Enjoy this brief demonstration of the fit observed data module (i.e., a simple mediation model).
Also, please see Fit Latent Data.
# Number of observations
n <- 1000
# Coefficient for a path (x -> m)
a <- .75
# Coefficient for b path (m -> y)
b <- .80
# Coefficient for total effect (x -> y)
c <- .60
# Coefficient for indirect effect (x -> m -> y)
ab <- a * b
# Coefficient for direct effect (x -> y)
cd <- c - ab
# Compute x, m, y values
set.seed(100)
x <- rnorm(n)
m <- a * x + sqrt(1 - a^2) * rnorm(n)
eps <- 1 - (cd^2 + b^2 + 2*a * cd * b)
y <- cd * x + b * m + eps * rnorm(n)
data <- data.frame(y = y,
x = x,
m = m)set.seed(102)
## create 5 y variables, 5 x variables and 5 m variables with jittered data
jitter.data <- lapply(1:5, function (i) {
apply(data , 2 , function (x) jitter(x , amount=1))
})
# Create data frame with jittered observed variables
jitter.data <- data.frame( sapply(jitter.data, "[" ,,1) , # First column (y)
sapply(jitter.data, "[" ,,2) , # Second column (x)
sapply(jitter.data, "[" ,,3) # Third column (m)
)
# Add column names
colnames(jitter.data) <- c( paste0("y",1:5) , paste0("x",1:5) , paste0("m",1:5))model <- '
y =~ y1+y2+y3+y4+y5
x =~ x1+x2+x3+x4+x5
m =~ m1+m2+m3+m4+m5
# direct effect
y ~ c*x
# mediator
m ~ a*x
y ~ b*m
# indirect effect (a*b)
ab := a*b
# total effect
cd := c + (a*b)
'
jitter.fit <- lavaan::sem(model, data = jitter.data)
lavaan::summary(jitter.fit)
#> lavaan 0.6-10 ended normally after 34 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 33
#>
#> Number of observations 1000
#>
#> Model Test User Model:
#>
#> Test statistic 53.272
#> Degrees of freedom 87
#> P-value (Chi-square) 0.998
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> y =~
#> y1 1.000
#> y2 0.998 0.030 33.055 0.000
#> y3 0.993 0.030 33.196 0.000
#> y4 0.940 0.030 31.517 0.000
#> y5 1.015 0.030 33.374 0.000
#> x =~
#> x1 1.000
#> x2 1.013 0.026 39.481 0.000
#> x3 0.986 0.026 38.610 0.000
#> x4 0.993 0.026 38.412 0.000
#> x5 0.983 0.026 38.157 0.000
#> m =~
#> m1 1.000
#> m2 0.963 0.026 37.703 0.000
#> m3 0.975 0.026 38.157 0.000
#> m4 1.009 0.026 38.710 0.000
#> m5 0.970 0.025 38.053 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> y ~
#> x (c) -0.014 0.029 -0.466 0.641
#> m ~
#> x (a) 0.791 0.030 26.756 0.000
#> y ~
#> m (b) 0.758 0.035 21.965 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .y1 0.303 0.016 18.350 0.000
#> .y2 0.330 0.018 18.705 0.000
#> .y3 0.321 0.017 18.641 0.000
#> .y4 0.351 0.018 19.322 0.000
#> .y5 0.328 0.018 18.558 0.000
#> .x1 0.304 0.017 17.798 0.000
#> .x2 0.318 0.018 17.879 0.000
#> .x3 0.329 0.018 18.276 0.000
