Free mediation models with pamlj

The jamovi module pamlj offers three ways to set up a mediation power analysis: the simple model (3 variables), the complex templates (two/three parallel mediators, serial mediation) and the free interface, which lets you describe any recursive mediation model by typing its equations in a model syntax. This vignette shows how to use the free interface from R, via pamlj::pamlmed() with mode = "medmodels".

The free interface is useful when your model does not match one of the predefined templates: several independent variables, an arbitrary number of mediators, serial and parallel pathways combined, and so on. From the equations, pamlj builds the path model, finds every indirect (mediated) effect, and computes the power parameters for each of them.

In R, the model is passed as a single string to the code argument (one equation per line, separated by \n). Every other argument — aim, n, power, test, sig.level, … — works as in the other mediation modes. The results table is returned in obj$powertab$asDF.

Model syntax

The model is described with one regression equation per endogenous variable (every variable that is predicted by others), written in the familiar lm()/lavaan style. The classical simple mediation reads:

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m ~ .3*x
           y ~ .2*x + .4*m",
  aim   = "n",
  power = .80,
  test  = "joint"
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF
#>      effect  n   a   b cprime   es power sig.level
#> 1 x → m → y 87 0.3 0.4    0.2 0.12   0.8      0.05

Each line reads as outcome ~ predictors. Here x predicts the mediator m with a standardized coefficient of .3, and y is predicted by both x (direct effect c' = .2) and m (b = .4). The mediated effect x → m → y is the product of the path coefficients, .3 * .4 = .12 (the es / ME column), and the table reports the sample size required to detect it with power .80.

A few rules apply to the syntax:

Every predictor needs a numeric coefficient, passed as value*variable (e.g. .4*m). Coefficients are completely standardized path coefficients and their absolute value must lie between .001 and .99.
One equation per outcome. A variable can appear on the left-hand side of at most one equation; list all its predictors on that single line.
No intercepts, interactions or categorical terms. The model works on standardized coefficients between single variables, so intercept tokens (1) are ignored and interaction (x:z) or categorical ([...]) terms are not allowed.
The model must be a proper mediation model: it has to be recursive (no feedback loops) and contain at least one indirect path (a variable that is both an outcome and a predictor). If these conditions are not met, the function reports what is missing instead of running.

Indirect effects

pamlj automatically enumerates every indirect effect in the model — each simple directed path of at least two edges that runs from an exogenous variable (a source, never predicted by others) to a final outcome (a sink, that predicts nothing). Each such effect becomes one row of the results table, labelled with its path in the effect column.

A model with two parallel mediators therefore yields two indirect effects:

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + .2*m1 + .3*m2",
  aim   = "power",
  n     = 200,
  test  = "joint"
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF[, c("effect", "a", "b", "es", "power")]
#>       effect   a   b   es     power
#> 1 x → m1 → y 0.4 0.2 0.08 0.8193946
#> 2 x → m2 → y 0.5 0.3 0.15 0.9826651

For every indirect effect the table reports the first path coefficient (a), the remainder of the product (b), the mediated effect size es (ME, the product of all coefficients along the path) and the requested quantity (here the power at n = 200).

The power of each indirect effect depends on the test selected. With the joint significance test (Yzerbyt et al., 2018), two effects that share their weakest path can require the same sample size even when their other coefficients differ, because the shared (weaker) coefficient drives the power.

Correlated variables

Variables can be correlated with the cor() directive, written on its own line. The most common use is two parallel mediators whose residuals are correlated:

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + .2*m1 + .3*m2
           cor(m1,m2) = .3",
  aim   = "power",
  n     = 200,
  test  = "joint",
  inspect_cors = TRUE
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF[, c("effect", "es", "power")]
#>       effect   es     power
#> 1 x → m1 → y 0.08 0.7771818
#> 2 x → m2 → y 0.15 0.9712231

cor(m1,m2) = .3 sets the correlation between m1 and m2 to .3, following the lavaan convention m1 ~~ .3*m2. The separator can be either : or =, so cor(m1,m2):.3 is equivalent. Several cor() lines can be specified. If the requested correlations imply an impossible (non positive-definite) covariance matrix, the function reports it.

Setting inspect_cors = TRUE adds a table of the model-implied correlations among all variables, which is handy to check what actually entered the model:

res$implied_cors$asDF
#>   variable    x   m1   m2    y
#> 1        x 1.00 0.40 0.50 0.33
#> 2       m1 0.40 1.00 0.50 0.39
#> 3       m2 0.50 0.50 1.00 0.45
#> 4        y 0.33 0.39 0.45 1.00

Minimum detectable effect and sensitivity analysis

When aim = "es", pamlj solves for the minimum detectable effect (MDE): given a fixed sample size and the desired power, it finds the smallest indirect effect the design can detect. Because a mediated effect is a product of path coefficients, the answer depends on which coefficient is allowed to change while the others stay fixed.

