LRA: Latent Rank Analysis

Description

A general function for estimating Latent Rank Analysis across different response types. This function automatically dispatches to the appropriate method based on the response type:

For binary data (LRA.binary): Analysis using either SOM or GTM method
For ordinal data (LRA.ordinal): Analysis using the GTM method with category thresholds
For rated data (LRA.rated): Analysis using the GTM method with rating categories

Latent Rank Analysis identifies underlying rank structures in test data and assigns examinees to these ranks based on their response patterns.

Usage

LRA(U, ...)
# S3 method for default
LRA(U, na = NULL, Z = NULL, w = NULL, ...)
# S3 method for binary
LRA(
  U,
  nrank = 2,
  method = "GTM",
  mic = FALSE,
  maxiter = 100,
  BIC.check = FALSE,
  seed = NULL,
  verbose = FALSE,
  ...
)
# S3 method for ordinal
LRA(
  U,
  nrank = 2,
  mic = FALSE,
  maxiter = 100,
  trapezoidal = 0,
  eps = 1e-04,
  verbose = TRUE,
  ...
)
# S3 method for rated
LRA(
  U,
  nrank = 2,
  mic = FALSE,
  maxiter = 100,
  trapezoidal = 0,
  eps = 1e-04,
  minFreqRatio = 0,
  verbose = TRUE,
  ...
)

Value

A list of class "exametrika" and the specific subclass (e.g., "LRA", "LRAordinal", "LRArated") containing the following common elements:

msg: A character string indicating the model type.
testlength: Length of the test (number of items).
nobs: Sample size (number of rows in the dataset).
Nrank: Number of latent ranks specified.
N_Cycle: Number of EM algorithm iterations performed.
TRP: Test Reference Profile vector showing expected scores at each rank.
LRD: Latent Rank Distribution vector showing the number of examinees at each rank.
RMD: Rank Membership Distribution vector showing the sum of probabilities for each rank.
Students: Rank Membership Profile matrix showing the posterior probabilities of examinees belonging to each rank, along with their estimated ranks and odds ratios.
ItemFitIndices: Fit indices for each item. See also ItemFit.
TestFitIndices: Overall fit indices for the test. See also TestFit.

Each subclass returns additional specific elements, detailed in their respective documentation.

For binary data (LRA.binary), the returned list additionally includes:

IRP: Item Reference Profile matrix showing the probability of correct response for each item across different ranks.
IRPIndex: Item Response Profile indices including the location parameters B and Beta, slope parameters A and Alpha, and monotonicity indices C and Gamma.

For ordinal data (LRA.ordinal), the returned list additionally includes:

msg: A character string indicating the model type.
ScoreReport: Descriptive statistics of test performance, including sample size, test length, central tendency, variability, distribution characteristics, and reliability.
ItemReport: Basic statistics for each item including category proportions and item-total correlations.
ICBR: Item Category Boundary Reference matrix showing cumulative probabilities for rank-category combinations.
ICRP: Item Category Reference Profile matrix showing probability of response in each category by rank.
ScoreRankCorr: Spearman's correlation between test scores and estimated ranks.
RankQuantCorr: Spearman's correlation between estimated ranks and quantile groups.
ScoreRank: Contingency table of raw scores by estimated ranks.
ScoreMembership: Expected rank memberships for each raw score.
RankQuantile: Cross-tabulation of rank frequencies and quantile groups.
MembQuantile: Cross-tabulation of rank membership probabilities and quantile groups.
CatQuant: Response patterns across item categories and quantile groups.