#> .x4 0.341 0.019 18.360 0.000
#> .x5 0.343 0.019 18.464 0.000
#> .m1 0.351 0.019 18.848 0.000
#> .m2 0.329 0.017 18.884 0.000
#> .m3 0.321 0.017 18.704 0.000
#> .m4 0.322 0.017 18.468 0.000
#> .m5 0.321 0.017 18.746 0.000
#> .y 0.165 0.014 12.069 0.000
#> x 1.048 0.060 17.520 0.000
#> .m 0.420 0.028 15.216 0.000
#>
#> Defined Parameters:
#> Estimate Std.Err z-value P(>|z|)
#> ab 0.599 0.032 18.475 0.000
#> cd 0.586 0.027 22.025 0.000bayesian.jitter.fit <- bfw(project.data = jitter.data,
observed = "x1,x2,x3,x4,x5,
m1,m2,m3,m4,m5,
y1,y2,y3,y4,y5",
latent.names = "Independent,Mediator,Dependent",
additional = "indirect <- xm * my , total <- xy + (xm * my)",
additional.names = "AB,C",
jags.model = "fit",
jags.seed = 102,
silent = TRUE)
round(bayesian.jitter.fit$summary.MCMC[,3:7],3)
#> Mode ESS HDIlo HDIhi n
#> lam[1]: X1 0.999 0 1.000 1.000 1000
#> lam[2]: X2 1.019 1309 0.966 1.067 1000
#> lam[3]: X3 0.988 1420 0.941 1.042 1000
#> lam[4]: X4 0.998 1440 0.946 1.048 1000
#> lam[5]: X5 0.990 1402 0.932 1.035 1000
#> lam[6]: M1 0.999 0 1.000 1.000 1000
#> lam[7]: M2 0.964 1164 0.912 1.013 1000
#> lam[8]: M3 0.970 1049 0.927 1.026 1000
#> lam[9]: M4 1.013 1075 0.959 1.062 1000
#> lam[10]: M5 0.969 1117 0.921 1.022 1000
#> lam[11]: Y1 0.999 0 1.000 1.000 1000
#> lam[12]: Y2 1.004 1136 0.939 1.059 1000
#> lam[13]: Y3 0.994 1139 0.933 1.050 1000
#> lam[14]: Y4 0.941 1270 0.883 0.999 1000
#> lam[15]: Y5 1.023 1149 0.954 1.074 1000
#> error[1]: X1 0.304 4691 0.273 0.340 1000
#> error[2]: X2 0.321 4464 0.285 0.355 1000
#> error[3]: X3 0.327 4637 0.296 0.366 1000
#> error[4]: X4 0.343 5200 0.305 0.379 1000
#> error[5]: X5 0.346 4795 0.309 0.383 1000
#> error[6]: M1 0.352 5155 0.315 0.388 1000
#> error[7]: M2 0.329 5256 0.296 0.365 1000
#> error[8]: M3 0.319 4979 0.288 0.355 1000
#> error[9]: M4 0.321 4903 0.291 0.358 1000
#> error[10]: M5 0.322 5080 0.288 0.356 1000
#> error[11]: Y1 0.300 4509 0.272 0.337 1000
#> error[12]: Y2 0.333 4645 0.298 0.367 1000
#> error[13]: Y3 0.317 5032 0.288 0.356 1000
#> error[14]: Y4 0.350 5481 0.317 0.388 1000
#> error[15]: Y5 0.329 4929 0.294 0.364 1000
#> cov[1,1]: Independent vs. Independent 1.044 884 0.968 1.114 1000
#> cov[2,1]: Mediator vs. Independent 0.822 949 0.781 0.872 1000
#> cov[3,1]: Dependent vs. Independent 0.609 936 0.574 0.647 1000
#> cov[1,2]: Independent vs. Mediator 0.822 949 0.781 0.872 1000
#> cov[2,2]: Mediator vs. Mediator 1.071 669 0.994 1.151 1000
#> cov[3,2]: Dependent vs. Mediator 0.798 776 0.756 0.848 1000
#> cov[1,3]: Independent vs. Dependent 0.609 936 0.574 0.647 1000
#> cov[2,3]: Mediator vs. Dependent 0.798 776 0.756 0.848 1000
