Package 'exametrika'

Description

This function computes Tau-Equivalent Measurement, also known as Cronbach's alpha coefficient, for a given data set.

Usage

AlphaCoefficient(x, na = NULL, Z = NULL, w = NULL)

Arguments

Value

For a correlation/covariance matrix input, returns a single numeric value representing the alpha coefficient. For a data matrix input, returns a list with three components:

• AlphaCov: Alpha coefficient calculated from covariance matrix

• AlphaPhi: Alpha coefficient calculated from phi coefficient matrix

• AlphaTetrachoric: Alpha coefficient calculated from tetrachoric correlation matrix

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of a test. Psychometrika, 16,297–334.

Description

This function returns the alpha coefficient when the specified item is excluded.

Usage

AlphaIfDel(x, delItem = NULL, na = NULL, Z = NULL, w = NULL)

Arguments

Description

Prior distribution function with guessing parameter

Usage

asymprior(c, alp, bet)

Arguments

Description

performs biclustering, rankclustering, and their confirmatory models.

Usage

Biclustering(
  U,
  ncls = 2,
  nfld = 2,
  Z = NULL,
  w = NULL,
  na = NULL,
  method = "B",
  conf = NULL,
  mic = FALSE,
  maxiter = 100,
  verbose = TRUE
)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

number of classes you set

BRM

Bicluster Reference Matrix

FRP

Field Reference Profile

FRPIndex

Index of FFP includes the item location parameters B and Beta, the slope parameters A and Alpha, and the monotonicity indices C and Gamma.

TRP

Test Reference Profile

FMP

Field Membership Profile

Students

Class/Rank Membership Profile matrix.The s-th row vector of $\hat{M}_R$, $\hat{m}_R$, is the rank membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes. It also includes the rank with the maximum estimated membership probability, as well as the rank-up odds and rank-down odds.

LRD

Latent Rank Distribution. see also plot.exametrika

LCD

Latent Class Distribution. see also plot.exametrika

LFD

Latent Field Distribution. see also plot.exametrika

RMD

Rank Membership Distribution.

TestFitIndices

Overall fit index for the test.See also TestFit

Examples

# Perform Biclustering with Binary method (B)
# Analyze data with 5 fields and 6 classes
Biclustering(J35S515, nfld = 5, ncls = 6, method = "B")

# Perform Biclustering with Rank method (R)
# Store results for further analysis and visualization
result.Ranklusteing <- Biclustering(J35S515, nfld = 5, ncls = 6, method = "R")

# Display the Bicluster Reference Matrix (BRM) as a heatmap
plot(result.Ranklusteing, type = "Array")

# Plot Field Reference Profiles (FRP) in a 2x3 grid
# Shows the probability patterns for each field
plot(result.Ranklusteing, type = "FRP", nc = 2, nr = 3)

# Plot Rank Membership Profiles (RMP) for students 1-9 in a 3x3 grid
# Shows the posterior probability distribution of rank membership for each student
plot(result.Ranklusteing, type = "RMP", students = 1:9, nc = 3, nr = 3)

Description

Bicluster Network Model: BINET is a model that combines the Bayesian network model and Biclustering. BINET is very similar to LDB and LDR. The most significant difference is that in LDB, the nodes represent the fields, whereas in BINET, they represent the class. BINET explores the local dependency structure among latent classes at each latent field, where each field is a locus.

Usage

BINET(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  conf = NULL,
  ncls = NULL,
  nfld = NULL,
  g_list = NULL,
  adj_list = NULL,
  adj_file = NULL,
  verbose = FALSE
)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

Optimal number of classes.

Nfield

Optimal number of fields.

crr

Correct Response Rate

ItemLabel

Label of Items

FieldLabel

Label of Fields

all_adj

Integrated Adjacency matrix used to plot graph.

all_g

Integrated graph object used to plot graph.see also plot.exametrika

adj_list

List of Adjacency matrix used in the model

params

A list of the estimated conditional probabilities. It indicates which path was obtained from which parent node(class) to which child node(class), held by parent, child, and field. The item Items contained in the field is in fld. Named chap includes the conditional correct response answer rate of the child node, while pap contains the pass rate of the parent node.

PSRP

Response pattern by the students belonging to the parent classes of Class c. A more comprehensible arrangement of params.

LCD

Latent Class Distribution. see also plot.exametrika

LFD

Latent Field Distribution. see also plot.exametrika

CMD

Class Membership Distribution.

FRP

Marginal bicluster reference matrix.

FRPIndex

Index of FFP includes the item location parameters B and Beta, the slope parameters A and Alpha, and the monotonicity indices C and Gamma.

TRP

Test Reference Profile

LDPSR

A rearranged set of parameters for output. It includes the field the items contained within that field, and the conditional correct response rate of parent nodes(class) and child node(class).

FieldEstimated

Given vector which correspondence between items and the fields.

Students

Rank Membership Profile matrix.The s-th row vector of $\hat{M}_R$, $\hat{m}_R$, is the rank membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes.

NextStage

The next class that easiest for students to move to, its membership probability, class-up odds, and the field required for more.

MG_FitIndices

Multigroup as Null model.See also TestFit

SM_FitIndices

Saturated Model as Null model.See also TestFit

Examples

# Example: Bicluster Network Model (BINET)
# BINET combines Bayesian network model and Biclustering to explore
# local dependency structure among latent classes at each field

# Create field configuration vector based on field assignments
conf <- c(
  1, 5, 5, 5, 9, 9, 6, 6, 6, 6, 2, 7, 7, 11, 11, 7, 7,
  12, 12, 12, 2, 2, 3, 3, 4, 4, 4, 8, 8, 12, 1, 1, 6, 10, 10
)

# Create edge data for network structure between classes
edges_data <- data.frame(
  "From Class (Parent) >>>" = c(
    1, 2, 3, 4, 5, 7, 2, 4, 6, 8, 10, 6, 6, 11, 8, 9, 12
  ),
  ">>> To Class (Child)" = c(
    2, 4, 5, 5, 6, 11, 3, 7, 9, 12, 12, 10, 8, 12, 12, 11, 13
  ),
  "At Field (Locus)" = c(
    1, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 7, 8, 8, 9, 9, 12
  )
)

# Save edge data to temporary CSV file
tmp_file <- tempfile(fileext = ".csv")
write.csv(edges_data, file = tmp_file, row.names = FALSE)

# Fit Bicluster Network Model
result.BINET <- BINET(
  U = J35S515,
  ncls = 13, # Maximum class number from edges (13)
  nfld = 12, # Maximum field number from conf (12)
  conf = conf, # Field configuration vector
  adj_file = tmp_file # Path to the CSV file
)

# Clean up temporary file
unlink(tmp_file)

# Display model results
print(result.BINET)

# Visualize different aspects of the model
plot(result.BINET, type = "Array") # Show bicluster structure
plot(result.BINET, type = "TRP") # Test Response Profile
plot(result.BINET, type = "LRD") # Latent Rank Distribution
plot(result.BINET,
  type = "RMP", # Rank Membership Profiles
  students = 1:9, nc = 3, nr = 3
)
plot(result.BINET,
  type = "FRP", # Field Reference Profiles
  nc = 3, nr = 2
)
plot(result.BINET,
  type = "LDPSR", # Locally Dependent Passing Student Rates
  nc = 3, nr = 2
)

Description

A biserial correlation is a correlation between dichotomous-ordinal and continuous variables.

Usage

Biserial_Correlation(i, t)

Arguments

Value

The biserial correlation coefficient between the two variables.

Description

Binary pattern maker

Usage

BitRespPtn(n)

Arguments

Details

if n <- 1, return 0,1 if n <- 2, return 00,01,10,11 and so on.

Value

binary patterns

Description

performs Bayesian Network Model with specified graph structure

Usage

BNM(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  g = NULL,
  adj_file = NULL,
  adj_matrix = NULL
)

Arguments

Details

This function performs a Bayesian network analysis on the relationships between items. This corresponds to Chapter 8 of the text. It uses the igraph package for graph visualization and checking the adjacency matrix. You need to provide either a graph object or a CSV file where the graph structure is specified.

