Polychoric Matrix

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Psychometrics (legacy hub)

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Estimates correlations between ORDINAL items (Likert scales, binary items) under the assumption of an underlying continuous bivariate normal distribution.

Pearson correlations on ordinal items are ATTENUATED — the discrete nature of the response truncates true correlations toward zero. Polychoric (for ordinal × ordinal) and tetrachoric (for binary × binary) correlations recover the underlying continuous correlation. Reports the polychoric matrix, convergence diagnostics, and matrix positivity (positive-definite required for downstream EFA / CFA).

Worked example

How do we correlate ordinal (Likert) items properly?

Polychoric correlations estimate the latent continuous correlation behind ordinal items — more appropriate than Pearson for Likert data before factoring.

Result

Polychoric correlations among items ranged .35–.68, higher than the Pearson values, yielding a cleaner factor solution.

How you'd report it (APA)

Polychoric correlations were used for the ordinal items (range .35–.68) prior to factor analysis.

When to use it

  • EFA / CFA input prep for ordinal Likert items
    30-item depression scale with 4-point Likert responses.
  • Tetrachoric matrix for binary-item analysis
    60-item multiple-choice math test (0/1 scoring) on n=400.

When NOT to — use instead

  • Continuous (interval/ratio) items
    Pearson is appropriate for continuous data — polychoric is unnecessary and adds estimator noise. → KMO + Bartlett
  • Likert with 7+ response options
    5+ point Likert can be treated as approximately continuous; Pearson is acceptable. → KMO + Bartlett
  • Single item or pair (not a matrix)
    Polychoric MATRIX needs ≥ 3 items. → AVE + CR + HTMT
  • Sample too small (n < 200)
    Polychoric estimates are noisy on small n. → KMO + Bartlett

Assumptions (and what to do if they fail)

Observed responses arise from latent continuous variablesmedium

Check: See the assumption diagnostics in the workspace.

If violated: The workspace flags this and suggests a robust or nonparametric alternative.

Items use the same underlying scale (or an ordered Likert)medium

Check: See the assumption diagnostics in the workspace.

If violated: The workspace flags this and suggests a robust or nonparametric alternative.

Listwise exclusion of incomplete pairsmedium

Check: See the assumption diagnostics in the workspace.

If violated: The workspace flags this and suggests a robust or nonparametric alternative.

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