#> cov[3,3]: Dependent vs. Dependent 0.759 683 0.701 0.827 1000
#> beta[2,1]: Mediator vs. Independent 0.788 1308 0.735 0.850 1000
#> beta[3,1]: Dependent vs. Independent -0.014 1008 -0.072 0.043 1000
#> beta[3,2]: Dependent vs. Mediator 0.760 630 0.696 0.828 1000
#> indirect[1]: AB 0.593 917 0.535 0.662 1000
#> total[1]: C 0.587 1713 0.534 0.640 1000
#> Fit (Predicted) 80.578 5928 69.388 97.268 1000
#> Fit (Simulated) 119.033 10000 92.518 152.408 1000
#> Fit (Discrepancy) -38.274 9228 -71.546 -5.316 1000
#> PPP 0.991 20 0.988 0.992 1000biased.sigma <-matrix(c(1,1,0,1,1,0,0,0,1),3,3)
set.seed(101)
noise <- MASS::mvrnorm(n=2,
mu=c(200, 300, 0),
Sigma=biased.sigma,
empirical=FALSE)
colnames(noise) <- c("y","x","m")
biased.data <- rbind(data,noise)
jitter.noise <- noise[,rep(1:3,each=5)]
colnames(jitter.noise) <- c( paste0("y",1:5) , paste0("x",1:5) , paste0("m",1:5))
biased.jitter.data <- rbind(jitter.data, jitter.noise)biased.jitter.fit <- lavaan::sem(model, data = biased.jitter.data)
lavaan::summary(biased.jitter.fit)
#> lavaan 0.6-10 ended normally after 254 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 33
#>
#> Number of observations 1002
#>
#> Model Test User Model:
#>
#> Test statistic 53.395
#> Degrees of freedom 87
#> P-value (Chi-square) 0.998
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> y =~
#> y1 1.000
#> y2 1.000 0.003 356.708 0.000
#> y3 1.000 0.003 358.478 0.000
#> y4 0.999 0.003 352.757 0.000
#> y5 1.000 0.003 358.036 0.000
#> x =~
#> x1 1.000
#> x2 1.000 0.002 538.595 0.000
#> x3 1.000 0.002 535.008 0.000
#> x4 1.000 0.002 529.637 0.000
#> x5 1.000 0.002 530.683 0.000
#> m =~
#> m1 1.000
#> m2 0.967 0.026 37.620 0.000
#> m3 0.972 0.026 37.466 0.000
#> m4 1.010 0.026 38.362 0.000
#> m5 0.967 0.026 37.414 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> y ~
#> x (c) 0.666 0.002 319.095 0.000
#> m ~
#> x (a) 0.004 0.003 1.501 0.133
#> y ~
#> m (b) 0.229 0.021 10.733 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .y1 0.301 0.017 18.170 0.000
#> .y2 0.332 0.018 18.564 0.000
#> .y3 0.326 0.018 18.490 0.000
#> .y4 0.346 0.018 18.724 0.000
#> .y5 0.328 0.018 18.509 0.000
#> .x1 0.303 0.017 17.775 0.000
#> .x2 0.320 0.018 18.032 0.000
#> .x3 0.329 0.018 18.144 0.000
#> .x4 0.341 0.019 18.307 0.000
#> .x5 0.339 0.019 18.276 0.000
#> .m1 0.349 0.019 17.965 0.000
#> .m2 0.319 0.018 17.852 0.000
#> .m3 0.327 0.018 17.932 0.000
#> .m4 0.319 0.018 17.435 0.000
#> .m5 0.326 0.018 17.959 0.000
#> .y 0.348 0.020 17.389 0.000
#> x 180.449 8.075 22.346 0.000
#> .m 1.072 0.063 17.123 0.000
#>
#> Defined Parameters:
#> Estimate Std.Err z-value P(>|z|)