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

crr

correct response ratio

TestFitIndices

Overall fit index for the test.See also TestFit

adj

Adjacency matrix

\

param

Learned Parameters

CCRR_table

Correct Response Rate tables

Examples

# Create a Directed Acyclic Graph (DAG) structure for item relationships
# Each row represents a directed edge from one item to another
DAG <-
  matrix(
    c(
      "Item01", "Item02", # Item01 influences Item02
      "Item02", "Item03", # Item02 influences Item03
      "Item02", "Item04", # Item02 influences Item04
      "Item03", "Item05", # Item03 influences Item05
      "Item04", "Item05" # Item04 influences Item05
    ),
    ncol = 2, byrow = TRUE
  )

# Convert the DAG matrix to an igraph object for network analysis
g <- igraph::graph_from_data_frame(DAG)
g

# Create adjacency matrix from the graph
# Shows direct connections between items (1 for connection, 0 for no connection)
adj_mat <- as.matrix(igraph::as_adjacency_matrix(g))
print(adj_mat)

# Fit Bayesian Network Model using the specified adjacency matrix
# Analyzes probabilistic relationships between items based on the graph structure
result.BNM <- BNM(J5S10, adj_matrix = adj_mat)
result.BNM

Description

A general function that returns the model fit indices.

Usage

calcFitIndices(chi_A, chi_B, df_A, df_B, nobs)

Arguments

Value

NFI

Normed Fit Index. Lager values closer to 1.0 indicate a better fit.

RFI

Relative Fit Index. Lager values closer to 1.0 indicate a better fit.

IFI

Incremental Fit Index. Lager values closer to 1.0 indicate a better fit.

TLI

Tucker-Lewis Index. Lager values closer to 1.0 indicate a better fit.

CFI

Comparative Fit Index. Lager values closer to 1.0 indicate a better fit.

RMSEA

Root Mean Square Error of Approximation. Smaller values closer to 0.0 indicate a better fit.

AIC

Akaike Information Criterion. A lower value indicates a better fit.

CAIC

Consistent AIC.A lower value indicates a better fit.

BIC

Bayesian Information Criterion. A lower value indicates a better fit.

Description

The conditional correct response rate (CCRR) represents the ratio of the students who passed Item C (consequent item) to those who passed Item A (antecedent item). This function is applicable only to binary response data.

Usage

CCRR(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of conditional correct response rates with exametrika class. Each element (i,j) represents the probability of correctly answering item j given that item i was answered correctly.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
# Calculate CCRR using sample dataset J5S10
CCRR(J5S10)

Description

The correct response rate (CRR) is one of the most basic and important statistics for item analysis. This is an index of item difficulty and a measure of how many students out of those who tried an item correctly responded to it. This function is applicable only to binary response data.

The CRR for each item is calculated as:

$p_j = \frac{\sum_{i=1}^n z_{ij}u_{ij}}{\sum_{i=1}^n z_{ij}}$

where $z_{ij}$ is the missing indicator and $u_{ij}$ is the response.

Usage

crr(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of weighted correct response rates for each item. Values range from 0 to 1, where higher values indicate easier items (more students answered correctly).

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# Simple binary data
U <- matrix(c(1, 0, 1, 1, 0, 1), ncol = 2)
crr(U)

# using sample datasaet
crr(J15S500)

Description

This function calculates the overall alpha and omega coefficients for the given data matrix. It also computes the alpha coefficient for each item, assuming that item is excluded.

Usage

CTT(U, na = NULL, Z = NULL, w = NULL)

Arguments

Value

Returns a list of class c("exametrika", "CTT") containing two data frames:

• Reliability: A data frame with overall reliability coefficients (Alpha and Omega) calculated using different correlation matrices (Covariance, Phi, and Tetrachoric)

• ReliabilityExcludingItem: A data frame showing alpha coefficients when each item is excluded, calculated using different correlation matrices

Examples

# using sample dataset
CTT(J15S500)

Description

This function serves the role of formatting the data prior to the analysis.

Usage

dataFormat(data, na = NULL, id = 1, Z = NULL, w = NULL)

Arguments

Value

U

For binary response data. A matrix with rows representing the sample size and columns representing the number of items, where elements are either 0 or 1. $u_{ij}=1$ indicates that student i correctly answered item j, while $u_{ij}=0$ means that student i answered item j incorrectly. If the data contains NA values, any value can be filled in the matrix U, represented by the following missing value index matrix Z. However, in this function, -1 is assigned.

Q

For polytomous response data. A matrix with rows representing the sample size and columns representing the number of items, where elements are non-negative integers. When input data is in factor format, the factor levels are converted to consecutive integers starting from 1, and the original factor labels are stored in factor_labels.

ID

The ID label given by the designated column or function.

ItemLabel

The item names given by the provided column names or function.

Z

Missing indicator matrix. $z_{ij}=1$ indicates that item j is presented to Student i, while $z_{ij}=0$ indicates item j is NOT presented to Student i.

w

Item weight vector

response.type

Character string indicating the type of response data: "binary" for binary responses or "polytomous" for polytomous responses.

factor_labels

List containing the original factor labels when polytomous responses are provided as factors. NULL if no factor data is present.

categories

Numeric vector containing the number of response categories for each item. For binary data, all elements are 2.

Description

A function to reshape long data into a dataset suitable for exametrika.

Usage

dataFormat.long(
  data,
  na = NULL,
  Sid = NULL,
  Qid = NULL,
  Resp = NULL,
  w = NULL,
  response.type = NULL
)

Arguments

Value

U

For binary response data. A matrix with rows representing the sample size and columns representing the number of items, where elements are either 0 or 1. $u_{ij}=1$ indicates that student i correctly answered item j, while $u_{ij}=0$ means that student i answered item j incorrectly.

Q

For polytomous response data. A matrix with rows representing the sample size and columns representing the number of items, where elements are non-negative integers. When input data is in factor format, the factor levels are converted to consecutive integers starting from 1, and the original factor labels are stored in factor_labels.

ID

The ID label given by the designated column or function.

ItemLabel

The item names given by the provided column names or function.

Z

Missing indicator matrix. $z_{ij}=1$ indicates that item j is presented to Student i, while $z_{ij}=0$ indicates item j is NOT presented to Student i.

w

Item weight vector

response.type

Character string indicating the type of response data: "binary" for binary responses or "polytomous" for polytomous responses.

factor_labels

List containing the original factor labels when polytomous responses are provided as factors. NULL if no factor data is present.

categories

Numeric vector containing the number of response categories for each item. For binary data, all elements are 2.

Description

The dimensionality is the number of components the test is measuring.

Usage

Dimensionality(U, na = NULL, Z = NULL, w = NULL)

Arguments

Value

Returns a list of class c("exametrika", "Dimensionality") containing:

• Component: Sequence of component numbers

• Eigenvalue: Eigenvalues of the tetrachoric correlation matrix

• PerOfVar: Percentage of variance explained by each component

• CumOfPer: Cumulative percentage of variance explained

Description

output for Field Analysis

Usage

FieldAnalysis(x, digits = 4)

Arguments

Value

Returns a list of class c("exametrika", "Biclustering", "FieldAnalysis") containing:

FieldAnalysisMatrix

A matrix showing field analysis results with rows representing items and columns showing:

• CRR: Correct Response Rate

• LFE: Latent Field Estimation

• Field1...FieldN: Field membership values

Description

Item Information Function for 2PLM

Usage

IIF2PLM(a, b, theta)

Arguments

Value

Returns a numeric vector representing the item information at each ability level theta. The information is calculated as: $I(\theta) = a^2P(\theta)(1-P(\theta))$

Description

Item Information Function for 3PLM

Usage

IIF3PLM(a, b, c, theta)

Arguments

Value

Returns a numeric vector representing the item information at each ability level theta. The information is calculated as: $I(\theta) = \frac{a^2(1-P(\theta))(P(\theta)-c)^2}{(1-c)^2P(\theta)}$

Description

Inter-Item Analysis returns various metrics for analyzing relationships between pairs of items. This function is applicable only to binary response data. The following metrics are calculated:

• JSS: Joint Sample Size

• JCRR: Joint Correct Response Rate

• IL: Item Lift

• MI: Mutual Information

• Phi: Phi Coefficient

• Tetrachoric: Tetrachoric Correlation

Each metric is returned in matrix form where element (i,j) represents the relationship between items i and j.

