PREPARATION
createCopula <- function(P){
H.p <- stats::ecdf(P) # 创建经验累积分布函数
H.p <- H.p(P) # 计算经验分布函数的
H.p <- ifelse(H.p==0, 0.0000001, H.p) # 避免0
H.p <- ifelse(H.p==1, 0.9999999, H.p) # 避免1
U.p <- H.p # 赋值处理后的概
p.star <- stats::qnorm(U.p) # 正态分布变
return(p.star) # 返回结果
}
bootstrapedSignificance <- function(dataset, bootstrapresults, numIndependentVariables, numCopulas){
for (i in 1:nrow(summary(bootstrapresults))){ # 循环遍历结果
t <- summary(bootstrapresults)[i, "original"] / summary(bootstrapresults)[i, "bootSE"] # 计算t
# df = n (number of observations) - k (number of independent variables + copulas) - 1
pvalue <- 2 * pt(-abs(t), df=nrow(dataset)-numIndependentVariables-numCopulas-1) # 计算p
cat("Pr(>|t|)", rownames(summary(bootstrapresults))[i], ": ", pvalue, "\n") # 打印p
}
} # 结束循环
Loading data
#data=import("https://drive.google.com/uc?id=1X9i6LDjVZhA_jQOaAnlnw8yhPsqfbM-G&export=download")%>%as.data.table() #spss
data=import("https://drive.google.com/uc?id=1v-jxaj6A7NgTYFATUBy-8zSSZMXxmHe_&export=download")%>%as.data.table() #csv
#setnames(data, old = c("ATT", "EOU", "UF", "BI"), new = c("ATT.raw", "EOU.raw", "UF.raw", "BI.raw"))
Calculate factor scores for latent variables
使用sem/SPSS的标准化后μ=0，sd=1的潜变量分数进行分析（以SPSS为例）： -
降维主成分分 - 固定因子数量 - 旋转（采用最优斜交法
SPSS 代码：
DATASET ACTIVATE DataSet1. /* 数据准备
/* 降维主成分分析
FACTOR
/VARIABLES Att1 Att2 Att3 EOU1 EOU2 EOU3 UF1 UF2 UF3 BI1 BI2 BI3
/MISSING LISTWISE /ANALYSIS Att1 Att2 Att3 EOU1 EOU2 EOU3 UF1 UF2 UF3
BI1 BI2 BI3
/PRINT INITIAL EXTRACTION ROTATION
/CRITERIA MINEIGEN(1) ITERATE(25)
/EXTRACTION PC /CRITERIA ITERATE(25)
/* 固定因子数量
/* 旋转（采用最优斜交法）
/ROTATION PROMAX(4) /SAVE REG(ALL) /METHOD=CORRELATION.
model <- '
# 定义四个因子
ATT =~ Att1 + Att2 + Att3
EOU =~ EOU1 + EOU2 + EOU3
UF =~ UF1 + UF2 + UF3
BI =~ BI1 + BI2 + BI3
'
fit <- cfa(model, data=data, estimator="MLM", missing="listwise", rotation="promax")
summary(fit, fit.measures=TRUE, standardized=TRUE) # 显示标准化结果和拟合指标
scores <- lavPredict(fit, type="lv") # 获取潜在因子得分
head(scores)%>%print_table # 查看前几行因子得分
#colnames(scores) <- c("ATT", "EOU", "UF", "BI")
data <- cbind(data, scores)
Describe(data)
#View(data)
Run
the he Kolmogorov-Smirnov and Shapiro-Wilk test with Lilliefors
correction
检验几个变量是否符合正态分布，需要使用非正态分布的数据进行估计（Tests
of Normality Kolmogorov-Smirnov Shapiro-Wilk）
library(KScorrect)
#LcKS(data$ATT, "pnorm", nreps = 4999)
#LcKS(data$UF, "pnorm", nreps = 4999)
#LcKS(data$BI, "pnorm", nreps = 4999)
ks.test(data$ATT, "pnorm", mean = mean(data$ATT), sd = sd(data$ATT))
##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: data$ATT
## D = 0.12, p-value = 3e-04
## alternative hypothesis: two-sided
ks.test(data$UF, "pnorm", mean = mean(data$UF), sd = sd(data$UF))
##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: data$UF
## D = 0.15, p-value = 3e-06
## alternative hypothesis: two-sided
ks.test(data$EOU, "pnorm", mean = mean(data$BI), sd = sd(data$BI))
##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: data$EOU
