Profile Analysis — Diagnostic Profile Comparisons

Dr. R. Düsing · Osnabrück University
Running Example
Person APatient after neuropsychological assessment (stanine profile, 7 subtests) Person BNorm sample (or profile from prior assessment / retest) Attributese.g. working memory, processing speed, verbal comprehension … ScaleStanine (M=5, SD=2) — or switch to IQ, T, z, scaled scores
How similar are two test-score profiles — and where does a difference come from: overall level, the spread of the subtest values, or the shape of the profile? Different metrics answer these questions differently.
① Profile Plot
② Profile Statistics
③ Profile Comparison
Concepts
Elevation, Scatter, Shape
Every profile decomposes into three independent components (Cronbach & Gleser 1953): elevation = mean level, scatter S² = variability of the subtest values around one's own level, shape = pattern of peaks/troughs independent of level and scatter. A profile difference can stem from just one of these components — the metrics below weight them differently.
D² — the Total Distance, Decomposed
The squared Euclidean distance measures the overall difference and decomposes additively into the three components: D² = k·ΔM̄² + k·ΔS² + 2k·S₁S₂(1−r) (level + scatter + shape portion). This shows not just that two profiles differ, but in what way. D² = 0 → identical profiles.
Why Simple Correlation Is Deceptive
The ordinary Pearson correlation across the subtests measures only shape: two profiles with an identical pattern but a constant level gap get r ≈ 1 — even though the person is consistently weaker throughout. In clinical diagnostics this is often misleading; that's why special profile metrics exist that also account for level.
Cattell & McCrae Indices
rₚ (Cattell 1949) is a distance-based profile correlation: rₚ = (2Eₖ − d²z)/(2Eₖ + d²z) — unlike simple r, it responds to level differences (drops when A is generally below norm), but depends on k and scatter. Iₚₐ / rₚₐ (McCrae 1993) are normed (≈ 0 at chance similarity) and allow high shape similarity even with a level difference present.
ICC_de (Griffin & Gonzalez)
Pearson correlation on a double-entered dataset (column 1 = [P₁;P₂], column 2 = [P₂;P₁]). This makes ICC_de the most sensitive to level differences — suitable when the absolute level fully matters, e.g. in retest or progress comparisons after therapy. Conceptually the intraclass correlation at the profile level.
Which Metric, When?
D²: a global, decomposable distance measure — shows the source of the difference. rₚ (Cattell): established in clinical diagnostics, but k-/scatter-dependent. Iₚₐ/rₚₐ (McCrae): clear zero reference and norming. ICC_de: when level differences should fully count. Closely related to the intraclass correlation and reliability logic. → ICC Lab · → CTT Foundations
Profile Analysis — Help
Example

Person A is a patient after neuropsychological assessment, Person B is the norm-sample profile (or a pretest profile). Both show stanine scores on 7 subtests (e.g. working memory, processing speed, verbal comprehension). Question: How similar are the profiles — and where does a difference come from?

Elevation, Scatter, Shape

Every profile can be decomposed into three independent components (Cronbach & Gleser, 1953):

Elevation M̄ = mean level = (1/k)·Σxᵢ. A difference in elevation → Person A scores overall higher or lower than B.

Scatter S² = variability of the subtest values around one's own level = (1/k)·Σ(xᵢ−M̄)². High scatter = uneven profile.

Shape / scatter pattern = pattern of peaks/troughs, independent of level and scatter. Two profiles can have the same elevation and scatter but completely different shapes.

In the example: if A is uniformly 1 stanine point below B (a pure level difference), shape metrics should still show high similarity.

D² after Cronbach & Gleser (1953)

The squared Euclidean distance D² measures the overall difference between two profiles and decomposes additively into the three components:

D² = k·ΔM̄² + k·ΔS² + 2k·S₁·S₂·(1−r)

In the example: D²=0 → identical profiles. D²=7 (with k=7 attributes) corresponds to an average difference of 1 scale point per attribute. Each component's contribution shows what drives the main difference.

rₚ after Cattell (1949)

Profile correlation on z-standardized values. Eₖ = expected value of the χ² distribution with k degrees of freedom (median via Wilson-Hilferty). Level differences enter via d²_z.

rₚ = (2Eₖ − d²_z) / (2Eₖ + d²_z)

rₚ is sensitive to level differences (unlike simple correlation). In the example: if A is generally below norm (M̄_A<M̄_B), rₚ drops even if the profile shape is identical.

Iₚₐ / rₚₐ after McCrae (1993)

Iₚₐ measures profile similarity on z-standardized values, normed so that Iₚₐ≈0 at chance similarity. Mᵢ = (zA,i+zB,i)/2 is the mean z score per attribute.

Iₚₐ = (k + 2ΣMᵢ² − d²_z) / √(10k)

rₚₐ transforms Iₚₐ into a correlational index ∈ (−1, 1]:

rₚₐ = Iₚₐ / √((k−2) + Iₚₐ²)

In the example: rₚₐ ≈ 0.90 = high shape similarity is possible even with a level difference present.

ICC_de after Griffin & Gonzalez (1995)

Pearson correlation on a double-entered dataset: column 1 = [P₁; P₂], column 2 = [P₂; P₁]. Sensitive to level differences because the symmetrization increases the variance when the profiles sit at different levels.

ICC_de = (MSB − MSW) / (MSB + MSW)

In the example: pure shape similarity (same scatter, same elevation) → ICC_de = high Pearson r on the doubled dataset. A level difference pulls ICC_de down more than rₚₐ.

Which Metric, When?

D² works well as a global distance measure — easy to interpret, decomposes into components. rₚ (Cattell) is established in clinical diagnostics, but depends on k and scatter. Iₚₐ / rₚₐ (McCrae) have a clearer zero reference and norming. ICC_de (Griffin) is especially suitable when level differences should be fully accounted for — e.g. in a retest comparison after therapy.

References

Cronbach, L. J. & Gleser, G. C. (1953). Assessing similarity between profiles. Psychological Bulletin, 50(6), 456–473.
Cattell, R. B. (1949). rₚ and other coefficients of pattern similarity. Psychometrika, 14(4), 279–298.
McCrae, R. R. (1993). Moderated analyses of longitudinal personality stability. Journal of Personality and Social Psychology, 65(3), 577–585.
Griffin, D. & Gonzalez, R. (1995). Correlational analysis of dyad-level data in the exchangeable case. Psychological Bulletin, 118(3), 430–439.