Usage

InterItemAnalysis(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A list of class "exametrika" and "IIAnalysis" containing the following matrices:

JSS

Joint Sample Size matrix

JCRR

Joint Correct Response Rate matrix

IL

Item Lift matrix

MI

Mutual Information matrix

Phi

Phi Coefficient matrix

Tetrachoric

Tetrachoric Correlation matrix

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
InterItemAnalysis(J15S500)

Description

The purpose of this method is to find the optimal number of classes C, and optimal number of fields F. It can be found in a single run of the analysis, but it takes a long computation time when the sample size S is large. In addition, this method incorporates the Chinese restaurant process and Gibbs sampling. In detail, See Section 7.8 in Shojima(2022).

Usage

IRM(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  gamma_c = 1,
  gamma_f = 1,
  max_iter = 100,
  stable_limit = 5,
  minSize = 20,
  EM_limit = 20,
  seed = 123,
  verbose = TRUE
)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

Optimal number of classes.

Nfield

Optimal number of fields.

BRM

Bicluster Reference Matrix

FRP

Field Reference Profile

FRPIndex

Index of FFP includes the item location parameters B and Beta, the slope parameters A and Alpha, and the monotonicity indices C and Gamma.

TRP

Test Reference Profile

FMP

Field Membership Profile

Students

Rank Membership Profile matrix.The s-th row vector of $\hat{M}_R$, $\hat{m}_R$, is the rank membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes. It also includes the rank with the maximum estimated membership probability, as well as the rank-up odds and rank-down odds.

LRD

Latent Rank Distribution. see also plot.exametrika

LFD

Latent Field Distribution. see also plot.exametrika

RMD

Rank Membership Distribution.

TestFitIndices

Overall fit index for the test.See also TestFit

Examples

# Fit an Infinite Relational Model (IRM) to determine optimal number of classes and fields
# gamma_c and gamma_f are concentration parameters for the Chinese Restaurant Process
result.IRM <- IRM(J35S515, gamma_c = 1, gamma_f = 1, verbose = TRUE)

# Display the Bicluster Reference Matrix (BRM) as a heatmap
# Shows the discovered clustering structure of items and students
plot(result.IRM, type = "Array")

# Plot Field Reference Profiles (FRP) in a 3-column grid
# Shows the probability patterns for each automatically determined field
plot(result.IRM, type = "FRP", nc = 3)

# Plot Test Reference Profile (TRP)
# Shows the overall response pattern across all fields
plot(result.IRM, type = "TRP")

Description

A function for estimating item parameters using the EM algorithm.

Usage

IRT(U, model = 2, na = NULL, Z = NULL, w = NULL, verbose = TRUE)

Arguments

Details

Apply the 2, 3, and 4 parameter logistic models to estimate the item and subject populations. The 4PL model can be described as follows.

$P(\theta,a_j,b_j,c_j,d_j)= c_j + \frac{d_j -c_j}{1+exp\{-a_j(\theta - b_j)\}}$

$a_j, b_j, c_j$, and $d_j$ are parameters related to item j, and are parameters that adjust the logistic curve. $a_j$ is called the slope parameter, $b_j$ is the location, $c_j$ is the lower asymptote, and $d_j$ is the upper asymptote parameter. The model includes lower models, and among the 4PL models, the case where $d=1$ is the 3PL model, and among the 3PL models, the case where $c=0$ is the 2PL model.

Value

model

number of item parameters you set.

testlength

Length of the test. The number of items included in the test.

nobs

Sample size. The number of rows in the dataset.

params

Matrix containing the estimated item parameters

Q3mat

Q3-matrix developed by Yen(1984)

itemPSD

Posterior standard deviation of the item parameters

ability

Estimated parameters of students ability

ItemFitIndices

Fit index for each item.See also ItemFit

TestFitIndices

Overall fit index for the test.See also TestFit

References

Yen, W. M. (1984) Applied Psychological Measurement, 8, 125-145.

Examples

# Fit a 3-parameter IRT model to the sample dataset
result.IRT <- IRT(J15S500, model = 3)

# Display the first few rows of estimated student abilities
head(result.IRT$ability)

# Plot Item Characteristic Curves (ICC) for items 1-6 in a 2x3 grid
plot(result.IRT, type = "ICC", items = 1:6, nc = 2, nr = 3)

# Plot Item Information Curves (IIC) for items 1-6 in a 2x3 grid
plot(result.IRT, type = "IIC", items = 1:6, nc = 2, nr = 3)

# Plot the Test Information Curve (TIC) for all items
plot(result.IRT, type = "TIC")

Description

The Item-Total Biserial Correlation computes the biserial correlation between each item and the total score. This function is applicable only to binary response data.

This correlation provides a measure of item discrimination, indicating how well each item distinguishes between high and low performing examinees.

Usage

ITBiserial(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of item-total biserial correlations. Values range from -1 to 1, where:

• Values near 1: Strong positive discrimination

• Values near 0: No discrimination

• Negative values: Potential item problems

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

The biserial correlation is generally preferred over the point-biserial correlation when the dichotomization is artificial (i.e., when the underlying trait is continuous).

Examples

# using sample dataset
ITBiserial(J15S500)

Description

The item entropy is an indicator of the variability or randomness of the responses. This function is applicable only to binary response data.

The entropy value represents the uncertainty or information content of the response pattern for each item, measured in bits. Maximum entropy (1 bit) occurs when correct and incorrect responses are equally likely (p = 0.5).

Usage

ItemEntropy(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Details

The item entropy is calculated as:

$e_j = -p_j\log_2p_j-(1-p_j)\log_2(1-p_j)$

where $p_j$ is the correct response rate for item j.

The entropy value has the following properties:

• Maximum value of 1 bit when p = 0.5 (most uncertainty)

• Minimum value of 0 bits when p = 0 or 1 (no uncertainty)

• Higher values indicate more balanced response patterns

• Lower values indicate more predictable response patterns

Value

A numeric vector of entropy values for each item, measured in bits. Values range from 0 to 1, where:

• 1: maximum uncertainty (p = 0.5)

• 0: complete certainty (p = 0 or 1)

• Values near 1 indicate items with balanced response patterns

• Values near 0 indicate items with extreme response patterns

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# using sample dataset
ItemEntropy(J5S10)

Description

A general function that returns the model fit indices.

Usage

ItemFit(U, Z, ell_A, nparam)

Arguments

Value

model_log_like

log likelihood of analysis model

bench_log_like

log likelihood of benchmark model

null_log_like

log likelihood of null model

model_Chi_sq

Chi-Square statistics for analysis model

null_Chi_sq

Chi-Square statistics for null model

model_df

degrees of freedom of analysis model

null_df

degrees of freedom of null model

NFI

Normed Fit Index. Lager values closer to 1.0 indicate a better fit.

RFI

Relative Fit Index. Lager values closer to 1.0 indicate a better fit.

IFI

Incremental Fit Index. Lager values closer to 1.0 indicate a better fit.

TLI

Tucker-Lewis Index. Lager values closer to 1.0 indicate a better fit.

CFI

Comparative Fit Index. Lager values closer to 1.0 indicate a better fit.

RMSEA

Root Mean Square Error of Approximation. Smaller values closer to 0.0 indicate a better fit.

AIC

Akaike Information Criterion. A lower value indicates a better fit.

CAIC

Consistent AIC.A lower value indicates a better fit.

BIC

Bayesian Information Criterion. A lower value indicates a better fit.

