Implementing the Prowise Learn Update Algorithm
In meow: Unified Framework for Computer Adaptive Testing Simulations

knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(meow)

The Prowise Learn algorithm (Vermeiren et al., 2025) extends the Elo-style Maths Garden updates (vignette("maths-garden-update")) with paired item updates that counteract rating drift --- the tendency for item difficulty estimates to slide systematically over time.

Mathematical foundation

Abilities are updated exactly as in Maths Garden:

$$\theta_j^{new} = \theta_j + K_\theta \sum_{i \in I_j} (S_{ij} - E(S_{ij})).$$

Item difficulties, however, are updated in consecutive pairs of items administered to the same respondent. For a pair (previous item, current item),

$$\kappa = 0.5\,\big(K_b (S_{now} - E_{now}) - K_b (S_{prev} - E_{prev})\big), \qquad b_{now} \mathrel{+}= \kappa, \quad b_{prev} \mathrel{-}= \kappa.$$

Because each pair adds $+\kappa$ to one item and $-\kappa$ to the other, the total difficulty mass is conserved, so items keep their relative positions and do not drift en masse. Expected responses use the Rasch model, $E(S_{ij}) = 1 / (1 + e^{-(\theta_j - b_i)})$.

Implementation in `meow`

Paired updates are inherently order dependent, so update_prowise_learn() uses meow_long(R, admin), which returns the administered responses ordered by respondent and then by administration order. Consecutive within-respondent rows form the pairs; the per-item contributions are aggregated with tapply():

update_prowise_learn <- function(pers, item, R, admin, K_theta = 0.1, K_b = 0.1) {
  long  <- meow_long(R, admin)
  E_Sij <- stats::plogis(pers$theta[long$id] - item$b[long$item])

  # ability update (as in Maths Garden)
  dtheta <- tapply(long$resp - E_Sij, long$id, sum)
  pers$theta[as.integer(names(dtheta))] <-
    pers$theta[as.integer(names(dtheta))] + K_theta * dtheta

  # paired item updates over consecutive administrations
  n <- nrow(long)
  if (n >= 2) {
    nxt <- 2:n; prv <- 1:(n - 1)
    pair <- which(long$id[nxt] == long$id[prv])
    if (length(pair) > 0) {
      now <- nxt[pair]; pre <- prv[pair]
      kappa <- 0.5 * (K_b * (long$resp[now] - E_Sij[now]) -
                      K_b * (long$resp[pre] - E_Sij[pre]))
      add_now <- tapply(kappa,  long$item[now], sum)
      add_pre <- tapply(-kappa, long$item[pre], sum)
      item$b[as.integer(names(add_now))] <- item$b[as.integer(names(add_now))] + add_now
      item$b[as.integer(names(add_pre))] <- item$b[as.integer(names(add_pre))] + add_pre
    }
  }
  list(pers = pers, item = item)
}

Using it

sim <- meow(
  select_fun  = select_max_info,
  update_fun  = update_prowise_learn,
  data_loader = data_simple_1pl,
  data_args   = list(N_persons = 100, N_items = 50),
  update_args = list(K_theta = 0.05, K_b = 0.05)
)
head(sim$results[, 1:3])

Practical notes

Paired updates require respondents to answer at least two items, so the item difficulties only begin to move once administration is under way.
Effectiveness depends on the administration order; this is exactly why the matrix admin carries the order of administration.
As with Maths Garden, keep learning rates modest and consider bounding the estimates for stability.