Fit Latent Data

Øystein Olav Skaar

2019-02-04

Fit Latent Data

Enjoy this brief demonstration of the fit latent data module (i.e., a simple mediation model).

Also, please see Fit Observed Data.

First we simulate some 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)

Next we run a standard mediation analysis using lavaan

model <- "
# 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)"

fit <- lavaan::sem(model, data = data)
lavaan::summary(fit)
#> lavaan 0.6-3 ended normally after 26 iterations
#> 
#>   Optimization method                           NLMINB
#>   Number of free parameters                          5
#> 
#>   Number of observations                          1000
#> 
#>   Estimator                                         ML
#>   Model Fit Test Statistic                       0.000
#>   Degrees of freedom                                 0
#> 
#> Parameter Estimates:
#> 
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#>   Standard Errors                             Standard
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   y ~                                                 
#>     x          (c)    0.022    0.018    1.242    0.214
#>   m ~                                                 
#>     x          (a)    0.759    0.020   38.118    0.000
#>   y ~                                                 
#>     m          (b)    0.752    0.018   40.944    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .y                 0.142    0.006   22.361    0.000
#>    .m                 0.421    0.019   22.361    0.000
#> 
#> Defined Parameters:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     ab                0.570    0.020   27.899    0.000
#>     cd                0.593    0.019   31.357    0.000

Now for the Bayesian model

bayesian.fit <- bfw::bfw(project.data = data,
                    latent = "x,m,y",
                    saved.steps = 50000,
                    latent.names = "Independent,Mediator,Dependent",
                    additional = "indirect <- xm * my , total <- xy + (xm * my)",
                    additional.names = "AB,C",
                    jags.model = "fit",
                    silent = TRUE)

round(bayesian.fit$summary.MCMC[,3:7],3)
#>                                       Mode   ESS  HDIlo HDIhi    n
#> beta[2,1]: Mediator vs. Independent  0.760 48988  0.720 0.799 1000
#> beta[3,1]: Dependent vs. Independent 0.024 13042 -0.012 0.058 1000
#> beta[3,2]: Dependent vs. Mediator    0.751 13230  0.715 0.786 1000
#> indirect[1]: AB                      0.571 21431  0.531 0.611 1000
#> total[1]: C                          0.591 49074  0.555 0.630 1000

Time for some noise

biased.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)

Rerun the lavaan model

biased.fit <- lavaan::sem(model, data = biased.data)
lavaan::summary(biased.fit)
#> lavaan 0.6-3 ended normally after 31 iterations
#> 
#>   Optimization method                           NLMINB
#>   Number of free parameters                          5
#> 
#>   Number of observations                          1002
#> 
#>   Estimator                                         ML
#>   Model Fit Test Statistic                       0.000
#>   Degrees of freedom                                 0
#> 
#> Parameter Estimates:
#> 
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#>   Standard Errors                             Standard
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   y ~                                                 
#>     x          (c)    0.665    0.001  499.371    0.000
#>   m ~                                                 
#>     x          (a)    0.004    0.002    1.528    0.127
#>   y ~                                                 
#>     m          (b)    0.250    0.018   14.152    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .y                 0.320    0.014   22.383    0.000
#>    .m                 1.028    0.046   22.383    0.000
#> 
#> Defined Parameters:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     ab                0.001    0.001    1.519    0.129
#>     cd                0.666    0.001  457.040    0.000

Run the Bayesian model with robust estimates

biased.bfit <- bfw::bfw(project.data = data,
                    latent = "x,m,y",
                    saved.steps = 50000,
                    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 = 101,
                    silent = TRUE)

round(biased.bfit$summary.MCMC[,3:7],3)
#>                                       Mode   ESS  HDIlo HDIhi    n
#> beta[2,1]: Mediator vs. Independent  0.763 31178  0.721 0.799 1000
#> beta[3,1]: Dependent vs. Independent 0.022  7724 -0.014 0.057 1000
#> beta[3,2]: Dependent vs. Mediator    0.751  7772  0.714 0.786 1000
#> indirect[1]: AB                      0.572 12913  0.531 0.610 1000
#> total[1]: C                          0.590 31362  0.557 0.631 1000