Description

Item Information Function for 4PLM

Usage

ItemInformationFunc(a = 1, b, c = 0, d = 1, theta)

Arguments

Value

Returns a numeric vector representing the item information at each ability level theta. The information is calculated based on the first derivative of the log-likelihood of the 4PL model with respect to theta.

Description

The lift is a commonly used index in a POS data analysis. The item lift of Item k to Item j is defined as follow: $l_{jk} = \frac{p_{k\mid j}}{p_k}$ This function is applicable only to binary response data.

Usage

ItemLift(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of item lift values with exametrika class. Each element (j,k) represents the lift value of item k given item j, which indicates how much more likely item k is to be correct given that item j was answered correctly.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

References

Brin, S., Motwani, R., Ullman, J., & Tsur, S. (1997). Dynamic itemset counting and implication rules for market basket data. In Proceedings of ACM SIGMOD International Conference on Management of Data (pp. 255–264). https://dl.acm.org/doi/10.1145/253262.253325

Examples

# example code
# Calculate ItemLift using sample dataset J5S10
ItemLift(J5S10)

Description

Item Odds are defined as the ratio of Correct Response Rate to Incorrect Response Rate:

$O_j = \frac{p_j}{1-p_j}$

where $p_j$ is the correct response rate for item j. This function is applicable only to binary response data.

The odds value represents how many times more likely a correct response is compared to an incorrect response. For example, an odds of 2 means students are twice as likely to answer correctly as incorrectly.

Usage

ItemOdds(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of odds values for each item. Values range from 0 to infinity, where:

• odds > 1: correct response more likely than incorrect

• odds = 1: equally likely

• odds < 1: incorrect response more likely than correct

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# using sample dataset
ItemOdds(J5S10)

Description

This function calculates statistics for each item.

Usage

ItemStatistics(U, na = NULL, Z = NULL, w = NULL)

Arguments

Value

NR

Number of Respondents

CRR

Correct Response Rate denoted as $p_j$.

ODDs

Item Odds is the ratio of the correct response rate to the incorrect response rate. Defined as $o_j = \frac{p_j}{1-p_j}$

Threshold

Item Threshold is a measure of difficulty based on a standard normal distribution.

Entropy

Item Entropy is an indicator of the variability or randomness of the responses. Defined as $e_j=-p_j \log_2 p_j - (1-p_j)\log_2(1-p_j)$

ITCrr

Item-total Correlation is a Pearson's correlation fo an item with the number of Number-Right score.

Examples

# using sample dataset
ItemStatistics(J15S500)

Description

Item threshold is a measure of difficulty based on a standard normal distribution. This function is applicable only to binary response data.

The threshold is calculated as:

$\tau_j = \Phi^{-1}(1-p_j)$

where $\Phi^{-1}$ is the inverse standard normal distribution function and $p_j$ is the correct response rate for item j.

Higher threshold values indicate more difficult items, as they represent the point on the standard normal scale above which examinees tend to answer incorrectly.

Usage

ItemThreshold(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of threshold values for each item on the standard normal scale. Typical values range from about -3 to 3, where:

• Positive values indicate difficult items

• Zero indicates items of medium difficulty (50% correct)

• Negative values indicate easy items

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# using sample dataset
ItemThreshold(J5S10)

Description

Item-Total correlation (ITC) is a Pearson's correlation of an item with the Number-Right Score (NRS) or total score. This function is applicable only to binary response data.

The ITC is a measure of item discrimination, indicating how well an item distinguishes between high and low performing examinees.

Usage

ItemTotalCorr(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Details

The correlation is calculated between:

• Each item's responses (0 or 1)

• The total test score (sum of correct responses)

Higher positive correlations indicate items that better discriminate between high and low ability examinees.

Value

A numeric vector of item-total correlations. Values typically range from -1 to 1, where:

• Values near 1: Strong positive discrimination

• Values near 0: No discrimination

• Negative values: Potential item problems (lower ability students performing better than higher ability students)

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Values below 0.2 might indicate problematic items that should be reviewed. Values above 0.3 are generally considered acceptable.

Examples

# using sample dataset
ItemTotalCorr(J15S500)

Description

J12S5000.Rdata

Usage

J12S5000

Format

A data frame with 5000 students and 12 items

Source

http://shojima.starfree.jp/tde/index.htm

Description

J15S500.Rdata

Usage

J15S500

Format

A data frame with 500 students and 15 items

Source

http://shojima.starfree.jp/tde/index.htm

Description

J20S400.Rdata

Usage

J20S400

Format

A data frame with 400 students and 20 items

Source

http://shojima.starfree.jp/tde/index.htm

Description

J35S515.Rdata

Usage

J35S515

Format

A data frame with 515 students and 35 items

Source

http://shojima.starfree.jp/tde/index.htm

Description

J5S10.Rdata

Usage

J5S10

Format

A data frame with 5 students and 10 items

Source

http://shojima.starfree.jp/tde/index.htm

Description

The joint correct response rate (JCRR) is the rate of students who passed both items. This function is applicable only to binary response data.

Usage

JCRR(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of joint correct response rates with exametrika class. Each element (i,j) represents the proportion of students who correctly answered both items i and j.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
# Calculate JCRR using sample dataset J5S10
JCRR(J5S10)

Description

The joint sample size is a matrix whose elements are the number of individuals who responded to each pair of items.

Usage

JointSampleSize(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

Returns a matrix of class c("exametrika", "matrix") where each element (i,j) represents the number of students who responded to both item i and item j. The diagonal elements represent the total number of responses for each item.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Description

A function for estimating LCA using the EM algorithm.

Usage

LCA(U, ncls = 2, na = NULL, Z = NULL, w = NULL, maxiter = 100)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

number of classes you set

TRP

Test Reference Profile matrix. The TRP is the column sum vector of estimated class reference matrix, $\hat{\Pi}_c$

LCD

Latent Class Distribution table.see also plot.exametrika

CMD

Class Membership Distribution table. see also plot.exametrika

Students

Class Membership Profile matrix.The s-th row vector of $\hat{M}_c$, $\hat{m}_c$, is the class membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes. The last column indicates the latent class estimate.

IRP

Item Reference Profile matrix.The IRP of item j is the j-th row vector in the class reference matrix, $\hat{\pi}_c$

ItemFitIndices

Fit index for each item.See also ItemFit

TestFitIndices

Overall fit index for the test.See also TestFit

Examples

# Fit a Latent Class Analysis model with 5 classes to the sample dataset
result.LCA <- LCA(J15S500, ncls = 5)

# Display the first few rows of student class membership probabilities
head(result.LCA$Students)

# Plot Item Response Profiles (IRP) for items 1-6 in a 2x3 grid
plot(result.LCA, type = "IRP", items = 1:6, nc = 2, nr = 3)

# Plot Class Membership Probabilities (CMP) for students 1-9 in a 3x3 grid
plot(result.LCA, type = "CMP", students = 1:9, nc = 3, nr = 3)

# Plot Test Response Profile (TRP) showing response patterns across all classes
plot(result.LCA, type = "TRP")

# Plot Latent Class Distribution (LCD) showing the size of each latent class
plot(result.LCA, type = "LCD")

Description

A function that extracts only the estimation of graph parameters after the rank estimation is completed.

Usage

LD_param_est(tmp, adj_list, classRefMat, ncls, smoothpost)

Arguments

Description

Latent dependence Biclustering, which incorporates biclustering and a Bayesian network model.

Usage

LDB(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  ncls = 2,
  method = "R",
  conf = NULL,
  g_list = NULL,
  adj_list = NULL,
  adj_file = NULL,
  verbose = FALSE
)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

Optimal number of classes.

Nfield

Optimal number of fields.

crr

Correct Response Rate

ItemLabel

Label of Items

FieldLabel

Label of Fields

adj_list

List of Adjacency matrix used in the model

g_list

List of graph object used in the model

IRP

List of Estimated Parameters. This object is three-dimensional PIRP array, where each dimension represents the number of rank,number of field, and Dmax. Dmax denotes the maximum number of correct response patterns for each field.

