The Aggregated Latent Profile Index: Measuring Person Profile Differentiation Within a Bootstrap-Validated Latent Profile Space

Abstract

Multivariate assessment profiles contain two conceptually distinct sources of variation: overall level and profile shape. Existing approaches recover some aspects of this structure, but none jointly establishes the replicability of latent pattern dimensions and provides a principled, population-referenced summary of person profile differentiation independent of overall level. We introduce the Aggregated Latent Profile Index (ALPI), a variance-weighted Euclidean distance that quantifies the degree to which an individual’s profile departs from a flat population reference within a bootstrap-validated latent profile space. ALPI is derived from the Aggregated Latent Profile Space (ALPS), a framework that combines parallel analysis with bootstrap stability diagnostics – principal angles between subspaces and Tucker’s congruence coefficients – to identify replicable latent pattern dimensions, then assembles the retained dimensions into a K-dimensional latent space through singular value decomposition, into which individuals and variables are jointly projected. In a simulation benchmark, ALPS recovered the known four-factor population structure as three replicable ipsatized pattern dimensions, confirming that the pipeline performs as intended when the true structure is known. In an application to normative WAIS–IV data (n = 900), ALPS identified a stable three-dimensional pattern space, and ALPI distinguished individuals with identical Full-Scale IQ – ranging from the 7th to the 85th percentile – on the basis of their ipsatized subtest configurations. ALPI provides assessment researchers and clinicians with a single, measurement-principled index of person profile differentiation that is grounded in replicable latent structure and independent of overall score level.

Keywords

Aggregated Latent Profile Index (ALPI)bootstrap stability person profile measurement

Introduction

Multivariate assessment data – such as cognitive subtest scores, behavioral ratings, and symptom scales – contain two analytically distinct sources of variation: overall level, reflecting a person’s average standing across variables, and configurational shape, reflecting the relative pattern of elevations and depressions across variables. Profile analysis is primarily concerned with the latter. In psychological and educational measurement, however, profile shape remains difficult to summarize in a way that is psychometrically principled, substantively interpretable, and demonstrably stable across samples.

Existing approaches address parts of this problem but not its full measurement implications. Principal component analysis (PCA; Jolliffe & Cadima, 2016) and factor analysis (FA; Brown, 2015) recover dimensions of shared variation, but they do not separate level from shape, do not formally evaluate the replicability of recovered pattern dimensions across samples, and do not yield scalar indices of individual profile differentiation. Latent profile analysis (LPA; Nylund et al., 2007) partitions individuals into discrete latent classes based on profile similarity, but its categorical representation does not provide a continuous profile space, dimension-specific person coordinates, or a quantitative summary of the degree of pattern differentiation within and across individuals. Multivariate regression (Rencher & Christensen, 2012) treats profiles as predictors or outcomes rather than as measurement objects in their own right and therefore does not recover latent pattern dimensions or quantify profile extremeness independently of overall level.

The Biplot Paradigm and Its Limits

Gabriel’s (1971) biplot introduced a useful geometric representation of multivariate profiles by jointly displaying individuals and variables in a low-dimensional space derived from principal components. Because distances, angles, and cosines can be interpreted directly, biplots offer an intuitively appealing framework for profile interpretation. From a measurement perspective, however, their usefulness is limited by their dependence on two-dimensional display. When profile structure spans more than two substantively meaningful dimensions – as is common in psychological and educational assessment – the biplot framework does not provide a principled way to integrate higher-order dimensions into a unified measurement space, nor does it provide an aggregated person-oriented summary across those validated dimensions.

Factor-Analytic Profile Analysis and Its Scope

The basic factor-analytic profile analysis (FAPA) paradigm was introduced by Kim (2013, 2024) as a framework for identifying latent pattern dimensions underlying multivariate profiles. The original formulation (Kim, 2013) used maximum-likelihood FA to extract core pattern components from ipsatized profiles (i.e., profiles row-centered by subtracting each person’s mean score across variables, thereby removing overall level and retaining only configurational shape; see equation (1)), with parallel analysis (PA) determining the number of statistically meaningful dimensions. Subsequent development (Kim, 2024) broadened this framework by incorporating PCA and singular value decomposition (SVD), while retaining ipsatization as the device for separating level from shape and PA as the criterion for dimensionality. In this sense, FAPA established an important psychometric foundation for profile measurement by modeling systematic shape variation after the removal of overall level effects. An open-source implementation is available in the FAPA R package (Kim, 2026b).

However, existing FAPA applications leave two important measurement problems unresolved. First, PA provides an upper bound on dimensionality relative to sampling noise, but it does not establish whether the resulting latent pattern dimensions are replicable across samples. For person-oriented interpretation, such replicability is essential. Second, FAPA has not yet provided a unified latent space in which individuals can be projected and compared across multiple validated dimensions, nor has it introduced a scalar index summarizing the overall degree of person profile differentiation relative to a common reference geometry. These limitations motivate the present study.

Person-Oriented and Predictive Alternatives

LPA and multivariate regression remain widely used with multivariate profile data, but they address inferential goals that differ from those of measurement-oriented profile analysis. LPA emphasizes classification into latent subgroups rather than continuous representation of individual differences in profile shape. Although useful for identifying qualitatively distinct subpopulations, it does not provide a continuous coordinate system for representing persons, dimension-specific indices of alignment with latent profile dimensions, or a scalar measure of overall profile differentiation. Multivariate regression, in turn, is designed for prediction or association rather than structural representation of profiles. It does not treat multivariate profiles as measurement objects, does not identify latent dimensions of shape, and does not quantify person profile pattern extremeness independently of overall level. Neither approach is designed to support continuous person-oriented measurement within a validated latent profile space.

