irt_coef
Stata command for y*-standardized coefficients in item response theory models
irt_coef is a Stata command that calculates y*-standardized coefficients for binary and ordinal item response theory (IRT) models. The discrimination parameters of an IRT model are on the log-odds (or probit) scale of each item, which makes their size hard to judge; standardizing on the latent propensity y* behind each item puts the coefficients on a common scale for interpreting them and for comparing across items and models. The raw coefficient, its standard error, and p-value are reported alongside the standardized coefficient.
Both types of coefficients are covered in my graduate course slides on item response theory: Item Response Theory slides. All of the materials for the course, including example code, are on the Latent Variable Modeling page.
Installation
net install irt_coef, from("https://tdmize.github.io/data") replaceTo read the help file in Stata, type help irt_coef; it is also on this site.
Citation
Please cite the use of irt_coef as:
Mize, Trenton D. 2024. “irt_coef: Stata command for y*-standardized coefficients in item response theory models.” https://www.trentonmize.com/software/irt_coef
Examples
After the irt command
The example fits a graded response model to ten ordinal items measuring mastery (a sense of control over one’s life) in Add Health Wave IV.
use "https://tdmize.github.io/data/data/lvm_ah4", clear(Add Health Wave IV | Latent Variable Modeling Class | )
. irt grm nocontrol handle yourway overwhelm nofutureR cantchangeR ///
othersdetR interfereR lowcontrolR cantsolveRFitting fixed-effects model:
Iteration 0: Log likelihood = -61673.715
Iteration 1: Log likelihood = -60212.937
Iteration 2: Log likelihood = -60187.957
Iteration 3: Log likelihood = -60187.915
Iteration 4: Log likelihood = -60187.915
Fitting full model:
Iteration 0: Log likelihood = -59216.094 (not concave)
Iteration 1: Log likelihood = -56183.755
Iteration 2: Log likelihood = -54851.956
Iteration 3: Log likelihood = -54726.048
Iteration 4: Log likelihood = -54724.395
Iteration 5: Log likelihood = -54724.39
Graded response model Number of obs = 5,113
Log likelihood = -54724.39
------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
nocontrol |
Discrim | -0.702 0.034 -20.896 0.000 -0.767 -0.636
Diff |
>=1 | 1.325 0.073 1.183 1.468
>=2 | -0.502 0.049 -0.598 -0.405
>=3 | -3.183 0.150 -3.477 -2.889
=4 | -5.032 0.244 -5.511 -4.553
-------------+----------------------------------------------------------------
handle |
Discrim | 0.943 0.037 25.197 0.000 0.870 1.017
Diff |
>=1 | -4.272 0.173 -4.611 -3.934
>=2 | -3.356 0.128 -3.607 -3.105
>=3 | -1.510 0.061 -1.630 -1.390
=4 | 0.567 0.041 0.487 0.646
-------------+----------------------------------------------------------------
yourway |
Discrim | 0.938 0.036 26.075 0.000 0.867 1.008
Diff |
>=1 | -4.103 0.160 -4.416 -3.790
>=2 | -2.699 0.100 -2.895 -2.504
>=3 | -0.231 0.036 -0.301 -0.161
=4 | 2.258 0.085 2.092 2.424
-------------+----------------------------------------------------------------
overwhelm |
Discrim | -0.756 0.034 -22.071 0.000 -0.823 -0.689
Diff |
>=1 | 1.185 0.064 1.059 1.310
>=2 | -0.818 0.053 -0.922 -0.714
>=3 | -3.191 0.142 -3.468 -2.913
=4 | -5.020 0.233 -5.478 -4.563
-------------+----------------------------------------------------------------
nofutureR |
Discrim | 1.301 0.043 29.963 0.000 1.216 1.387
Diff |
>=2 | -4.112 0.155 -4.415 -3.808
>=3 | -2.450 0.075 -2.597 -2.302
>=4 | -1.610 0.050 -1.709 -1.511
=5 | 1.227 0.043 1.142 1.313
-------------+----------------------------------------------------------------
cantchangeR |
Discrim | 1.950 0.059 33.233 0.000 1.835 2.065
Diff |
>=2 | -3.245 0.101 -3.443 -3.047
>=3 | -2.057 0.051 -2.157 -1.957
>=4 | -1.318 0.035 -1.385 -1.250
=5 | 1.027 0.031 0.965 1.088
-------------+----------------------------------------------------------------
othersdetR |
Discrim | 2.049 0.063 32.477 0.000 1.925 2.172
Diff |
>=2 | -3.261 0.102 -3.462 -3.061
>=3 | -2.228 0.055 -2.337 -2.119
>=4 | -1.532 0.038 -1.607 -1.458
=5 | 0.580 0.026 0.529 0.630
-------------+----------------------------------------------------------------
interfereR |
Discrim | 1.329 0.042 31.677 0.000 1.246 1.411
Diff |
>=2 | -3.304 0.106 -3.512 -3.097
>=3 | -1.099 0.039 -1.174 -1.023
>=4 | -0.218 0.028 -0.272 -0.164
=5 | 2.143 0.063 2.019 2.267
-------------+----------------------------------------------------------------
lowcontrolR |
Discrim | 2.376 0.070 33.719 0.000 2.238 2.514
Diff |
>=2 | -3.007 0.088 -3.180 -2.835
>=3 | -1.900 0.043 -1.985 -1.815
>=4 | -1.096 0.028 -1.151 -1.040
=5 | 1.078 0.029 1.020 1.136
-------------+----------------------------------------------------------------
cantsolveR |
Discrim | 2.551 0.082 31.055 0.000 2.390 2.712
Diff |
>=2 | -3.140 0.099 -3.333 -2.946
>=3 | -2.304 0.055 -2.412 -2.197
>=4 | -1.546 0.035 -1.615 -1.478
=5 | 0.729 0.025 0.679 0.778
------------------------------------------------------------------------------irt_coef then calculates the y*-standardized coefficients. The help option adds a legend beneath the table explaining each column.
