Person assessment with PPass, also for eRm models

PPass() (Person Parameter assessment) estimates the abilities, computes the person fit and returns both in one table, together with the other person variables. It accepts

Data with item parameters

fourpl_df contains an id, the sex and the responses of 60 persons to 12 items of a 3PL test. items names the item columns (positions or names); all other columns are kept as person variables. mod chooses the model, and the item parameters must match it.

library(PP)
data(fourpl_df)
res <- PPass(fourpl_df, items = 3:14,
             thres = attr(fourpl_df, "diffpar"), slopes = attr(fourpl_df, "slopes"),
             lowerA = attr(fourpl_df, "lowerA"))
res
#> Person assessment: 60 persons, 12 items 
#> Model: 3PL model | WLE estimates
#> Status: 60 ok 
#> Person fit: lz, lzstar, infit, outfit 
#> Flagged (|lzstar| > 1.96): 4 persons
#> 
#>         id sex estimate    SE status     lz lzstar infit    in_t outfit   ou_t
#> 1  LVL0694   w   -0.936 0.767     ok  1.019  1.026 0.692 -1.0907 0.5342 -0.866
#> 2  BBU1225   w    1.811 1.022     ok  1.101  1.149 0.377 -1.5365 0.2061 -0.526
#> 3  MJN2028   w    0.969 0.987     ok  1.270  1.324 0.288 -1.7568 0.2011 -1.204
#> 4  TSU0771   m    1.224 1.008     ok -1.039 -1.157 1.487  0.9781 1.9539  1.107
#> 5  XDS0698   w    0.252 0.925     ok  1.187  1.224 0.436 -1.3918 0.3219 -1.257
#> 6  BOS1292   w   -4.730 2.118     ok  0.998  0.637 0.108 -1.5050 0.0884 -1.196
#> 7  KFF1422   w    2.618 1.016     ok  0.545  0.580 0.765 -0.4927 0.3216  0.181
#> 8  DCQ0198   w    1.232 1.008     ok -1.099 -1.221 1.480  0.9681 2.2503  1.295
#> 9  FTT1492   w   -2.377 0.913     ok  0.182  0.103 0.973  0.0586 0.7565 -0.157
#> 10 GCP0645   m   -0.400 0.802     ok  0.922  0.948 0.678 -0.8861 0.5105 -0.932
#> ... 50 more persons
#> Not shown: lz_unst, in_chisq, in_df, in_pv, ou_chisq, ou_df, ou_pv

By default, PPass() computes WLE estimates and all four person fit statistics (lz, lz*, infit and outfit). Other arguments of PP_4pl(), PP_gpcm() and PPall() can be passed on, e.g. type = "mle" or a prior for type = "map". summary() shows the distribution of the estimates, the person fit and the persons flagged by lz* (two-sided, as in the summary of Pfit()):

s <- summary(res)
#> Person assessment: 60 persons, 12 items 
#> Model: 3PL model | WLE estimates
#> Status: 60 ok 
#> Person fit: lz, lzstar, infit, outfit 
#> Flagged (|lzstar| > 1.96): 4 persons
#> 
#> Estimates (status ok):
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -4.730  -1.278  -0.339  -0.313   0.969   3.624 
#> 
#> Person fit:
#>  statistic  n   mean    sd     min  max flagged
#>         lz 60  0.245 1.012 -3.0334 1.33       3
#>     lzstar 60  0.192 1.093 -3.2532 1.34       4
#>      infit 60  0.870 0.430  0.1080 2.13      NA
#>       in_t 60 -0.292 1.013 -1.7568 2.28       2
#>     outfit 60  0.853 0.828  0.0884 4.29      NA
#>       ou_t 60 -0.163 0.980 -1.3185 3.01       3
#> 
#> Flagged persons (lzstar):
#>  person      id sex estimate lzstar
#>      26 HBH0927   m    -0.57  -3.25
#>      60 RWQ0486   w    -2.16  -3.07
#>      43 WUS0752   w     0.78  -3.04
#>      39 UHY1090   m     2.88  -2.07

plot() shows the person fit map: lz* against the estimate, with the critical values of summary() and the flagged persons labelled, here by their id. which = "precision" shows the measurement precision of the test and which = "profile" the answers of single persons (see ?"pp-plots").

