Compute Person Parameters for the 1/2/3/4-PL model 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 vector or a numeric matrix which contains the threshold parameter (also known as difficulty parameter or beta parameter) for each item. A matrix with one column per item is also accepted; its first row of zeros (as in
PP_gpcm) is optional.- 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.- lowerA
A numeric vector, which contains the lower asymptote parameters (kind of guessing parameter) for each item. A single value is used for all items. If
NULL(default), all lower asymptotes are 0.- upperA
numeric vector, which contains the upper asymptote parameters for each item. A single value is used for all items. If
NULL(default), all upper asymptotes 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
With this function you can estimate:
1-PL model (Rasch model) by submitting: the data matrix, item difficulties and nothing else, since the 1-PL model is merely a 4-PL model with: any slope = 1, any lower asymptote = 0 and any upper asymptote = 1!
2-PL model by submitting: the data matrix, item difficulties and slope parameters. Lower and upper asymptotes are automatically set to 0 and 1 respectively.
3-PL model by submitting anything except the upper asymptote parameters
4-PL model by submitting all parameters.
The probability function of the 4-PL model is: $$P(x_{ij} = 1 | \hat \alpha_i, \hat\beta_i, \hat\gamma_i, \hat\delta_i, \theta_j ) = \hat\gamma_i + (\hat\delta_i-\hat\gamma_i) \frac{exp(\hat \alpha_i (\theta_{j} - \hat\beta_{i}))}{\,1 + exp(\hat\alpha_i (\theta_{j} - \hat\beta_{i}))}$$
In our case \(\theta\) is to be estimated, and the four item parameters are assumed as fixed (usually these are estimates of a former scaling procedure).
The 3-PL model is the same, except that \(\delta_i = 1, \forall i\).
In the 2-PL model \(\delta_i = 1, \gamma_i = 0, \forall i\).
In the 1-PL model \(\delta_i = 1, \gamma_i = 0, \alpha_i = 1, \forall i\).
The robust estimation method, applies a Huber-type estimator (Schuster & Yuan, 2011), which downweights responses to items which provide little information for the ability estimation. First a residuum is estimated and on this basis, the weight for each observation is computed.
residuum: $$r_i = \alpha_i(\theta - \beta_i)$$
weight:
$$w(r_i) = 1 \rightarrow if\, |r_i| \leq H$$ $$w(r_i) = H/|r| \rightarrow if\, |r_i| > H$$
References
Baker, Frank B., and Kim, Seock-Ho (2004). Item Response Theory - Parameter Estimation Techniques. CRC-Press.
Barton, M. A., & Lord, F. M. (1981). An Upper Asymptote for the Three-Parameter Logistic Item-Response Model.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F.M. & Novick, M.R. (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Magis, D. (2013). A note on the item information function of the four-parameter logistic model. Applied Psychological Measurement, 37(4), 304-315.
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.
Warm, Thomas A. (1989). Weighted Likelihood Estimation Of Ability In Item Response Theory. Psychometrika, 54, 427-450.
Yen, Y.-C., Ho, R.-G., Liao, W.-W., Chen, L.-J., & Kuo, C.-C. (2012). An empirical evaluation of the slip correction in the four parameter logistic models with computerized adaptive testing. Applied Psychological Measurement, 36, 75-87.
Examples
## real data: an adaptive intelligence test with known item difficulties (1PL)
data(pp_amt)
d <- as.matrix(pp_amt$daten_amt[, -(1:7)])
res <- PP_4pl(respm = d, thres = pp_amt$betas[, 2], type = "wle")
summary(res)
#> PP Version: 1.0.0
#>
#> Call: PP_4pl(respm = d, thres = pp_amt$betas[, 2], type = "wle")
#> - job started @ Wed Oct 14 13:06:24 2026
#>
#> Estimation type: wle
#>
#> Status of the persons:
#>
#> ok extreme boundary no_response not_converged
#> 710 0 0 0 0
#>
#> Newton steps: at most 7
#> -------------------------------------
#> estimate SE nsteps status
#> 1 1.4099 0.4178 4 ok
#> 2 0.3516 0.3729 3 ok
#> 3 1.0745 0.3741 4 ok
#> 4 0.3204 0.6900 3 ok
#> 5 1.3880 0.4644 3 ok
#> 6 0.9457 0.4185 4 ok
#> 7 1.8276 0.4108 4 ok
#> 8 -0.2274 0.3964 4 ok
#> 9 1.4426 0.4074 4 ok
#> 10 1.6430 0.4033 4 ok
#> 11 -0.4802 0.4759 3 ok
#> 12 -0.3591 0.4086 3 ok
#> 13 -0.1911 0.4137 3 ok
#> 14 0.5781 0.4177 3 ok
#> 15 -0.8898 0.4215 4 ok
#> --------> output truncated <--------
## simulated data: 2PL, 3PL and 4PL
set.seed(1522)
b <- seq(-3, 3, length = 12)
sl <- round(runif(12, 0.5, 1.5), 2)
la <- round(runif(12, 0, 0.25), 2)
ua <- round(runif(12, 0.8, 1), 2)
x <- sim_4pl(beta = b, alpha = sl, lowerA = la, upperA = ua, theta = rnorm(50))
res2pl <- PP_4pl(respm = x, thres = b, slopes = sl, type = "mle")
res3pl <- PP_4pl(respm = x, thres = b, slopes = sl, lowerA = la, type = "wle")
res4pl <- PP_4pl(respm = x, thres = b, slopes = sl, lowerA = la, upperA = ua, type = "map")
# persons with all answers right (or wrong) have no finite ML estimate
table(res2pl$estimates$status)
#>
#> ok extreme boundary no_response not_converged
#> 50 0 0 0 0
head(res3pl$estimates)
#> estimate SE nsteps status
#> 1 0.2323310 0.9439296 2 ok
#> 2 -0.4027227 1.0044541 2 ok
#> 3 -2.2338872 1.0789771 8 ok
#> 4 0.4016054 0.9324362 4 ok
#> 5 0.8125907 0.9142177 4 ok
#> 6 0.1237144 0.9524919 4 ok
coef(res4pl)[1:5]
#> [1] 0.23729052 0.01516983 -0.98313162 0.22643916 0.63615728
## EAP with person-specific priors
res_eap <- PP_4pl(respm = x, thres = b, slopes = sl, type = "eap",
mu = rep(c(-0.5, 0.5), 25), sigma2 = 0.8)
## robust estimation: the example of Schuster & Yuan (2011)
AWM <- matrix(0, 7, 7)
diag(AWM) <- 1
coef(PP_4pl(respm = AWM, thres = -3:3, type = "mle"))
#> [1] -2.94088 -2.94088 -2.94088 -2.94088 -2.94088 -2.94088 -2.94088
coef(PP_4pl(respm = AWM, thres = -3:3, type = "robust"))
#> [1] -2.746716 -2.746657 -4.039526 -4.254544 -4.417379 -4.548895 -4.659856