# IRT.truescore: Converts a theta Score into a True Score tau ( theta) In TAM: Test Analysis Modules

## Description

Converts a θ score into an unweighted true score τ ( θ)=∑_i ∑_h h P_i ( θ ) . In addition, a weighted true score τ ( θ)=∑_i ∑_h q_{ih} P_i ( θ ) can also be computed by specifying item-category weights q_{ih} in the matrix `Q`.

## Usage

 `1` ```IRT.truescore(object, iIndex=NULL, theta=NULL, Q=NULL) ```

## Arguments

 `object` Object for which the `CDM::IRT.irfprob` S3 method is defined `iIndex` Optional vector with item indices `theta` Optional vector with θ values `Q` Optional weighting matrix

## Value

Data frame containing θ values and corresponding true scores τ( θ ) .

See also `sirt::truescore.irt` for a conversion function for generalized partial credit models.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48``` ```############################################################################# # EXAMPLE 1: True score conversion for a test with polytomous items ############################################################################# data(data.Students, package="CDM") dat <- data.Students[, paste0("mj",1:4) ] # fit partial credit model mod1 <- TAM::tam.mml( dat,control=list(maxiter=20) ) summary(mod1) # true score conversion tmod1 <- TAM::IRT.truescore( mod1 ) round( tmod1, 4 ) # true score conversion with user-defined theta grid tmod1b <- TAM::IRT.truescore( mod1, theta=seq( -8,8, len=33 ) ) # plot results plot( tmod1\$theta, tmod1\$truescore, type="l", xlab=expression(theta), ylab=expression(tau( theta ) ) ) points( tmod1b\$theta, tmod1b\$truescore, pch=16, col="brown" ) ## Not run: ############################################################################# # EXAMPLE 2: True scores with different category weightings ############################################################################# data(data.timssAusTwn.scored) dat <- data.timssAusTwn.scored # extract item response data dat <- dat[, grep("M03", colnames(dat) ) ] # select items with do have maximum score of 2 (polytomous items) ind <- which( apply( dat, 2, max, na.rm=TRUE )==2 ) I <- ncol(dat) # define Q-matrix with scoring variant Q <- matrix( 1, nrow=I, ncol=1 ) Q[ ind, 1 ] <- .5 # score of 0.5 for polyomous items # estimate model mod1 <- TAM::tam.mml( dat, Q=Q, irtmodel="PCM2", control=list( nodes=seq(-10,10,len=61) ) ) summary(mod1) # true score with scoring (0,1,2) which is the default of the function tmod1 <- TAM::IRT.truescore(mod1) # true score with user specified weighting matrix Q <- mod1\$B[,,1] tmod2 <- TAM::IRT.truescore(mod1, Q=Q) ## End(Not run) ```