plot(res, label = "id")

lz* of every person against the WLE estimate, flagged persons highlighted and labelled with their id

The result is a data.frame. It can be filtered, merged or written to a file like any other data.frame; as.data.frame() drops the extra information.

res[res$sex == "w" & res$status == "ok", c("id", "estimate", "SE", "lzstar")][1:5, ]
#>        id estimate    SE lzstar
#> 1 LVL0694   -0.936 0.767  1.026
#> 2 BBU1225    1.811 1.022  1.149
#> 3 MJN2028    0.969 0.987  1.324
#> 5 XDS0698    0.252 0.925  1.224
#> 6 BOS1292   -4.730 2.118  0.637
knitr::kable(head(as.data.frame(res)[, c("id", "sex", "estimate", "SE", "lzstar", "infit")]),
             digits = 3)

id	sex	estimate	SE	lzstar	infit
LVL0694	w	-0.936	0.767	1.026	0.692
BBU1225	w	1.811	1.022	1.149	0.377
MJN2028	w	0.969	0.987	1.324	0.288
TSU0771	m	1.224	1.008	-1.157	1.487
XDS0698	w	0.252	0.925	1.224	0.436
BOS1292	w	-4.730	2.118	0.637	0.108

JKpp() and PV() work directly on the result, without estimating again:

head(JKpp(res)$estimates)
#>     estimate        SE  L excluded status
#> 1 -0.9723726 0.6735435 12        0     ok
#> 2  1.8740917 0.6208916 12        0     ok
#> 3  1.0570466 0.4916653 12        0     ok
#> 4  1.5539111 1.6822352 12        0     ok
#> 5  0.3442215 0.5400380 11        0     ok
#> 6 -5.8298151 0.5859734 12        0     ok

The model follows from the item parameters: here slopes and lowerA make it a 3PL model. A threshold matrix with several thresholds per item gives the PCM (all slopes 1) or the GPCM; together with lowerA or upperA it gives a test with both dichotomous (4PL) and polytomous (GPCM) items, estimated with PPall().

Models fitted with eRm

eRm estimates item parameters by conditional maximum likelihood; eRm::person.parameter() then gives maximum likelihood estimates of the abilities. PPass() takes the fitted object and adds WLE, MAP, EAP and robust estimates, the person fit, the jackknife and plausible values.

library(eRm)
rm1 <- RM(raschdat1)
res_rm <- PPass(rm1)
res_rm
#> Person assessment: 100 persons, 30 items 
#> Model: Rasch model (eRm::RM) | WLE estimates
#> Status: 100 ok 
#> Person fit: lz, lzstar, infit, outfit 
#> Flagged (|lzstar| > 1.96): 4 persons
#> 
#>    estimate    SE status        lz  lzstar infit     in_t outfit    ou_t
#> 1     -3.29 0.868     ok  0.411679  0.2080 0.722 -0.16277  0.478 -0.2043
#> 2     -2.72 0.688     ok  0.000319 -0.8777 0.946  0.07782  1.003  0.2821
#> 3     -2.33 0.596     ok  0.180246 -0.0421 0.954  0.03047  0.736 -0.2103
#> 4     -2.02 0.539     ok  0.518854  0.9215 0.845 -0.35622  0.661 -0.4994
#> 5     -2.02 0.539     ok  0.134508 -0.0476 0.952 -0.02333  0.857 -0.0800
#> 6     -2.02 0.539     ok -0.079114 -0.5862 1.034  0.21354  0.932  0.0643
#> 7     -1.76 0.499     ok  0.611982  1.0803 0.843 -0.47085  0.665 -0.6300
#> 8     -1.54 0.471     ok -0.229133 -0.6356 0.977 -0.00897  1.846  1.7622
#> 9     -1.54 0.471     ok  0.055611 -0.0814 1.029  0.19749  0.831 -0.2891
#> 10    -1.54 0.471     ok -0.081792 -0.3488 1.011  0.12597  1.023  0.1941
#> ... 90 more persons
#> Not shown: lz_unst, in_chisq, in_df, in_pv, ou_chisq, ou_df, ou_pv

