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:
|
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.
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: $$P(x_{ij} = 1 | \hat \alpha_i, \hat\beta_i, \hat\gamma_i, \hat\delta_i, \theta_j ) = \hat\gamma_i + (\hat\delta_i-\hat\gamma_i) \frac{exp(\hat \alpha_i (\theta_{j} - \hat\beta_{i}))}{\,1 + exp(\hat\alpha_i (\theta_{j} - \hat\beta_{i}))}$$
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:
$$w(r_i) = 1 \rightarrow if\, |r_i| \leq H$$ $$w(r_i) = H/|r| \rightarrow if\, |r_i| > H$$
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.
Manuel Reif
################# 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")#> Estimating: 1pl model ... #> type = wle #> Estimation finished!#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = d, thres = pp_amt$betas[, 2], type = "wle") #> - job started @ Mon May 24 13:27:52 2021 #> #> Estimation type: wle #> #> Number of iterations: 4 #> ------------------------------------- #> estimate SE #> [1,] 1.4099 0.4178 #> [2,] 0.3516 0.3729 #> [3,] 1.0745 0.3741 #> [4,] 0.3205 0.6900 #> [5,] 1.3879 0.4644 #> [6,] 0.9457 0.4185 #> [7,] 1.8277 0.4109 #> [8,] -0.2274 0.3964 #> [9,] 1.4426 0.4074 #> [10,] 1.6430 0.4033 #> [11,] -0.4802 0.4759 #> [12,] -0.3591 0.4086 #> [13,] -0.1911 0.4137 #> [14,] 0.5781 0.4177 #> [15,] -0.8898 0.4215 #> --------> output truncated <--------rd_res1 <- PP_4pl(respm = d, thres = pp_amt$betas[,2], theta_start = 0,type = "wle")#> Estimating: 1pl model ... #> type = wle #> Estimation finished!#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = d, thres = pp_amt$betas[, 2], theta_start = 0, type = "wle") #> - job started @ Mon May 24 13:27:52 2021 #> #> Estimation type: wle #> #> Number of iterations: 40 #> ------------------------------------- #> estimate SE #> [1,] 1.4099 0.4178 #> [2,] 0.3516 0.3729 #> [3,] 1.0745 0.3741 #> [4,] 0.3205 0.6900 #> [5,] 1.3879 0.4644 #> [6,] 0.9457 0.4185 #> [7,] 1.8277 0.4109 #> [8,] -0.2274 0.3964 #> [9,] 1.4426 0.4074 #> [10,] 1.6430 0.4033 #> [11,] -0.4802 0.4759 #> [12,] -0.3591 0.4086 #> [13,] -0.1911 0.4137 #> [14,] 0.5781 0.4177 #> [15,] -0.8898 0.4215 #> --------> output truncated <--------### 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")#> Estimating: 1pl model ... #> type = mle #> Estimation finished!# WLE res1plwle <- PP_4pl(respm = awm,thres = diffpar,type = "wle")#> Estimating: 1pl model ... #> type = wle #> Estimation finished!# MAP estimation res1plmap <- PP_4pl(respm = awm,thres = diffpar,type = "map")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 1pl model ... #> type = map #> Estimation finished!# EAP estimation res1pleap <- PP_4pl(respm = awm,thres = diffpar,type = "eap")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 1pl model ... #> type = eap #> Estimation finished!# robust estimation res1plrob <- PP_4pl(respm = awm,thres = diffpar,type = "robust")#> Estimating: 1pl model ... #> type = robust #> Estimation finished!#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = awm, thres = diffpar, type = "mle") #> - job started @ Mon May 24 13:27:53 2021 #> #> Estimation type: mle #> #> Number of iterations: 5 #> ------------------------------------- #> estimate SE #> [1,] 0.5903 0.7715 #> [2,] 3.5841 1.1379 #> [3,] 0.5903 0.7715 #> [4,] 0.5903 0.7715 #> [5,] 0.5903 0.7715 #> [6,] 0.0000 0.7668 #> [7,] 0.5903 0.7715 #> [8,] -1.1966 0.7880 #> [9,] 0.5903 0.7715 #> [10,] -1.1966 0.7880#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = awm, thres = diffpar, type = "wle") #> - job started @ Mon May 24 13:27:53 2021 #> #> Estimation type: wle #> #> Number of iterations: 4 #> ------------------------------------- #> estimate SE #> [1,] 0.5778 0.7713 #> [2,] 3.2839 1.0488 #> [3,] 0.5778 0.7713 #> [4,] 0.5778 0.7713 #> [5,] 0.5778 0.7713 #> [6,] 0.0000 0.7668 #> [7,] 0.5778 0.7713 #> [8,] -1.1667 