### Example output

```Loading required package: CDM
**********************************
** CDM 7.0-12 (2018-11-06 16:23:12)
** Cognitive Diagnostic Models  **
**********************************

* TAM 2.13-15 (2018-09-30 14:35:48)
....................................................
Processing Data      2019-01-09 12:12:36
* Response Data: 2400 Persons and  4 Items
* Numerical integration with 21 nodes
* Created Design Matrices   ( 2019-01-09 12:12:36 )
* Calculated Sufficient Statistics   ( 2019-01-09 12:12:36 )
....................................................
Iteration 1     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 20499.2631
Maximum item intercept parameter change: 0.562054
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.486617
....................................................
Iteration 2     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 20105.4515 | Absolute change: 393.8116 | Relative change: 0.0195873
Maximum item intercept parameter change: 0.206128
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.35712
....................................................
Iteration 3     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19944.2336 | Absolute change: 161.2179 | Relative change: 0.00808343
Maximum item intercept parameter change: 0.207467
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.360083
....................................................
Iteration 4     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19822.6875 | Absolute change: 121.5461 | Relative change: 0.00613167
Maximum item intercept parameter change: 0.193255
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.356844
....................................................
Iteration 5     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19733.5628 | Absolute change: 89.1247 | Relative change: 0.0045164
Maximum item intercept parameter change: 0.171963
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.340186
....................................................
Iteration 6     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19669.3356 | Absolute change: 64.2272 | Relative change: 0.00326534
Maximum item intercept parameter change: 0.152592
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.314329
....................................................
Iteration 7     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19624.3371 | Absolute change: 44.9986 | Relative change: 0.002293
Maximum item intercept parameter change: 0.13107
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.280392
....................................................
Iteration 8     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19593.3973 | Absolute change: 30.9398 | Relative change: 0.00157909
Maximum item intercept parameter change: 0.111203
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.243661
....................................................
Iteration 9     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19572.5719 | Absolute change: 20.8253 | Relative change: 0.00106401
Maximum item intercept parameter change: 0.092884
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.206905
....................................................
Iteration 10     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19558.7926 | Absolute change: 13.7794 | Relative change: 0.00070451
Maximum item intercept parameter change: 0.076677
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.172435
....................................................
Iteration 11     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19549.7968 | Absolute change: 8.9957 | Relative change: 0.00046015
Maximum item intercept parameter change: 0.062737
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.141668
....................................................
Iteration 12     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19543.9844 | Absolute change: 5.8124 | Relative change: 0.0002974
Maximum item intercept parameter change: 0.050949
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.114982
....................................................
Iteration 13     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19540.2487 | Absolute change: 3.7357 | Relative change: 0.00019118
Maximum item intercept parameter change: 0.041233
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.092517
....................................................
Iteration 14     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19537.855 | Absolute change: 2.3937 | Relative change: 0.00012252
Maximum item intercept parameter change: 0.033314
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.073949
....................................................
Iteration 15     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19536.3194 | Absolute change: 1.5356 | Relative change: 7.86e-05
Maximum item intercept parameter change: 0.026862
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.058895
....................................................
Iteration 16     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19535.332 | Absolute change: 0.9874 | Relative change: 5.055e-05
Maximum item intercept parameter change: 0.021656
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.046795
....................................................
Iteration 17     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19534.6945 | Absolute change: 0.6375 | Relative change: 3.263e-05
Maximum item intercept parameter change: 0.01742
