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PP: Person Parameter Estimation

Source: R/PP-package.R
PP-package.Rd

The PP package estimates person parameters (abilities) in item response theory when the item parameters are known and fixed, e.g. for items from an item bank given to a new sample of persons.

Details

Models

the 1PL (Rasch), 2PL, 3PL and 4PL model (PP_4pl), the partial credit and generalized partial credit model (PP_gpcm) and tests that mix both (PPall).

Estimates

maximum likelihood (MLE), weighted likelihood (WLE; Warm, 1989), maximum a posteriori (MAP), expected a posteriori (EAP) and a robust Huber-type estimate, each with a standard error and a status per person.

Further analyses

jackknife estimates and standard errors (JKpp), plausible values (PV) and the person fit indices lz, lz*, infit and outfit (Pfit).

In one step

PPass estimates and computes the person fit for a data.frame with item responses and other person variables, or for a model fitted with the eRm package (RM, LLTM, RSM, PCM, LRSM, LPCM).

Simulation

sim_4pl, sim_gpcm and sim_mixed.

The numerical core is written in C++. Every person is estimated with a safeguarded Newton method, EAP estimates and plausible values come from the posterior on a grid. The vignettes give an introduction: vignette("intro_pp", package = "PP"), vignette("intro_pfit", package = "PP") and vignette("ppass_erm", package = "PP").

References

Barton, M. A., & Lord, F. M. (1981). An upper asymptote for the three-parameter logistic item-response model. ETS Research Report Series, 1981(1), i–8.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.

Drasgow, F., Levine, M. V., & Williams, E. A. (1985). Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38(1), 67–86.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.

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.

Snijders, T. A. B. (2001). Asymptotic null distribution of person fit statistics with estimated person parameter. Psychometrika, 66(3), 331–342.

Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45(3), 373–391.

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427–450.

See also

Useful links:

Author

Maintainer: Manuel Reif manuel.reif@gmail.com

Authors:

Examples

# 12 items of a 2PL test with known parameters, 100 persons
set.seed(1522)
b  <- seq(-3, 3, length = 12)
sl <- round(runif(12, 0.5, 1.5), 2)
x  <- sim_4pl(beta = b, alpha = sl, lowerA = 0, upperA = 1, theta = rnorm(100))

res <- PP_4pl(respm = x, thres = b, slopes = sl, type = "wle")
head(res$estimates)
#>     estimate        SE nsteps status
#> 1 -0.9304929 0.8369461      4     ok
#> 2 -2.7893053 0.8908146      5     ok
#> 3  3.2224542 1.0747279      5     ok
#> 4  0.1683472 0.7848443      3     ok
#> 5 -1.6555101 0.8532582      4     ok
#> 6  1.2207987 0.7839490      4     ok

# estimates and person fit in one table
PPass(data.frame(id = 1:100, x), items = 2:13, thres = b, slopes = sl)
#> Person assessment: 100 persons, 12 items 
#> Model: 2PL model | WLE estimates
#> Status: 100 ok 
#> Person fit: lz, lzstar, infit, outfit 
#> Flagged (|lzstar| > 1.96): 1 persons
#> 
#>    id estimate    SE status     lz lzstar infit   in_t outfit   ou_t
#> 1   1   -0.930 0.837     ok -2.167 -2.248 1.883  1.863  2.200  1.301
#> 2   2   -2.789 0.891     ok  0.689  0.751 0.678 -0.712  0.316  0.371
#> 3   3    3.222 1.075     ok  0.785  0.885 0.513 -0.821  0.209  1.707
#> 4   4    0.168 0.785     ok -0.659 -0.664 1.206  0.659  1.309  0.613
#> 5   5   -1.656 0.853     ok  0.902  0.935 0.597 -0.968  0.343 -0.290
#> 6   6    1.221 0.784     ok -0.805 -0.837 1.349  0.969  1.115  0.601
#> 7   7   -0.109 0.796     ok  0.269  0.265 0.949 -0.020  0.664 -0.191
#> 8   8    0.350 0.780     ok  0.548  0.548 0.792 -0.506  0.644 -0.103
#> 9   9   -0.599 0.820     ok  0.688  0.694 0.749 -0.580  0.488 -0.504
#> 10 10   -0.367 0.808     ok -0.454 -0.474 1.052  0.260  1.700  0.990
#> ... 90 more persons
#> Not shown: lz_unst, in_chisq, in_df, in_pv, ou_chisq, ou_df, ou_pv