Profile Analysis via Multidimensional Scaling

Implements Profile Analysis via Multidimensional Scaling (PAMS) for the identification of population-level core response profiles from cross-sectional and longitudinal person-score data. Each person profile is decomposed into a level component (the person mean) and a pattern component (ipsatized subscores). PAMS uses nonmetric multidimensional scaling via the SMACOF algorithm to identify a small number of core profiles that represent the central response patterns in a sample of any size. Bootstrap standard errors and bias-corrected and accelerated (BCa) confidence intervals for individual core profile coordinates are estimated, enabling significance testing of coordinates that is not available in other profile analysis methods such as cluster profile analysis or latent profile analysis. Person-level weights, R-squared values, and correlations with core profiles are also estimated, allowing individual profiles to be interpreted in terms of the core profile structure. PAMS can be applied to both cross-sectional data and longitudinal data, where core trajectory profiles describe how response patterns change over time. Methods are described in Kim and Kim (2024) , de Leeuw and Mair (2009) , and Kruskal (1964) .

pams: Profile Analysis via Multidimensional Scaling

Overview

PAMS implements Profile Analysis via Multidimensional Scaling for the identification of population-level core response profiles from cross-sectional and longitudinal person-score data.

In a typical social-science dataset each row is a person profile — a vector of scores across J related subscales. PAMS decomposes each profile into two components:

• Level: the person's mean across all subscales, capturing overall elevation.
• Pattern: the ipsatized subscores (deviations from the person mean), capturing the shape of the profile — its peaks and valleys across subscales.

PAMS then uses nonmetric multidimensional scaling (via the SMACOF algorithm) on the J × J inter-variable proximity matrix to identify a small number of core profiles — the central response patterns in the population. Because the proximity matrix is J × J rather than I × I, PAMS scales to any sample size, unlike cluster profile analysis.

The key inferential contribution of PAMS in R is the estimation of bootstrap standard errors and BCa confidence intervals for every core profile coordinate, enabling researchers to test which subscale peaks and valleys are statistically significant. This capability is not available in other profile analysis methods such as cluster profile analysis (CPA) or latent profile analysis (LPA). Both CPA and LPA also recover only level information, whereas PAMS recovers both level and pattern.

PAMS applies equally to cross-sectional and longitudinal data. For longitudinal data the input variables are the subscale scores stacked across time points, and the resulting core profiles are trajectory profiles that show how response patterns evolve over time.

Installation

# Install the development version from GitHub:
# install.packages("devtools")
devtools::install_github("sekangakim/pams")

Quick Start

library(pams)
library(smacof)

# ---- Cross-sectional example ------------------------------------------------

# Load your data (persons x subscales)
cross_data <- read.csv("Cross-sectional.csv", header = FALSE)
colnames(cross_data) <- c("OV1","NS2","VA3","LP4","PP5","SR6","VS7","GI8",
                           "CF9","NR10","NP11","NW12","VA13","PR14","AS15",
                           "ON16","PC17","MW18")

# Step 1: inspect stress across dimensionalities to choose nprofile
smacofSym(dist(t(cross_data)), ndim = 2, type = "ordinal")$stress  # 0.069
smacofSym(dist(t(cross_data)), ndim = 3, type = "ordinal")$stress  # 0.047
smacofSym(dist(t(cross_data)), ndim = 4, type = "ordinal")$stress  # 0.027
# Three-dimensional solution chosen (stress <= 0.05, Kruskal 1964)

# Step 2: run PAMS with 2,000 bootstrap samples
set.seed(1)
result <- BootSmacof(
  testdata    = cross_data,
  participant = 1:10,          # persons selected for individual assessment
  mds         = "smacof",
  type        = "ordinal",
  distance    = "euclid",
  nprofile    = 3,
  direction   = c(-1, 1, 1),  # flip dimensions to aid interpretation
  cl          = 0.95,
  nBoot       = 2000,
  testname    = colnames(cross_data)
)

# Step 3: inspect results
result$MDS$stress          # stress of original fit
result$WeightmeanR2        # mean R^2 across all persons

# Core profile coordinates with BCa CIs
round(result$MDSsummary[[1]], 3)  # Core Profile 1
round(result$MDSsummary[[2]], 3)  # Core Profile 2
round(result$MDSsummary[[3]], 3)  # Core Profile 3

# Weights and core-profile correlations for selected persons
round(result$Weight[1:10, ], 2)

Key Function

BootSmacof()

The core function of the package. It fits a nonmetric MDS solution to the J × J inter-variable distance matrix of the input data, bootstraps the solution to produce empirical sampling distributions of core profile coordinates, and computes BCa confidence intervals. It also estimates person-level weights, R-squared values, and correlations with core profiles for all participants, with optional bootstrap CIs for a selected subset.

Returns a named list with components:

Worked Examples

Full worked examples for both cross-sectional and longitudinal data — including dimensionality selection, direction checking, BCa CI plots, individual person assessment, and residual assumption checks — are provided in the package vignette:

vignette("PAMS_analysis", package = "pams")

Cross-sectional example

Data: Woodcock-Johnson IV Cognitive Ability Battery (n = 1,650, ages 18–35, 18 cognitive subscales). Three core profiles are extracted and labelled by their CHC factor structure peaks and valleys:

1. High Cognitive Processing Speed with Low Long-Term Retrieval
2. High Working Memory with Low Cognitive Processing Speed
3. High Fluid Reasoning with Low Auditory Processing

Longitudinal example

Data: Eating Disorder Inventory-2 (n = 1,261 female anorexic patients, 11 subscales at Pre and Post treatment). Three core trajectory profiles describe differential treatment response patterns across symptom indicators, with BCa CIs used to identify which symptom improvements are statistically significant after treatment.

Comparison with Related Methods

Citation

If you use PAMS in your research, please cite:

Kim, S.-K., & Kim, D. (2024). Utility of profile analysis via multidimensional scaling in R for the study of person response profiles in cross-sectional and longitudinal data. The Quantitative Methods for Psychology, 20(3), 230–247. https://doi.org/10.20982/tqmp.20.3.p230

For the theoretical foundation of PAMS please also cite:

Davison, M. L. (1996). Multidimensional scaling interest and aptitude profiles: Idiographic dimensions, nomothetic factors. Presidential address to Division 5, American Psychological Association, Toronto.

For the SMACOF algorithm used in estimation:

de Leeuw, J., & Mair, P. (2009). Multidimensional scaling using majorization: SMACOF in R. Journal of Statistical Software, 31(3), 1–30. https://doi.org/10.18637/jss.v031.i03

Related Package

SEPA (Subprofile Extraction via Pattern Analysis) is a companion package by the same authors. While PAMS identifies population-level core profiles of response patterns, SEPA locates individual profiles in a two-dimensional SVD biplot space and quantifies each person's alignment with cognitive ability domains via direction cosines. The two packages address complementary scientific questions and can be used together.

License

MIT © Se-Kang Kim & Donghoh Kim