Choosing the coefficient to vary

By default, pamlj works on each indirect effect separately and resizes its first edge (the path leaving the independent variable, the a path). This needs no extra input — just set aim = "es":

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + .2*m1 + .3*m2",
  aim   = "es",
  n     = 250,
  power = .80,
  test  = "joint"
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF[, c("effect", "a", "b", "es", "power")]
#>       effect         a   b         es power
#> 1 x → m1 → y 0.1913634 0.2 0.03827269   0.8
#> 2 x → m2 → y 0.1759490 0.3 0.05278470   0.8

To target a specific coefficient instead — for instance the b path of a mediator, or a coefficient shared by several indirect effects — give it a symbolic label in the syntax and select it with the test: directive. A label is a letter placed before the value, written label*value*variable:

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + a*.2*m1 + .3*m2
           test: a",
  aim   = "es",
  n     = 250,
  power = .80,
  test  = "joint"
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF[, c("effect", "a", "b", "es", "power")]
#>       effect   a         b         es     power
#> 1 x → m1 → y 0.4 0.1754476 0.07017906 0.7999768
#> 2 x → m2 → y 0.5 0.3000000 0.15000000 0.9948623

Here the coefficient of m1 → y is labelled a, and test: a (equivalently test = a) tells pamlj to find the minimum detectable effect by resizing that coefficient. If the label is not defined, or the labelled coefficient is not part of any indirect effect, the analysis falls back to the per-effect default.

How the search works

pamlj resizes the selected coefficient — keeping every other coefficient at its specified value — until the power of the indirect effect(s) that travel through that coefficient reaches the desired level. When the coefficient lies on more than one indirect path (a shared edge), the search drives the weakest of those effects to the target, so that every affected effect reaches at least the requested power. Effects that do not pass through the coefficient are left unchanged.

The balanced solution

Resizing a single coefficient cannot always reach the requested power. The power of an indirect effect, seen as a function of one of its coefficients, is unimodal: it rises, reaches a peak, then falls back as the coefficient approaches 1 (the mediator becomes collinear with its predictor and the other coefficients’ standard errors explode). The other coefficients on the path therefore cap the achievable power: if the requested power sits above that peak, no value of the single coefficient can deliver it.

When this happens, pamlj switches to a balanced solution: instead of moving one coefficient, it grows all the coefficients of the path together to a common value, which removes the bottleneck and can reach any power below 1. A message explains what happened, and the solution is recognisable in the table because the a and b columns of an affected effect become equal (every edge on the path is set to the same value):

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + .2*m1 + .3*m2",
  aim   = "es",
  n     = 100,
  power = .90,
  test  = "joint"
)
#> 
#> Joint-significance power is computed from the distribution of each path coefficient.
res$powertab$asDF[, c("effect", "a", "b", "es", "power")]
#>       effect         a         b        es power
#> 1 x → m1 → y 0.3374171 0.3374171 0.1138503   0.9
#> 2 x → m2 → y 0.3453462 0.3453462 0.1192640   0.9

If even the balanced solution cannot reach the target at the given sample size, the table reports the largest detectable effect together with the maximum power it yields, and suggests increasing the sample size. In practice the balanced solution is most useful as a feasibility check: when it appears, the chosen sample size is too small for the requested power whatever single coefficient you adjust, and the common value shows how large every path would have to be to make the design work.

Tests

The free interface supports the same tests as the other mediation models, set with the test argument:

"joint" — joint significance (Yzerbyt et al., 2018), the default and recommended method;
"sobel" — the Sobel test (Sobel, 1982);
"parametric" / "simulation" — the Monte Carlo confidence-interval methods (MacKinnon, Lockwood & Williams, 2004). The number of simulations is set with mcR, and set_seed = TRUE with seed = <n> makes the result reproducible.

res <- pamlj::pamlmed(
  mode  = "medmodels",
  code  = "m1 ~ .4*x
           m2 ~ .5*x
           y  ~ .1*x + .2*m1 + .3*m2",
  aim   = "power",
  n     = 200,
  test  = "sobel"
)
#> 
#> Sobel test method is based on power parameters approximation. Please use joint-significance or bootstrap confidence intervals for more accurate results
res$powertab$asDF[, c("effect", "es", "power")]
#>       effect   es     power
#> 1 x → m1 → y 0.08 0.7430981
#> 2 x → m2 → y 0.15 0.9549460

References

Yzerbyt, V., Muller, D., Batailler, C., & Judd, C. M. (2018). New recommendations for testing indirect effects in mediational models: The need to report and test component paths. Journal of Personality and Social Psychology, 115(6), 929–943.

Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312.

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99–128.

Marcello Gallucci

2026-07-27