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.037109
....................................................
Iteration 18     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19534.2805 | Absolute change: 0.414 | Relative change: 2.119e-05
Maximum item intercept parameter change: 0.014093
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.029415
....................................................
Iteration 19     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19534.0093 | Absolute change: 0.2712 | Relative change: 1.388e-05
Maximum item intercept parameter change: 0.011381
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.02315
....................................................
Iteration 20     2019-01-09 12:12:36
E Step
M Step Intercepts   |----
Deviance = 19533.8292 | Absolute change: 0.1801 | Relative change: 9.22e-06
Maximum item intercept parameter change: 0.009215
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.018311
....................................................
Item Parameters
xsi.index xsi.label     est
1          1  mj1_Cat1 -4.2876
2          2  mj1_Cat2 -2.6936
3          3  mj1_Cat3  0.1857
4          4  mj2_Cat1 -3.7349
5          5  mj2_Cat2 -1.0935
6          6  mj2_Cat3  1.3039
7          7  mj3_Cat1 -2.1422
8          8  mj3_Cat2  0.3272
9          9  mj3_Cat3  1.9774
10        10  mj4_Cat1 -2.6496
11        11  mj4_Cat2 -0.0615
12        12  mj4_Cat3  1.8041
...................................
Regression Coefficients
[,1]
[1,]    0

Variance:
[,1]
[1,] 4.755

EAP Reliability:
[1] 0.841

-----------------------------
Start:  2019-01-09 12:12:36
End:  2019-01-09 12:12:36
Time difference of 0.2715771 secs

------------------------------------------------------------
TAM 2.13-15 (2018-09-30 14:35:48)
sh: 1: cannot create /dev/null: Permission denied
R version 3.4.4 (2018-03-15) x86_64, linux-gnu | nodename=ip-172-31-1-5 | login=unknown

Date of Analysis: 2019-01-09 12:12:36
Time difference of 0.2715771 secs
Computation time: 0.2715771

Multidimensional Item Response Model in TAM

IRT Model 1PL

Call:
TAM::tam.mml(resp = dat, control = list(maxiter = 20))

------------------------------------------------------------
Number of iterations= 20
Numeric integration with 21 integration points

Deviance= 19533.83
Log likelihood= -9766.91
Number of persons= 2400
Number of persons used= 2371
Number of items= 4
Number of estimated parameters= 13
Item threshold parameters= 12
Item slope parameters  = 0
Regression parameters  = 0
(Co)Variance parameters= 1

AIC = 19560  | penalty= 26    | AIC=-2*LL + 2*p
AIC3 = 19573  | penalty= 39    | AIC3=-2*LL + 3*p
BIC = 19635  | penalty= 101.02    | BIC=-2*LL + log(n)*p
aBIC = 19594  | penalty= 59.7    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC)
CAIC = 19648  | penalty= 114.02    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC)
AICc = 19560  | penalty= 26.15    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC)

------------------------------------------------------------
EAP Reliability
[1] 0.841
------------------------------------------------------------
Covariances and Variances
[,1]
[1,] 4.755
------------------------------------------------------------
Correlations and Standard Deviations (in the diagonal)
[,1]
[1,] 2.181
------------------------------------------------------------
Regression Coefficients
[,1]
[1,]    0
------------------------------------------------------------
Item Parameters -A*Xsi
item    N     M xsi.item AXsi_.Cat1 AXsi_.Cat2 AXsi_.Cat3 B.Cat1.Dim1
mj1  mj1 2351 2.260   -2.260     -4.278     -6.966     -6.779           1
mj2  mj2 2345 1.885   -1.171     -3.727     -4.816     -3.513           1
mj3  mj3 2334 1.474    0.056     -2.136     -1.807      0.168           1
mj4  mj4 2346 1.595   -0.300     -2.643     -2.702     -0.899           1
B.Cat2.Dim1 B.Cat3.Dim1
mj1           2           3
mj2           2           3
mj3           2           3
mj4           2           3

Item Parameters Xsi
xsi se.xsi
mj1_Cat1 -4.288  0.133
mj1_Cat2 -2.694  0.076
mj1_Cat3  0.186  0.056
mj2_Cat1 -3.735  0.099
mj2_Cat2 -1.094  0.061
mj2_Cat3  1.304  0.062
mj3_Cat1 -2.142  0.070
mj3_Cat2  0.327  0.060
mj3_Cat3  1.977  0.072
mj4_Cat1 -2.650  0.076
mj4_Cat2 -0.062  0.059
mj4_Cat3  1.804  0.069
theta truescore
1   -6.0    0.3110
2   -5.4    0.5332
3   -4.8    0.8803
4   -4.2    1.3799
5   -3.6    2.0333
6   -3.0    2.8093
7   -2.4    3.6597
8   -1.8    4.5441
9   -1.2    5.4475
10  -0.6    6.3782
11   0.0    7.3439
12   0.6    8.3246
13   1.2    9.2658
14   1.8   10.0967
15   2.4   10.7596
16   3.0   11.2366
17   3.6   11.5499
18   4.2   11.7423
19   4.8   11.8551
20   5.4   11.9193
21   6.0   11.9554
Warning message:
In TAM::IRT.truescore(mod1, theta = seq(-8, 8, len = 33)) :
The true score cannot be computed for some theta values.
Use a larger theta grid in the fitted model.

....................................................
Processing Data      2019-01-09 12:12:37
* Response Data: 1769 Persons and  11 Items
* Numerical integration with 61 nodes
* Created Design Matrices   ( 2019-01-09 12:12:37 )
* Calculated Sufficient Statistics   ( 2019-01-09 12:12:37 )
....................................................
Iteration 1     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 23601.1264
Maximum item intercept parameter change: 0.726289
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.700586
....................................................
Iteration 2     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22573.6347 | Absolute change: 1027.492 | Relative change: 0.04551734
