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title: "Model equations: serial mediation"
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vignette: >
  %\VignetteIndexEntry{Model equations: serial mediation}
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---

**Purpose:** definitions and equations for `CN` (serial mediation).
For a first analysis, start with [Choose an analysis](AnalysisGuide.html#choose-your-model)
or [Get started](GetStarted.html). `GenerateModelCN()` creates the model syntax
used for this structure.

Every earlier mediator predicts every later mediator. With three mediators, this includes M1 to M3 as well as M1 to M2 and M2 to M3.

<p align="center">
<img src="Wb.png" alt="serial within-subject mediation model" width="100%">
</p>

## Difference scores and centered averages

For mediator $i$, $M_{i1}$ and $M_{i2}$ are its measurements in conditions 1
and 2. Define

$$
M_{\mathrm{diff},i}=M_{i2}-M_{i1},\qquad
\widetilde M_{\mathrm{avg},i}=\frac{M_{i1}+M_{i2}}{2},\qquad
M_{\mathrm{avg},i}=\widetilde M_{\mathrm{avg},i}-\overline{\widetilde M}_{\mathrm{avg},i}.
$$

The bar denotes the sample mean used for centering. The outcome difference
is $Y_{\mathrm{diff}}=Y_2-Y_1$. `PrepareData()` constructs these variables:
`diff` is not centered, whereas `avg` is centered.

## Regression equations

The equations below show the regression part of the model without covariates
or an external moderator. All $N$ mediators predict the outcome difference:

$$
Y_{\mathrm{diff}}=c'+\sum_{i=1}^{N}\left(b_iM_{\mathrm{diff},i}+d_iM_{\mathrm{avg},i}\right)+e.
$$

The mediator equations determine the structure:

$$
M_{\mathrm{diff},i}=a_i+\sum_{j<i}\left(b_{ji}M_{\mathrm{diff},j}+d_{ji}M_{\mathrm{avg},j}\right)+\epsilon_i.
$$

The intercepts $a_i$ and $c'$ represent direct condition effects in their
respective equations; the model syntax labels $c'$ as `cp`. Each $b_i$ is
the coefficient of a mediator difference, and $d_i$ is the coefficient of
its centered average. For a mediator-to-mediator connection, $b_{ji}$ and
$d_{ji}$ are the corresponding coefficients from mediator $j$ to mediator $i$
(syntax labels `b_j_i` and `d_j_i`).

The $d$ coefficients capture differences between condition-specific slopes.
The average component is not itself an external moderator. Moderation by a
separate variable $W$ requires specifying `W` and `MP`.

## Indirect pathways

Every mediator has a single-mediator indirect pathway $IE_i=a_i b_i$.
A longer directed pathway is the product of its starting $a$ coefficient,
its mediator-to-mediator $b$ coefficients and its final outcome $b$ coefficient.
The centered-average $d$ coefficients remain in the fitted regressions but
do not enter these indirect-effect products.

For three mediators, the complete set is:

| Mediator route to Y | Indirect effect | Result identifier |
|---|---|---|
| M1 → Y | $a_1 b_1$ | `indirect_1` |
| M2 → Y | $a_2 b_2$ | `indirect_2` |
| M3 → Y | $a_3 b_3$ | `indirect_3` |
| M1 → M2 → Y | $a_1 b_{12} b_2$ | `indirect_1_2` |
| M1 → M3 → Y | $a_1 b_{13} b_3$ | `indirect_1_3` |
| M2 → M3 → Y | $a_2 b_{23} b_3$ | `indirect_2_3` |
| M1 → M2 → M3 → Y | $a_1 b_{12} b_{23} b_3$ | `indirect_1_2_3` |

There are **seven** specific indirect effects. The M1 → M3 → Y route
uses `b_1_3` and is included even though it skips M2.

## Total effects and contrasts

Sum every complete indirect pathway to obtain the total indirect effect:

$$
IE_{\mathrm{total}}=\sum_{p\in\mathcal P}IE_p,\qquad
TE=c'+IE_{\mathrm{total}}.
$$

Here $\mathcal P$ is the set of directed condition-to-outcome pathways through
one or more mediators. To compare two such pathways, define a signed contrast:

$$
\Delta_{p,q}=IE_p-IE_q.
$$

A positive contrast means that the first indirect effect is numerically larger;
it does not necessarily have a larger absolute magnitude. For example,
$-0.1-(-0.3)=0.2$, although $|-0.1|<|-0.3|$. A contrast estimate alone does
not establish a difference. Compute its CI from the difference of the effects
within each matched joint draw, using `wsmed_contrasts()`; do not subtract the
endpoints of two separate CIs. See [Read and export results](ResultsGuide.html).

## Condition-specific slopes

On the unstandardized scale, the outcome slopes in conditions 1 and 2 are

$$
g_{i1}=b_i-d_i/2,\qquad g_{i2}=b_i+d_i/2.
$$

The same relation applies to each included mediator-to-mediator connection,
using $b_{ji}$ and $d_{ji}$. These are path coefficients, not additional
indirect effects.

## Standardization and moderation

`GenerateModel*()` constructs unstandardized model syntax. With a fitted
`wsMed` result, use `summary(result, standardized = TRUE)` to request
standardized effects and paths, or set `standardized = TRUE` in `wsMed()`
to make that scale the reporting default.

Let $s_{D M_i}$, $s_{A M_i}$ and $s_{D Y}$ be the marginal model-implied SDs
of the mediator differences, centered averages and outcome difference. Then

$$
a_i^*=a_i/s_{D M_i},\quad b_i^*=b_i s_{D M_i}/s_{D Y},\quad
d_i^*=d_i s_{A M_i}/s_{D Y},\quad c'^*=c'/s_{D Y}.
$$

Mediator-to-mediator slopes use the corresponding source and destination SDs.
Intermediate scales cancel in each complete indirect-effect product, giving
$IE_p^*=IE_p/s_{D Y}$. With moderation, evaluate the conditional paths first:
$IE_p^*(w)=IE_p(w)/s_{D Y}$. All moderator values or groups share the same
marginal endpoint scale. Continuous moderator probes retain their raw units.

Standardized CIs transform coefficients and scales jointly in each draw.
Do not divide raw CI endpoints by a single fitted SD, or reconstruct
condition-specific slopes as $b_i^*\pm d_i^*/2$: the two standardized
coefficients use different predictor scales. See
[Standardized effects](StandardizedModeratedMediation.html) for conditional
effects, interaction coefficients and inference.