The Aggregated Latent Profile Index and Its Enabling Framework

To address this gap, we introduce the Aggregated Latent Profile Index (ALPI), a variance-weighted Euclidean distance that quantifies the degree of person profile differentiation relative to a fixed, population-referenced geometry and independently of overall score level. ALPI is derived from the Aggregated Latent Profile Space (ALPS), a framework that provides the validated latent basis ALPI requires. ALPS proceeds in three stages. First, PA is combined with bootstrap-based subspace stability diagnostics – principal angles between subspaces and Tucker’s congruence coefficients – to identify latent pattern dimensions that are both statistically supported and reproducible across samples. Second, the retained dimensions are assembled into a K-dimensional latent profile space via SVD, within which individuals and variables are jointly represented through orthogonal projection. Third, each individual’s K-dimensional coordinate vector is aggregated into the scalar ALPI. Although ALPI emerges as the final stage of the ALPS pipeline, it is conceptually distinct from the space itself: ALPS is the validated latent basis, and ALPI is the scalar measurement index derived from that basis. We therefore treat them as related but separable contributions – ALPS as the enabling framework and ALPI as the person profile differentiation index it makes possible. The technical distinction is formalized in Step 3 of the method. Together, ALPS and ALPI yield a unified framework for multivariate profile measurement that is statistically defensible, geometrically interpretable, and explicitly person-oriented.

To evaluate whether ALPI performs as intended when the latent structure is known, we first examine its behavior in a controlled simulation setting. We then apply ALPS and ALPI to normative WAIS–IV data to assess whether the validated latent profile space supports interpretable person profile differentiation among individuals with identical Full-Scale IQ.

Contributions of the Present Paper

The present study makes four contributions. First, it introduces ALPI as a new person-oriented measurement index that quantifies the degree of profile differentiation within a bootstrap-validated latent profile space, independent of an overall score level. Second, it introduces ALPS as the enabling framework for ALPI, providing a formal procedure for evaluating dimensional replicability through bootstrap principal angles and Tucker’s congruence coefficients and assembling retained dimensions into a validated latent space that supports both dimension-specific person coordinates and ALPI computation. Third, it evaluates the recovery of known latent profile structure in a controlled simulation benchmark. Fourth, it provides a fully reproducible open-source implementation and an empirical demonstration using normative WAIS–IV data.

Method

Overview of Study Design

The present study included two components designed to introduce and evaluate ALPI as a person profile differentiation index. First, a simulation study evaluated whether the pipeline that produces ALPI could recover replicable latent profile dimensions under controlled conditions where the population structure was known. Second, an empirical application to the WAIS–IV normative dataset examined whether ALPI yields interpretable measurement of person profile differentiation in real assessment data. In both components, ALPS serves as the validated latent basis from which ALPI is derived.

Simulation Study Design

Before computing ALPI from the WAIS–IV normative data, we conducted a simulation study to evaluate whether the proposed pipeline could recover known latent profile structure under controlled conditions. A dataset with n = 900 cases and p = 18 variables was generated from a multivariate normal population whose marginal distributions approximated WAIS–IV scaled-score distributions (M = 10, SD = 3). The population correlation matrix was specified to reflect the published four-factor WAIS–IV structure (Wechsler, 2008), with within-factor correlations of .55 and between-factor correlations of .30, corresponding to Perceptual Reasoning, Processing Speed, Verbal Comprehension, and Working Memory.

The simulated data were analyzed using the same pipeline applied to the empirical data. Profiles were ipsatized by row-centering, a PA was conducted using 2,000 permutations with retention based on the 95th-percentile null eigenvalue, and bootstrap stability was evaluated using 2,000 resamples. In each resample, principal components were re-estimated, aligned to the full-sample solution by orthogonal Procrustes rotation, and evaluated using principal angles between subspaces and Tucker’s congruence coefficients, with retention criteria defined in Step 1 below. This simulation provided a proof-of-concept benchmark for whether the ALPI pipeline recovers replicable latent profile dimensions suitable for person-oriented measurement when the population structure is known.

Empirical Data Structure and Notation

The empirical application used the WAIS–IV normative dataset (ages 16–70; n = 900; Wechsler, 2008), restricted to the 18 primary subtests spanning perceptual reasoning, processing speed, verbal comprehension, working memory, and related domains. FSIQ is a composite score derived from a weighted combination of subtest scaled scores and expressed on a standard score metric ( $M = 100, SD = 15$ ), serving as the primary index of overall cognitive ability. Let $X \in R^{n \times p}$ denote the matrix of subtest scores, where n is the number of individuals, and $p = 18$ is the number of subtests. Descriptive statistics are reported in Table 1.

Table 1.

Descriptive Statistics of Subtests and Full Scale IQ (FSIQ)

Subtest	$n$	$M$	$SD$	$Min$	$Max$
1_bd	900	10.12	3.03	3	19
2_mr	900	10.14	3.03	2	19
3_fw	900	10.26	3.03	3	19
4_pc	900	10.09	2.97	2	18
5_ss	900	10.20	3.06	2	19
6_cd	900	10.12	3.01	1	19
7_ca	900	10.05	2.99	1	19
8_vc	900	10.19	3.03	3	19
9_in	900	9.99	3.03	4	19
10_co	900	10.26	3.17	2	19
11_si	900	10.08	2.90	2	19
12_ar	900	10.08	2.97	3	19
13_ds	900	10.08	2.91	3	19
14_ln	900	10.20	2.98	1	19
15_vp	900	10.24	3.05	1	19
16_df	900	9.99	2.87	2	18
17_dsb	900	10.21	2.92	1	19
18_dq	900	10.09	2.91	1	19
FSIQ	900	100.62	14.86	63	137

Note. 1 = block design; 2 = matrix reasoning; 3 = figure weights; 4 = picture completion; 5 = symbol search; 6 = coding; 7 = cancelation; 8 = vocabulary; 9 = information; 10 = comprehension; 11 = similarities; 12 = arithmetic; 13 = digit span; 14 = letter–number sequencing; 15 = visual puzzles; 16 = digit span forward; 17 = digit span backward; 18 = digit span sequencing.