Maximum item intercept parameter change: 0.395613
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.751357
....................................................
Iteration 3     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22260.453 | Absolute change: 313.1817 | Relative change: 0.01406897
Maximum item intercept parameter change: 0.081896
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.692569
....................................................
Iteration 4     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22141.9141 | Absolute change: 118.5389 | Relative change: 0.0053536
Maximum item intercept parameter change: 0.05851
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.528348
....................................................
Iteration 5     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22096.2882 | Absolute change: 45.6259 | Relative change: 0.00206487
Maximum item intercept parameter change: 0.040523
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.363251
....................................................
Iteration 6     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22079.2655 | Absolute change: 17.0227 | Relative change: 0.00077098
Maximum item intercept parameter change: 0.028215
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.234557
....................................................
Iteration 7     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22072.8619 | Absolute change: 6.4035 | Relative change: 0.00029011
Maximum item intercept parameter change: 0.020153
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.146099
....................................................
Iteration 8     2019-01-09 12:12:37
E Step
M Step Intercepts   |----
Deviance = 22070.3307 | Absolute change: 2.5313 | Relative change: 0.00011469
Maximum item intercept parameter change: 0.014899
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.089323
....................................................
Iteration 9     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22069.2274 | Absolute change: 1.1032 | Relative change: 4.999e-05
Maximum item intercept parameter change: 0.011416
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.054204
....................................................
Iteration 10     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22068.6733 | Absolute change: 0.5541 | Relative change: 2.511e-05
Maximum item intercept parameter change: 0.009049
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.033027
....................................................
Iteration 11     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22068.3511 | Absolute change: 0.3222 | Relative change: 1.46e-05
Maximum item intercept parameter change: 0.007363
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.020267
....................................................
Iteration 12     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22068.1397 | Absolute change: 0.2114 | Relative change: 9.58e-06
Maximum item intercept parameter change: 0.006125
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.012693
....................................................
Iteration 13     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.9905 | Absolute change: 0.1492 | Relative change: 6.76e-06
Maximum item intercept parameter change: 0.005174
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.008143
....................................................
Iteration 14     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.8807 | Absolute change: 0.1098 | Relative change: 4.97e-06
Maximum item intercept parameter change: 0.00442
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.005395
....................................................
Iteration 15     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.7983 | Absolute change: 0.0824 | Relative change: 3.74e-06
Maximum item intercept parameter change: 0.003806
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.003715
....................................................
Iteration 16     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.7358 | Absolute change: 0.0625 | Relative change: 2.83e-06
Maximum item intercept parameter change: 0.003294
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.002664
....................................................
Iteration 17     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.6882 | Absolute change: 0.0476 | Relative change: 2.16e-06
Maximum item intercept parameter change: 0.002862
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.001992
....................................................
Iteration 18     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.6518 | Absolute change: 0.0364 | Relative change: 1.65e-06
Maximum item intercept parameter change: 0.002492
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.001546
....................................................
Iteration 19     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.624 | Absolute change: 0.0278 | Relative change: 1.26e-06
Maximum item intercept parameter change: 0.002174
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.001239
....................................................
Iteration 20     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.6027 | Absolute change: 0.0213 | Relative change: 9.6e-07
Maximum item intercept parameter change: 0.001898
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.001018
