# Detail: SEM (Linear) Parametric Model

This page describes the **SEM (linear SEM)** model type in the **Parametric Model** and
**Instantiated Model** boxes. These models are **linear Gaussian structural equation models**
with path coefficients and Gaussian error terms.

```{figure} ../../_static/images/tetrad-interface/box-by-box/sem-pm.png
:name: tetrad-sem-pm-screenshot
:alt: SEM Parametric Model

SEM Parametric Model
```

## When to use SEM models

Use the SEM model family when:

- Your variables are **continuous** (or treated as such).
- You want a **linear** model of the form  
  \( X = \sum_{Y \in 	ext{Parents}(X)} b_{YX} Y + arepsilon_X \),  
  with \( arepsilon_X \) Gaussian (possibly correlated).

Common use cases include:

- Covariance-structure modeling based on a graph.
- Evaluating search algorithms that assume linear Gaussian SEMs.
- Connection to standard SEM fit indices (\(\chi^2\), RMSEA, CFI, etc.).

## Main panel layout

For SEM parametric models, the main panel typically shows:

- A **parameter table** listing:
  - Regression/path coefficients for edges in the graph.
  - Error variances (and optionally covariances).
- Indicators for whether parameters are **free** or **fixed**.
- Optional constraints or labels on parameters.

For **instantiated** SEM models (after estimation), you may see:

- Estimated parameter values and standard errors.
- Global fit indices (\(\chi^2\), df, RMSEA, CFI, SRMR, BIC, etc.), when supported.
- Residual covariance information.

## Typical workflow

1. **Create an SEM parametric model**
   - Start from a directed graph (often a DAG or SEM-style graph) in the *Graph* box.
   - In the *Parametric Model* box, choose **New → SEM (linear)** to create a model whose structure
     matches the graph.

2. **Specify or inspect parameters**
   - Review default path coefficients and error variances.
   - Fix or free parameters as needed (e.g., setting certain paths to fixed values or zero).
   - Optionally impose equality or other constraints, if supported.

3. **Estimate from data**
   - Pass the SEM parametric model and a dataset to the *Estimator* box.
   - Choose an SEM estimator (e.g., ML) and compute parameter estimates and fit indices.
   - The results appear in an **Instantiated Model**.

4. **Use with Simulation or Compare**
   - Use the fitted SEM as a data-generating model in the *Simulation* box.
   - Use *Compare* and *Model Fit* to evaluate how well the SEM describes observed data versus
     alternative models.

## Tips and caveats

- Check **model identification**; non-identified models can give unstable or meaningless estimates.
- Make sure the graph structure used to create the SEM matches your theoretical assumptions
  (e.g., no cycles in a standard SEM).
- For mixed or strongly nonlinear relationships, consider **Hybrid** or **Generalized** models.