## D = 0.072, p-value = 0.1
## alternative hypothesis: two-sided
##
## Shapiro-Wilk normality test
##
## data: data$ATT
## W = 0.95, p-value = 3e-08
##
## Shapiro-Wilk normality test
##
## data: data$UF
## W = 0.96, p-value = 3e-07
##
## Shapiro-Wilk normality test
##
## data: data$EOU
## W = 0.97, p-value = 2e-05
Variables in our analysis
Variables in the model
X1-X3: ATT; EOU; UF
Y: BI
PRIMARY ANALYSIS
Overview
OLS analysis Calculate copulas based on createCopula() function Model
comparisons (7 models for 3 X in this example) - Include Copula for ATT
(Model 1) - Include Copula for UF (Model 2)
OLS
analysis
stdModel <- lm (BI ~ ATT + UF + EOU,data=data)
GLM_summary(stdModel);
##
## General Linear Model (OLS Regression)
##
## Model Fit:
## F(3, 291) = 51.39, p = 1e-26 ***
## R² = 0.34631 (Adjusted R² = 0.33957)
##
## Unstandardized Coefficients:
## Outcome Variable: BI
## N = 295
## ──────────────────────────────────────────────────────────────────
## b S.E. t p [95% CI of b] VIF
## ──────────────────────────────────────────────────────────────────
## (Intercept) -0.000 (0.047) -0.001 .999 [-0.093, 0.093]
## ATT 0.218 (0.058) 3.735 <.001 *** [ 0.103, 0.333] 1.516
## UF 0.373 (0.061) 6.100 <.001 *** [ 0.253, 0.493] 1.664
## EOU 0.100 (0.062) 1.616 .107 [-0.022, 0.222] 1.709
## ──────────────────────────────────────────────────────────────────
##
## Standardized Coefficients (β):
## Outcome Variable: BI
## N = 295
## ─────────────────────────────────────────────────────────────────────
## β S.E. t p [95% CI of β] r(partial) r(part)
## ─────────────────────────────────────────────────────────────────────
## ATT 0.218 (0.058) 3.735 <.001 *** [ 0.103, 0.333] 0.214 0.177
## UF 0.373 (0.061) 6.100 <.001 *** [ 0.253, 0.493] 0.337 0.289
## EOU 0.100 (0.062) 1.616 .107 [-0.022, 0.222] 0.094 0.077
## ─────────────────────────────────────────────────────────────────────
calculate copulas for independent variables within model
added(data, {ATT_star = createCopula(ATT)
UF_star = createCopula(UF)
EOU_star = createCopula(EOU)
})
dat=as.data.table(data)
COPULA ANALYSIS BASED ON CODE OF HULT ET AL. (2018)
Include Copula for ATT (Model 1)
# RESULTS RECOMMEND SETTING IT TO 10000
bootrounds = 10000
# Calculate Results
# Normal regression
copulaResults1 <- lm (BI ~ ATT + UF + EOU + ATT_star + 0,data=dat)
GLM_summary(copulaResults1)#%>%print_table()
##
## General Linear Model (OLS Regression)
##
## Model Fit:
## F(4, 291) = 38.64, p = 6e-26 ***
## R² = 0.34689 (Adjusted R² = 0.33791)
##
## Unstandardized Coefficients:
## Outcome Variable: BI
## N = 295
## ─────────────────────────────────────────────────────────────
## b S.E. t p [95% CI of b] VIF
## ─────────────────────────────────────────────────────────────
## ATT 0.161 (0.127) 1.268 .206 [-0.089, 0.411]
## UF 0.373 (0.061) 6.108 <.001 *** [ 0.253, 0.494] 7.173
## EOU 0.097 (0.062) 1.564 .119 [-0.025, 0.220] 1.664
## ATT_star 0.055 (0.109) 0.506 .613 [-0.159, 