irt_coef, helpy* standardized coefficients (and raw coefficient) from IRT model N=5113
| Std Coef Coef Std. Err. P>|z|
-------------+-----------------------------------------------
nocontrol | -0.361 -0.702 0.034 0.000
handle | 0.461 0.943 0.037 0.000
yourway | 0.459 0.938 0.036 0.000
overwhelm | -0.385 -0.756 0.034 0.000
nofutureR | 0.583 1.301 0.043 0.000
cantchangeR | 0.732 1.950 0.059 0.000
othersdetR | 0.749 2.049 0.063 0.000
interfereR | 0.591 1.329 0.042 0.000
lowcontrolR | 0.795 2.376 0.070 0.000
cantsolveR | 0.815 2.551 0.082 0.000
Std Coef : y* standardized IRT regression coefficient
Coef : IRT raw regression coefficient
NOTE : SE and p-value based on raw regression coefficientAfter gsem
An identical model can be fit with gsem. Because gsem lets the latent variable take any name, the latent() option tells irt_coef which one to use (after irt it is always Theta, and the option is not needed).
gsem (Mastery -> nocontrol handle yourway overwhelm nofutureR cantchangeR ///
othersdetR interfereR lowcontrolR cantsolveR, ologit), var(Mastery@1)Fitting fixed-effects model:
Iteration 0: Log likelihood = -60187.915
Iteration 1: Log likelihood = -60187.915
Refining starting values:
Grid node 0: Log likelihood = -57791.913
Fitting full model:
Iteration 0: Log likelihood = -57791.913 (not concave)
Iteration 1: Log likelihood = -55506.534
Iteration 2: Log likelihood = -54756.929
Iteration 3: Log likelihood = -54724.67
Iteration 4: Log likelihood = -54724.391
Iteration 5: Log likelihood = -54724.39
Generalized structural equation model Number of obs = 5,113
Response: nocontrol Number of obs = 5,107
Family: Ordinal
Link: Logit
Response: handle Number of obs = 5,111
Family: Ordinal
Link: Logit
Response: yourway Number of obs = 5,108
Family: Ordinal
Link: Logit
Response: overwhelm Number of obs = 5,106
Family: Ordinal
Link: Logit
Response: nofutureR Number of obs = 5,101
Family: Ordinal
Link: Logit
Response: cantchangeR Number of obs = 5,097
Family: Ordinal
Link: Logit
Response: othersdetR Number of obs = 5,103
Family: Ordinal
Link: Logit
Response: interfereR Number of obs = 5,101
Family: Ordinal
Link: Logit
Response: lowcontrolR Number of obs = 5,096
Family: Ordinal
Link: Logit
Response: cantsolveR Number of obs = 5,103
Family: Ordinal
Link: Logit
Log likelihood = -54724.39
( 1) [/]var(Mastery) = 1
------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
nocontrol |
Mastery | 0.702 0.034 20.896 0.000 0.636 0.767
-------------+----------------------------------------------------------------
handle |
Mastery | -0.943 0.037 -25.197 0.000 -1.017 -0.870
-------------+----------------------------------------------------------------
yourway |
Mastery | -0.938 0.036 -26.075 0.000 -1.008 -0.867
-------------+----------------------------------------------------------------
overwhelm |
Mastery | 0.756 0.034 22.071 0.000 0.689 0.823
-------------+----------------------------------------------------------------
nofutureR |
Mastery | -1.301 0.043 -29.963 0.000 -1.387 -1.216
-------------+----------------------------------------------------------------
cantchangeR |
Mastery | -1.950 0.059 -33.233 0.000 -2.065 -1.835
-------------+----------------------------------------------------------------
othersdetR |
Mastery | -2.049 0.063 -32.477 0.000 -2.172 -1.925
-------------+----------------------------------------------------------------
interfereR |
Mastery | -1.329 0.042 -31.677 0.000 -1.411 -1.246
