This is a demonstration of regularized mediation analysis (*regmed*) for mediation

The user user-level functions in the *regmed* package are:

`regmed.grid()`

penalized model for grid of lambdas`trim.best()`

get best model, trim off mediators with no effect, refit model`regmed.fit()`

fit penalized model with specified lambda`plot.regmed.grid()`

plot results of regmed.grid`plot.regmed()`

plot results of a single model fit, from either trim.best() or regmed.fit()

```
require("regmed")
```

The data used in this example was created from selecting four methylyation loci as examples from the publicly available data E-GEOD-77445 - Genome-wide DNA methylation levels. The “exposure” variable is childhood trauma, the “outcome” variable is cortisol stress reactivity, and the four methylation loci are potential mediators to be evaluated.

This example data has only four potential mediators. In practice, the number of mediators can be much larger than the sample size. In this case, one can use sure independence screening (Fan, J., & Lv, J. , 2008, Sure independence screening for ultrahigh dimensional feature space. J. R. Statist. Soc.B, 70, 849-911). This approach is based on ranking marginal correlations and then selecting the highest ranked values such that the number of parameters is less than the sample size. Because mediation depends on the two correlations, r(x,m) and r(m,y), where m is a potential mediator, one can rank the absolute values of their products, |r(x,m) * r(m,y)|, and then select the highest ranked values to determine which potential mediators to include in the penalized mediation models, such that the number parameters in the model is less than the sample size.

```
data(regmed_example)
y <- regmed_example$y
x <- regmed_example$x
med <- regmed_example[, -c(1, 2)]
```

Fit penalized models for a grid of lambda's

```
fit.grid <- regmed.grid(x, med, y, lambda.vec = c(seq(from = 1, to = 0, by = -0.1)),
frac.lasso = 0.8)
```

```
plot.regmed.grid(fit.grid)
```

Get best model, trim to remove mediators with near zero effect, and refit with specified lambda (default lambda = 0).

```
fit.trim <- trim.best(fit.grid)
summary.regmed(fit.trim)
```

```
Call:
regmed.fit(x = x, mediator = med[, c("med.cg01644731", "med.cg06890779"),
drop = FALSE], y = y, lambda = 0, frac.lasso = 0.8)
Coefficients:
alpha beta alpha.beta
med.cg01644731 0.3186614 -0.3050835 -0.09721834
med.cg06890779 -0.3715008 0.2071779 -0.07696674
sum of alpha*beta = -0.1741851
delta = -0.1036191
sum of delta + alpha*beta = -0.2778042
var(x) = 1
var(y) = 0.936029
```

```
plot(fit.trim, cex = 0.6)
```

This example subsets to the mediators in fit.trim, and uses *regmed()* on the selected
subset of mediators.

```
## choose subset of mediators that are in fit.trim
which.med <- colnames(med) %in% dimnames(fit.trim$alpha)[[1]]
med.selected <- med[, which.med]
trauma = x
cortisol = y
fit.lam2 <- regmed.fit(trauma, med.selected, cortisol, lambda = 0.2, frac.lasso = 0.8)
summary(fit.lam2)
```

```
Call:
regmed.fit(x = trauma, mediator = med.selected, y = cortisol,
lambda = 0.2, frac.lasso = 0.8)
Coefficients:
alpha beta alpha.beta
med.cg01644731 0.2029702 -0.1965572 -0.03989525
med.cg06890779 -0.2557726 0.1039875 -0.02659716
sum of alpha*beta = -0.06649241
delta = -0.1363947
sum of delta + alpha*beta = -0.2028871
var(x) = 1
var(y) = 0.8880832
```

```
fit.lam0 <- regmed.fit(trauma, med.selected, cortisol, lambda = 0, frac.lasso = 0.8)
summary(fit.lam0)
```

```
Call:
regmed.fit(x = trauma, mediator = med.selected, y = cortisol,
lambda = 0, frac.lasso = 0.8)
Coefficients:
alpha beta alpha.beta
med.cg01644731 0.3186614 -0.3050835 -0.09721834
med.cg06890779 -0.3715008 0.2071779 -0.07696674
sum of alpha*beta = -0.1741851
delta = -0.1036191
sum of delta + alpha*beta = -0.2778042
var(x) = 1
var(y) = 0.936029
```

We show the plot method can display different sized text and line types,
along with using the variable names from the call to *regmed.fit()*.

```
plot(fit.lam2, lty = 2, lwd = 3, cex = 0.7)
```

When using trim.best() with default lambda = 0, the returned object is an unpenalized structural equation model fit. For this model, the lavaan R package can be used to estimate standard errors of the model parameters, although these are approximations that do not consider that the terms in the model were first chosen by the penalized model.

```
library(lavaan)
```

```
This is lavaan 0.6-5
```

```
lavaan is BETA software! Please report any bugs.
```

```
## choose subset of mediators that are in fit.trim
which.med <- colnames(med) %in% dimnames(fit.trim$alpha)[[1]]
med.selected <- med[, which.med]
mediator.names <- dimnames(med.selected)[[2]]
## setup lavaan model
med.model <- lavaan.model(y.name = "y", x.name = "x", med.name = mediator.names,
medcov = fit.trim$MedCov)
## setup data for lavaan
dat <- data.frame(cbind(scale(x, center = TRUE, scale = TRUE), scale(y, center = TRUE,
scale = TRUE), scale(med.selected, center = TRUE, scale = TRUE)))
names(dat) <- c("x", "y", dimnames(med.selected)[[2]])
## fit sem with lavaan
fit.lavaan <- sem(model = med.model, data = dat)
summary(fit.lavaan)
```

```
lavaan 0.6-5 ended normally after 14 iterations
Estimator ML
Optimization method NLMINB
Number of free parameters 6
Number of observations 85
Model Test User Model:
Test statistic 0.030
Degrees of freedom 3
P-value (Chi-square) 0.999
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard errors Standard
Regressions:
Estimate Std.Err z-value P(>|z|)
y ~
x -0.098 0.109 -0.894 0.372
med.cg01644731 -0.309 0.102 -3.029 0.002
med.cg06890779 0.211 0.104 2.029 0.042
med.cg01644731 ~
x 0.319 0.103 3.083 0.002
med.cg06890779 ~
x -0.371 0.101 -3.666 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
.med.cg01644731 ~~
.med.cg06890779 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.med.cg01644731 0.898
.med.cg06890779 0.862
.y 0.793 0.122 6.519 0.000
```

The output from lavaan sem model shows the regressions for y regressed on x and the mediators. The estimates and std.err for these coefficents are the direct effect of x on y (delta), and the the beta coeffients for the mediators. The regression of each mediator on x gives the estimate of alpha and its std.err. The covariances of the mediators are fixed (so no std.err), see reference by Schaid and Sinnwell (2020). The estimate of the residual variance of y is shown in the section of “Variances”, along with its std.err.

Schaid DJ, Sinnwell JP. (2020) Penalized Models for Analysis of Multiple Mediators. Genet Epidemiol, to appear.

Rosseel Y. (2012) lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1–36. http://www.jstatsoft.org/v48/i02/.