0.270] 1.724
## ─────────────────────────────────────────────────────────────
##
## Standardized Coefficients (β):
## Outcome Variable: BI
## N = 295
## ──────────────────────────────────────────────────────────────────────────
## β S.E. t p [95% CI of β] r(partial) r(part)
## ──────────────────────────────────────────────────────────────────────────
## UF 0.373 (0.061) 6.108 <.001 *** [ 0.253, 0.494] 0.338 0.290
## EOU 0.097 (0.062) 1.564 .119 [-0.025, 0.220] 0.091 0.074
## ATT_star 0.063 (0.124) 0.506 .613 [-0.182, 0.308] 0.030 0.024
## ──────────────────────────────────────────────────────────────────────────
# Bootstrap Standard Errors
library(carData)
bootCopulaResults1 <- Boot(copulaResults1, R=bootrounds)
print(bootCopulaResults1)
##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot::boot(data = dd, statistic = boot.f, R = R, .fn = f, parallel = p_type,
## ncpus = ncores, cl = cl2)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 0.16090 0.013828 0.16573
## t2* 0.37324 -0.003888 0.07710
## t3* 0.09726 0.010750 0.08952
## t4* 0.05520 -0.016226 0.14897
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults1, 3, 1)
## Pr(>|t|) ATT : 0.3324
## Pr(>|t|) UF : 2.106e-06
## Pr(>|t|) EOU : 0.2782
## Pr(>|t|) ATT_star : 0.7112
#3 = Number of independent variables in the regression model
#1 = Number of Gaussian copulas in the regression model
Include Copula for UF (Model 2)
# Normal copula regression
copulaResults2 <- lm (BI ~ ATT + UF + EOU + UF_star + 0,data=dat)
summary(copulaResults2)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + UF_star + 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.545 -0.516 0.025 0.395 4.004
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.21840 0.05854 3.73 0.00023 ***
## UF 0.36230 0.12475 2.90 0.00396 **
## EOU 0.09976 0.06208 1.61 0.10918
## UF_star 0.00936 0.09575 0.10 0.92220
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.814 on 291 degrees of freedom
## Multiple R-squared: 0.346, Adjusted R-squared: 0.337
## F-statistic: 38.5 on 4 and 291 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults2 <- Boot(copulaResults2, R=bootrounds)
summary(bootCopulaResults2)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.218 | -0.00374 | 0.0628 | 0.214 |
| 1e+04 | 0.362 | 0.0173 | 0.244 | 0.382 |
| 1e+04 | 0.0998 | 0.0138 | 0.0889 | 0.115 |
| 1e+04 | 0.00936 | -0.0164 | 0.186 | -0.00502 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults2, 3, 1)
## Pr(>|t|) ATT : 0.0005846
## Pr(>|t|) UF : 0.1386
## Pr(>|t|) EOU : 0.263
## Pr(>|t|) UF_star : 0.9599
Include Copula for EOU (Model 3)
# Normal copula regression
copulaResults3 <- lm (BI ~ ATT + UF + EOU + EOU_star + 0,data=dat)
summary(copulaResults3)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + EOU_star + 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5532 -0.5385 -0.0174 0.3752 3.1049
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.2128 0.0577 3.69 0.00027 ***
