irt (version 0.2.9)

prob_sum_score: Calculate summed-score probabilities

Description

This function calculates all summed-score probabilities of a given theta value(s) using recursive algorithm described in Thissen, Pommerich, Billeaud and Williams (1995). This function is the extension of the recursive algorithm proposed by Lord and Wingersky (1984) to polytomous items.

Usage

prob_sum_score(ip, theta, theta_pdf = NULL)

Value

A matrix containing the probabilities of each possible sum score. Each row represent a sum score and each column represent the theta value provided by theta argument.

Arguments

ip

An Itempool-class object. Item pool parameters can be composed of any combination of unidimensional dichotomous or polytomous items.

theta

A numeric vector representing the theta values at which the sum score probabilities will be calculated.

theta_pdf

A numeric vector with the same length of theta argument representing the density values of each theta value. The resulting probabilities will be weighted by these values. The default value is NULL where the resulting probabilities will not be weighted.

Author

Emre Gonulates

References

Kolen, M. J., & Brennan, R. L. (2014). Test equating, scaling, and linking: Methods and practices. Springer Science & Business Media.

Lord, F. M., & Wingersky, M. S. (1984). Comparison of IRT true-score and equipercentile observed-score" equatings". Applied Psychological Measurement, 8(4), 453-461.

Thissen, D., Pommerich, M., Billeaud, K., & Williams, V. S. (1995). Item response theory for scores on tests including polytomous items with ordered responses. Applied Psychological Measurement, 19(1), 39-49.

Examples

### Example with weighting ###
ip <- generate_ip(model = sample(c("GPCM", "2PL"), 10, TRUE))
theta <- c(-3, -1.2, 0.5, 3)
prob_sum_score(ip, theta = theta)
# Most probable sum scores:
apply(prob_sum_score(ip, theta = theta), MARGIN = 2, which.max) - 1
if (FALSE) {
   plot(ip, type = "tcc", suppress_plot = TRUE) +
     ggplot2::geom_vline(xintercept = theta, lty = "dashed")
}
### Example from Kolen and Brennan (2014) ###
# Item parameters from Kolen and Brennan (2014), p.175, Table 6.1.
ip <- itempool(a = c(1.30, .6, 1.7),
               b = c(-1.30, -.10, .9),
               c = c(.1, .17, .18),
               D = 1.7)
prob(ip, theta = c(-2, 1))
# IRT observed score distribution using recursive formula from
# Kolen and Brennan (2014), p.200, Table 6.4.
# Numbers are not exactly the same as Kolen and Brennan since due to
# rounding applied to the numbers in the book.
prob_sum_score(ip, theta = -2)


### Example from Thissen, Pommerich, Billeaud and Williams (1995) ###
# Replicating Thissen et al. (1995) example, p.43-44, Table 1.
i1 <- item(a = .5, b = -1)
i2 <- item(a = 1, b = 0)
i3 <- item(a = 1.5, b = 1)
ip <- c(i1, i2, i3) # combine items to form an item pool
theta <- -3:3 # Quadrature points

prob_sum_score(ip, theta)

# Item parameters in Table 2
i1 <- item(a = 1.87, b = c(.65, 1.97, 3.14), model = "GRM")
i2 <- item(a = 2.66, b = c(.12, 1.57, 2.69), model = "GRM")
i3 <- item(a = 1.24, b = c(.08, 2.03, 4.30), model = "GRM")
ip <- c(i1, i2, i3)
delta <- 0.01
theta <- seq(-3, 3, delta)

x <- prob_sum_score(ip = ip, theta = theta, theta_pdf = dnorm(theta))

# Figure 1
plot(x = theta, y = x[2, ], type = "l", ylab = "Posterior Density",
     xlab = "Theta",
     main = paste0("Posterior Distribution for all Examinees Obtaining ",
                   "a Summed Score of 1"))

# Table 3, column "Modeled Score Group Proportion"
rowSums(x)/sum(rowSums(x))