 6 4.435e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1结构方程建模
model <- '
# measurement model
ind60 =~ x1 + x2 + x3
dem60 =~ y1 + y2 + y3 + y4
dem65 =~ y5 + y6 + y7 + y8
# regressions
dem60 ~ ind60
dem65 ~ ind60 + dem60
# residual correlations
y1 ~~ y5
y2 ~~ y4 + y6
y3 ~~ y7
y4 ~~ y8
y6 ~~ y8
'
fit <- sem(model, data = PoliticalDemocracy)
summary(fit, standardized = TRUE)## lavaan 0.6-12 ended normally after 68 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 31
##
## Number of observations 75
##
## Model Test User Model:
##
## Test statistic 38.125
## Degrees of freedom 35
## P-value (Chi-square) 0.329
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ind60 =~
## x1 1.000 0.670 0.920
## x2 2.180 0.139 15.742 0.000 1.460 0.973
## x3 1.819 0.152 11.967 0.000 1.218 0.872
## dem60 =~
## y1 1.000 2.223 0.850
## y2 1.257 0.182 6.889 0.000 2.794 0.717
## y3 1.058 0.151 6.987 0.000 2.351 0.722
## y4 1.265 0.145 8.722 0.000 2.812 0.846
## dem65 =~
## y5 1.000 2.103 0.808
## y6 1.186 0.169 7.024 0.000 2.493 0.746
## y7 1.280 0.160 8.002 0.000 2.691 0.824
## y8 1.266 0.158 8.007 0.000 2.662 0.828
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## dem60 ~
## ind60 1.483 0.399 3.715 0.000 0.447 0.447
## dem65 ~
## ind60 0.572 0.221 2.586 0.010 0.182 0.182
## dem60 0.837 0.098 8.514 0.000 0.885 0.885
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .y1 ~~
## .y5 0.624 0.358 1.741 0.082 0.624 0.296
## .y2 ~~
## .y4 1.313 0.702 1.871 0.061 1.313 0.273
## .y6 2.153 0.734 2.934 0.003 2.153 0.356
## .y3 ~~
## .y7 0.795 0.608 1.308 0.191 0.795 0.191
## .y4 ~~
## .y8 0.348 0.442 0.787 0.431 0.348 0.109
## .y6 ~~
## .y8 1.356 0.568 2.386 0.017 1.356 0.338
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .x1 0.082 0.019 4.184 0.000 0.082 0.154
## .x2 0.120 0.070 1.718 0.086 0.120 0.053
## .x3 0.467 0.090 5.177 0.000 0.467 0.239
## .y1 1.891 0.444 4.256 0.000 1.891 0.277
## .y2 7.373 1.374 5.366 0.000 7.373 0.486
## .y3 5.067 0.952 5.324 0.000 5.067 0.478
## .y4 3.148 0.739 4.261 0.000 3.148 0.285
## .y5 2.351 0.480 4.895 0.000 2.351 0.347
## .y6 4.954 0.914 5.419 0.000 4.954 0.443
## .y7 3.431 0.713 4.814 0.000 3.431 0.322
## .y8 3.254 0.695 4.685 0.000 3.254 0.315
## ind60 0.448 0.087 5.173 0.000 1.000 1.000
## .dem60 3.956 0.921 4.295 0.000 0.800 0.800
## .dem65 0.172 0.215 0.803 0.422 0.039 0.039fitMeasures(fit)## npar fmin chisq df
## 31.000 0.254 38.125 35.000
## pvalue baseline.chisq baseline.df baseline.pvalue
## 0.329 730.654 55.000 0.000
## cfi tli nnfi rfi
## 0.995 0.993 0.993 0.918
## nfi pnfi ifi rni
## 0.948 0.603 0.996 0.995
## logl unrestricted.logl aic bic
## -1547.791 -1528.728 3157.582 3229.424
## ntotal bic2 rmsea rmsea.ci.lower
## 75.000 3131.720 0.035 0.000
## rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
## 0.092 0.611 0.276 0.276
## srmr srmr_bentler srmr_bentler_nomean crmr
## 0.044 0.044 0.044 0.049
## crmr_nomean srmr_mplus srmr_mplus_nomean cn_05
## 0.049 0.045 0.045 98.970
## cn_01 gfi agfi pgfi
## 113.803 0.923 0.854 0.489
## mfi ecvi
## 0.979 1.335library(lavaanPlot)
lavaanPlot(model = fit,coefs = TRUE,stand = TRUE, sig = 0.05,stars = c("regress",'latent'))#交叉滞后模型 - t1:x,y;t2:x,y - 传统的交叉滞后模型类似于路径分析，投稿时可能受到质疑，要求进行随机截距的交叉滞后模型 - 随机截距的交叉滞后模型：https://jeroendmulder.github.io/RI-CLPM/lavaan.html
