Detail: N-tad Explorer

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

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.