## UF 0.3718 0.0604 6.16 2.4e-09 ***
## EOU -0.3099 0.1632 -1.90 0.05850 .
## EOU_star 0.4011 0.1480 2.71 0.00711 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.804 on 291 degrees of freedom
## Multiple R-squared: 0.362, Adjusted R-squared: 0.354
## F-statistic: 41.4 on 4 and 291 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults3 <- Boot(copulaResults3, R=bootrounds)
summary(bootCopulaResults3)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.213 | -0.00151 | 0.0618 | 0.211 |
| 1e+04 | 0.372 | 0.00313 | 0.0728 | 0.376 |
| 1e+04 | -0.31 | 0.00749 | 0.386 | -0.284 |
| 1e+04 | 0.401 | 0.00764 | 0.346 | 0.379 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults3, 3, 1)
## Pr(>|t|) ATT : 0.0006527
## Pr(>|t|) UF : 5.933e-07
## Pr(>|t|) EOU : 0.4224
## Pr(>|t|) EOU_star : 0.2478
Include Copula for ATT and UF (Model 4)
# Normal copula regression
copulaResults4 <- lm (BI ~ ATT + UF + EOU + ATT_star + UF_star + 0,data=dat)
summary(copulaResults4)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + ATT_star + UF_star + 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.567 -0.516 0.025 0.387 3.971
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.1599 0.1316 1.22 0.2252
## UF 0.3765 0.1282 2.94 0.0036 **
## EOU 0.0973 0.0624 1.56 0.1196
## ATT_star 0.0560 0.1128 0.50 0.6198
## UF_star -0.0029 0.0990 -0.03 0.9766
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.815 on 290 degrees of freedom
## Multiple R-squared: 0.347, Adjusted R-squared: 0.336
## F-statistic: 30.8 on 5 and 290 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults4 <- Boot(copulaResults4, R=bootrounds)
summary(bootCopulaResults4)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.16 | 0.0097 | 0.153 | 0.168 |
| 1e+04 | 0.377 | 0.0133 | 0.241 | 0.397 |
| 1e+04 | 0.0973 | 0.0168 | 0.0915 | 0.116 |
| 1e+04 | 0.056 | -0.0149 | 0.134 | 0.0471 |
| 1e+04 | -0.0029 | -0.0146 | 0.185 | -0.0191 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults4, 3, 2)
## Pr(>|t|) ATT : 0.296
## Pr(>|t|) UF : 0.1196
## Pr(>|t|) EOU : 0.2882
## Pr(>|t|) ATT_star : 0.676
## Pr(>|t|) UF_star : 0.9875
Include Copula for ATT and EOU (Model 5)
# Normal copula regression
copulaResults5 <- lm (BI ~ ATT + UF + EOU + ATT_star + EOU_star + 0,data=dat)
summary(copulaResults5)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + ATT_star + EOU_star + 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5343 -0.5372 -0.0196 0.3755 3.1061