ALPI is derived from ipsatized profiles that separate the overall level from the configurational shape. Following conventions of the FAPA paradigm (Kim, 2013, 2024), this separation is achieved through ipsatization:

X = m 1^{⊤} + X^{*},

(1)

where $m$ is the $n \times 1$ vector of individual row means, 1 is a p-dimensional vector of ones, and $X^{*}$ contains the resulting row-centered (or ipsatized) profiles. ALPI is derived from $X^{*}$ , thereby ensuring that person profile differentiation is quantified independent of between-person differences in overall performance level.

Step 1: Establishing the Replicable Dimensional Basis for ALPI

ALPI requires latent dimensions that are both statistically supported and demonstrably reproducible across samples. Computing ALPI from dimensions that are not cross-sample replicable would conflate genuine profile differentiation with sampling idiosyncrasy. The following validation procedure establishes which dimensions meet this requirement by combining dimensionality assessment with explicit bootstrap stability screening.

Parallel Analysis

Let $λ_{1} \geq \dots \geq λ_{p}$ denote the eigenvalues of the covariance matrix of $X^{*}$ , corresponding to the scaled squared singular values from the SVD:

X^{*} = U Σ V^{T},

(2)

where $U (n \times p)$ contains person weights indexing individual alignment with each latent dimension, $Σ$ is a diagonal matrix of singular values, and $V (p \times p)$ contains the variable loadings (Gollob, 1968a, 1968b; Kim, 2024). PA (Horn, 1965) compares the observed eigenvalue spectrum to that from permuted random data:

K_{PA} = max {k : λ_{k} > {\tilde{λ}}_{k}},

(3)

where ${\tilde{λ}}_{k}$ denotes the 95th-percentile reference eigenvalue. PA identifies dimensions exceeding sampling noise but does not establish cross-sample replicability – a necessary but insufficient condition for the dimensions used to compute ALPI.

Principal Angles Between Subspaces

To evaluate subspace stability, the $K_{PA}$ -dimensional latent space estimated from the full sample,

S = span (v_{1}, \dots, v_{K_{PA}}),

(4)

is compared with its bootstrap counterpart $S^{(b)}$ . Principal angles

0 \leq θ_{1} \leq \dots \leq θ_{K_{PA}} \leq \frac{π}{2}

(5)

quantify concordance between subspaces, with smaller angles indicating more stable recovery of the latent profile space across resamples and larger angles indicating weaker replicability (Björck & Golub, 1973; Davis & Kahan, 1970).

Tucker’s Congruence Coefficients

Component-level stability is evaluated using Tucker’s congruence coefficient between the full-sample loading vector $v_{k}$ and its bootstrap estimate $v_{k}^{(b)}$ ,

φ_{k} = \frac{v_{k}^{⊤} v_{k}^{(b)}}{∥ v_{k} ∥ ∥ v_{k}^{(b)} ∥} .

(6)

Following Lorenzo-Seva and ten Berge (2006), values of $φ \geq . 85$ are taken as evidence of acceptable component replicability. Tucker’s coefficient serves as an index of whether the direction of a latent profile dimension is recovered consistently enough to support computation and interpretation of ALPI.

Final Dimensionality

A dimension is retained for ALPI computation if it simultaneously exceeds sampling noise under PA and meets both bootstrap stability criteria:

K = {k \leq K_{PA} : Q_{. 05} (φ_{k}) \geq . 85 and Q_{. 95} (θ_{k}) \leq 20^{°}} .

(7)

Consistent with this dual-criterion logic, the lower tail of the $φ$ distribution ( $φ_{5}$ ) and the upper tail of the $θ$ distribution ( $θ_{9} 5$ ) serve as the operative retention bounds, since these represent the worst-case bootstrap resamples for each diagnostic, respectively – ensuring that the dimensional basis for ALPI is not only statistically detectable but sufficiently replicable to support person-oriented measurement.

Step 2: Constructing the Latent Profile Space (ALPS) Required for ALPI

ALPI is a distance measure within a validated latent space. That space – the ALPS – is constructed from the SVD in equation (2). Retaining the K validated components gives the loading matrix $V_{K} = [v_{1}, \dots, v_{K}]$ , and the ALPS is defined as the subspace

L = span (v_{1}, \dots, v_{K}) .

(8)

In measurement terms, ALPS represents the validated latent basis for profile shape variation after the overall level has been removed – the space within which ALPI is subsequently defined and computed.

Domain Coordinates

Each subtest $j$ is assigned a $K$ -dimensional coordinate vector

g_{j} = V_{K}^{⊤} e_{j},

(9)

where $e_{j}$ is the canonical basis vector selecting the j-th variable. These coordinates describe how each subtest contributes to the retained latent profile dimensions and provide the basis for interpreting the substantive meaning of the dimensions along which ALPI is computed.

Individual Projection

For individual $i$ , let $x_{i}^{*} \in R^{p}$ denote the ipsatized profile. Its coordinates in ALPS are:

z_{i} = V_{K}^{⊤} x_{i}^{*} = (z_{i 1}, z_{i 2}, \dots, z_{iK})^{⊤} .