....................................................
Iteration 21     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.5864 | Absolute change: 0.0163 | Relative change: 7.4e-07
Maximum item intercept parameter change: 0.001659
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000853
....................................................
Iteration 22     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.5739 | Absolute change: 0.0125 | Relative change: 5.7e-07
Maximum item intercept parameter change: 0.00145
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000724
....................................................
Iteration 23     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.5644 | Absolute change: 0.0096 | Relative change: 4.3e-07
Maximum item intercept parameter change: 0.001269
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000621
....................................................
Iteration 24     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.5571 | Absolute change: 0.0073 | Relative change: 3.3e-07
Maximum item intercept parameter change: 0.00111
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000537
....................................................
Iteration 25     2019-01-09 12:12:37
E Step
M Step Intercepts   |---
Deviance = 22067.5515 | Absolute change: 0.0056 | Relative change: 2.5e-07
Maximum item intercept parameter change: 0.000971
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000466
....................................................
Iteration 26     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5472 | Absolute change: 0.0043 | Relative change: 1.9e-07
Maximum item intercept parameter change: 0.00085
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000405
....................................................
Iteration 27     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5439 | Absolute change: 0.0033 | Relative change: 1.5e-07
Maximum item intercept parameter change: 0.000743
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000355
....................................................
Iteration 28     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5414 | Absolute change: 0.0025 | Relative change: 1.1e-07
Maximum item intercept parameter change: 0.00065
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000309
....................................................
Iteration 29     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5394 | Absolute change: 0.0019 | Relative change: 9e-08
Maximum item intercept parameter change: 0.000569
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.00027
....................................................
Iteration 30     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.538 | Absolute change: 0.0015 | Relative change: 7e-08
Maximum item intercept parameter change: 0.000498
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000236
....................................................
Iteration 31     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5368 | Absolute change: 0.0011 | Relative change: 5e-08
Maximum item intercept parameter change: 0.000436
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000206
....................................................
Iteration 32     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.536 | Absolute change: 9e-04 | Relative change: 4e-08
Maximum item intercept parameter change: 0.000382
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.00018
....................................................
Iteration 33     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5353 | Absolute change: 7e-04 | Relative change: 3e-08
Maximum item intercept parameter change: 0.000334
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000158
....................................................
Iteration 34     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5348 | Absolute change: 5e-04 | Relative change: 2e-08
Maximum item intercept parameter change: 0.000293
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000138
....................................................
Iteration 35     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5344 | Absolute change: 4e-04 | Relative change: 2e-08
Maximum item intercept parameter change: 0.000256
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000121
....................................................
Iteration 36     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5341 | Absolute change: 3e-04 | Relative change: 1e-08
Maximum item intercept parameter change: 0.000224
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 0.000106
....................................................
Iteration 37     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5339 | Absolute change: 2e-04 | Relative change: 1e-08
Maximum item intercept parameter change: 0.000196
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 9.3e-05
....................................................
Iteration 38     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5337 | Absolute change: 2e-04 | Relative change: 1e-08
Maximum item intercept parameter change: 0.000172
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 8.1e-05
....................................................
Iteration 39     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5336 | Absolute change: 1e-04 | Relative change: 1e-08
Maximum item intercept parameter change: 0.000151
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 7.1e-05
....................................................
