Compute Modified Item Response Probabilities

Description

Compute Modified Item Response Probabilities

Usage

compute_Modified_probabilities(results, theta_all)

Arguments

Value

A numeric matrix of predicted probabilities with dimensions length(theta_all) rows by nrow(results) columns. Each cell contains the probability of a correct response for a given ability level and item, calculated using the logistic function: exp(\theta - \beta) / (1 + exp(\theta - \beta)), where \beta is the item threshold. Columns are named after the items.

Examples

set.seed(123)
sample_data <- matrix(sample(c(0, 1), 100, replace = TRUE), ncol = 10)
prepared <- prepare_data(sample_data)
irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)

abilities <- seq(0, 10, length.out = 5)

probs <- compute_Modified_probabilities(irt_results, abilities)

print(head(probs))

Extract Rasch Item Difficulties in Original Order

Description

Extract Rasch Item Difficulties in Original Order

Usage

extract_rasch_difficulties_ordered(final_logit_matrix)

Arguments

Value

A data frame containing the calculated difficulty parameters for each item, with the following columns:

• Item_Number: The sequential index of the item.

• Item_Name: The name or label of the item.

• Difficulty_Logit: The item difficulty parameter (beta) expressed on the logit scale, rounded to four decimal places. Higher values indicate more difficult items.

Examples

set.seed(123)
sample_data <- matrix(sample(c(0, 1), 100, replace = TRUE), ncol = 10)
colnames(sample_data) <- paste0("Q", 1:10)
prepared <- prepare_data(sample_data)
irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)
probs <- compute_Modified_probabilities(irt_results, prepared$total_score)
logits <- rasch_logit(probs)

difficulty_table <- extract_rasch_difficulties_ordered(logits)

print(difficulty_table)

Fit Binary IRT Model using GLM

Description

Fit Binary IRT Model using GLM

Usage

fit_binary_irt(data_mat, total_score)

Arguments

Value

A data frame with one row per item and the following columns:

• Item: The name of the item.

• Intercept: The estimated intercept parameter from the GLM logit model.

• Slope: The estimated slope parameter (discrimination) from the GLM logit model.

• threshold: The calculated item difficulty (also known as the beta parameter), computed as -Intercept / Slope. This represents the point on the ability scale where the probability of a correct response is 0.5.

Examples

set.seed(42)
sample_data <- matrix(sample(c(0, 1), 50, replace = TRUE), ncol = 5)
colnames(sample_data) <- paste0("Item", 1:5)

prepared <- prepare_data(sample_data)

irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)

print(irt_results)

Plot Item Characteristic Curves (ICC)

Description

Plot Item Characteristic Curves (ICC)

Usage

plot_item_curves(theta_all, Modified_prob_matrix, results)

Arguments

Value

No return value, called for side effects. It generates a series of plots showing the Item Characteristic Curves (ICC) for each item in the model.

Examples

set.seed(123)
sample_data <- matrix(sample(c(0, 1), 100, replace = TRUE), ncol = 10)
prepared <- prepare_data(sample_data)
irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)

theta_range <- seq(0, 10, length.out = 100)
probs <- compute_Modified_probabilities(irt_results, theta_range)

# Plot the curves
plot_item_curves(theta_range, probs, irt_results)

Plot Rasch Item Curves

Description

Plot Rasch Item Curves

Usage

plot_rasch_curves(prob_matrix, final_logit_matrix)

Arguments

Value

No return value, called for side effects. It generates a series of plots showing the Rasch Item Characteristic Curves (ICC) with difficulty indicators.

Examples

set.seed(123)
sample_data <- matrix(sample(c(0, 1), 100, replace = TRUE), ncol = 10)
prepared <- prepare_data(sample_data)
irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)
probs <- compute_Modified_probabilities(irt_results, prepared$total_score)
logits <- rasch_logit(probs)

# Plot the Rasch curves
plot_rasch_curves(probs, logits)

Prepare Data for IRT Analysis

Description

Prepare Data for IRT Analysis

Usage

prepare_data(data)

Arguments

Value

A list containing the following two components:

• matrix: A numeric matrix of the original data, converted to numeric format and potentially reordered by total score.

• total_score: A numeric vector representing the sum of correct responses (row sums) for each person, sorted in descending order if the original data was unsorted.

Examples

set.seed(123)
raw_data <- data.frame(
  item1 = c(1, 0, 1, 1),
  item2 = c(0, 0, 1, 1),
  item3 = c(1, 1, 1, 0)
)

prepared <- prepare_data(raw_data)

print(prepared$matrix)
print(prepared$total_score)

Compute Logit and Row Means

Description

Compute Logit and Row Means

Usage

rasch_logit(prob_matrix)

Arguments

Value

A numeric matrix of logit transformations (log-odds) with dimensions nrow(prob_matrix) rows by ncol(prob_matrix) + 1 columns. The first columns contain the logit values for each item, and the final column, named Row_Mean_Logit, contains the mean logit across all items for each person, which serves as an estimate of Student Ability.

Examples

set.seed(123)
sample_data <- matrix(sample(c(0, 1), 100, replace = TRUE), ncol = 10)
prepared <- prepare_data(sample_data)
irt_results <- fit_binary_irt(prepared$matrix, prepared$total_score)
probs <- compute_Modified_probabilities(irt_results, prepared$total_score)

logit_results <- rasch_logit(probs)

print(head(logit_results))