-------------+----------------------------------------------------------------
lowcontrolR |
Mastery | -2.376 0.070 -33.719 0.000 -2.514 -2.238
-------------+----------------------------------------------------------------
cantsolveR |
Mastery | -2.551 0.082 -31.054 0.000 -2.712 -2.390
-------------+----------------------------------------------------------------
/nocontrol |
cut1 | -0.930 0.034 -0.996 -0.864
cut2 | 0.352 0.031 0.291 0.413
cut3 | 2.233 0.048 2.139 2.327
cut4 | 3.531 0.079 3.376 3.686
-------------+----------------------------------------------------------------
/handle |
cut1 | -4.031 0.093 -4.214 -3.848
cut2 | -3.166 0.067 -3.297 -3.035
cut3 | -1.424 0.040 -1.502 -1.346
cut4 | 0.535 0.034 0.469 0.601
-------------+----------------------------------------------------------------
/yourway |
cut1 | -3.848 0.087 -4.018 -3.678
cut2 | -2.532 0.054 -2.637 -2.426
cut3 | -0.216 0.033 -0.281 -0.152
cut4 | 2.118 0.048 2.025 2.211
-------------+----------------------------------------------------------------
/overwhelm |
cut1 | -0.896 0.034 -0.962 -0.829
cut2 | 0.619 0.033 0.555 0.682
cut3 | 2.412 0.051 2.312 2.513
cut4 | 3.796 0.088 3.623 3.968
-------------+----------------------------------------------------------------
/nofutureR |
cut1 | -5.351 0.150 -5.644 -5.058
cut2 | -3.188 0.068 -3.321 -3.055
cut3 | -2.096 0.051 -2.195 -1.996
cut4 | 1.598 0.045 1.508 1.687
-------------+----------------------------------------------------------------
/cantchangeR |
cut1 | -6.328 0.176 -6.673 -5.983
cut2 | -4.011 0.091 -4.190 -3.832
cut3 | -2.570 0.066 -2.700 -2.439
cut4 | 2.002 0.059 1.887 2.117
-------------+----------------------------------------------------------------
/othersdetR |
cut1 | -6.682 0.193 -7.061 -6.303
cut2 | -4.564 0.107 -4.774 -4.355
cut3 | -3.139 0.078 -3.293 -2.986
cut4 | 1.187 0.051 1.088 1.287
-------------+----------------------------------------------------------------
/interfereR |
cut1 | -4.390 0.101 -4.588 -4.193
cut2 | -1.460 0.043 -1.544 -1.375
cut3 | -0.290 0.037 -0.362 -0.217
cut4 | 2.847 0.063 2.723 2.971
-------------+----------------------------------------------------------------
/lowcontrolR |
cut1 | -7.145 0.203 -7.544 -6.747
cut2 | -4.513 0.108 -4.724 -4.303
cut3 | -2.603 0.074 -2.747 -2.459
cut4 | 2.561 0.074 2.416 2.706
-------------+----------------------------------------------------------------
/cantsolveR |
cut1 | -8.010 0.258 -8.516 -7.505
cut2 | -5.879 0.154 -6.180 -5.578
cut3 | -3.945 0.106 -4.154 -3.737
cut4 | 1.859 0.067 1.728 1.990
-------------+----------------------------------------------------------------
var(Mastery)| 1.000 (constrained)
------------------------------------------------------------------------------irt_coef, latent(Mastery)y* standardized coefficients (and raw coefficient) from IRT model N=5113
| Std Coef Coef Std. Err. P>|z|
-------------+-----------------------------------------------
nocontrol | 0.361 0.702 0.034 0.000
handle | -0.461 -0.943 0.037 0.000
yourway | -0.459 -0.938 0.036 0.000
overwhelm | 0.385 0.756 0.034 0.000
nofutureR | -0.583 -1.301 0.043 0.000
cantchangeR | -0.732 -1.950 0.059 0.000
othersdetR | -0.749 -2.049 0.063 0.000
interfereR | -0.591 -1.329 0.042 0.000
lowcontrolR | -0.795 -2.376 0.070 0.000
cantsolveR | -0.815 -2.551 0.082 0.000 The sort option orders the items by the size of the standardized coefficient, and decimals() sets the number of decimal places; see the help file for the full list of options.