0.7869 #> [9,] 0.5778 0.7713 #> [10,] -1.1667 0.7869#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = awm, thres = diffpar, type = "map") #> - job started @ Mon May 24 13:27:53 2021 #> #> Estimation type: map #> #> Number of iterations: 3 #> ------------------------------------- #> estimate SE #> [1,] 0.3706 0.7686 #> [2,] 1.9069 0.8297 #> [3,] 0.3706 0.7686 #> [4,] 0.3706 0.7686 #> [5,] 0.3706 0.7686 #> [6,] 0.0000 0.7668 #> [7,] 0.3706 0.7686 #> [8,] -0.7436 0.7745 #> [9,] 0.3706 0.7686 #> [10,] -0.7436 0.7745## 2PL model ##### # MLE res2plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "mle")#> Estimating: 2pl model ... #> type = mle #> Estimation finished!# WLE res2plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "wle")#> Estimating: 2pl model ... #> type = wle #> Estimation finished!# MAP estimation res2plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "map")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 2pl model ... #> type = map #> Estimation finished!# EAP estimation res2pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "eap")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 2pl model ... #> type = eap #> Estimation finished!# robust estimation res2plrob <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,type = "robust")#> Estimating: 2pl model ... #> type = robust #> Estimation finished!## 3PL model ##### # MLE res3plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "mle")#> Estimating: 3pl model ... #> type = mle #> Estimation finished!# WLE res3plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "wle")#> Estimating: 3pl model ... #> type = wle #> Estimation finished!# MAP estimation res3plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,type = "map")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 3pl model ... #> type = map #> Estimation finished!# EAP estimation res3pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la, type = "eap")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 3pl model ... #> type = eap #> Estimation finished!## 4PL model ##### # MLE res4plmle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "mle")#> Estimating: 4pl model ... #> type = mle #> Estimation finished!# WLE res4plwle <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "wle")#> Estimating: 4pl model ... #> type = wle #> Estimation finished!# MAP estimation res4plmap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "map")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 4pl model ... #> type = map #> Estimation finished!# EAP estimation res4pleap <- PP_4pl(respm = awm,thres = diffpar, slopes = sl,lowerA = la,upperA=ua,type = "eap")#> Warning: all mu's are set to 0!#> Warning: all sigma2's are set to 1!#> Estimating: 4pl model ... #> type = eap #> Estimation finished!## 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")#> Estimating: 1pl model ... #> type = mle #> Estimation finished!#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = AWM, thres = diffpar, type = "mle") #> - job started @ Mon May 24 13:27:53 2021 #> #> Estimation type: mle #> #> Number of iterations: 4 #> ------------------------------------- #> estimate SE #> [1,] -2.9409 1.2524 #> [2,] -2.9409 1.2524 #> [3,] -2.9409 1.2524 #> [4,] -2.9409 1.2524 #> [5,] -2.9409 1.2524 #> [6,] -2.9409 1.2524 #> [7,] -2.9409 1.2524res1plrob <- PP_4pl(respm = AWM,thres = diffpar, type = "robust")#> Estimating: 1pl model ... #> type = robust #> Estimation finished!#> PP Version: 0.6.3.11 #> #> Call: PP_4pl(respm = AWM, thres = diffpar, type = "robust") #> - job started @ Mon May 24 13:27:53 2021 #> #> Estimation type: robust #> #> Number of iterations: 10 #> ------------------------------------- #> estimate SE #> [1,] -2.7467 1.2114 #> [2,] -2.7467 1.2114 #> [3,] -4.0393 1.6545 #> [4,] -4.2548 1.7791 #> [5,] -4.4176 1.8855 #> [6,] -4.5492 1.9800 #> [7,] -4.6600 2.0658