LFD

Latent Field Distribution. see also plot.exametrika

LRD

Latent Rank Distribution. see also plot.exametrika

FRP

Marginal Field Reference Matrix

FRPIndex

Index of FFP includes the item location parameters B and Beta, the slope parameters A and Alpha, and the monotonicity indices C and Gamma.

CCRR_table

This table is a rearrangement of IRP into a data.frame format for output, consisting of combinations of rank ,field and PIRP.

TRP

Test Reference Profile

RMD

Rank Membership Distribution.

FieldEstimated

Given vector which correspondence between items and the fields.

ClassEstimated

An index indicating which class a student belongs to, estimated by confirmatory Ranklustering.

Students

Rank Membership Profile matrix.The s-th row vector of $\hat{M}_R$, $\hat{m}_R$, is the rank membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes. It also includes the rank with the maximum estimated membership probability, as well as the rank-up odds and rank-down odds.

TestFitIndices

Overall fit index for the test.See also TestFit

Examples

# Example: Latent Dirichlet Bayesian Network model
# Create field configuration vector based on field assignments
conf <- c(
  1, 6, 6, 8, 9, 9, 4, 7, 7, 7, 5, 8, 9, 10, 10, 9, 9,
  10, 10, 10, 2, 2, 3, 3, 5, 5, 6, 9, 9, 10, 1, 1, 7, 9, 10
)

# Create edge data for the network structure between fields
edges_data <- data.frame(
  "From Field (Parent) >>>" = c(
    6, 4, 5, 1, 1, 4, # Class/Rank 2
    3, 4, 6, 2, 4, 4, # Class/Rank 3
    3, 6, 4, 1, # Class/Rank 4
    7, 9, 6, 7 # Class/Rank 5
  ),
  ">>> To Field (Child)" = c(
    8, 7, 8, 7, 2, 5, # Class/Rank 2
    5, 8, 8, 4, 6, 7, # Class/Rank 3
    5, 8, 5, 8, # Class/Rank 4
    10, 10, 8, 9 # Class/Rank 5
  ),
  "At Class/Rank (Locus)" = c(
    2, 2, 2, 2, 2, 2, # Class/Rank 2
    3, 3, 3, 3, 3, 3, # Class/Rank 3
    4, 4, 4, 4, # Class/Rank 4
    5, 5, 5, 5 # Class/Rank 5
  )
)

# Save edge data to temporary CSV file
tmp_file <- tempfile(fileext = ".csv")
write.csv(edges_data, file = tmp_file, row.names = FALSE)

# Fit Latent Dirichlet Bayesian Network model
result.LDB <- LDB(
  U = J35S515,
  ncls = 5, # Number of latent classes
  conf = conf, # Field configuration vector
  adj_file = tmp_file # Path to the CSV file
)

# Clean up temporary file
unlink(tmp_file)

# Display model results
print(result.LDB)

# Visualize different aspects of the model
plot(result.LDB, type = "Array") # Show bicluster structure
plot(result.LDB, type = "TRP") # Test Response Profile
plot(result.LDB, type = "LRD") # Latent Rank Distribution
plot(result.LDB,
  type = "RMP", # Rank Membership Profiles
  students = 1:9, nc = 3, nr = 3
)
plot(result.LDB,
  type = "FRP", # Field Reference Profiles
  nc = 3, nr = 2
)
# Field PIRP Profile showing correct answer counts for each rank and field
plot(result.LDB, type = "FieldPIRP")

Description

performs local dependence latent lank analysis(LD_LRA) by Shojima(2011)

Usage

LDLRA(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  ncls = 2,
  method = "R",
  g_list = NULL,
  adj_list = NULL,
  adj_file = NULL,
  verbose = FALSE
)

Arguments

Details

This function is intended to perform LD-LRA. LD-LRA is an analysis that combines LRA and BNM, and it is used to analyze the network structure among items in the latent rank. In this function, structural learning is not performed, so you need to provide item graphs for each rank as separate files. The file format for this is plain text CSV that includes edges (From, To) and rank numbers.

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

crr

correct response ratio

adj_list

adjacency matrix list

g_list

graph list

referenceMatrix

Learned Parameters.A three-dimensional array of patterns where item x rank x pattern.

IRP

Marginal Item Reference Matrix

IRPIndex

IRP Indices which include Alpha, Beta, Gamma.

TRP

Test Reference Profile matrix.

LRD

latent Rank/Class Distribution

RMD

Rank/Class Membership Distribution

TestFitIndices

Overall fit index for the test.See also TestFit

Estimation_table

Estimated parameters tables.

CCRR_table

Correct Response Rate tables

Studens

Student information. It includes estimated class membership, probability of class membership, RUO, and RDO.

Examples

# Create sample DAG structure with different rank levels
# Format: From, To, Rank
DAG_dat <- matrix(c(
  "From", "To", "Rank",
  "Item01", "Item02", "1", # Simple structure for Rank 1
  "Item01", "Item02", "2", # More complex structure for Rank 2
  "Item02", "Item03", "2",
  "Item01", "Item02", "3", # Additional connections for Rank 3
  "Item02", "Item03", "3",
  "Item03", "Item04", "3"
), ncol = 3, byrow = TRUE)

# Method 1: Directly use graph and adjacency lists
g_list <- list()
adj_list <- list()

for (i in 1:3) {
  adj_R <- DAG_dat[DAG_dat[, 3] == as.character(i), 1:2, drop = FALSE]
  g_tmp <- igraph::graph_from_data_frame(
    d = data.frame(
      From = adj_R[, 1],
      To = adj_R[, 2]
    ),
    directed = TRUE
  )
  adj_tmp <- igraph::as_adjacency_matrix(g_tmp)
  g_list[[i]] <- g_tmp
  adj_list[[i]] <- adj_tmp
}

# Fit Local Dependence Latent Rank Analysis
result.LDLRA1 <- LDLRA(J12S5000,
  ncls = 3,
  g_list = g_list,
  adj_list = adj_list
)

# Plot Item Reference Profiles (IRP) in a 4x3 grid
# Shows the probability patterns of correct responses for each item across ranks
plot(result.LDLRA1, type = "IRP", nc = 4, nr = 3)

# Plot Test Reference Profile (TRP)
# Displays the overall pattern of correct response probabilities across ranks
plot(result.LDLRA1, type = "TRP")

# Plot Latent Rank Distribution (LRD)
# Shows the distribution of students across different ranks
plot(result.LDLRA1, type = "LRD")

Description

The four-parameter logistic model is a model where one additional parameter d, called the upper asymptote parameter, is added to the 3PLM.

Usage

LogisticModel(a = 1, b, c = 0, d = 1, theta)

Arguments

Value

Returns a numeric vector of probabilities between c and d, representing the probability of a correct response given the ability level theta. The probability is calculated using the formula: $P(\theta) = c + \frac{(d-c)}{1 + e^{-a(\theta-b)}}$

Description

A function for estimating LRA by SOM/GTM

Usage

LRA(
  U,
  nrank = 2,
  na = NULL,
  Z = NULL,
  w = NULL,
  method = "GTM",
  mic = FALSE,
  maxiter = 100,
  BIC.check = FALSE,
  seed = NULL
)

Arguments

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

Nclass

number of classes you set

TRP

Test Reference Profile matrix. The TRP is the column sum vector of estimated class reference matrix, $\hat{\Pi}_c$

LCD

Latent Class Distribution table.see also plot.exametrika

CMD

Class Membership Distribution table. see also plot.exametrika

Students

Class Membership Profile matrix.The s-th row vector of $\hat{M}_c$, $\hat{m}_c$, is the class membership profile of Student s, namely the posterior probability distribution representing the student's belonging to the respective latent classes. It also includes the rank with the maximum estimated membership probability, as well as the rank-up odds and rank-down odds.

IRP

Item Reference Profile matrix.The IRP of item j is the j-th row vector in the class reference matrix, $\hat{\pi}_c$

IRPIndex

The IRP information includes the item location parameters B and Beta, the slope parameters A and Alpha, and the monotonicity indices C and Gamma.

