Compute person parameters for the GPCM and choose between five common estimation techniques: ML, WL, MAP, EAP and a robust estimation. All item parameters are treated as fixed.
Arguments
- respm
An integer matrix (or a data.frame), which contains the responses of the examinees. A persons x items matrix is expected. Missing responses are
NA.- thres
A numeric matrix which contains the threshold parameter for each item. If the first row of the matrix is not set to zero (only zeroes in the first row) - then a row-vector with zeroes is added by default.
- slopes
A numeric vector, which contains the slope parameters for each item - one parameter per item is expected. A single value is used for all items. If
NULL(default), all slopes are 1.- theta_start
A vector which contains a starting value for each person. If NULL is submitted, the starting values are set automatically. If a scalar is submitted, this start value is used for each person.
- mu
A numeric vector of location parameters for each person, the normal prior for MAP and EAP estimation and for
PV. A single value is used for all persons. If nothing is submitted this is set to 0 for each person. MLE, WLE and robust estimation have no prior: a value given there is ignored with a warning, and the result storesNA.- sigma2
A numeric vector of variance parameters for each person, the normal prior for MAP and EAP estimation and for
PV. A single value is used for all persons. If nothing is submitted this is set to 1 for each person. MLE, WLE and robust estimation have no prior: a value given there is ignored with a warning, and the result storesNA.- type
Which maximization should be applied? There are five valid entries possible: "mle", "wle", "map", "eap" and "robust". To choose between the methods, or just to get a deeper understanding the papers mentioned below are quite helpful. The default is
"wle"which is a good choice in many cases. With"eap"the estimate is the mean and the SE the standard deviation of the posterior, computed on a grid of ability values (see PV). Plausible values (PV) need the prior of"eap"or"map".- maxsteps
The maximum number of Newton steps per person. Default = 100.
- exac
How accurate are the estimates supposed to be? The iteration of a person stops when the step is smaller than
exac. Default is 0.001.- H
In case
type = "robust"a Huber ability estimate is performed, andHmodulates how fast the downweighting takes place (for more Details read Schuster & Yuan 2011).- ctrl
No longer used. Its options
killdupliandskipcheckof version 0.6.4 saved time with the old estimation routine; since 1.0.0 every person is estimated separately, and options given are ignored with a warning.- range
The interval
c(lower, upper)in which mle, wle, map and robust estimates are searched. If the estimating equation has no root inside, the estimate is set to the bound it points to and its status is"boundary". Defaultc(-6, 6).- verbose
If
TRUE, a message names the model and the estimation type. DefaultFALSE.
Value
An object of class ppeo (and fourpl, gpcm or gpcm4pl), a list with
estimatesA data.frame with one row per person, in the order of
respm:estimate,SE,nsteps(Newton steps of the person; 0 for eap) andstatus, a factor with the levels"ok": converged insiderange,"extreme": mle or robust estimate of a person with all answers in the lowest or highest category, the estimate is-InforInfand the SENA,"boundary": the estimating equation has no root insiderange, the estimate is the bound,"no_response": the person answered no item, estimate and SE areNA(eap: prior mean and SD),"not_converged": no convergence withinmaxsteps.
A warning reports persons with status
"boundary"or"not_converged".type,callThe estimation type and the call.
modelThe checked and prepared inputs.
resPPThe result in the layout of earlier versions:
resPP$resPPis the persons x 2 matrix of estimates and SEs,resPP$nstepsthe largest number of steps.
Use coef() for the estimates, as.data.frame() for the table and plot() for the measurement precision, the person fit map or person profiles (see pp-plots).
Details
Please note, that robust estimation with (Huber ability estimate) polytomous items is still experimental!
The probability choosing the k-th category is as follows:
$$P(x_{ij} = k | \hat \alpha_i, \hat\beta_{iv}, \theta_j) = \frac{exp(\sum_{v=0}^{(k-1)}\hat \alpha_{i}(\theta_j - \hat \beta_{iv}))}{\,\sum_{c=0}^{m_i - 1}exp(\sum_{v=0}^{c}\hat \alpha_{i}(\theta_j - \hat \beta_{iv})))}$$
In our case \(\theta\) is to be estimated. The item parameters are assumed as fixed (usually these are estimates of a former scaling procedure).