The maximum likelihood estimates are the same as those of eRm:

res_mle <- PPass(rm1, type = "mle", fitindices = "lzstar")
pp <- person.parameter(rm1)
ok <- res_mle$status == "ok"
max(abs(res_mle$estimate[ok] - pp$theta.table[ok, "Person Parameter"]))
#> [1] 2.457956e-05

eRm stores the parameters of the item categories as easiness parameters. PPass() converts them into PP thresholds, so the eRm package is not even needed to run PPass() on a saved object. Partial credit and rating scale models are estimated with PP_gpcm(), with all slopes equal to 1:

set.seed(2751)
thres <- rbind(c(-2, -1.2, -0.5, 0), c(-1, 0.3, 1, 1.5), c(0.5, NA, NA, 2))
x <- sim_gpcm(thres = thres, alpha = 1, theta = rnorm(300))
res_pcm <- PPass(PCM(x), type = "map")
summary(res_pcm$estimate)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> -1.5928 -0.5035  0.1379  0.0808  0.4635  1.9687

With several time points (mpoints) and groups (groupvec), every combination of item, time point and group is a virtual item with its own parameters. A person answers the virtual items of their group and gets one ability estimate across all time points. eRm gives no person parameters for models with group contrasts; PPass() does:

x2 <- sim_gpcm(thres = thres, alpha = 1, theta = rnorm(300, 0.5))
lpcm <- LPCM(cbind(x, x2), mpoints = 2, groupvec = rep(1:2, 150))
res_lpcm <- PPass(lpcm)
res_lpcm
#> Person assessment: 300 persons, 16 virtual items 
#> Model: linear partial credit model (eRm::LPCM), 2 time points, 2 groups | WLE estimates
#> Status: 300 ok 
#> Person fit: lz, lzstar, infit, outfit 
#> Flagged (|lzstar| > 1.96): 7 persons
#> 
#>    group estimate    SE status     lz lzstar infit   in_t outfit    ou_t
#> 1      1  -1.0074 0.496     ok  0.260  0.271 0.872 -0.140  0.845 -0.1578
#> 2      2  -0.3677 0.454     ok  0.218  0.220 0.833 -0.285  0.972  0.0761
#> 3      1   0.2085 0.453     ok -1.223 -1.260 1.284  0.736  1.256  0.6987
#> 4      2   1.0699 0.500     ok  1.077  1.404 0.328 -1.768  0.377 -1.3477
#> 5      1   0.0114 0.450     ok  1.326  1.328 0.334 -1.975  0.374 -1.8644
#> 6      2   0.4210 0.458     ok  0.444  0.463 0.618 -0.835  0.668 -0.7103
#> 7      1   0.0114 0.450     ok -0.654 -0.656 1.053  0.264  1.164  0.5115
#> 8      2   0.2201 0.453     ok  0.782  0.797 0.578 -0.982  0.562 -1.0909
#> 9      1  -0.3797 0.454     ok -1.543 -1.634 1.398  0.986  1.586  1.3135
#> 10     2   0.0231 0.450     ok  1.377  1.379 0.477 -1.368  0.478 -1.4254
#> ... 290 more persons
#> Not shown: lz_unst, in_chisq, in_df, in_pv, ou_chisq, ou_df, ou_pv

Plausible values need a prior, so they require an EAP or MAP estimate, such as res_pcm above:

set.seed(1)
pv <- PV(res_pcm, npv = 5)
head(pv$pvdraws, 3)
#>             PV1         PV2        PV3        PV4        PV5
#> [1,] -1.6410665 -1.13229619 -1.7724358 -0.2457368 -1.0406880
#> [2,] -0.6591065  0.09483645  0.2394663  0.1457675 -0.3621865
#> [3,]  2.1276217  1.43667885  0.8140052 -0.1673230  1.8893809

Person assessment with PPass, also for eRm models

Manuel Reif

Data with item parameters

Models fitted with eRm