(10)

This projection places each individual within the validated latent profile space. The resulting coordinate vector z_i is the direct input to ALPI computation and permits direct comparison of profile shape among persons who may be indistinguishable on overall score level, including those with identical FSIQ.

The ALPS-reconstructed profile for individual $i$ is obtained by projecting the individual’s coordinate vector $z_{i}$ back into the original $p$ -dimensional variable space: ${\hat{x}}_{i}^{*} = V_{K} z_{i} = V_{K} V K^{T} x_{i}^{*}$ . This back-projection yields a $p$ -dimensional vector on the same ipsatized scale as the original row-centered profiles, representing the best low-dimensional approximation of $x_{i}^{*}$ within the K-dimensional ALPS subspace. The discrepancy $x_{i}^{*} - {\hat{x}}_{i}^{*}$ constitutes the residual profile variation not captured by the retained ALPS dimensions. Figures 1 and 2 display both $x_{i}^{*}$ (observed ipsatized profile) and ${\hat{x}}_{i}^{*}$ (ALPS-reconstructed profile) on this common ipsatized scale.

Figure 1.

ALPS-reconstructed and observed profiles for individual 206 (high ALPI).

Figure 2.

ALPS-reconstructed and observed profiles for individual 949 (low ALPI).

Population-Referenced Aggregated Latent Profile

The population-referenced aggregated latent profile is defined as:

g = V_{K} w = \sum_{k = 1}^{K} w_{k} v_{k},

(11)

where $w = (w_{1}, \dots, w_{K})^{⊤}$ are variance-based weights (defined below). Because $g$ is estimated once from the full sample and does not depend on a given individual, it provides a fixed population-referenced profile geometry that serves as the reference anchor against which ALPI quantifies person profile differentiation.

Step 3: Computing ALPI

ALPI ( $α_{i}$ ) is the central measurement quantity of the present framework. To distinguish the space from the index it produces, we refer to the K-dimensional latent profile space as ALPS and the scalar person profile differentiation index derived from it as ALPI ( $α_{i}$ ). ALPI for individual $i$ is defined as:

α_{i} = {(\sum_{k = 1}^{K} w_{k} z_{ik}^{2})}^{1 / 2},

(12)

where $z_{ik}$ is the standardized score of person $i$ on ALPS dimension $k$ , and $w_{k}$ is the variance-based weight assigned to that dimension. Geometrically, $α_{i}$ is a variance-weighted Euclidean distance from the origin in the $K$ -dimensional ALPS space, where the origin represents a flat ipsatized profile. From a measurement perspective, $α_{i}$ indexes the overall degree of profile differentiation across the replicability-validated latent dimensions, independent of FSIQ. Larger values indicate greater profile differentiation; high values may reflect differentiated strengths, focal weaknesses, or combinations of both.

Variance-Based Weights

Let $λ_{k} = s_{k}^{2}$ denote the squared singular value for dimension $k$ . Variance-based weights were defined as

w_{k} = \frac{λ_{k}}{\sum_{j = 1}^{K} λ_{j}},

(13)

such that $w_{k} \geq 0$ and $\sum_{k = 1}^{K} w_{k} = 1$ . These weights allow dimensions contributing more validated pattern variance to contribute proportionally more to ALPI, so that the index reflects the dimensional structure of the latent profile space rather than treating all dimensions equally.

Standardized Index

Because $α_{i}$ is unbounded and lacks an intrinsic reference scale, two standardized forms are reported: a robust MAD-based form and a conventional z-standardization:

{\tilde{α}}_{i} = \frac{α_{i} - median (α)}{MAD (α)} and z (α_{i}) = \frac{α_{i} - α}{s_{α}},

(14)

where $α = (α_{1}, \dots, α_{N})$ , $s_{α}$ is the sample standard deviation; and $MAD (α)$ is scaled by 1.4826 for consistency with the population standard deviation under normality. Specifically, $MAD (α) = 1.4826 \times median (| α_{i} - median (α) |)$ , where the inner median measures the typical absolute deviation of each individual’s ALPI from the sample median, and the scaling constant renders this estimate asymptotically equivalent to $σ$ (the population standard deviation) for Gaussian data, permitting direct comparison with z-standardized values. The MAD-based form was preferred because it is resistant to skewness and outliers and does not require distributional assumptions. Standardized ALPI values are interpreted relative to the normative distribution: values within 0.5 SD of the center indicate weak profile differentiation; values 0.5–1.0 SD above indicate moderate differentiation; values 1.0–1.5 SD above indicate strong differentiation; and values ≥ 2 SD above indicate extreme differentiation. These descriptors serve as interpretive guides for profile measurement rather than as diagnostic cut points.

Reproducibility

ALPI as introduced here is implemented as the continuous pipeline (calsi_workflow()) of the broader open-source alsi R package (Kim, 2026a), which extends the ALPI framework to binary and ordinal data through Multiple Correspondence Analysis (MCA) and Homogeneity Analysis by Means of Alternating Least Squares (HOMALS) pipelines, respectively. The package is available on CRAN, and development materials are maintained on GitHub. The calsi_workflow() function provides an integrated pipeline encompassing ipsatization, PA, bootstrap dual-criterion stability screening, SVD-based latent space construction, individual projection, and ALPI computation. Annotated R scripts reproducing all analyses reported here are provided in the Supplemental Material.

