PP_4pl {PP} | R Documentation |
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.
PP_4pl(respm, thres, slopes = NULL, lowerA = NULL, upperA = NULL, theta_start = NULL, mu = NULL, sigma2 = NULL, type = "wle", maxsteps = 40, exac = 0.001, H = 1, ctrl = list())
respm |
An integer matrix, which contains the examinees responses. A persons x items matrix is expected. |
thres |
A numeric vector or a numeric matrix which contains the threshold parameter (also known as difficulty parameter or beta parameter) for each item. If a matrix is submitted, the first row must contain only zeroes! |
slopes |
A numeric vector, which contains the slope parameters for each item - one parameter per item is expected. |
lowerA |
A numeric vector, which contains the lower asymptote parameters (kind of guessing parameter) for each item. |
upperA |
numeric vector, which contains the upper asymptote parameters for each item. |
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 in case of MAP or EAP estimation. If nothing is submitted this is set to 0 for each person for MAP estimation. |
sigma2 |
A numeric vector of variance parameters for each person in case of MAP or EAP estimation. If nothing is submitted this is set to 1 for each person for MAP estimation. |
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 |
maxsteps |
The maximum number of steps the NR Algorithm will take. Default = 100. |
exac |
How accurate are the estimates supposed to be? Default is 0.001. |
H |
In case |
ctrl |
more controls:
|
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 und 1 respectively.
3-PL model by submitting anything except the upper asymptote parameters
4-PL model —> submit all parameters ...
The probability function of the 4-PL model is:
P(x_{ij} = 1 | \hat α_i, \hatβ_i, \hatγ_i, \hatδ_i, θ_j ) = \hatγ_i + (\hatδ_i-\hatγ_i) \frac{exp(\hat α_i (θ_{j} - \hatβ_{i}))}{\,1 + exp(\hatα_i (θ_{j} - \hatβ_{i}))}
In our case θ 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 δ_i = 1, \forall i.
In the 2-PL model δ_i = 1, γ_i = 0, \forall i.
In the 1-PL model δ_i = 1, γ_i = 0, α_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 = α_i(θ - β_i)
weight:
w(r_i) = 1 \rightarrow if\, |r_i| ≤q H
w(r_i) = H/|r| \rightarrow if\, |r_i| > H
The function returns a list with the estimation results and pretty much everything which has been submitted to fit the model. The estimation results can be found in OBJ$resPP
. The core result is a number_of_persons x 2 matrix, which contains the ability estimate and the SE for each submitted person.
Manuel Reif
Baker, Frank B., and Kim, Seock-Ho (2004). Item Response Theory - Parameter Estimation Techniques. CRC-Press.
Barton, M. A., & Lord, F. M. (1981). An Upper Asymptote for the Three-Parameter Logistic Item-Response Model.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F.M. & Novick, M.R. (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Magis, D. (2013). A note on the item information function of the four-parameter logistic model. Applied Psychological Measurement, 37(4), 304-315.
Samejima, Fumiko (1993). The bias function of the maximum likelihood estimate of ability for the dichotomous response level. Psychometrika, 58, 195-209.
Samejima, Fumiko (1993). An approximation of the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete. Psychometrika, 58, 119-138.
Schuster, C., & Yuan, K. H. (2011). Robust estimation of latent ability in item response models. Journal of Educational and Behavioral Statistics, 36(6), 720-735.
Warm, Thomas A. (1989). Weighted Likelihood Estimation Of Ability In Item Response Theory. Psychometrika, 54, 427-450.
Yen, Y.-C., Ho, R.-G., Liao, W.-W., Chen, L.-J., & Kuo, C.-C. (2012). An empirical evaluation of the slip correction in the four parameter logistic models with computerized adaptive testing. Applied Psychological Measurement, 36, 75-87.
PPass, PPall, PP_gpcm, JKpp, PV
################# 4 PL ############################################################# ### real data ########## data(pp_amt) d <- as.matrix(pp_amt$daten_amt[,-(1:7)]) rd_res <- PP_4pl(respm = d, thres = pp_amt$betas[,2], type = "wle") summary(rd_res) rd_res1 <- PP_4pl(respm = d, thres = pp_amt$betas[,2], theta_start = 0,type = "wle") summary(rd_res1) ### fake data ########## # smaller ... faster set.seed(1522) # intercepts diffpar <- seq(-3,3,length=12) # slope parameters 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) # response matrix awm <- matrix(sample(0:1,10*12,replace=TRUE),ncol=12) ## 1PL model ##### # MLE res1plmle <- PP_4pl(respm = awm,thres = diffpar,type = "mle") # WLE res1plwle <- PP_4pl(respm = awm,thres = diffpar,type = "wle") # MAP estimation res1plmap <- PP_4pl(respm = awm,thres = diffpar,type = "map") # EAP estimation res1pleap <- PP_4pl(respm = awm,thres = diffpar,type = "eap") # robust estimation res1plrob <- PP_4pl(respm = awm,thres = diffpar,type = "robust") # summarize results summary(res1plmle) summary(res1plwle) summary(res1plmap) ## 2PL model ##### # MLE res2plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "mle") # WLE res2plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "wle") # MAP estimation res2plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "map") # EAP estimation res2pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "eap") # robust estimation res2plrob <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "robust") ## 3PL model ##### # MLE res3plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "mle") # WLE res3plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "wle") # MAP estimation res3plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "map") # EAP estimation res3pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la, type = "eap") ## 4PL model ##### # MLE res4plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "mle") # WLE res4plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "wle") # MAP estimation res4plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "map") # EAP estimation res4pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "eap") ## A special on robust estimation: # it reproduces the example given in Schuster & Ke-Hai 2011: diffpar <- c(-3,-2,-1,0,1,2,3) AWM <- matrix(0,7,7) diag(AWM) <- 1 res1plmle <- PP_4pl(respm = AWM,thres = diffpar, type = "mle") summary(res1plmle) res1plrob <- PP_4pl(respm = AWM,thres = diffpar, type = "robust") summary(res1plrob)