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.2692 0.1318 2.04 0.0420 *
## UF 0.3714 0.0605 6.14 2.7e-09 ***
## EOU -0.3342 0.1712 -1.95 0.0519 .
## ATT_star -0.0549 0.1153 -0.48 0.6347
## EOU_star 0.4277 0.1584 2.70 0.0073 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.805 on 290 degrees of freedom
## Multiple R-squared: 0.363, Adjusted R-squared: 0.352
## F-statistic: 33 on 5 and 290 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults5 <- Boot(copulaResults5, R=bootrounds)
summary(bootCopulaResults5)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.269 | 0.0248 | 0.172 | 0.283 |
| 1e+04 | 0.371 | 0.00134 | 0.0718 | 0.375 |
| 1e+04 | -0.334 | -0.00853 | 0.399 | -0.321 |
| 1e+04 | -0.0549 | -0.0279 | 0.156 | -0.0637 |
| 1e+04 | 0.428 | 0.0273 | 0.364 | 0.416 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults5, 3, 2)
## Pr(>|t|) ATT : 0.1188
## Pr(>|t|) UF : 4.278e-07
## Pr(>|t|) EOU : 0.4023
## Pr(>|t|) ATT_star : 0.7255
## Pr(>|t|) EOU_star : 0.2415
#3 = Number of independent variables in the regression model
#2 = Number of Gaussian copulas in the regression model
Include Copula for UF and EOU (Model 6)
# Normal copula regression
copulaResults6 <- lm (BI ~ ATT + UF + EOU + UF_star + EOU_star + 0,data=dat)
summary(copulaResults6)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + UF_star + EOU_star + 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.543 -0.536 0.015 0.368 3.222
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.2055 0.0579 3.55 0.00046 ***
## UF 0.5125 0.1332 3.85 0.00015 ***
## EOU -0.3903 0.1766 -2.21 0.02788 *
## UF_star -0.1241 0.1047 -1.18 0.23704
## EOU_star 0.4847 0.1638 2.96 0.00334 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.803 on 290 degrees of freedom
## Multiple R-squared: 0.365, Adjusted R-squared: 0.355
## F-statistic: 33.4 on 5 and 290 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults6 <- Boot(copulaResults6, R=bootrounds)
summary(bootCopulaResults6)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.205 | -0.00259 | 0.0608 | 0.203 |
| 1e+04 | 0.512 | 0.0395 | 0.211 | 0.555 |
| 1e+04 | -0.39 | -0.0403 | 0.436 | -0.378 |
| 1e+04 | -0.124 | -0.0336 | 0.17 | -0.149 |
| 1e+04 | 0.485 | 0.0575 | 0.402 | 0.477 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults6, 3, 2)
## Pr(>|t|) ATT : 0.0008218
## Pr(>|t|) UF : 0.01552
## Pr(>|t|) EOU : 0.3713
## Pr(>|t|) UF_star : 0.4664
## Pr(>|t|) EOU_star : 0.2291
Include Copula for ATT, UF and EOU (Model 7)
# Normal copula regression
copulaResults7 <- lm (BI ~ ATT + UF + EOU + ATT_star + UF_star + EOU_star + 0,data=dat)
summary(copulaResults7)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + ATT_star + UF_star + EOU_star +
## 0, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.529 -0.533 0.013 0.368 3.218
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ATT 0.2465 0.1332 1.85 0.06527 .
## UF 0.5075 0.1342 3.78 0.00019 ***
## EOU -0.4052 0.1822 -2.22 0.02690 *
## ATT_star -0.0397 0.1161 -0.34 0.73266
## UF_star -0.1199 0.1056 -1.14 0.25696
## EOU_star 0.5012 0.1710 2.93 0.00365 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.805 on 289 degrees of freedom
## Multiple R-squared: 0.366, Adjusted R-squared: 0.353
## F-statistic: 27.8 on 6 and 289 DF, p-value: <2e-16
# Bootstrap standard errors
bootCopulaResults7 <- Boot(copulaResults7, R=bootrounds)
summary(bootCopulaResults7)
| R | original | bootBias | bootSE | bootMed |
| 1e+04 | 0.246 | 0.02 | 0.165 | 0.257 |