Results

Simulation Study

Prior to the empirical application, we evaluated whether the pipeline that produces ALPI recovers a known latent structure – specifically, whether it identifies replicable latent dimensions suitable for computing a valid person profile differentiation index. A dataset of $n = 900$ cases and $p = 18$ variables was generated from a multivariate normal population with marginal distributions matching WAIS–IV scaled scores ( $M = 10$ , $SD = 3$ ) and a correlation structure consistent with the published four-factor WAIS–IV model (Wechsler, 2008): within-factor correlations of .55 and between-factor correlations of .30, yielding four population dimensions corresponding to Perceptual Reasoning, Processing Speed, Verbal Comprehension, and Working Memory. PA (2,000 permutations) identified $K_{PA} = 3$ components exceeding the 95th-percentile null eigenvalue threshold (Table 2, Panel A). Bootstrap stability diagnostics (2,000 resamples) confirmed that all three dimensions met both retention criteria simultaneously ( $φ$ mean $\geq . 99$ ; median principal angle $\leq 7 . 7^{°}$ ), jointly accounting for 41.4% of total ipsatized pattern variance (Table 3, Panel A). These findings indicate that the ALPI pipeline recovers replicable latent geometry when the population structure is known, establishing the dimensional foundation on which ALPI is subsequently computed.

Table 2.

Parallel Analysis: Simulation (Panel A) and Ipsatized WAIS–IV Covariance Matrix (Panel B), 2,000 Permutations.

Panel A – Simulation (n = 900, four-factor population structure).
Dim	Eigenvalue	% variance	PA 95th percentile	Retain?
1	16.65	16.8	7.33	Yes
2	12.94	13.0	6.96	Yes
3	11.50	11.6	6.70	Yes
Panel B – WAIS–IV normative data (n = 900).
Dim	Eigenvalue	% variance	PA 95th percentile	Retain?
1	15.61	17.7	8.01	Yes
2	12.87	14.6	6.72	Yes
3	7.92	9.0	6.34	Yes
4	6.46	7.3	6.07	Yes
5	5.90	6.7	5.80	Yes

Table 3.

Bootstrap Stability of Candidate Dimensions: Simulation (Panel A) and WAIS–IV Normative Data (Panel B), 2,000 Resamples.

Panel A – Simulation (n = 900, four-factor population structure).
Dim	Eigenvalue	% var	φ mean	φ₅	φ₉₅	Angle median	Angle₉₅
1	16.65	16.8	.996	.994	.998	3.8°	5.1°
2	12.94	13.0	.994	.990	.997	5.5°	7.1°
3	11.50	11.6	.993	.988	.997	7.7°	9.7°
Panel B – WAIS–IV normative data (n = 900).
Dim	Eigenvalue	% var	φ mean	φ₅	φ₉₅	Angle median	Angle₉₅
1	15.61	17.7	.977	.925	.997	2.4°	3.5°
2	12.87	14.	.973	.921	.995	3.9°	5.5°
3	7.92	9.0%	.960	.908	.988	6.7°	9.5°
4	6.46	7.3	.849	.421	.976	10.8°	15.3°
5	5.90	6.7	.827	.365	.970	18.8°	35.4°

Empirical Application: Ipsatized WAIS–IV Profiles

Parallel Analysis

The simulation study provided a benchmark showing that ipsatization removes the general-level dimension and yields three replicable pattern dimensions from the four-factor population structure (Table 2, Panel A). PA of the ipsatized WAIS–IV data identified an upper bound of five candidate components, with each of the first five observed eigenvalues exceeding the corresponding 95th-percentile null reference value (Table 2, Panel B). These results indicate that the empirical data contain more candidate structure than is expected by chance alone, although additional stability filtering is required to determine which dimensions are reproducible.

Replicability of Candidate Dimensions

To determine which PA-supported dimensions were reproducible, 2,000 bootstrap resamples were generated, principal components were recomputed in each resample, the resulting loading matrices were Procrustes-aligned to the full-sample solution, and stability was evaluated using principal angles between subspaces and Tucker’s congruence coefficients (Table 3).

Two features of Table 3 deserve emphasis. First, the simulation panel contains only three dimensions because PA itself returned $K_{PA} = 3$ : ipsatization removes the general-level dimension, leaving three replicable pattern dimensions from the four-factor population structure. In contrast, the empirical WAIS–IV data initially yielded five candidate dimensions, which were subsequently reduced to three by the bootstrap stability filter. This contrast shows that the stability criterion functions as an independent and consequential screening step in the empirical application rather than merely reproducing the parallel-analysis result. Second, stability was stronger in the simulation than in the real data, as expected given the clean block-structured covariance used in data generation. Nonetheless, in the empirical data, Dimensions 1–3 were consistently recovered across resamples, with median Tucker congruence coefficients exceeding .96 and median principal angles no larger than $6 . 7^{°}$ (95th percentile $\leq 9 . 5^{°}$ ). By contrast, Dimensions 4 and 5 showed substantially lower congruence and markedly larger principal angles, indicating unstable and nonreplicable structure. Accordingly, $K = 3$ validated dimensions were retained as the latent basis for ALPI computation, jointly accounting for 41.3% of total ipsatized WAIS–IV pattern variance.