#> ab 0.001 0.001 1.488 0.137
#> cd 0.666 0.002 309.325 0.000bayesian.biased.jitter.fit <- bfw(project.data = biased.jitter.data,
observed = "x1,x2,x3,x4,x5,
m1,m2,m3,m4,m5,
y1,y2,y3,y4,y5",
latent.names = "Independent,Mediator,Dependent",
additional = "indirect <- xm * my , total <- xy + (xm * my)",
additional.names = "AB,C",
jags.model = "fit",
run.robust = TRUE,
jags.seed = 103,
silent = TRUE)
round(bayesian.biased.jitter.fit$summary.MCMC[,3:7],3)
#> Mode ESS HDIlo HDIhi n
#> lam[1]: X1 0.999 0 1.000 1.000 1002
#> lam[2]: X2 1.011 813 0.961 1.068 1002
#> lam[3]: X3 0.983 846 0.933 1.037 1002
#> lam[4]: X4 0.987 833 0.938 1.048 1002
#> lam[5]: X5 0.973 906 0.925 1.032 1002
#> lam[6]: M1 0.999 0 1.000 1.000 1002
#> lam[7]: M2 0.961 633 0.911 1.014 1002
#> lam[8]: M3 0.977 624 0.925 1.026 1002
#> lam[9]: M4 1.014 588 0.958 1.060 1002
#> lam[10]: M5 0.969 630 0.921 1.022 1002
#> lam[11]: Y1 0.999 0 1.000 1.000 1002
#> lam[12]: Y2 0.998 765 0.940 1.062 1002
#> lam[13]: Y3 1.012 816 0.943 1.068 1002
#> lam[14]: Y4 0.938 936 0.876 0.999 1002
#> lam[15]: Y5 1.023 717 0.962 1.090 1002
#> error[1]: X1 0.223 1986 0.186 0.255 1002
#> error[2]: X2 0.230 2228 0.200 0.275 1002
#> error[3]: X3 0.249 2177 0.214 0.287 1002
#> error[4]: X4 0.263 2074 0.223 0.299 1002
#> error[5]: X5 0.271 2583 0.232 0.308 1002
#> error[6]: M1 0.343 3209 0.309 0.381 1002
#> error[7]: M2 0.324 2957 0.289 0.359 1002
#> error[8]: M3 0.312 2928 0.282 0.352 1002
#> error[9]: M4 0.315 2939 0.283 0.352 1002
#> error[10]: M5 0.319 3258 0.284 0.352 1002
#> error[11]: Y1 0.221 2172 0.192 0.260 1002
#> error[12]: Y2 0.259 2553 0.226 0.297 1002
#> error[13]: Y3 0.248 2543 0.209 0.279 1002
#> error[14]: Y4 0.284 2934 0.247 0.322 1002
#> error[15]: Y5 0.250 2396 0.215 0.288 1002
#> cov[1,1]: Independent vs. Independent 1.051 543 0.974 1.126 1002
#> cov[2,1]: Mediator vs. Independent 0.825 597 0.782 0.872 1002
#> cov[3,1]: Dependent vs. Independent 0.604 603 0.566 0.643 1002
#> cov[1,2]: Independent vs. Mediator 0.825 597 0.782 0.872 1002
#> cov[2,2]: Mediator vs. Mediator 1.070 377 0.997 1.159 1002
#> cov[3,2]: Dependent vs. Mediator 0.797 541 0.746 0.838 1002
#> cov[1,3]: Independent vs. Dependent 0.604 603 0.566 0.643 1002
#> cov[2,3]: Mediator vs. Dependent 0.797 541 0.746 0.838 1002
#> cov[3,3]: Dependent vs. Dependent 0.761 458 0.701 0.832 1002
#> beta[2,1]: Mediator vs. Independent 0.787 715 0.732 0.847 1002
#> beta[3,1]: Dependent vs. Independent -0.024 621 -0.080 0.041 1002
#> beta[3,2]: Dependent vs. Mediator 0.750 420 0.684 0.825 1002
#> indirect[1]: AB 0.588 604 0.528 0.660 1002
#> total[1]: C 0.572 1358 0.523 0.629 1002