x1,y1,x2,y2
autoregressive effects
x2~x1 y2~y1 # cross-lagged effects y2~x1 x2~y1
covariance
x1y1 x2y2
variance
x1x1 x2x2 y1y1 y2y2
# cross lagged panel modelling
dat <- read.table("https://gitee.com/vv_victorwei/r-language-data-analysis/raw/master/%E7%9B%B8%E5%85%B3%E5%88%86%E6%9E%90/RICLPM.dat", col.names = c("x1", "x2", "x3", "x4", "x5", "y1", "y2", "y3", "y4", "y5"))
CLPM <-'# autoregressive and cross-lagged effects
x2+y2~x1+y1
x3+y3~x2+y2
x4+y4~x3+y3
x5+y5~x4+y4
# covariance
x1~~y1
x2~~y2
x3~~y3
x4~~y4
x5~~y5
# variance
x1~~x1
y1~~y1
x2~~x2
y2~~y2
x3~~x3
y3~~y3
x4~~x4
y4~~y4
x5~~x5
y5~~y5
'
CLPM.fit <- lavaan(CLPM, data = dat, missing = 'ML') ## Warning in lav_partable_check(lavpartable, categorical = lavoptions$.categorical, : lavaan WARNING: automatically added intercepts are set to zero:
## [x2 y2 x3 y3 x4 y4 x5 y5 x1 y1]summary(CLPM.fit, standardized = T)## lavaan 0.6-12 ended normally after 96 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 31
##
## Number of observations 1189
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 2098.471
## Degrees of freedom 34
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## x2 ~
## x1 0.398 0.022 17.734 0.000 0.398 0.516
## y1 0.140 0.017 8.263 0.000 0.140 0.241
## y2 ~
## x1 0.359 0.039 9.293 0.000 0.359 0.262
## y1 0.538 0.029 18.456 0.000 0.538 0.520
## x3 ~
## x2 0.500 0.028 18.043 0.000 0.500 0.485
## y2 0.169 0.016 10.832 0.000 0.169 0.291
## y3 ~
## x2 0.336 0.042 7.934 0.000 0.336 0.195
## y2 0.605 0.024 25.386 0.000 0.605 0.623
## x4 ~
## x3 0.449 0.026 16.967 0.000 0.449 0.517
## y3 0.070 0.016 4.459 0.000 0.070 0.136
## y4 ~
## x3 0.471 0.044 10.590 0.000 0.471 0.253
## y3 0.664 0.027 25.032 0.000 0.664 0.597
## x5 ~
## x4 0.439 0.028 15.510 0.000 0.439 0.447
## y4 0.096 0.013 7.273 0.000 0.096 0.210
## y5 ~
## x4 0.205 0.043 4.725 0.000 0.205 0.096
## y4 0.762 0.020 37.764 0.000 0.762 0.765
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## x1 ~~
## y1 0.112 0.006 19.961 0.000 0.112 0.710
## .x2 ~~
## .y2 0.019 0.002 10.367 0.000 0.019 0.315
## .x3 ~~
## .y3 0.019 0.002 11.030 0.000 0.019 0.338
## .x4 ~~
## .y4 0.018 0.002 10.074 0.000 0.018 0.305
## .x5 ~~
## .y5 0.013 0.002 7.972 0.000 0.013 0.238
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .x2 0.000 0.000 0.000
## .y2 0.000 0.000 0.000
## .x3 0.000 0.000 0.000
## .y3 0.000 0.000 0.000
## .x4 0.000 0.000 0.000
## .y4 0.000 0.000 0.000
## .x5 0.000 0.000 0.000
## .y5 0.000 0.000 0.000
## x1 0.000 0.000 0.000
## y1 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## x1 0.120 0.005 24.382 0.000 0.120 1.000
## y1 0.210 0.009 24.382 0.000 0.210 1.000
## .x2 0.036 0.001 24.382 0.000 0.036 0.500
## .y2 0.105 0.004 24.382 0.000 0.105 0.468
## .x3 0.038 0.002 24.382 0.000 0.038 0.498
## .y3 0.088 0.004 24.382 0.000 0.088 0.416
## .x4 0.035 0.001 24.382 0.000 0.035 0.621
## .y4 0.100 0.004 24.382 0.000 0.100 0.380
## .x5 0.036 0.001 24.382 0.000 0.036 0.646
## .y5 0.083 0.003 24.382 0.000 0.083 0.320fitMeasures(CLPM.fit)## npar fmin chisq df
## 31.000 0.882 2098.471 34.000
## pvalue baseline.chisq baseline.df baseline.pvalue
## 0.000 3166.400 45.000 0.000
## cfi tli nnfi rfi
## 0.339 0.125 0.125 0.123
## nfi pnfi ifi rni
## 0.337 0.255 