ALPI-Based Profile Measurement of WAIS–IV Data

Interpretation of Retained ALPS Dimensions

Inspection of subtest coordinates within the three-dimensional ALPS space yielded clear and interpretable latent pattern axes. ALPS-1 contrasted Processing Speed subtests (Coding, Symbol Search) and Working Memory measures (Digit Span) with Verbal Comprehension subtests (Vocabulary, Information, Similarities), broadly opposing the Processing Speed Index (PSI) and Working Memory Index (WMI) factors of the published WAIS–IV structure (Wechsler, 2008) against the Verbal Comprehension Index (VCI) factor. ALPS-2 opposed Perceptual Reasoning subtests (Block Design, Visual Puzzles, Picture Completion) – the core PRI factor – to sequencing-based Working Memory tasks (Digit Span Backward, Digit Span Sequencing), reflecting a contrast between visuospatial reasoning and sequential processing demands. ALPS-3 contrasted Fluid Reasoning subtests (Matrix Reasoning, Figure Weights) with Verbal Knowledge measures (Comprehension, Information), capturing a distinction within the broad reasoning domain between fluid problem-solving and crystallized verbal knowledge. These three dimensions therefore correspond closely to the established four-factor WAIS–IV structure, with ipsatization collapsing the general factor and the bootstrap stability filter further consolidating the five PA-supported dimensions into three cross-sample-replicable pattern axes. From a measurement perspective, this alignment confirms that ALPS recovers substantively meaningful and interpretable cognitive profile dimensions rather than statistical artifacts.

Individual Projection, Coordinates, and Profile Differentiation

Individual coordinates in ALPS were obtained by projection onto $V_{K}$ , as described in the Method. As expected, individuals matched on FSIQ were often separated in ALPS space, reflecting distinct cognitive configurations beyond overall ability level. To illustrate this point, four individuals (IDs 60, 206, 667, and 949) with identical FSIQ = 100 were selected; their ALPS coordinates and ALPI values are shown in Table 4. The purpose of this illustration is to show that ALPI provides a principled, population-referenced summary of profile differentiation that is not available from FSIQ or from direct inspection of raw profiles: individuals who are indistinguishable on overall ability can be clearly differentiated on the degree and structure of their cognitive profile patterning.

Table 4.

ALPS Coordinates ( $z_{i}$ ) and ALPI ( $α_{i}$ ) for Four Individuals With FSIQ = 100.

ID	$z_{i 1}$	$z_{i 2}$	$z_{i 3}$	$α_{i}$
60	1.59	6.49	–1.93	4.10
206	4.51	6.25	–2.46	4.88
667	4.91	−2.33	–1.53	3.57
949	0.91	1.60	1.39	1.30

Each individual’s coordinate vector, $z_{i} = (z_{i 1}, z_{i 2}, z_{i 3})^{⊤}$ , indexes alignment along the three retained latent pattern dimensions, whereas the ALPI, $α_{i}$ , summarizes the overall degree of profile differentiation across those dimensions. The absolute values $∣ z_{ik} ∣$ reflect the strength of contribution of Dimension $k$ , the sign indicates the direction of alignment, and the squared terms $z_{ik}^{2}$ determine the relative contribution of each dimension to $α_{i}$ . In this sense, the coordinate vector describes the shape and direction of a person’s profile differentiation, whereas $α_{i}$ summarizes its overall magnitude.

Individual 60 exhibited a dominant displacement along ALPS-2 ( $z_{i 2} = 6.49$ ), with smaller positive and negative contributions from ALPS-1 and ALPS-3, respectively. This configuration yielded $α_{60} = 4.10$ , indicating pronounced but dimensionally concentrated profile differentiation.

Individual 206 showed large positive coordinates on ALPS-1 ( $z_{i 1} = 4.51$ ) and ALPS-2 ( $z_{i 2} = 6.25$ ), together with a substantial negative coordinate on ALPS-3 ( $z_{i 3} = - 2.46$ ). Because substantial displacement occurred across multiple validated dimensions, this individual showed the highest overall profile differentiation in the group ().

Individual 667 was characterized by strong positive alignment on ALPS-1 ( $z_{i 1} = 4.91$ ), with comparatively smaller contributions from ALPS-2 and ALPS-3. This yielded a moderate aggregated magnitude ( $α_{667} = 3.57$ ) and a distinct orientation relative to Individuals 60 and 206.

Individual 949 displayed uniformly small coordinates across all three dimensions ( $∣ z_{ik} ∣ \leq 1.60$ ), placing this individual near the origin of ALPS space and producing the smallest aggregated index ( $α_{949} = 1.30$ ). This configuration reflects a relatively flat and weakly differentiated profile.

Standardized ALPI Interpretation

Although the four illustrative individuals shared identical Full-Scale IQ scores, their locations in ALPS space and corresponding degrees of profile differentiation differed substantially. Table 5 reports raw, standardized, and robustly scaled ALPI values, together with percentile ranks based on the empirical distribution of $α_{i}$ . Larger standardized values reflect stronger departure from a flat ipsatized profile across the validated ALPS dimensions. Thus, ALPI is the only available index that simultaneously (a) restricts profile differentiation to replicable latent dimensions, (b) expresses that differentiation as a single population-referenced scalar, and (c) does so independently of overall score level – information that is not recoverable from FSIQ or from direct inspection of raw or ipsatized profiles.

Table 5.

FSIQ, Raw, Standardized, and Robust ALPI Values and Percentile Ranks for Four Illustrative Individuals.

ID	FSIQ	$α_{i}$	z( $α_{i}$ )	${\tilde{α}}_{i}$ (MAD)	Percentile (α)
60	100	4.10	0.55	0.68	∼73rd
206	100	4.88	1.07	1.20	∼85th
667	100	3.57	0.20	0.33	∼62nd
949	100	1.30	−1.31	−1.18	∼7th

Individual 206 ( ${\tilde{α}}_{i} \approx 1.20$ ; approximately the 85th percentile) exhibited the most strongly differentiated configuration, distributed across all three ALPS dimensions. Individual 949 ( ${\tilde{α}}_{i} \approx - 1.18$ ; approximately the 7th percentile) showed minimal patterning and remained close to the origin of the latent profile space. Importantly, ALPI is not simply a re-expression of raw profile variability. Rather, it is a summary of profile differentiation within the validated latent space so that higher values reflect stronger systematic patterning captured by the retained dimensions, not merely visually striking but potentially nonreplicable irregularities in the observed profile. Examining the coordinate vector $z_{i}$ alongside $α_{i}$ , therefore, clarifies both the magnitude and the dimensional composition of an individual’s profile differentiation, information that is not available from FSIQ or raw subtest means alone.

