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Fit Latent Data

Øystein Olav Skaar

2022-02-22

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 0.6-10 ended normally after 1 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                         5
#>                                                       
#>   Number of observations                          1000
#>                                                       
#> Model Test User Model:
#>                                                       
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> 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 0.6-10 ended normally after 31 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                         5
#>                                                       
#>   Number of observations                          1002
#>                                                       
#> Model Test User Model:
#>                                                       
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> 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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.