| 1e+04 | 0.508 | 0.0367 | 0.215 | 0.553 |
| 1e+04 | -0.405 | -0.0401 | 0.446 | -0.393 |
| 1e+04 | -0.0397 | -0.0254 | 0.15 | -0.0497 |
| 1e+04 | -0.12 | -0.0326 | 0.173 | -0.147 |
| 1e+04 | 0.501 | 0.0648 | 0.415 | 0.497 |
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(data, bootCopulaResults7, 3, 3)
## Pr(>|t|) ATT : 0.1371
## Pr(>|t|) UF : 0.01881
## Pr(>|t|) EOU : 0.364
## Pr(>|t|) ATT_star : 0.7919
## Pr(>|t|) UF_star : 0.4896
## Pr(>|t|) EOU_star : 0.2285
model_summary(list(stdModel,copulaResults1,copulaResults2,copulaResults3,copulaResults4,copulaResults5,copulaResults6,copulaResults7))
##
## Model Summary
##
## ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## (1) BI (2) BI (3) BI (4) BI (5) BI (6) BI (7) BI (8) BI
## ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## (Intercept) -0.000
## (0.047)
## ATT 0.218 *** 0.161 0.218 *** 0.213 *** 0.160 0.269 * 0.205 *** 0.246
## (0.058) (0.127) (0.059) (0.058) (0.132) (0.132) (0.058) (0.133)
## UF 0.373 *** 0.373 *** 0.362 ** 0.372 *** 0.377 ** 0.371 *** 0.512 *** 0.508 ***
## (0.061) (0.061) (0.125) (0.060) (0.128) (0.060) (0.133) (0.134)
## EOU 0.100 0.097 0.100 -0.310 0.097 -0.334 -0.390 * -0.405 *
## (0.062) (0.062) (0.062) (0.163) (0.062) (0.171) (0.177) (0.182)
## ATT_star 0.055 0.056 -0.055 -0.040
## (0.109) (0.113) (0.115) (0.116)
## UF_star 0.009 -0.003 -0.124 -0.120
## (0.096) (0.099) (0.105) (0.106)
## EOU_star 0.401 ** 0.428 ** 0.485 ** 0.501 **
## (0.148) (0.158) (0.164) (0.171)
## ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## R^2 0.346 0.347 0.346 0.362 0.347 0.363 0.365 0.366
## Adj. R^2 0.340 0.338 0.337 0.354 0.336 0.352 0.355 0.353
## Num. obs. 295 295 295 295 295 295 295 295
## ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## Note. * p < .05, ** p < .01, *** p < .001.
COPULA ANALYSIS BASED ON RENDO
Include Copula for ATT (Model 1)
library(REndo)
copulaResults2 <- copulaCorrection(BI~ATT + UF + EOU|continuous(ATT),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(ATT),
## data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.0059 0.0492 -0.0982 0.0945
## ATT 0.1587 0.1518 -0.1255 0.4766
## UF 0.3743 0.0744 0.2278 0.5148
## EOU 0.0988 0.0892 -0.0690 0.2772
## Number of bootstraps: 1000
##
## Continuous endogenous variables: ATT
##
## Further parameters estimated during model fitting:
## rho sigma
## 0.0706 0.8091
## (see help file for details)
##
## Initial parameter values:
## (Intercept)=0 ATT=0.218 UF=0.373 EOU=0.1 rho=0 sigma=0
##
## The value of the log-likelihood function: 355.7
## AIC: -699.4 , BIC: -677.3
## KKT1: TRUE KKT2: TRUE Optimx Convergence Code: 0
Include Copula for UF (Model 2)
copulaResults2 <- copulaCorrection(BI~ATT + UF + EOU|continuous(UF),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(UF),
## data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.00276 0.05304 -0.11000 0.09441
## ATT 0.21811 0.06250 0.10081 0.34394
## UF 0.35285 0.25006 -0.17270 0.77888
## EOU 0.10057 0.09024 -0.06355 0.28593
## Number of bootstraps: 1000
##
## Continuous endogenous variables: UF
##
## Further parameters estimated during model fitting:
## rho sigma
## 0.0207 0.8080
## (see help file for details)
##
## Initial parameter values:
## (Intercept)=0 ATT=0.218 UF=0.373 EOU=0.1 rho=0 sigma=0
##
## The value of the log-likelihood function: 355.8
## AIC: -699.7 , BIC: -677.6
## KKT1: FALSE KKT2: TRUE Optimx Convergence Code: 0
Include Copula for EOU (Model 3)
# Normal copula regression
copulaResults3 <- copulaCorrection(BI~ATT + UF + EOU|continuous(EOU),num.boots=1000, data=data)