0.341 0.339
## logl unrestricted.logl aic bic
## -504.583 544.652 1071.166 1228.673
## ntotal bic2 rmsea rmsea.ci.lower
## 1189.000 1130.205 0.226 0.218
## rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
## 0.234 0.000 0.128 0.071
## srmr srmr_bentler srmr_bentler_nomean crmr
## 0.833 0.833 0.810 0.466
## crmr_nomean srmr_mplus srmr_mplus_nomean cn_05
## 0.254 0.573 0.473 28.538
## cn_01 gfi agfi pgfi
## 32.764 0.747 0.517 0.391
## mfi ecvi
## 0.420 1.817dat <- read.table("https://gitee.com/vv_victorwei/r-language-data-analysis/raw/master/%E7%9B%B8%E5%85%B3%E5%88%86%E6%9E%90/RICLPM.dat", col.names = c("x1", "x2", "x3", "x4", "x5", "y1", "y2", "y3", "y4", "y5"))
RICLPM <- '
# Create between components (random intercepts)
RIx =~ 1*x1 + 1*x2 + 1*x3 + 1*x4 + 1*x5
RIy =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
# Create within-person centered variables
wx1 =~ 1*x1
wx2 =~ 1*x2
wx3 =~ 1*x3
wx4 =~ 1*x4
wx5 =~ 1*x5
wy1 =~ 1*y1
wy2 =~ 1*y2
wy3 =~ 1*y3
wy4 =~ 1*y4
wy5 =~ 1*y5
# Estimate the lagged effects between the within-person centered variables.
wx2 + wy2 ~ wx1 + wy1
wx3 + wy3 ~ wx2 + wy2
wx4 + wy4 ~ wx3 + wy3
wx5 + wy5 ~ wx4 + wy4
# Estimate the covariance between the within-person centered variables at the first wave.
wx1 ~~ wy1 # Covariance
# Estimate the covariances between the residuals of the within-person centered variables (the innovations).
wx2 ~~ wy2
wx3 ~~ wy3
wx4 ~~ wy4
wx5 ~~ wy5
# Estimate the variance and covariance of the random intercepts.
RIx ~~ RIx
RIy ~~ RIy
RIx ~~ RIy
# Estimate the (residual) variance of the within-person centered variables.
wx1 ~~ wx1 # Variances
wy1 ~~ wy1
wx2 ~~ wx2 # Residual variances
wy2 ~~ wy2
wx3 ~~ wx3
wy3 ~~ wy3
wx4 ~~ wx4
wy4 ~~ wy4
wx5 ~~ wx5
wy5 ~~ wy5
'
RICLPM.fit <- lavaan(RICLPM, data = dat, missing = 'ML', meanstructure = T, int.ov.free = T)
summary(RICLPM.fit, standardized = T)## lavaan 0.6-12 ended normally after 115 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 44
##
## Number of observations 1189
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 25.806
## Degrees of freedom 21
## P-value (Chi-square) 0.214
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RIx =~
## x1 1.000 0.096 0.390
## x2 1.000 0.096 0.473
## x3 1.000 0.096 0.475
## x4 1.000 0.096 0.461
## x5 1.000 0.096 0.465
## RIy =~
## y1 1.000 0.178 0.569
## y2 1.000 0.178 0.558
## y3 1.000 0.178 0.535
## y4 1.000 0.178 0.525
## y5 1.000 0.178 0.533
## wx1 =~
## x1 1.000 0.227 0.921
## wx2 =~
## x2 1.000 0.179 0.881
## wx3 =~
## x3 1.000 0.178 0.880
## wx4 =~
## x4 1.000 0.185 0.887
## wx5 =~
## x5 1.000 0.183 0.885
## wy1 =~
## y1 1.000 0.257 0.822
## wy2 =~
## y2 1.000 0.265 0.830
## wy3 =~
## y3 1.000 0.281 0.845
## wy4 =~
## y4 1.000 0.288 0.851
## wy5 =~
## y5 1.000 0.282 0.846
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## wx2 ~
## wx1 0.232 0.028 8.314 0.000 0.294 0.294
## wy1 0.009 0.026 0.329 0.742 0.012 0.012
## wy2 ~
## wx1 0.174 0.045 3.888 0.000 0.149 0.149
## wy1 0.004 0.046 0.092 0.927 0.004 0.004
## wx3 ~
## wx2 0.241 0.037 6.509 0.000 0.242 0.242
## wy2 0.026 0.024 1.082 0.279 0.039 0.039
## wy3 ~
## wx2 0.156 0.054 2.871 0.004 0.099 0.099
## wy2 0.262 0.039 6.747 0.000 0.247 0.247
## wx4 ~
## wx3 0.279 0.038 7.267 0.000 0.269 0.269
## wy3 0.010 0.023 0.431 0.666 0.015 0.015
## wy4 ~
## wx3 0.185 0.055 3.367 0.001 0.114 0.114
## wy3 0.296 0.035 8.362 0.000 0.288 0.288
## wx5 ~
## wx4 0.290 0.035 8.244 0.000 0.293 0.293
## wy4 -0.004 0.022 -0.186 0.852 -0.006 -0.006
## wy5 ~
## wx4 0.124 0.048 2.612 0.009 0.082 0.082
## wy4 0.392 0.031 12.644 0.000 0.400 0.400
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## wx1 ~~