Profile Reconstruction, Aggregated Geometry, and the Utility of ALPI

Figures 1 and 2 display three complementary representations for Individuals 206 (high ALPI) and 949 (low ALPI): the observed ipsatized profile, the ALPS-reconstructed profile, and the population-referenced aggregated latent profile. These three representations serve different interpretive functions. The observed ipsatized profile preserves the person’s full pattern of relative strengths and weaknesses after the removal of the overall level. The ALPS-reconstructed profile represents that person’s best low-dimensional approximation within the validated latent profile space, that is, the portion of the observed pattern that is captured by replicable latent dimensions. The population-referenced aggregated latent profile is a fixed reference pattern estimated once from the full sample and displayed identically for all individuals. It does not represent any one person’s profile; rather, it summarizes the dominant weighted direction of shared profile variation in the population-defined ALPS space.

This distinction is important for practical assessment. Observed profiles can always be inspected directly, but raw inspection alone does not distinguish between pattern features that reflect stable, population-referenced latent structure and features that reflect residual or idiosyncratic variation. The ALPS-reconstructed profile addresses this problem by isolating the component of an individual’s profile that lies within the validated latent space. In this sense, it functions as a measurement-oriented summary of the person’s systematic profile differentiation, whereas discrepancies between the reconstructed and observed profiles indicate residual pattern variation not captured by the retained ALPS dimensions. ALPI then quantifies the overall magnitude of that validated differentiation in a single scalar index.

For Individual 206 (Figure 1), the observed ipsatized profile shows marked differentiation across subtests, and the ALPS-reconstructed profile preserves a substantial portion of that structure. This agreement indicates that the person’s observed profile is not merely irregular at the raw-score level but is meaningfully aligned with the validated latent dimensions defining ALPS. The large value of $α_{206}$ is therefore interpretable as evidence of strong, systematic profile differentiation within the common latent space rather than as a simple descriptive fluctuation in observed scores. Relative to the nearly flat population-referenced aggregated latent profile, Individual 206 shows pronounced departure, and that departure is distributed across multiple dimensions rather than concentrated on a single axis. Discrepancies between the reconstructed and observed ipsatized profiles reflect remaining person-specific variation outside the retained three-dimensional ALPS solution, which accounted for 41.3% of total ipsatized pattern variance.

For Individual 949 (Figure 2), the observed and reconstructed profiles both remain close to the flat population-referenced geometry, consistent with uniformly small ALPS coordinates and a low ALPI value. Here, the practical interpretation is different from that of Individual 206. The low $α_{949}$ indicates not simply that the observed profile looks visually less dramatic, but that little of the person’s ipsatized pattern reflects strong systematic differentiation along the validated latent dimensions. In applied terms, this individual’s profile is comparatively weakly differentiated within the common measurement space despite having the same FSIQ as the other illustrative cases.

Taken together, Figures 1 and 2 clarify the utility of ALPI for assessment-oriented interpretation. The coordinate vector $z_{i}$ shows where a person lies in the latent profile space, the reconstructed profile shows what portion of the observed pattern is captured by replicable latent structure, and $α_{i}$ shows how strongly differentiated that person is overall within that space. Thus, ALPI is not a substitute for profile inspection; rather, it supplements direct inspection by quantifying the magnitude of validated person profile differentiation relative to a fixed, population-referenced geometry. This information is not available from FSIQ alone and is not obtained by simple inspection of observed raw-score profiles.

Discussion

The present study introduced ALPI as a person profile differentiation index derived from ALPS, a framework in which latent profile dimensions are explicitly validated for cross-sample replicability before being used to represent and summarize individual profiles. In the empirical application, the WAIS–IV normative data supported a stable three-dimensional latent profile space – reproduced across 2,000 bootstrap resamples – that distinguished individuals with identical Full-Scale IQ on the basis of their ipsatized subtest configurations. ALPI values ranged from the 7th to the 85th percentile among individuals sharing the same overall ability level, demonstrating that overall score level conceals substantial and measurable differences in profile organization.

Reconstructed Profiles, Measurement Error, and the Interpretive Advantage of ALPI

A fundamental interpretive advantage of ALPI lies in the basis on which it is computed: unlike indices derived from observed profiles, ALPI is grounded exclusively in ALPS-reconstructed profiles – filtered representations that separate reliable latent structure from measurement noise. Observed ipsatized profiles capture a person’s full pattern of relative strengths and weaknesses, but they do so with contamination from random fluctuation and measurement error. Because observed profiles are not filtered through any replicability criterion, visually salient features may reflect sampling variability or instrument imprecision rather than stable individual characteristics. ALPS-reconstructed profiles address this problem directly. They represent each individual’s profile as projected onto dimensions that have been retained only after surviving both statistical and cross-sample replicability tests – first passing PA against sampling noise, then confirming stable recovery across 2,000 bootstrap resamples via principal angles and Tucker’s congruence coefficients. In this sense, ALPS-reconstructed profiles isolate the portion of observed profile variation that reflects a reliable, population-referenced latent structure. For assessment interpretation, this distinction matters: a reconstructed profile offers a more defensible basis for clinical judgment than a raw observed profile because it separates systematic profile differentiation from measurement noise.

