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tppcm

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 a_il:

P(X_i = k | θ) ∝ exp( Σ_{l ≤ k} (a_il θ − d_il) ), 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.

Installation

# install.packages("remotes")
remotes::install_github("jiewenTsai/tppcm", build_vignettes = TRUE)

Features

• Step designs for TAM. 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, a_il = α_i γ_l) is fitted with xxirt_tppcm(). Any other equality pattern goes through index.
• Parameter table. irt_pars() prints step discriminations and difficulties with standard errors and a model-fit header.
• Model fit. m2() gives limited-information fit statistics (M2, RMSEA, SRMSR, CFI, TLI) for any step design.
• Step information. step_info() splits item information into the contributions of each step.
• Tests between models. 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.
• Rank-1. xxirt_tppcm(dat, design = ~ item + step) fits a_il = α_i γ_l and non-negativity bounds through sirt::xxirt().
• Differential step functioning. On a multigroup fit, 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.
• Components. get_parts() extracts the quantities behind these functions (scores, information, Jacobian, posterior and more).
• Simulation. sim_tppcm() generates responses from a TPPCM.

Workflow

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 steps

Documentation

Two 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").

References

• Yu, M.-N. (1991). A two-parameter partial credit model (Doctoral dissertation). University of Illinois at Urbana-Champaign.
• Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
• Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159-176.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.