Iteration 40     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5335 | Absolute change: 1e-04 | Relative change: 0
Maximum item intercept parameter change: 0.000132
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 6.2e-05
....................................................
Iteration 41     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5334 | Absolute change: 1e-04 | Relative change: 0
Maximum item intercept parameter change: 0.000115
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 5.4e-05
....................................................
Iteration 42     2019-01-09 12:12:37
E Step
M Step Intercepts   |--
Deviance = 22067.5333 | Absolute change: 1e-04 | Relative change: 0
Maximum item intercept parameter change: 0.000101
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 4.8e-05
....................................................
Iteration 43     2019-01-09 12:12:37
E Step
M Step Intercepts   |-
Deviance = 22067.5333 | Absolute change: 0 | Relative change: 0
Maximum item intercept parameter change: 8.8e-05
Maximum item slope parameter change: 0
Maximum regression parameter change: 0
Maximum variance parameter change: 4.2e-05
....................................................
Item Parameters
xsi.index      xsi.label     est
1          1        M032166 -1.4881
2          2        M032721 -0.1519
3          3        M032757 -0.8105
4          4       M032760A -0.1056
5          5       M032760B  0.7601
6          6       M032760C  1.2492
7          7        M032761  0.5487
8          8        M032692  0.4326
9          9        M032626 -0.8406
10        10        M032595 -1.3811
11        11        M032673 -1.3387
12        12  M032757_step1  1.9865
13        13 M032760A_step1  2.3451
14        14  M032761_step1  0.6156
15        15  M032692_step1  2.2930
...................................
Regression Coefficients
[,1]
[1,]    0

Variance:
[,1]
[1,] 4.658

EAP Reliability:
[1] 0.844

-----------------------------
Start:  2019-01-09 12:12:36
End:  2019-01-09 12:12:37
Time difference of 0.7887216 secs

------------------------------------------------------------
TAM 2.13-15 (2018-09-30 14:35:48)
sh: 1: cannot create /dev/null: Permission denied
R version 3.4.4 (2018-03-15) x86_64, linux-gnu | nodename=ip-172-31-1-5 | login=unknown

Date of Analysis: 2019-01-09 12:12:37
Time difference of 0.7887216 secs
Computation time: 0.7887216

Multidimensional Item Response Model in TAM

IRT Model PCM2

Call:
TAM::tam.mml(resp = dat, irtmodel = "PCM2", Q = Q, control = list(nodes = seq(-10,
10, len = 61)))

------------------------------------------------------------
Number of iterations= 43
Numeric integration with 61 integration points

Deviance= 22067.53
Log likelihood= -11033.77
Number of persons= 1769
Number of persons used= 1769
Number of items= 11
Number of estimated parameters= 16
Item threshold parameters= 15
Item slope parameters  = 0
Regression parameters  = 0
(Co)Variance parameters= 1

AIC = 22100  | penalty= 32    | AIC=-2*LL + 2*p
AIC3 = 22116  | penalty= 48    | AIC3=-2*LL + 3*p
BIC = 22187  | penalty= 119.65    | BIC=-2*LL + log(n)*p
aBIC = 22136  | penalty= 68.78    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC)
CAIC = 22203  | penalty= 135.65    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC)
AICc = 22100  | penalty= 32.31    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC)

------------------------------------------------------------
EAP Reliability
[1] 0.844
------------------------------------------------------------
Covariances and Variances
[,1]
[1,] 4.658
------------------------------------------------------------
Correlations and Standard Deviations (in the diagonal)
[,1]
[1,] 2.158
------------------------------------------------------------
Regression Coefficients
[,1]
[1,]    0
------------------------------------------------------------
Item Parameters -A*Xsi
item    N     M xsi.item AXsi_.Cat1 AXsi_.Cat2 B.Cat1.Dim1
M032166   M032166 1769 0.695   -1.488     -1.488         NA         1.0
M032721   M032721 1769 0.510   -0.152     -0.152         NA         1.0
M032757   M032757 1769 1.414   -0.810      1.176     -1.621         0.5
M032760A M032760A 1769 1.037   -0.106      2.240     -0.211         0.5
M032760B M032760B 1769 0.385    0.760      0.760         NA         1.0
M032760C M032760C 1769 0.323    1.249      1.249         NA         1.0
M032761   M032761 1769 0.717    0.549      1.164      1.097         0.5
M032692   M032692 1769 0.748    0.433      2.726      0.865         0.5
M032626   M032626 1769 0.607   -0.841     -0.841         NA         1.0
M032595   M032595 1769 0.681   -1.381     -1.381         NA         1.0
M032673   M032673 1769 0.676   -1.339     -1.339         NA         1.0
B.Cat2.Dim1
M032166            2
M032721            2
M032757            1
M032760A           1
M032760B           2
M032760C           2
M032761            1
M032692            1
M032626            2
M032595            2
M032673            2

Item Parameters Xsi
xsi se.xsi
M032166        -1.488  0.066
M032721        -0.152  0.063
M032757        -0.811  0.034
M032760A       -0.106  0.032
M032760B        0.760  0.066
M032760C        1.249  0.068
M032761         0.549  0.035
M032692         0.433  0.033
M032626        -0.841  0.064
M032595        -1.381  0.065
M032673        -1.339  0.065
M032757_step1   1.987  0.119
M032760A_step1  2.345  0.134
M032761_step1   0.616  0.068
M032692_step1   2.293  0.134
```

TAM documentation built on June 25, 2021, 5:13 p.m.