The Aggregated Profile as a Clinical Reference Anchor

The population-referenced aggregated latent profile plays a specific and important role within this framework. It is estimated once from the full normative sample as the variance-weighted dominant direction of shared profile variation across the retained ALPS dimensions. In the present data, this profile was approximately flat, indicating that the normative population does not exhibit strong systematic differentiation in any particular direction after level effects are removed. This flatness is precisely what makes the aggregated profile useful as a clinical reference anchor: it represents the population baseline of profile patterning against which individual deviations are evaluated. An individual’s ALPS-reconstructed profile can then be read relative to this fixed reference – not relative to any arbitrary group mean or norm-referenced standard – so that elevations and depressions in specific subtest domains carry interpretive meaning grounded in the validated latent structure of the full normative sample. Because ALPI is defined as a variance-weighted Euclidean distance from the origin of this validated space, it directly quantifies how far each individual’s reconstructed profile departs from this population-referenced anchor – making the aggregated profile not merely a descriptive reference but the geometric foundation on which ALPI rests.

ALPI as a Clinical Screening Tool and the Two-Step Interpretive Workflow

For applied psychologists and clinicians, ALPI supports a natural two-step interpretive workflow. In the first step, ALPI ( $α_{i}$ ) provides a single scalar summary – a Gestalt index – of how strongly an individual’s reconstructed profile departs from the population-referenced aggregated profile across all retained dimensions simultaneously. A large $α_{i}$ value signals that the individual’s profile is substantially differentiated within the validated latent space, warranting closer examination. A small $α_{i}$ value indicates that the individual’s profile remains close to the population reference, suggesting relatively uniform cognitive organization at the latent level. Critically, this global judgment is based on replicability-validated latent dimensions rather than on raw observed scores, making it more objective and less susceptible to the measurement noise that inflates apparent differentiation in observed profiles.

In the second step, the clinician inspects the ALPS-reconstructed profile directly – as illustrated in Figures 1 and 2– to identify which specific subtest domains are elevated, depressed, or unchanged relative to the aggregated reference profile. This domain-level inspection identifies the directional content and clinical meaning of the differentiation that α_i has already quantified in aggregate. Together, the two steps provide complementary information: $α_{i}$ answers how much an individual is differentiated, while the reconstructed profile answers where and in what direction. Neither quantity alone is sufficient; their combination enables the kind of person profile interpretation that is both measurement-principled and clinically actionable.

Limitations

Several limitations warrant consideration. The empirical demonstration rests on a single normative dataset, and the coherence of the retained ALPS dimensions is specific to the WAIS–IV. The operating characteristics of ALPI – including its sampling behavior and sensitivity to the number of retained dimensions under varying sample sizes and signal-to-noise conditions – have not yet been comprehensively characterized. The retention thresholds ( $φ \geq . 85, Lorenzo - Seva ⩕ ten Berge, 2006; θ \leq 20^{°}$ , Knyazev & Argentati, 2006) were chosen on substantive rather than empirically optimized grounds. In addition, ALPI reflects differentiation within the validated latent subspace only; when residual variance is substantial, observed profiles may contain interpretable features beyond what ALPS represents – a general consequence of any low-dimensional approximation that is especially consequential when K is small relative to p.

Future Directions

More comprehensive simulation studies characterizing the sampling distribution and operating characteristics of ALPI across varying dimensionality, sample size, and profile structures represent the most important next step for establishing ALPI as a general profile-measurement index. Longitudinal extensions – tracking ALPS-reconstructed profile changes over time within a fixed latent space – and multigroup extensions – aligning ALPS spaces across clinical or demographic groups to evaluate measurement equity – are especially promising directions. The open-source alsi package, which implements the ALPS workflow for continuous, binary, and ordinal data, provides a computational foundation for these developments.

Conclusion

ALPI provides assessment researchers and clinicians with a principled, population-referenced index of person profile differentiation that is grounded in replicable latent structure and independent of overall score level. It is computed from ALPS-reconstructed profiles – representations that filter observed profile variation through cross-sample validated latent dimensions, separating systematic profile differentiation from measurement noise. The population-referenced aggregated latent profile provides the geometric anchor against which ALPI quantifies individual departure, and direct inspection of the reconstructed profile then identifies which specific subtest domains drive that departure. Together, ALPI and the ALPS framework convert multivariate profile interpretation from descriptive inspection of observed scores into measurement-principled, clinically actionable person-oriented assessment – wherever multivariate profiles are collected and where the goal is defensible, replicable individual interpretation.

Supplemental Material

sj-docx-1-epm-10.1177_00131644261451487 – Supplemental material for The Aggregated Latent Profile Index: Measuring Person Profile Differentiation Within a Bootstrap-Validated Latent Profile Space

Supplemental material, sj-docx-1-epm-10.1177_00131644261451487 for The Aggregated Latent Profile Index: Measuring Person Profile Differentiation Within a Bootstrap-Validated Latent Profile Space by Se-Kang Kim in Educational and Psychological Measurement

Footnotes

ORCID iD

Se-Kang Kim

Ethical Considerations

This study analyzed archival, de-identified human data. Ethical approval was not required.

Consent to Participate

Informed consent to participate was not required because this study analyzed archival, de-identified data.

Consent for Publication

Not applicable.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statements

Data and R code used in this study are included in the Supplemental Material. The open-source R package used in this study is publicly available on CRAN and GitHub: .

Supplemental Material

Supplemental material for this article is available online.

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