## Detail: N-tad Explorer

The **N-tad Explorer** is a data-analysis tool for finding **rank-deficient covariance blocks** in a dataset.

```{figure} ../../_static/images/tetrad-interface/box-by-box/ntad-explorer.png
:name: tetrad-ntad-explorer-screenshot
:alt: N-tad Explorer dialog in the Data box.

N-tad Explorer dialog for listing rank-deficient tetrads, sextads, and higher-order N-tads.
```

Given a dataset and a subset of variables, it:

- enumerates disjoint pairs of variable blocks (A, B) of size m by m (for example, m = 2 for tetrads, m = 3 for sextads),
- tests whether the cross-covariance block Sigma_AB has **rank smaller than m**, using
  regularized canonical correlation analysis (rCCA) and Wilks’ Lambda,
- lists all rank-deficient blocks found, together with an estimated rank and a p-value.

This is useful for exploring **latent variable structure** (for example, Multiple Indicator Models) and for
identifying interesting tetrads, sextads, and higher-order N-tads suggested by the data.

### Basic workflow

1. **Create an N-tad Explorer box**

   - Start from a **Data** box containing the dataset you want to analyze.
   - Add an **N-tad Explorer** box connected to that Data box.
   - Open the N-tad Explorer box to bring up the dialog.

2. **Select variables**

   The left side of the dialog shows two lists:

   - **Available variables** – all variables in the input dataset,
   - **Selected variables** – the variables that will be used to form N-tads.

   Use the arrow buttons between the lists:

   - **>** moves the selected variables from *Available* to *Selected*,
   - **<** moves them back,
   - **>>** moves *all* variables to *Selected*,
   - **<<** clears the selection.

   Only the variables in the **Selected variables** list are used when forming blocks.

3. **Set N-tad parameters**

   In the **N-tad parameters** panel, specify:

   - **Block size m**  
     The size of each block A and B, so each N-tad uses 2m variables:
       - m = 2 → tetrads (2 by 2 blocks),
       - m = 3 → sextads (3 by 3 blocks),
       - m = 4 → octets (4 by 4 blocks), etc.

   - **Max results**  
     The maximum number of rank-deficient blocks to list.  
     This limits the amount of output when many N-tads satisfy the test.

   - **Alpha**  
     The significance level for the Wilks rank test used to estimate the rank.  
     Typical values are 0.05 or 0.01.

4. **Run the search**

   Click **Find N-tads** to:

   - construct all disjoint pairs (A, B) of size m from the selected variables,
   - estimate the rank of Sigma_AB using Wilks Lambda,
   - compute a p-value for the hypothesis that the rank is at most m - 1,
   - keep only those blocks that are **rank-deficient** (estimated rank less than m).

   The results appear in the table on the right-hand side.

5. **Inspect and sort results**

   Each row of the table corresponds to one rank-deficient block and contains:

   - **Block A** – comma-separated list of variables in block A,
   - **Block B** – comma-separated list of variables in block B,
   - **Block size** – the value of m used,
   - **Rank** – the estimated rank of Sigma_AB,
   - **p-value** – p-value for the null hypothesis that rank(Sigma_AB) is less than or equal to m - 1.

   You can click on any **column header** to sort the table by that column:

   - first click: sort ascending,
   - second click: sort descending,
   - third click: return to the original order.

### Interpretation

- A row with **Block size = m** and **Rank = r < m** indicates that the corresponding
  m by m cross-covariance block Sigma_AB is estimated to have rank r
  rather than full rank m.

- A **small p-value** suggests strong evidence against the null hypothesis
  that the rank is at most m - 1; conversely, a **larger p-value** indicates that the
  data are compatible with a lower-rank structure at the specified alpha level.

- In practice, for Multiple Indicator Models and related latent-variable structures,
  particular rank patterns (for example, tetrads with rank 1 instead of 2) can reflect
  underlying constraints implied by the latent variables.

The N-tad Explorer does **not** modify the original dataset; it only summarizes which
blocks of variables show evidence of rank deficiency according to the chosen test.

### Tips and notes

- **Choice of variables.**  
  Restricting the Selected variables to plausible indicator sets can greatly reduce
  runtime and focus the results on blocks of interest.

- **Combinatorial growth.**  
  The number of candidate blocks grows quickly with:
  - the number of selected variables, and
  - the block size m.  
  Use **Max results** and a moderate block size to keep the output manageable.

- **Reproducibility.**  
  The list of results is stored with the N-tad Explorer box. If you save and reopen
  the project, the last computed results remain available in the table until you
  run the search again with different settings.

- **Exporting results.**  
  If desired, you can copy the rows from the table into an external tool (such as a
  spreadsheet or script) for further analysis, or cross-reference the listed blocks
  with latent variable models you are fitting elsewhere.

### Using N-tad Explorer with SEMs

To list tetrads, sextads, and octads implied by a fitted SEM:

1. Fit a SEM from a covariance matrix using a **SEM Est** or **SEM IM** box.
2. Create a **Data** box for the **implied covariance (measured variables)** from the SEM.
3. Attach an **N-tad Explorer** box to that implied covariance Data box.
4. Run N-tad Explorer with your chosen block size (m = 2, 3, 4, …).

In this setup, the rank-deficient blocks reported by N-tad Explorer correspond to N-tads
implied by the estimated SEM, rather than N-tads in the raw data.