## wy1 0.021 0.002 9.372 0.000 0.364 0.364
## .wx2 ~~
## .wy2 0.009 0.002 5.168 0.000 0.196 0.196
## .wx3 ~~
## .wy3 0.013 0.002 7.837 0.000 0.274 0.274
## .wx4 ~~
## .wy4 0.013 0.002 8.177 0.000 0.277 0.277
## .wx5 ~~
## .wy5 0.007 0.001 4.916 0.000 0.160 0.160
## RIx ~~
## RIy 0.010 0.001 7.992 0.000 0.587 0.587
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .x1 0.241 0.007 33.687 0.000 0.241 0.977
## .x2 0.173 0.006 29.331 0.000 0.173 0.851
## .x3 0.186 0.006 31.646 0.000 0.186 0.918
## .x4 0.117 0.006 19.288 0.000 0.117 0.559
## .x5 0.111 0.006 18.427 0.000 0.111 0.534
## .y1 0.336 0.009 37.099 0.000 0.336 1.076
## .y2 0.348 0.009 37.686 0.000 0.348 1.093
## .y3 0.319 0.010 33.098 0.000 0.319 0.960
## .y4 0.384 0.010 39.097 0.000 0.384 1.134
## .y5 0.388 0.010 40.056 0.000 0.388 1.162
## RIx 0.000 0.000 0.000
## RIy 0.000 0.000 0.000
## wx1 0.000 0.000 0.000
## .wx2 0.000 0.000 0.000
## .wx3 0.000 0.000 0.000
## .wx4 0.000 0.000 0.000
## .wx5 0.000 0.000 0.000
## wy1 0.000 0.000 0.000
## .wy2 0.000 0.000 0.000
## .wy3 0.000 0.000 0.000
## .wy4 0.000 0.000 0.000
## .wy5 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RIx 0.009 0.001 8.722 0.000 1.000 1.000
## RIy 0.032 0.003 12.351 0.000 1.000 1.000
## wx1 0.052 0.002 21.067 0.000 1.000 1.000
## wy1 0.066 0.004 17.985 0.000 1.000 1.000
## .wx2 0.029 0.001 20.793 0.000 0.911 0.911
## .wy2 0.068 0.004 17.503 0.000 0.977 0.977
## .wx3 0.030 0.001 20.467 0.000 0.935 0.935
## .wy3 0.072 0.003 21.324 0.000 0.918 0.918
## .wx4 0.032 0.001 21.445 0.000 0.925 0.925
## .wy4 0.074 0.003 21.876 0.000 0.884 0.884
## .wx5 0.031 0.001 21.680 0.000 0.915 0.915
## .wy5 0.065 0.003 22.446 0.000 0.813 0.813
## .x1 0.000 0.000 0.000
## .x2 0.000 0.000 0.000
## .x3 0.000 0.000 0.000
## .x4 0.000 0.000 0.000
## .x5 0.000 0.000 0.000
## .y1 0.000 0.000 0.000
## .y2 0.000 0.000 0.000
## .y3 0.000 0.000 0.000
## .y4 0.000 0.000 0.000
## .y5 0.000 0.000 0.000fitMeasures(RICLPM.fit)## npar fmin chisq df
## 44.000 0.011 25.806 21.000
## pvalue baseline.chisq baseline.df baseline.pvalue
## 0.214 3166.400 45.000 0.000
## cfi tli nnfi rfi
## 0.998 0.997 0.997 0.983
## nfi pnfi ifi rni
## 0.992 0.463 0.998 0.998
## logl unrestricted.logl aic bic
## 531.750 544.652 -975.499 -751.941
## ntotal bic2 rmsea rmsea.ci.lower
## 1189.000 -891.701 0.014 0.000
## rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
## 0.030 1.000 0.001 0.001
## srmr srmr_bentler srmr_bentler_nomean crmr
## 0.014 0.014 0.015 0.014
## crmr_nomean srmr_mplus srmr_mplus_nomean cn_05
## 0.015 0.013 0.014 1506.306
## cn_01 gfi agfi pgfi
## 1794.811 0.997 0.992 0.322
## mfi ecvi
## 0.998 0.096潜增长模型
model <- ' i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 '
fit <- growth(model, data=Demo.growth)
summary(fit)## lavaan 0.6-12 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