The model simplifies to the Partial Credit Model by setting \(\alpha_{i} = 1, \forall i\).
References
Baker, Frank B., and Kim, Seock-Ho (2004). Item Response Theory - Parameter Estimation Techniques. CRC-Press.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.
Muraki, Eiji (1992). A Generalized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16, 159-176.
Muraki, Eiji (1993). Information Functions of the Generalized Partial Credit Model. Applied Psychological Measurement, 17, 351-363.
Samejima, Fumiko (1993). The bias function of the maximum likelihood estimate of ability for the dichotomous response level. Psychometrika, 58, 195-209.
Samejima, Fumiko (1993). An approximation of the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete. Psychometrika, 58, 119-138.
Schuster, C., & Yuan, K. H. (2011). Robust estimation of latent ability in item response models. Journal of Educational and Behavioral Statistics, 36(6), 720-735.
Wang, S. and Wang, T. (2001). Precision of Warm's Weighted Likelihood Estimates for a Polytomous Model in Computerized Adaptive Testing. Applied Psychological Measurement, 25, 317-331.
Warm, Thomas A. (1989). Weighted Likelihood Estimation Of Ability In Item Response Theory. Psychometrika, 54, 427-450.
Examples
# six items with three categories: one column of thresholds per item
thres <- matrix(c(-2, -1.23, 1.11, 3.48, 1, 2, -1, -0.2, 0.5, 1.3, -0.8, 1.5), nrow = 2)
sl <- c(0.5, 1, 1.5, 1.1, 1, 0.98)
set.seed(1750)
x <- sim_gpcm(thres = thres, alpha = sl, theta = rnorm(30))
# GPCM, weighted likelihood
resgpcm <- PP_gpcm(respm = x, thres = thres, slopes = sl, type = "wle")
summary(resgpcm)
#> PP Version: 1.0.0
#>
#> Call: PP_gpcm(respm = x, thres = thres, slopes = sl, type = "wle")
#> - job started @ Wed Oct 14 13:06:25 2026
#>
#> Estimation type: wle
#>
#> Status of the persons:
#>
#> ok extreme boundary no_response not_converged
#> 30 0 0 0 0
#>
#> Newton steps: at most 5
#> -------------------------------------
#> estimate SE nsteps status
#> 1 -1.71563 0.9957 4 ok
#> 2 1.33733 0.6221 5 ok
#> 3 1.06287 0.6205 4 ok
#> 4 1.40762 0.6244 5 ok
#> 5 2.18275 0.7115 3 ok
#> 6 1.77794 0.6514 5 ok
#> 7 0.69604 0.6338 4 ok
#> 8 0.29161 0.6615 3 ok
#> 9 0.07075 0.6798 3 ok
#> 10 -0.65730 0.7579 4 ok
#> 11 1.06287 0.6205 4 ok
#> 12 1.02014 0.6212 4 ok
#> 13 -0.02115 0.6879 3 ok
#> 14 0.07075 0.6798 3 ok
#> 15 1.23916 0.6202 5 ok
#> --------> output truncated <--------
# partial credit model: all slopes 1 (the default), EAP
respcm <- PP_gpcm(respm = x, thres = thres, type = "eap")
head(respcm$estimates)
#> estimate SE nsteps status
#> 1 -1.4398282 0.6313135 0 ok
#> 2 0.6271971 0.5680080 0 ok
#> 3 0.6271971 0.5680080 0 ok
#> 4 0.9511157 0.5710108 0 ok
#> 5 1.6241091 0.5931807 0 ok
#> 6 1.2812317 0.5790233 0 ok
# items with a different number of categories: NA below the last threshold
thres4 <- rbind(thres, c(NA, NA, NA, NA, 1.7, 1))
x4 <- sim_gpcm(thres = thres4, alpha = sl, theta = rnorm(30))
head(PP_gpcm(respm = x4, thres = thres4, slopes = sl, type = "map")$estimates)
#> estimate SE nsteps status
#> 1 0.1099944 0.5332622 3 ok
#> 2 -0.3784739 0.5724116 3 ok
#> 3 -0.7512036 0.6033999 3 ok
#> 4 -0.7584931 0.6040396 4 ok
#> 5 -1.0960477 0.6357668 4 ok
#> 6 -1.3485548 0.6623468 4 ok