ItemFitIndices

Fit index for each item.See also ItemFit

TestFitIndices

Overall fit index for the test.See also TestFit

Examples

# Fit a Latent Rank Analysis model with 6 ranks to the sample dataset
result.LRA <- LRA(J15S500, nrank = 6)

# Display the first few rows of student rank membership profiles
# This shows posterior probabilities of students belonging to each rank
head(result.LRA$Students)

# Plot Item Reference Profiles (IRP) for items 1-6 in a 2x3 grid
# Shows the probability of correct response for each rank
plot(result.LRA, type = "IRP", items = 1:6, nc = 2, nr = 3)

# Plot Rank Membership Profiles (RMP) for students 1-9 in a 3x3 grid
# Shows the posterior probability distribution of rank membership for each student
plot(result.LRA, type = "RMP", students = 1:9, nc = 3, nr = 3)

# Plot Test Reference Profile (TRP)
# Shows the column sum vector of estimated rank reference matrix
plot(result.LRA, type = "TRP")

# Plot Latent Rank Distribution (LRD)
# Shows the distribution of students across different ranks
plot(result.LRA, type = "LRD")

Description

Function to limit the number of parent nodes

Usage

maxParents_penalty(vec, testlength, maxParents)

Arguments

Details

When generating an adjacency matrix using GA, the number of edges coming from a single node should be limited to 2 or 3. This is because if there are too many edges, it becomes difficult to interpret in practical applications. This function works to adjust the sampling of the randomly generated adjacency matrix so that the column sum of the upper triangular elements fits within the set limit.

Description

Mutual Information is a measure that represents the degree of interdependence between two items. This function is applicable only to binary response data. The measure is calculated using the joint probability distribution of responses between item pairs and their marginal probabilities.

Usage

MutualInformation(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of mutual information values with exametrika class. Each element (i,j) represents the mutual information between items i and j, measured in bits. Higher values indicate stronger interdependence between items.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
# Calculate Mutual Information using sample dataset J15S500
MutualInformation(J15S500)

Description

The Number-Right Score (NRS) function calculates the weighted sum of correct responses for each examinee. This function is applicable only to binary response data.

For each examinee, the score is computed as:

$NRS_i = \sum_{j=1}^J z_{ij}u_{ij}w_j$

where:

• $z_{ij}$ is the missing response indicator (0/1)

• $u_{ij}$ is the response (0/1)

• $w_j$ is the item weight

Usage

nrs(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector containing the Number-Right Score for each examinee. The score represents the weighted sum of correct answers, where:

• Maximum score is the sum of all item weights

• Minimum score is 0

• Missing responses do not contribute to the score

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# using sample dataset
nrs(J15S500)

Description

Log-likelihood function used in the Maximization Step (M-Step).

Usage

objective_function_IRT(lambda, model, qjtrue, qjfalse, quadrature)

Arguments

Description

This function computes Tau-Congeneric Measurement, also known as McDonald's tau coefficient, for a given data set.

Usage

OmegaCoefficient(x, na = NULL, Z = NULL, w = NULL)

Arguments

Value

For a correlation/covariance matrix input, returns a single numeric value representing the omega coefficient. For a data matrix input, returns a list with three components:

• OmegaCov: Omega coefficient calculated from covariance matrix

• OmegaPhi: Omega coefficient calculated from phi coefficient matrix

• OmegaTetrachoric: Omega coefficient calculated from tetrachoric correlation matrix

References

McDonald, R. P. (1999). Test theory: A unified treatment. Erlbaum.

Description

The Passage Rate for each student is calculated as their Number-Right Score (NRS) divided by the number of items presented to them. This function is applicable only to binary response data.

The passage rate is calculated as:

$P_i = \frac{\sum_{j=1}^J z_{ij}u_{ij}w_j}{\sum_{j=1}^J z_{ij}}$

where:

• $z_{ij}$ is the missing response indicator (0/1)

• $u_{ij}$ is the response (0/1)

• $w_j$ is the item weight

Usage

passage(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector containing the passage rate for each student. Values range from 0 to 1 (or maximum weight) where:

• 1: Perfect score on all attempted items

• 0: No correct answers

• NA: No items attempted

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

The passage rate accounts for missing responses by only considering items that were actually presented to each student. This provides a fair comparison between students who attempted different numbers of items.

Examples

# using sample dataset
passage(J15S500)

Description

The percentile function calculates each student's relative standing in the group, expressed as a percentile rank (1-100). This function is applicable only to binary response data.

The percentile rank indicates the percentage of scores in the distribution that fall below a given score. For example, a percentile rank of 75 means the student performed better than 75% of the group.

Usage

percentile(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of percentile ranks (1-100) for each student, where:

• 100: Highest performing student(s)

• 50: Median performance

• 1: Lowest performing student(s)

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Percentile ranks are calculated using the empirical cumulative distribution function of standardized scores. Tied scores receive the same percentile rank. The values are rounded up to the nearest integer to provide ranks from 1 to 100.

Examples

# using sample dataset
percentile(J5S10)

Description

The phi coefficient is the Pearson's product moment correlation coefficient between two binary items. This function is applicable only to binary response data. The coefficient ranges from -1 to 1, where 1 indicates perfect positive correlation, -1 indicates perfect negative correlation, and 0 indicates no correlation.

Usage

PhiCoefficient(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of phi coefficients with exametrika class. Each element (i,j) represents the phi coefficient between items i and j. The matrix is symmetric with ones on the diagonal.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
# Calculate Phi-Coefficient using sample dataset J15S500
PhiCoefficient(J15S500)

Description

The calculation results of the exametrika package have an exametrika class attribute. In addition, the class name of the analysis model is also assigned. The models are listed as follows: IRT, LCA, LRA, Biclustering, IRM, LDLRA, LDB, BINET. A plot is made for each model. Although the analysis results are visualized from various perspectives, they correspond by specifying the 'type' variable when plotting.

Usage

## S3 method for class 'exametrika'
plot(
  x,
  type = c("IIC", "ICC", "TIC", "IRP", "TRP", "LCD", "CMP", "FRP", "RMP", "LRD", "Array",
    "FieldPIRP", "LDPSR"),
  items = NULL,
  students = NULL,
  nc = 1,
  nr = 1,
  ...
)

Arguments

Details

• "IRT": Can only have types "ICC", "IIC", "TIC".

• "LCA": Can only have types "IRP", "FRP", "TRP", "LCD", "CMP".

• "LRA": Can only have types "IRP", "FRP", "TRP", "LRD", "RMP".

• "Biclustering": Can only have types "IRP", "FRP", "LCD", "LRD", "CMP", "RMP", "Array".

• "IRM": Can only have types "FRP", "TRP", "Array".

• "LDLRA": Can only have types "IRP", "TRP", "LRD", "RMP".

• "LDB": Can only have types "FRP", "TRP", "LRD", "RMP", "Array", "FieldPIRP".

• "BINET": Can only have types "FRP", "TRP", "LRD", "RMP", "Array", "LDPSR".

Value

Produces different types of plots depending on the class of the input object and the specified type:

• For IRT models: ICC (Item Characteristic Curves), IIC (Item Information Curves), or TIC (Test Information Curves)

• For LCA/LRA models: IRP (Item Reference Profile), TRP (Test Reference Profile), LCD/LRD (Latent Class/Rank Distribution), CMP/RMP (Class/Rank Membership Profile)

• For Biclustering/IRM models: Array plots showing clustering patterns

• For LDLRA/LDB/BINET models: Various network and profile plots specific to each model

The function returns NULL invisibly.

