Compute Maximum Possible Alpha Value

Description

This function calculates the maximum possible value of Cronbach's alpha by using a binary search algorithm with optimization methods. The function iteratively narrows the bounds of alpha until the desired tolerance level is reached.

Usage

compute_alpha_max(
  n_person,
  sigma_x_info,
  sigma_y_info,
  score_max = 1,
  alpha_lb = 0,
  alpha_ub = 1,
  tol = 0.001,
  num_try = 1,
  method = "GA",
  ...
)

Arguments

Details

This function finds the maximum possible Cronbach's alpha by using an iterative binary search algorithm. It evaluates the feasibility of each midpoint value of alpha by solving the corresponding optimization problem with the chosen method.

The optimization methods can be specified via the method parameter, and additional control parameters for the optimization methods can be passed through the ... argument. The function adjusts the upper and lower bounds of alpha until the tolerance criterion is met.

Value

A numeric value representing the maximum possible Cronbach's alpha.

See Also

compute_alpha_min, examine_alpha_bound

Compute Minimum Possible Alpha Value

Description

This function calculates the minimum possible value of Cronbach's alpha by using a binary search algorithm with optimization methods. The function iteratively narrows the bounds of alpha until the desired tolerance level is reached.

Usage

compute_alpha_min(
  n_person,
  sigma_x_info,
  sigma_y_info,
  score_max = 1,
  alpha_lb = 0,
  alpha_ub = 1,
  tol = 0.001,
  num_try = 1,
  method = "GA",
  ...
)

Arguments

Details

This function finds the minimum possible Cronbach's alpha by using an iterative binary search algorithm. It evaluates the feasibility of each midpoint value of alpha by solving the corresponding optimization problem with the chosen method.

The optimization methods can be specified via the method parameter, and additional control parameters for the optimization methods can be passed through the ... argument. The function adjusts the upper and lower bounds of alpha until the tolerance criterion is met.

Value

A numeric value representing the minimum possible Cronbach's alpha.

See Also

compute_alpha_max, examine_alpha_bound

Compute Exact Bounds of Cronbach's Alpha via Enumeration

Description

This function computes the minimum and maximum possible values of Cronbach's alpha by enumerating all possible values for unknown entries in the score matrix.

Usage

cronbach_alpha_enum(scores_mat, score_max)

Arguments

Details

This function works by enumerating all possible combinations of values for missing entries (represented by NA) in the scores_mat. It systematically explores all combinations of missing values from 0 to score_max using the expand.grid function. For each combination, it calculates Cronbach's alpha using the compute_cronbach_alpha function and keeps track of the minimum and maximum alpha values encountered.

The enumeration ensures that the function finds the exact minimum and maximum possible values of Cronbach's alpha given the possible missing score combinations. However, due to the exhaustive nature of enumeration, this function may become computationally expensive for large datasets or a high number of missing values.

Value

A numeric vector of length 2, where the first element is the minimum Cronbach's alpha and the second element is the maximum Cronbach's alpha.

See Also

cronbach_alpha_rough

Compute Rough Approximation of Cronbach's Alpha Bounds

Description

This function computes a rough approximation of the lower and upper bounds of Cronbach's alpha by performing random sampling or integer sampling for missing values in the score matrix.

Usage

cronbach_alpha_rough(scores_mat, score_max, num_try = 1000, int_only = FALSE)

Arguments

Details

This function performs random sampling to estimate the bounds of Cronbach's alpha for a given test score matrix with missing values. It first calculates the alpha assuming all missing values are either 0 or score_max. Then, it iteratively samples either integer values or continuous values (depending on the value of int_only) for the missing scores and recalculates the Cronbach's alpha. The minimum and maximum alphas observed over all iterations are returned as the estimated bounds.

Value

A numeric vector of length 2, where the first value is the estimated minimum Cronbach's alpha and the second value is the estimated maximum Cronbach's alpha.

See Also

cronbachs_alpha

Compute Lower and Upper Bound of Cronbach's Alpha

Description

This function computes the lower and upper bound of Cronbach's alpha using various methods such as enumeration, random sampling, or optimization algorithms. The function also supports rough approximations and allows integer-only or floating-point scores during sampling.

Usage

cronbachs_alpha(
  scores_mat,
  score_max,
  tol = 0.001,
  num_random = 1000,
  enum_all = FALSE,
  rough = FALSE,
  num_opt = 1,
  int_only = TRUE,
  method = "GA",
  ...
)

Arguments

Value

A list containing:

See Also

compute_alpha_min, compute_alpha_max, cronbach_alpha_enum, cronbach_alpha_rough, generate_scores_mat_bernoulli, qp_solver

Examples

# Example 1: Run `cronbachs_alpha` with a sample matrix
scores_mat <- matrix(c(
  NaN, 1, 0, 0, 0, 0, 0, 0, NaN, 0, 0, 0,
  2, 0, 0, 1, NaN, 0, 0, 0, 0, 0, 0, 0,
  1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1
), nrow = 10, ncol = 4, byrow = TRUE)

result <- cronbachs_alpha(scores_mat, score_max = 4, enum_all = FALSE)
print(result$alpha_min_opt)
print(result$alpha_max_opt)

# Example 2: Generate a Bernoulli matrix and compute Cronbach's alpha
score_max <- 2
scores_mat_bernoulli <- generate_scores_mat_bernoulli(50, 10, 20, score_max)
result <- cronbachs_alpha(scores_mat_bernoulli, score_max, enum_all = FALSE)
print(result$alpha_min_opt)
print(result$alpha_max_opt)

# Example 3: Using a predefined dataset from missalpha
scores_df <- missalpha::sample
scores_mat <- as.matrix(scores_df)
result <- cronbachs_alpha(scores_mat, score_max = 4, enum_all = FALSE)
print(result$alpha_min_opt)
print(result$alpha_max_opt)

Display All Possible Parameter Combinations for Cronbach's Alpha

Description

This function computes and displays all possible combinations of Cronbach's alpha bounds for various parameter settings, including combinations of optimization methods, rough approximation, random sampling, and enumeration.

