# Lord-Wingersky Recursion Formula This function computes the conditional distributions of number-correct (or observed) scores given either the probabilities of category responses for each item or a set of theta values, using the Lord and Wingersky recursion formula (1984). ```r lwrc(x = NULL, theta, prob = NULL, cats, D = 1) ``` ## Arguments - `x`: A data frame containing item metadata (e.g., item parameters, number of categories, IRT model types, etc.). See `est_irt()` or `simdat()` for more details about the item metadata. This data frame can be easily created using the `shape_df()` function. If `prob` is `NULL`, the item metadata in `x` is used in the recursion formula. See Details below. - `theta`: A vector of theta values at which the conditional distributions of observed scores are computed. This argument is required only when item metadata is provided via the `x` argument. - `prob`: A matrix containing the category response probabilities for each item. Each row corresponds to an item, and each column represents a score category. If items have different numbers of categories, empty cells should be filled with zeros or `NA` values. If `x` is `NULL`, this matrix is used in the recursion formula. - `cats`: A numeric vector specifying the number of score categories for each item. For example, a dichotomous item has two categories. This argument is required only when a probability matrix is provided via the `prob` argument. - `D`: A scaling constant used in IRT models to make the logistic function closely approximate the normal ogive function. A value of 1.7 is commonly used for this purpose. Default is 1. ## Returns When the `prob` argument is provided, the function returns a vector of probabilities for all possible observed scores on a test. When the `x` argument is specified, it returns a matrix of conditional probabilities for each observed score across all specified theta values. ## Details The Lord and Wingersky recursive algorithm provides an efficient method for calculating the compound probabilities of all possible number-correct (i.e., observed) scores on a test, based on IRT models. This algorithm is particularly useful for obtaining the IRT model-based distribution of observed scores. The conditional distributions of observed scores can be computed using either the item metadata specified in `x` or the category probability matrix specified in `prob`. ## Examples ```r ## Example 1: Using a matrix of category probabilities ## This example is from Kolen and Brennan (2004, p. 183) # Create a matrix of probabilities for correct and incorrect responses to three items probs <- matrix(c(.74, .73, .82, .26, .27, .18), nrow = 3, ncol = 2, byrow = FALSE) # Create a vector specifying the number of score categories for each item cats <- c(2, 2, 2) # Compute the conditional distribution of observed scores lwrc(prob = probs, cats = cats) ## Example 2: Using a matrix of category probabilities for a mixed-format test # Category probabilities for a dichotomous item p1 <- c(0.2, 0.8, 0, 0, 0) # Category probabilities for another dichotomous item p2 <- c(0.4, 0.6, NA, NA, NA) # Category probabilities for a polytomous item with five categories p3 <- c(0.1, 0.2, 0.2, 0.4, 0.1) # Category probabilities for a polytomous item with three categories p4 <- c(0.5, 0.3, 0.2, NA, NA) # Combine the probability vectors into a matrix p <- rbind(p1, p2, p3, p4) # Create a vector specifying the number of score categories for each item cats <- c(2, 2, 5, 3) # Compute the conditional distribution of observed scores lwrc(prob = p, cats = cats) ## Example 3: Using a data frame of item metadata for a mixed-format test # Import the "-prm.txt" output file from flexMIRT flex_prm <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ") # Read item parameters and convert them to item metadata x <- bring.flexmirt(file = flex_prm, "par")$Group1$full_df # Compute the conditional distribution of observed scores for a range of theta values lwrc(x = x, theta = seq(-4, 4, 0.2), D = 1) ``` ## References Kolen, M. J. & Brennan, R. L. (2004) **Test Equating, Scaling, and Linking** (2nd ed.). New York: Springer. Lord, F. & Wingersky, M. (1984). Comparison of IRT true score and equipercentile observed score equatings. **Applied Psychological Measurement, 8**(4), 453-461. ## Author(s) Hwanggyu Lim hglim83@gmail.com

lwrc function