Description

Output format for exametrika Class

Usage

## S3 method for class 'exametrika'
print(x, digits = 3, ...)

Arguments

Value

Prints a formatted summary of the exametrika object to the console, with content varying by object class:

• TestStatistics: Basic descriptive statistics of the test

• Dimensionality: Eigenvalue analysis results with scree plot

• ItemStatistics: Item-level statistics

• CTT: Classical Test Theory reliability measures

• IRT: Item parameters and fit indices

• LCA/LRA: Class/Rank profiles and model fit information

• Biclustering/IRM: Cluster profiles and model diagnostics

• Network models (LDLRA/LDB/BINET): Network visualizations and parameter estimates

Description

internal functions for PSD of Item parameters

Usage

PSD_item_params(model, Lambda, quadrature, marginal_posttheta)

Arguments

Description

The one-parameter logistic model is a model with only one parameter b. This model is a 2PLM model in which a is constrained to 1. This model is also called the Rasch model.

Usage

RaschModel(b, theta)

Arguments

Value

Returns a numeric vector of probabilities between 0 and 1, representing the probability of a correct response given the ability level theta. The probability is calculated using the formula: $P(\theta) = \frac{1}{1 + e^{-(\theta-b)}}$

Description

Prior distribution function with respect to the slope.

Usage

slopeprior(a, m, s, const = 1e-15)

Arguments

Description

to avoid overflow

Usage

softmax(x)

Arguments

Description

The standardized score (z-score) indicates how far a student's performance deviates from the mean in units of standard deviation. This function is applicable only to binary response data.

The score is calculated by standardizing the passage rates:

$Z_i = \frac{r_i - \bar{r}}{\sigma_r}$

where:

• $r_i$ is student i's passage rate

• $\bar{r}$ is the mean passage rate

• $\sigma_r$ is the standard deviation of passage rates

Usage

sscore(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A numeric vector of standardized scores for each student. The scores follow a standard normal distribution with:

• Mean = 0

• Standard deviation = 1

• Approximately 68% of scores between -1 and 1

• Approximately 95% of scores between -2 and 2

• Approximately 99% of scores between -3 and 3

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

The standardization allows for comparing student performance across different tests or groups. A positive score indicates above-average performance, while a negative score indicates below-average performance.

Examples

# using sample dataset
sscore(J5S10)

Description

The Stanine (Standard Nine) scoring system divides students into nine groups based on a normalized distribution. This function is applicable only to binary response data.

These groups correspond to the following percentile ranges:

• Stanine 1: lowest 4% (percentiles 1-4)

• Stanine 2: next 7% (percentiles 5-11)

• Stanine 3: next 12% (percentiles 12-23)

• Stanine 4: next 17% (percentiles 24-40)

• Stanine 5: middle 20% (percentiles 41-60)

• Stanine 6: next 17% (percentiles 61-77)

• Stanine 7: next 12% (percentiles 78-89)

• Stanine 8: next 7% (percentiles 90-96)

• Stanine 9: highest 4% (percentiles 97-100)

Usage

stanine(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A list containing two elements:

stanine

The score boundaries for each stanine level

stanineScore

The stanine score (1-9) for each student

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Stanine scores provide a normalized scale with:

• Mean = 5

• Standard deviation = 2

• Scores range from 1 to 9

• Score of 5 represents average performance

References

Angoff, W. H. (1984). Scales, norms, and equivalent scores. Educational Testing Service. (Reprint of chapter in R. L. Thorndike (Ed.) (1971) Educational Measurement (2nd Ed.). American Council on Education.

Examples

result <- stanine(J15S500)
# View score boundaries
result$stanine
# View individual scores
result$stanineScore

Description

Generating a DAG from data using a genetic algorithm.

Usage

StrLearningGA_BNM(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  seed = 123,
  population = 20,
  Rs = 0.5,
  Rm = 0.005,
  maxParents = 2,
  maxGeneration = 100,
  successiveLimit = 5,
  crossover = 0,
  elitism = 0,
  filename = NULL,
  verbose = TRUE
)

Arguments

Details

This function generates a DAG from data using a genetic algorithm. Depending on the size of the data and the settings, the computation may take a significant amount of computational time. For details on the settings or algorithm, see Shojima(2022), section 8.5

Value

adj

Optimal adjacency matrix

testlength

Length of the test. The number of items included in the test.

TestFitIndices

Overall fit index for the test.See also TestFit

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

crr

correct response ratio

TestFitIndices

Overall fit index for the test.See also TestFit

adj

Adjacency matrix

param

Learned Parameters

CCRR_table

Correct Response Rate tables

Examples

# Perform Structure Learning for Bayesian Network Model using Genetic Algorithm
# Parameters are set for balanced exploration and computational efficiency
StrLearningGA_BNM(J5S10,
  population = 20, # Size of population in each generation
  Rs = 0.5, # 50% survival rate for next generation
  Rm = 0.002, # 0.2% mutation rate for genetic diversity
  maxParents = 2, # Maximum of 2 parent nodes per item
  maxGeneration = 100, # Maximum number of evolutionary steps
  crossover = 2, # Use two-point crossover method
  elitism = 2 # Keep 2 best solutions in each generation
)

Description

Generating a DAG from data using a Population-Based Incremental Learning

Usage

StrLearningPBIL_BNM(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  seed = 123,
  population = 20,
  Rs = 0.5,
  Rm = 0.002,
  maxParents = 2,
  maxGeneration = 100,
  successiveLimit = 5,
  elitism = 0,
  alpha = 0.05,
  estimate = 1,
  filename = NULL,
  verbose = TRUE
)

Arguments

Details

This function performs structural learning using the Population-Based Incremental Learning model(PBIL) proposed by Fukuda et al.(2014) within the genetic algorithm framework. Instead of learning the adjacency matrix itself, the 'genes of genes' that generate the adjacency matrix are updated with each generation. For more details, please refer to Fukuda(2014) and Section 8.5.2 of the text(Shojima,2022).

Value

adj

Optimal adjacency matrix

testlength

Length of the test. The number of items included in the test.

TestFitIndices

Overall fit index for the test.See also TestFit

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

crr

correct response ratio

TestFitIndices

Overall fit index for the test.See also TestFit

param

Learned Parameters

CCRR_table

Correct Response Rate tables

References

Fukuda, S., Yamanaka, Y., & Yoshihiro, T. (2014). A Probability-based evolutionary algorithm with mutations to learn Bayesian networks. International Journal of Artificial Intelligence and Interactive Multimedia, 3, 7–13. DOI: 10.9781/ijimai.2014.311

Examples

# Perform Structure Learning for Bayesian Network Model using PBIL
# (Population-Based Incremental Learning)
StrLearningPBIL_BNM(J5S10,
  population = 20, # Size of population in each generation
  Rs = 0.5, # 50% survival rate for next generation
  Rm = 0.005, # 0.5% mutation rate for genetic diversity
  maxParents = 2, # Maximum of 2 parent nodes per item
  alpha = 0.05, # Learning rate for probability update
  estimate = 4 # Use rounded generational gene method
)

Description

Generating DAG list from data using Population-Based Incremental learning

Usage

StrLearningPBIL_LDLRA(
  U,
  Z = NULL,
  w = NULL,
  na = NULL,
  seed = 123,
  ncls = 2,
  method = "R",
  population = 20,
  Rs = 0.5,
  Rm = 0.002,
  maxParents = 2,
  maxGeneration = 100,
  successiveLimit = 5,
  elitism = 0,
  alpha = 0.05,
  estimate = 1,
  filename = NULL,
  verbose = TRUE
)

Arguments

Details

This function performs structural learning for each classes by using the Population-Based Incremental Learning model(PBIL) proposed by Fukuda et al.(2014) within the genetic algorithm framework. Instead of learning the adjacency matrix itself, the 'genes of genes' that generate the adjacency matrix are updated with each generation. For more details, please refer to Fukuda(2014) and Section 9.4.3 of the text(Shojima,2022).

Value

nobs

Sample size. The number of rows in the dataset.

testlength

Length of the test. The number of items included in the test.

crr

correct response ratio

adj_list

adjacency matrix list

g_list

graph list

referenceMatrix

Learned Parameters.A three-dimensional array of patterns where item x rank x pattern.

IRP

Marginal Item Reference Matrix

IRPIndex

IRP Indices which include Alpha, Beta, Gamma.