Usage

display_all(
  scores_mat,
  score_max,
  tol = 0.001,
  num_random = 1000,
  num_opt = 1,
  methods = c("GA", "DEoptim", "nloptr"),
  enum_all = FALSE,
  rough = TRUE
)

Arguments

Value

A list where each element is a result of the cronbachs_alpha function for a unique parameter combination, including computation time.

Check Feasibility of Alpha Bound for Optimization Problem

Description

This function checks whether a given value of alpha is a feasible solution to a min/max optimization problem using quadratic functions for sigma_x and sigma_y. The function supports different optimization methods and iteratively attempts to solve the problem.

Usage

examine_alpha_bound(
  alpha,
  n_person,
  sigma_x_info,
  sigma_y_info,
  alpha_type,
  score_max = 1,
  num_try = 1,
  method = "GA",
  ...
)

Arguments

Details

The function combines quadratic information from sigma_x_info and sigma_y_info to form a new optimization problem. The optimization checks whether the value of alpha is feasible for either a minimization or maximization problem, depending on the value of alpha_type.

The function supports multiple optimization methods, including Genetic Algorithm (GA), Differential Evolution (DEoptim), and Sequential Least Squares Programming (nloptr). Additional control parameters can be passed through the ... argument to fine-tune the optimization process.

For each iteration, the function calls qp_solver with the combined quadratic function and checks whether the objective function's value is feasible (i.e., less than or equal to 0).

Value

A list with the following elements:

See Also

qp_solver, compute_alpha_min, compute_alpha_max

Generate Bernoulli Distributed Scores Matrix with Missing Values

Description

This function generates a matrix of scores for a set of people and items, where the scores are generated using a Bernoulli distribution with person-specific probabilities. It also allows for some scores to be missing (represented by NA).

Usage

generate_scores_mat_bernoulli(n_person, n_item, n_missing, score_max = 1)

Arguments

Details

The function generates a score matrix where each person's score for each item is drawn from a Bernoulli distribution with a person-specific probability. A number of scores are set to NA to simulate missing values.

The probability of each person scoring on the items is determined by randomly generating a probability for each person using runif. The Bernoulli distribution is then used (via rbinom) to generate the scores, and NA values are assigned to randomly selected positions in the matrix based on n_missing.

Value

A matrix of size n_person by n_item containing generated scores (0 or score_max) with some values replaced by NA to simulate missing data.

Examples

# Generate a 10x5 score matrix with 10 missing values and maximum score of 1
scores_mat <- generate_scores_mat_bernoulli(10, 5, 10, score_max = 1)
print(scores_mat)

General Solver for Quadratic Programming Problems

Description

This function provides a general interface to solve quadratic programming problems using different optimization methods. It supports Genetic Algorithm (GA), Differential Evolution (DEoptim), and Sequential Least Squares Programming (SLSQP).

Usage

qp_solver(n, A, b, c, x_max = 1, method = "GA", print_message = FALSE, ...)

Arguments

Value

A list containing:

See Also

qp_solver_DEoptim, qp_solver_GA, qp_solver_nloptr

Solve Quadratic Programming Problem using DEoptim

Description

This function solves a quadratic programming problem using the Differential Evolution optimization method from the DEoptim package.

Usage

qp_solver_DEoptim(
  n,
  A,
  b,
  c,
  x_max = 1,
  print_message = FALSE,
  NP = 100,
  itermax = 100,
  ...
)

Arguments

Value

A list containing:

See Also

qp_solver, qp_solver_GA, qp_solver_nloptr

Solve Quadratic Programming Problem using GA

Description

This function solves a quadratic programming problem using the Genetic Algorithm (GA) from the GA package.

Usage

qp_solver_GA(
  n,
  A,
  b,
  c,
  x_max = 1,
  print_message = FALSE,
  maxiter = 1000,
  popSize = 50,
  pmutation = 0.2,
  elitism = 5,
  monitor = FALSE,
  seed = 123,
  ...
)

Arguments

Value

A list containing:

See Also

qp_solver, qp_solver_DEoptim, qp_solver_nloptr

Solve Quadratic Programming Problem using nloptr

Description

This function solves a quadratic programming problem using the Sequential Least Squares Programming (SLSQP) algorithm from the nloptr package.

Usage

qp_solver_nloptr(
  n,
  A,
  b,
  c,
  x_max = 1,
  print_message = FALSE,
  xtol_rel = 1e-08,
  maxeval = 10000,
  print_level = 0,
  ...
)

Arguments

Value

A list containing:

See Also

qp_solver, qp_solver_DEoptim, qp_solver_GA

Sample Dataset with Missing Values

Description

This dataset contains a matrix of scores with 50 rows and 4 columns, representing 50 individuals and 4 test items. Some entries are NA, indicating missing data.

Usage

sample

Format

A 50 x 4 matrix:

Rows

Each row represents an individual (total 50 individuals).

Columns

Each column represents a test item or score (total 4 items).

Missing values

Some entries are NA, representing missing data.

Source

Generated for demonstration purposes.

Examples

# Load the sample dataset
data(sample)

# Display the first few rows of the sample dataset
head(sample)