##
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lm (BI ~ ATT + UF + EOU + EOU_star + 0,data=dat)
##
## Call:
## lm(formula = BI ~ ATT + UF + EOU + EOU_star + 0, data = dat)
##
## Coefficients:
## ATT UF EOU EOU_star
## 0.213 0.372 -0.310 0.401
##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(EOU),
## data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.0456 0.0582 -0.1672 0.0682
## ATT 0.2122 0.0589 0.0913 0.3260
## UF 0.3711 0.0734 0.2349 0.5127
## EOU -0.3542 0.3190 -0.8780 0.2935
## Number of bootstraps: 1000
##
## Continuous endogenous variables: EOU
##
## Further parameters estimated during model fitting:
## rho sigma
## 0.488 0.914
## (see help file for details)
##
## Initial parameter values:
## (Intercept)=0 ATT=0.218 UF=0.373 EOU=0.1 rho=0 sigma=0
##
## The value of the log-likelihood function: 351.8
## AIC: -691.5 , BIC: -669.4
## KKT1: TRUE KKT2: TRUE Optimx Convergence Code: 0
Include Copula for ATT and UF (Model 4)
# Normal copula regression
copulaResults4 <- copulaCorrection(BI~ATT + UF + EOU|continuous(ATT,UF),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(ATT,
## UF), data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.005736 0.056025 -0.125370 0.101413
## ATT 0.158153 0.153801 -0.109631 0.493710
## UF 0.372380 0.264380 -0.149931 0.882807
## EOU 0.097088 0.090708 -0.058696 0.300519
## PStar.ATT 0.057897 0.135780 -0.265165 0.281614
## PStar.UF 0.000772 0.207164 -0.418408 0.396346
## Number of bootstraps: 1000
##
## Continuous endogenous variables: ATT, UF
Include Copula for ATT and EOU (Model 5)
# Normal copula regression
copulaResults5 <- copulaCorrection(BI~ATT + UF + EOU|continuous(ATT,EOU),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(ATT,
## EOU), data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.0432 0.0637 -0.1817 0.0710
## ATT 0.2561 0.1616 -0.0433 0.6254
## UF 0.3714 0.0712 0.2360 0.5079
## EOU -0.3736 0.3436 -0.9221 0.3683
## PStar.ATT -0.0427 0.1412 -0.3761 0.2060
## PStar.EOU 0.4656 0.2956 -0.0967 0.9817
## Number of bootstraps: 1000
##
## Continuous endogenous variables: ATT, EOU
Include Copula for UF and EOU (Model 6)
# Normal copula regression
copulaResults6 <- copulaCorrection(BI~ATT + UF + EOU|continuous(UF,EOU),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(UF,
## EOU), data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.0328 0.0651 -0.1686 0.0964
## ATT 0.2061 0.0642 0.0848 0.3316
## UF 0.4927 0.2152 0.0975 0.9589
## EOU -0.4130 0.3430 -1.0214 0.2918
## PStar.UF -0.1066 0.1681 -0.4881 0.1927
## PStar.EOU 0.5062 0.2999 -0.0602 1.0748
## Number of bootstraps: 1000
##
## Continuous endogenous variables: UF, EOU
Include Copula for ATT, UF and EOU (Model 7)
copulaResults7 <- copulaCorrection(BI~ATT + UF + EOU|continuous(ATT,UF,EOU),num.boots=1000, data=data)
##
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##
## Call:
## copulaCorrection(formula = BI ~ ATT + UF + EOU | continuous(ATT,
## UF, EOU), data = data, num.boots = 1000)
##
## Coefficients:
## Point Estimate Boots SE Lower Boots CI (95%) Upper Boots CI (95%)
## (Intercept) -0.0314 0.0682 -0.1648 0.1011
## ATT 0.2400 0.1675 -0.0339 0.5960
## UF 0.4894 0.2275 0.0461 0.9203
## EOU -0.4244 0.3600 -1.0673 0.2726
## PStar.ATT -0.0328 0.1495 -0.3706 0.1994
## PStar.UF -0.1039 0.1804 -0.4686 0.2353
## PStar.EOU 0.5189 0.3174 -0.0590 1.1065
## Number of bootstraps: 1000
##
## Continuous endogenous variables: ATT, UF, EOU