Parallel Analysis

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Psychometrics (legacy hub)

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Determines how many factors to retain in EFA by comparing the eigenvalues of the observed correlation matrix to the eigenvalues of correlation matrices from RANDOM data with the same n × k shape.

Retain factors whose observed eigenvalue exceeds the 95th-percentile (or mean) random eigenvalue. Reports the recommended number of factors, scree plot with random-data overlay, and per-factor observed-vs-random eigenvalue table. Widely regarded as the most accurate factor-retention method (Zwick & Velicer 1986); supersedes Kaiser's eigenvalue-> 1 rule and visual scree-plot inspection.

Worked example

How many factors should we retain?

Horn's parallel analysis compares observed eigenvalues to those from random data — retain factors whose eigenvalue beats the random benchmark.

Result

Three observed eigenvalues exceeded the 95th-percentile random values, supporting a three-factor solution.

How you'd report it (APA)

Parallel analysis supported retaining three factors.

When to use it

  • Pre-EFA factor-count determination (default use)
    30-item personality questionnaire on n=500.
  • Comparison with Kaiser / MAP / scree as a robustness check
    Item analysis on n=300, k=20.

When NOT to — use instead

  • Single-factor scale (no question of how many factors)
    Parallel analysis is for determining factor count when unknown. → Exploratory FA
  • Confirmatory factor analysis with pre-specified factor count
    PA is exploratory. → Confirmatory FA
  • Sample too small (n < 5 × k)
    PA needs adequate sample for both observed and random eigenvalue stability. → KMO + Bartlett
  • Reliability assessment (not factor count)
    PA is about retaining factors. → Cronbach's Alpha

Assumptions (and what to do if they fail)

Items measured on a continuous / Likert scalemedium

Check: See the assumption diagnostics in the workspace.

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

Correlation matrix is positive-definitemedium

Check: See the assumption diagnostics in the workspace.

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

Random-data simulations use the same n and p as the observed datamedium

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 response setsmedium

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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