For rated data (LRA.rated), the returned list additionally includes:

msg: A character string indicating the model type.
ScoreReport: Descriptive statistics of test performance, including sample size, test length, central tendency, variability, distribution characteristics, and reliability.
ItemReport: Basic statistics for each item including category proportions and item-total correlations.
ICRP: Item Category Reference Profile matrix showing probability of response in each category by rank.
ScoreRankCorr: Spearman's correlation between test scores and estimated ranks.
RankQuantCorr: Spearman's correlation between estimated ranks and quantile groups.
ScoreRank: Contingency table of raw scores by estimated ranks.
ScoreMembership: Expected rank memberships for each raw score.
RankQuantile: Cross-tabulation of rank frequencies and quantile groups.
MembQuantile: Cross-tabulation of rank membership probabilities and quantile groups.
ItemQuantileRef: Reference values for each item across quantile groups.
CatQuant: Response patterns across item categories and quantile groups.

Arguments

U: Either an object of class "exametrika" or raw data. When raw data is given, it is converted to the exametrika class with the dataFormat function.
...: Additional arguments passed to specific methods.
na: Values to be treated as missing values.
Z: Missing indicator matrix of type matrix or data.frame. 1 indicates observed values, 0 indicates missing values.
w: Item weight vector.
nrank: Number of latent ranks to estimate. Must be between 2 and 20.
method: For binary data only. Either "SOM" (Self-Organizing Maps) or "GTM" (Gaussian Topographic Mapping). Default is "GTM".
mic: Logical; if TRUE, forces Item Reference Profiles to be monotonically increasing. Default is FALSE.
maxiter: Maximum number of iterations for estimation. Default is 100.
BIC.check: For binary data with SOM method only. If TRUE, convergence is checked using BIC values. Default is FALSE.
seed: For binary data with SOM method only. Random seed for reproducibility.
verbose: Logical; if TRUE, displays detailed progress during estimation. Default is TRUE.
trapezoidal: Specifies the height of both tails when using a trapezoidal prior distribution. Must be less than 1/nrank. The default value is 0, which results in a uniform prior distribution.
eps: Convergence threshold for parameter updates. Default is 1e-4.
minFreqRatio: Minimum frequency ratio for response categories (default = 0). Categories with occurrence rates below this threshold will be excluded from analysis. For example, if set to 0.1, response categories that appear in less than 10% of responses for an item will be omitted.

Binary Data Method

LRA.binary analyzes dichotomous (0/1) response data using either Self-Organizing Maps (SOM) or Gaussian Topographic Mapping (GTM).

Ordinal Data Method

LRA.ordinal analyzes ordered categorical data with multiple thresholds, such as Likert-scale responses or graded items.

Rated Data Method

LRA.rated analyzes data with ratings assigned to each response, such as partially-credited items or preference scales where response categories have different weights.

Examples

Run this code

# \donttest{
# Binary data example
# Fit a Latent Rank Analysis model with 6 ranks to binary data
result.LRA <- LRA(J15S500, nrank = 6)

# Display the first few rows of student rank membership profiles
head(result.LRA$Students)

# Plot Item Reference Profiles (IRP) for the first 6 items
plot(result.LRA, type = "IRP", items = 1:6, nc = 2, nr = 3)

# Plot Test Reference Profile (TRP) showing expected scores at each rank
plot(result.LRA, type = "TRP")
# }

# \donttest{
# Ordinal data example
# Fit a Latent Rank Analysis model with 3 ranks to ordinal data
result.LRAord <- LRA(J15S3810, nrank = 3, mic = TRUE)

# Plot score distributions
plot(result.LRAord, type = "ScoreFreq")
plot(result.LRAord, type = "ScoreRank")

# Plot category response patterns for items 1-6
plot(result.LRAord, type = "ICBR", items = 1:6, nc = 3, nr = 2)
plot(result.LRAord, type = "ICRP", items = 1:6, nc = 3, nr = 2)
# }

# \donttest{
# Rated data example
# Fit a Latent Rank Analysis model with 10 ranks to rated data
result.LRArated <- LRA(J35S5000, nrank = 10, mic = TRUE)

# Plot score distributions
plot(result.LRArated, type = "ScoreFreq")
plot(result.LRArated, type = "ScoreRank")

# Plot category response patterns for items 1-6
plot(result.LRArated, type = "ICRP", items = 1:6, nc = 3, nr = 2)
# }

Run the code above in your browser using DataLab