irt_me
Stata command for marginal effects from item response theory models
irt_me is a Stata command that automates the calculation of marginal effects after item response theory (IRT) models. In an IRT model the latent variable is the independent variable and the observed items are the dependent variables, so a marginal effect quantifies the relationship between the latent variable and an item in the predicted probability metric – an alternative to interpreting the discrimination and difficulty parameters directly. irt_me works with every model the irt commands fit, with binary, ordinal, and nominal items as well as continuous and count items in gsem, and with a mix of item types in one model.
The approach is described in the slides from my presentation at the 2024 Italian Stata Conference: Marginal effects for IRT models. It is also covered in my graduate course slides on Item Response Theory; all of the materials for the course, including example code, are on the Latent Variable Modeling page.
Installation
net install irt_me, from("https://tdmize.github.io/data") replaceTo read the help file in Stata, type help irt_me; it is also on this site.
Citation
Please cite the use of irt_me as:
Mize, Trenton D. 2024. “Stata command for marginal effects from item response theory models.” https://www.trentonmize.com/software/irt_me
Examples
Load the data and fit the IRT model
The example uses ten binary depression items from Add Health Wave IV and a two-parameter logistic model.
use "https://tdmize.github.io/data/data/lvm_ah4", clear(Add Health Wave IV | Latent Variable Modeling Class | )
. irt 2pl botherB bluesB keepmindB depressB tiredB dislikedB sadB ///
feltgoodBR happyBR enjoylifeBRFitting fixed-effects model:
Iteration 0: Log likelihood = -24968.123
Iteration 1: Log likelihood = -24426.745
Iteration 2: Log likelihood = -24398.991
Iteration 3: Log likelihood = -24398.692
Iteration 4: Log likelihood = -24398.692
Fitting full model:
Iteration 0: Log likelihood = -23626.374 (not concave)
Iteration 1: Log likelihood = -21936.782
Iteration 2: Log likelihood = -21692.881
Iteration 3: Log likelihood = -21320.524
Iteration 4: Log likelihood = -21283.244
Iteration 5: Log likelihood = -21281.501
Iteration 6: Log likelihood = -21281.483
Iteration 7: Log likelihood = -21281.482
Two-parameter logistic model Number of obs = 5,113
Log likelihood = -21281.482
------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
botherB |
Discrim | 1.453 0.060 24.219 0.000 1.335 1.570
Diff | 0.295 0.028 10.613 0.000 0.240 0.349
-------------+----------------------------------------------------------------
bluesB |
Discrim | 3.198 0.163 19.655 0.000 2.879 3.517
Diff | 0.787 0.024 32.561 0.000 0.740 0.835
-------------+----------------------------------------------------------------
keepmindB |
Discrim | 1.024 0.049 20.967 0.000 0.929 1.120
Diff | -0.591 0.039 -14.994 0.000 -0.669 -0.514
-------------+----------------------------------------------------------------
depressB |
Discrim | 3.497 0.187 18.682 0.000 3.130 3.864
Diff | 0.603 0.022 27.706 0.000 0.561 0.646
-------------+----------------------------------------------------------------
tiredB |
Discrim | 0.738 0.043 17.274 0.000 0.654 0.821
Diff | -0.972 0.064 -15.261 0.000 -1.097 -0.848
-------------+----------------------------------------------------------------
dislikedB |
Discrim | 0.992 0.050 19.971 0.000 0.894 1.089
Diff | 1.363 0.063 21.748 0.000 1.240 1.486
-------------+----------------------------------------------------------------
sadB |
Discrim | 2.631 0.122 21.578 0.000 2.392 2.870
Diff | 0.046 0.021 2.208 0.027 0.005 0.086
-------------+----------------------------------------------------------------
feltgoodBR |
Discrim | 0.805 0.079 10.188 0.000 0.650 0.960
Diff | 4.055 0.342 11.845 0.000 3.384 4.726
-------------+----------------------------------------------------------------
happyBR |
Discrim | 1.894 0.159 11.946 0.000 1.583 2.204
Diff | 2.833 0.134 21.087 0.000 2.569 3.096
-------------+----------------------------------------------------------------
enjoylifeBR |
Discrim | 2.275 0.215 10.606 0.000 1.855 2.696
Diff | 2.799 0.128 21.934 0.000 2.549 3.049
------------------------------------------------------------------------------Calculate the marginal effects
By default irt_me reports, for each item, the change in the predicted probability for a one standard deviation increase in the latent variable, centered on its mean (from -0.5 to 0.5 on the standardized scale). The help option adds a legend beneath the table.
irt_me, helpMarginal Effects of + 1.000 Increase in Latent Variable (theta) N=5113
| PrStart PrEnd ME Est. Std. Err. P>|z|
-------------+-----------------------------------------------------------
botherB | 0.240 0.574 0.334 0.013 0.000
bluesB | 0.016 0.285 0.269 0.013 0.000
keepmindB | 0.523 0.754 0.230 0.010 0.000
depressB | 0.021 0.411 0.390 0.017 0.000
tiredB | 0.586 0.748 0.161 0.009 0.000
dislikedB | 0.136 0.298 0.162 0.007 0.000
sadB | 0.192 0.768 0.576 0.020 0.000
feltgoodBR | 0.025 0.054 0.029 0.002 0.000
happyBR | 0.002 0.012 0.010 0.002 0.000
enjoylifeBR | 0.001 0.005 0.005 0.001 0.000
PrStart : Pr(y=1) at theta = -0.500
PrEnd : Pr(y=1) at theta = 0.500
ME : PrEnd - PrStartFor an average person, a one standard deviation increase in latent depression is associated with a 0.334 increase in the probability of reporting being bothered by things last week (the botherB item).