##
## Number of observations 400
##
## Model Test User Model:
##
## Test statistic 8.069
## Degrees of freedom 5
## P-value (Chi-square) 0.152
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## i =~
## t1 1.000
## t2 1.000
## t3 1.000
## t4 1.000
## s =~
## t1 0.000
## t2 1.000
## t3 2.000
## t4 3.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## i ~~
## s 0.618 0.071 8.686 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .t1 0.000
## .t2 0.000
## .t3 0.000
## .t4 0.000
## i 0.615 0.077 8.007 0.000
## s 1.006 0.042 24.076 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .t1 0.595 0.086 6.944 0.000
## .t2 0.676 0.061 11.061 0.000
## .t3 0.635 0.072 8.761 0.000
## .t4 0.508 0.124 4.090 0.000
## i 1.932 0.173 11.194 0.000
## s 0.587 0.052 11.336 0.000# 增加预测因子
model <- ' i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4
i + s ~ x1 + x2 '
fit <- growth(model, data=Demo.growth)
summary(fit)## lavaan 0.6-12 ended normally after 31 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 13
##
## Number of observations 400
##
## Model Test User Model:
##
## Test statistic 10.873
## Degrees of freedom 9
## P-value (Chi-square) 0.285
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## i =~
## t1 1.000
## t2 1.000
## t3 1.000
## t4 1.000
## s =~
## t1 0.000
## t2 1.000
## t3 2.000
## t4 3.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## i ~
## x1 0.609 0.060 10.079 0.000
## x2 0.613 0.065 9.488 0.000
## s ~
## x1 0.263 0.029 9.157 0.000
## x2 0.517 0.031 16.868 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .i ~~
## .s 0.088 0.042 2.116 0.034
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .t1 0.000
## .t2 0.000
## .t3 0.000
## .t4 0.000
## .i 0.586 0.062 9.400 0.000
## .s 0.958 0.030 32.382 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .t1 0.597 0.085 7.067 0.000
## .t2 0.671 0.060 11.088 0.000
## .t3 0.599 0.065 9.254 0.000
## .t4 0.593 0.109 5.437 0.000
## .i 1.076 0.115 9.335 0.000
## .s 0.219 0.027 7.975 0.000利用lavaan来检验中介作用
library(haven)
math_data <- read_sav("https://gitee.com/vv_victorwei/r-language-data-analysis/raw/master/%E7%9B%B8%E5%85%B3%E5%88%86%E6%9E%90/regression.sav")
head(math_data)## # A tibble: 6 × 3
## age intelligence maths
## <dbl> <dbl> <dbl>
## 1 61 4 17
## 2 60 12 17
## 3 60 4 19
## 4 69 5 20
## 5 67 13 21
## 6 64 10 21library(lavaan)
model <- ' # direct effect
maths ~ c*age
# mediator
intelligence ~ a*age
maths ~ b*intelligence
# indirect effect (a*b)
ab := a*b
# total effect
total := c + (a*b)
'
fit <- sem(model, data = math_data,se = "bootstrap")
summary(fit)## lavaan 0.6-12 ended normally after 1 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 5
##
## Number of observations 173
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Bootstrap
## Number of requested bootstrap draws 1000
## Number of successful bootstrap draws 1000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## maths ~
## age (c) 0.386 0.207 1.864 0.062
## intelligence ~
## age (a) 0.477 0.109 4.388 0.000
## maths ~
## intellignc (b) 1.002 0.154 6.488 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .maths 88.234 8.318 10.607 0.000
## .intelligence 25.733 3.358 7.664 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## ab 0.478 0.125 3.813 0.000
## total 0.863 0.227 3.807 0.000