TRP

Test Reference Profile matrix.

LRD

latent Rank/Class Distribution

RMD

Rank/Class Membership Distribution

TestFitIndices

Overall fit index for the test.See also TestFit

Estimation_table

Estimated parameters tables.

CCRR_table

Correct Response Rate tables

Studens

Student information. It includes estimated class membership, probability of class membership, RUO, and RDO.

References

Fukuda, S., Yamanaka, Y., & Yoshihiro, T. (2014). A Probability-based evolutionary algorithm with mutations to learn Bayesian networks. International Journal of Artificial Intelligence and Interactive Multimedia, 3, 7–13. DOI: 10.9781/ijimai.2014.311

Examples

# Perform Structure Learning for LDLRA using PBIL algorithm
# This process may take considerable time due to evolutionary optimization
result.LDLRA.PBIL <- StrLearningPBIL_LDLRA(J35S515,
  seed = 123, # Set random seed for reproducibility
  ncls = 5, # Number of latent ranks
  method = "R", # Use rank model (vs. class model)
  elitism = 1, # Keep best solution in each generation
  successiveLimit = 15 # Convergence criterion
)

# Examine the learned network structure
# Plot Item Response Profiles showing item patterns across ranks
plot(result.LDLRA.PBIL, type = "IRP", nc = 4, nr = 3)

# Plot Test Response Profile showing overall response patterns
plot(result.LDLRA.PBIL, type = "TRP")

# Plot Latent Rank Distribution showing student distribution
plot(result.LDLRA.PBIL, type = "LRD")

Description

The StudentAnalysis function returns descriptive statistics for each individual student. Specifically, it provides the number of responses, the number of correct answers, the passage rate, the standardized score, the percentile, and the stanine.

Usage

StudentAnalysis(U, na = NULL, Z = NULL, w = NULL)

Arguments

Value

Returns a data frame containing the following columns for each student:

• ID: Student identifier

• NR: Number of responses

• NRS: Number-right score (total correct answers)

• PR: Passage rate (proportion correct)

• SS: Standardized score (z-score)

• Percentile: Student's percentile rank

• Stanine: Student's stanine score (1-9)

Examples

# using sample dataset
StudentAnalysis(J15S500)

Description

A general function that returns the model fit indices.

Usage

TestFit(U, Z, ell_A, nparam)

Arguments

Value

model_log_like

log likelihood of analysis model

bench_log_like

log likelihood of benchmark model

null_log_like

log likelihood of null model

model_Chi_sq

Chi-Square statistics for analysis model

null_Chi_sq

Chi-Square statistics for null model

model_df

degrees of freedom of analysis model

null_df

degrees of freedom of null model

NFI

Normed Fit Index. Lager values closer to 1.0 indicate a better fit.

RFI

Relative Fit Index. Lager values closer to 1.0 indicate a better fit.

IFI

Incremental Fit Index. Lager values closer to 1.0 indicate a better fit.

TLI

Tucker-Lewis Index. Lager values closer to 1.0 indicate a better fit.

CFI

Comparative Fit Index. Lager values closer to 1.0 indicate a better fit.

RMSEA

Root Mean Square Error of Approximation. Smaller values closer to 0.0 indicate a better fit.

AIC

Akaike Information Criterion. A lower value indicates a better fit.

CAIC

Consistent AIC.A lower value indicates a better fit.

BIC

Bayesian Information Criterion. A lower value indicates a better fit.

Description

A general function that returns the model fit indices.

Usage

TestFitSaturated(U, Z, ell_A, nparam)

Arguments

Value

model_log_like

log likelihood of analysis model

bench_log_like

log likelihood of benchmark model

null_log_like

log likelihood of null model

model_Chi_sq

Chi-Square statistics for analysis model

null_Chi_sq

Chi-Square statistics for null model

model_df

degrees of freedom of analysis model

null_df

degrees of freedom of null model

NFI

Normed Fit Index. Lager values closer to 1.0 indicate a better fit.

RFI

Relative Fit Index. Lager values closer to 1.0 indicate a better fit.

IFI

Incremental Fit Index. Lager values closer to 1.0 indicate a better fit.

TLI

Tucker-Lewis Index. Lager values closer to 1.0 indicate a better fit.

CFI

Comparative Fit Index. Lager values closer to 1.0 indicate a better fit.

RMSEA

Root Mean Square Error of Approximation. Smaller values closer to 0.0 indicate a better fit.

AIC

Akaike Information Criterion. A lower value indicates a better fit.

CAIC

Consistent AIC.A lower value indicates a better fit.

BIC

Bayesian Information Criterion. A lower value indicates a better fit.

Description

Test Information Function for 4PLM

Usage

TestInformationFunc(params, theta)

Arguments

Value

Returns a numeric vector representing the test information at each ability level theta. The test information is the sum of item information functions for all items in the test: $I_{test}(\theta) = \sum_{j=1}^n I_j(\theta)$

Description

Statistics regarding the total score.

Usage

TestStatistics(U, na = NULL, Z = NULL, w = NULL)

Arguments

Value

TestLength

Length of the test. The number of items included in the test.

SampleSize

Sample size. The number of rows in the dataset.

Mean

Average number of correct answers.

SEofMean

Standard error of mean

Variance

Variance

SD

Standard Deviation

Skewness

Skewness

Kurtosis

Kurtosis

Min

Minimum score

Max

Max score

Range

Range of score

Q1

First quartile. Same as the 25th percentile.

Median

Median.Same as the 50th percentile.

Q3

Third quartile. Same as the 75th percentile.

IQR

Interquartile range. It is calculated by subtracting the first quartile from the third quartile.

Stanine

see stanine

Examples

# using sample dataset
TestStatistics(J15S500)

Description

Tetrachoric Correlation is superior to the phi coefficient as a measure of the relation of an item pair. See Divgi, 1979; Olsson, 1979;Harris, 1988.

Usage

tetrachoric(x, y)

Arguments

Value

Returns a single numeric value of class "exametrika" representing the tetrachoric correlation coefficient between the two binary variables. The value ranges from -1 to 1, where:

• 1 indicates perfect positive correlation

• -1 indicates perfect negative correlation

• 0 indicates no correlation

References

Divgi, D. R. (1979). Calculation of the tetrachoric correlation coefficient. Psychometrika, 44, 169–172.

Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika,44, 443–460.

Harris, B. (1988). Tetrachoric correlation coefficient. In L. Kotz, & N. L. Johnson (Eds.), Encyclopedia of statistical sciences (Vol. 9, pp. 223–225). Wiley.

Description

Calculates the matrix of tetrachoric correlations between all pairs of items. Tetrachoric Correlation is superior to the phi coefficient as a measure of the relation of an item pair. This function is applicable only to binary response data.

Usage

TetrachoricCorrelationMatrix(U, na = NULL, Z = NULL, w = NULL, ...)

Arguments

Value

A matrix of tetrachoric correlations with exametrika class. Each element (i,j) represents the tetrachoric correlation between items i and j. The matrix is symmetric with ones on the diagonal.

Note

This function is implemented using a binary data compatibility wrapper and will raise an error if used with polytomous data.

Examples

# example code
TetrachoricCorrelationMatrix(J15S500)

Description

The three-parameter logistic model is a model where the lower asymptote parameter c is added to the 2PLM

Usage

ThreePLM(a, b, c, theta)

Arguments

Value

Returns a numeric vector of probabilities between c and 1, representing the probability of a correct response given the ability level theta. The probability is calculated using the formula: $P(\theta) = c + \frac{1-c}{1 + e^{-a(\theta-b)}}$

Description

The two-parameter logistic model is a classic model that defines the probability of a student with ability theta successfully answering item j, using both a slope parameter and a location parameter.

Usage

TwoPLM(a, b, theta)

Arguments

Value

Returns a numeric vector of probabilities between 0 and 1, representing the probability of a correct response given the ability level theta. The probability is calculated using the formula: $P(\theta) = \frac{1}{1 + e^{-a(\theta-b)}}$