Boundary-level diagnostics for the two-parameter partial credit model (TPPCM; Yu, 1991), built on top of TAM.
In the TPPCM every step l of item i has its own discrimination ail:
P(Xi = k | θ) ∝ exp( Σl ≤ k (ail θ − dil) ), k = 0, …, K−1.
The generalized partial credit model (GPCM; Muraki, 1992) is the special case with equal steps within items, and the partial credit model (PCM; Masters, 1982) has one common discrimination.
Estimation stays in TAM (and in sirt for the product form). tppcm takes the fitted object and adds what TAM does not report at the level of category boundaries: step discriminations (CBD), locations (CBL) and intercepts (IBD) with standard errors, tests between models, fit statistics, step information and tests of differential step functioning (DSF) across groups.
# install.packages("remotes")
remotes::install_github("jiewenTsai/tppcm", build_vignettes = TRUE)tppcm(dat, design = ) builds the design array for
tam.mml.3pl(). The design formula reads the step
discriminations as an item x step layout on the log scale:
~ item * step (saturated TPPCM, default),
~ item (GPCM), ~ step (steps common to the
items), ~ 1 (PCM); ~ item + step (Rank-1,
ail = αi γl) is fitted with
xxirt_tppcm(). Any other equality pattern goes through
index.irt_pars() prints
step discriminations and difficulties with standard errors and a
model-fit header.m2() gives
limited-information fit statistics (M2, RMSEA, SRMSR, CFI, TLI) for any
step design.step_info() splits
item information into the contributions of each step.wald_test()
tests equal steps within an item. score_test(),
item_test() and mi() test a fitted model
against a larger one without fitting it.xxirt_tppcm(dat, design = ~ item + step) fits
ail = αi γl and non-negativity bounds
through sirt::xxirt().dif_test(fit, g, by = "step") screens every category
boundary with a joint score (LM) test; dsf() decomposes
flagged steps into CBD and location differences by likelihood ratio
tests; dsf_design() builds configural, metric, scalar and
partial multigroup models (lavaan-style group.equal /
group.partial). tppcmtree() grows DIF trees
when the grouping is unknown. Tables of tests report Holm-adjusted
p_adj.get_parts() extracts the
quantities behind these functions (scores, information, Jacobian,
posterior and more).sim_tppcm() generates
responses from a TPPCM.Fit with TAM, then pass the fitted object to tppcm.
library(tppcm) # attaches TAM
set.seed(1)
a <- outer(c(0.8, 1.0, 1.2, 1.5, 1.0), c(1, 1.5, 0.7))
b <- matrix(c(-1, 0, 1), 5, 3, byrow = TRUE)
dat <- sim_tppcm(1000, disc = a, diff = b)
# 1. Fit the models with TAM
m_gpcm <- tam.mml.2pl(dat, irtmodel = "GPCM", verbose = FALSE)
m_tppcm <- tam.mml.3pl(dat, E = tppcm(dat), est.variance = FALSE, verbose = FALSE,
control = list(maxiter = 3000, max.increment = 0.1))
# 2. Parameters, fit and step-level tests
irt_pars(m_tppcm)
m2(m_tppcm)
wald_test(m_tppcm)
# 3. Where does the GPCM fail?
sp <- get_parts(m_gpcm)
score_test(sp)
item_test(sp)
mi(sp)
# 4. Rank-1: is there a step effect, an item x step interaction?
score_test(sp, against = ~ item + step) # GPCM vs Rank-1
m_rank1 <- xxirt_tppcm(dat, design = ~ item + step)
irt_pars(m_rank1, restricted = TRUE)
score_test(m_rank1) # Rank-1 vs saturated
# 5. Differential step functioning between two groups
g <- rep(c("A", "B"), each = 500)
m_mg <- tam.mml.3pl(dat, E = tppcm(dat), group = g, est.variance = FALSE, verbose = FALSE)
st <- dif_test(m_mg, g, by = "step") # screen every step (2 df)
flagged <- st$unit[st$p_adj < 0.05]
if (length(flagged)) dsf(m_mg, items = flagged) # CBD vs location of flagged stepsTwo vignettes:
vignette("tppcm-tutorial", package = "tppcm"): the
model and the package.vignette("tppcm-recipes", package = "tppcm"): short
recipes for every function.Traditional Chinese versions, with the same code, are in the GitHub
repository (not in the CRAN package): tppcm-tutorial-zh
and tppcm-recipes-zh.
Cite with citation("tppcm").