The same model with gsem
An identical model can be fit with gsem. Because gsem lets the latent variable take any name, the latent() option tells irt_me which one to use (after irt it is always Theta, and the option is not needed).
gsem (Depress -> botherB bluesB keepmindB depressB tiredB dislikedB sadB ///
feltgoodBR happyBR enjoylifeBR), logit var(Depress@1)Fitting fixed-effects model:
Iteration 0: Log likelihood = -24968.123
Iteration 1: Log likelihood = -24426.745
Iteration 2: Log likelihood = -24398.991
Iteration 3: Log likelihood = -24398.692
Iteration 4: Log likelihood = -24398.692
Refining starting values:
Grid node 0: Log likelihood = -23626.374
Fitting full model:
Iteration 0: Log likelihood = -23626.374 (not concave)
Iteration 1: Log likelihood = -21936.782
Iteration 2: Log likelihood = -21692.881
Iteration 3: Log likelihood = -21320.524
Iteration 4: Log likelihood = -21283.244
Iteration 5: Log likelihood = -21281.501
Iteration 6: Log likelihood = -21281.483
Iteration 7: Log likelihood = -21281.482
Generalized structural equation model Number of obs = 5,113
Response: botherB Number of obs = 5,112
Family: Bernoulli
Link: Logit
Response: bluesB Number of obs = 5,113
Family: Bernoulli
Link: Logit
Response: keepmindB Number of obs = 5,112
Family: Bernoulli
Link: Logit
Response: depressB Number of obs = 5,113
Family: Bernoulli
Link: Logit
Response: tiredB Number of obs = 5,113
Family: Bernoulli
Link: Logit
Response: dislikedB Number of obs = 5,111
Family: Bernoulli
Link: Logit
Response: sadB Number of obs = 5,113
Family: Bernoulli
Link: Logit
Response: feltgoodBR Number of obs = 5,109
Family: Bernoulli
Link: Logit
Response: happyBR Number of obs = 5,113
Family: Bernoulli
Link: Logit
Response: enjoylifeBR Number of obs = 5,113
Family: Bernoulli
Link: Logit
Log likelihood = -21281.482
( 1) [/]var(Depress) = 1
------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
botherB |
Depress | 1.453 0.060 24.219 0.000 1.335 1.570
_cons | -0.428 0.040 -10.747 0.000 -0.506 -0.350
-------------+----------------------------------------------------------------
bluesB |
Depress | 3.198 0.163 19.655 0.000 2.879 3.517
_cons | -2.518 0.116 -21.654 0.000 -2.746 -2.290
-------------+----------------------------------------------------------------
keepmindB |
Depress | 1.024 0.049 20.967 0.000 0.929 1.120
_cons | 0.606 0.036 16.926 0.000 0.536 0.676
-------------+----------------------------------------------------------------
depressB |
Depress | 3.497 0.187 18.682 0.000 3.130 3.864
_cons | -2.110 0.112 -18.875 0.000 -2.329 -1.891
-------------+----------------------------------------------------------------
tiredB |
Depress | 0.738 0.043 17.274 0.000 0.654 0.821
_cons | 0.717 0.034 21.348 0.000 0.651 0.783
-------------+----------------------------------------------------------------
dislikedB |
Depress | 0.992 0.050 19.971 0.000 0.894 1.089
_cons | -1.352 0.042 -32.001 0.000 -1.435 -1.269
-------------+----------------------------------------------------------------
sadB |
Depress | 2.631 0.122 21.578 0.000 2.392 2.870
_cons | -0.120 0.055 -2.199 0.028 -0.228 -0.013
-------------+----------------------------------------------------------------
feltgoodBR |
Depress | 0.805 0.079 10.188 0.000 0.650 0.960
_cons | -3.264 0.086 -37.936 0.000 -3.432 -3.095
-------------+----------------------------------------------------------------
happyBR |
Depress | 1.894 0.159 11.946 0.000 1.583 2.204
_cons | -5.364 0.248 -21.605 0.000 -5.851 -4.878
-------------+----------------------------------------------------------------
enjoylifeBR |
Depress | 2.275 0.215 10.606 0.000 1.855 2.696
_cons | -6.368 0.373 -17.077 0.000 -7.099 -5.637
-------------+----------------------------------------------------------------
var(Depress)| 1.000 (constrained)
------------------------------------------------------------------------------irt_me, latent(Depress)Marginal Effects of + 1.000 Increase in Latent Variable (theta) N=5113
| PrStart PrEnd ME Est. Std. Err. P>|z|
-------------+-----------------------------------------------------------
botherB | 0.240 0.574 0.334 0.013 0.000
bluesB | 0.016 0.285 0.269 0.013 0.000
keepmindB | 0.523 0.754 0.230 0.010 0.000
depressB | 0.021 0.411 0.390 0.017 0.000
tiredB | 0.586 0.748 0.161 0.009 0.000
dislikedB | 0.136 0.298 0.162 0.007 0.000
sadB | 0.192 0.768 0.576 0.020 0.000
feltgoodBR | 0.025 0.054 0.029 0.002 0.000
happyBR | 0.002 0.012 0.010 0.002 0.000
enjoylifeBR | 0.001 0.005 0.005 0.001 0.000 Marginal effects across the trimmed range of the latent variable
The range option calculates the change in each item’s probability across the trimmed range of the latent variable: from its 1st percentile to its 99th percentile, as predicted for the sample.
irt_me, range helpIf gsem was used to fit the model , the latent variable must be specified in the latent( ) option.Customized marginal effects
start() and end() set any starting and ending values on the standardized scale of the latent variable. Here the change is from two to three standard deviations above the mean: the effect of becoming more depressed among those who are already highly depressed.
irt_me, start(2) end(3) helpIf gsem was used to fit the model , the latent variable must be specified in the latent( ) option.