Hierarchical Item Response Theory Models.
hIRT: hierarchical item response theory (IRT) models
hIRT is an R package that implements a class of hierarchical item response theory (IRT) models where both the mean and the variance of the latent “ability parameters” may depend on observed covariates. The current implementation includes both the two-parameter latent trait model for binary data (hltm() and hltm2()) and the graded response model for ordinal data (hgrm() and hgrm2()). Both are fitted via the Expectation-Maximization (EM) algorithm. Asymptotic standard errors are derived from the observed information matrix.
Main Reference: Zhou, Xiang. 2019. “Hierarchical Item Response Models for Analyzing Public Opinion.” Political Analysis, 27(4): 481-502. Available at: https://doi.org/10.1017/pan.2018.63
Full paper with technical appendix is available at: https://scholar.harvard.edu/files/xzhou/files/Zhou2019_hIRT.pdf
Installation
You can install the released version of hIRT from CRAN with:
install.packages("hIRT")
And the development version from GitHub with:
# install.packages("devtools")
devtools::install_github("xiangzhou09/hIRT")
Example
The following example illustrates how the hgrm() function can be used to examine the effects of education and party affiliation on economic ideology, a latent variable gauged by a number of survey items in the American National Election Studies (ANES), 2008. Documentation of the dataset nes_econ2008 can be accessed by running ?nes_econ2008 in R after loading the hIRT package.
library(hIRT)
#> Registered S3 method overwritten by 'pryr':
#> method from
#> print.bytes Rcpp
# survey items used to measure economic ideology
y <- nes_econ2008[, -(1:3)]
# predictors for the mean of economic ideology
x <- model.matrix( ~ party * educ, nes_econ2008)
# predictors for the variance of economic ideology
z <- model.matrix( ~ party, nes_econ2008)
# fitting a hierarhical graded response model
nes_m1 <- hgrm(y, x, z)
#> ............
#> converged at iteration 12
nes_m1
#>
#> Call:
#> hgrm(y = y, x = x, z = z)
#>
#> Mean Regression:
#> Estimate Std_Error z_value p_value
#> x_(Intercept) -0.480 0.105 -4.570 0.000
#> x_partyindependent 0.386 0.086 4.473 0.000
#> x_partyRepublican 1.133 0.135 8.408 0.000
#> x_educ2 0.037 0.079 0.467 0.641
#> x_partyindependent:educ2 0.235 0.117 2.007 0.045
#> x_partyRepublican:educ2 0.428 0.148 2.886 0.004
#>
#> Variance Regression:
#> Estimate Std_Error z_value p_value
#> z_(Intercept) -0.097 0.139 -0.697 0.486
#> z_partyindependent 0.166 0.100 1.661 0.097
#> z_partyRepublican 0.172 0.126 1.373 0.170
#>
#> Log Likelihood: -16259.16
The output from hgrm is an object of class hIRT. The print() method for hIRT outputs the regression tables for the mean regression and the variance regression.
Extracting coefficients
The coef_item(), coef_mean(), and coef_var() functions can be used to extract coefficient tables for item parameters, the mean regression, and the variance regression respectively.
coef_item(nes_m1)
#> $health_ins7
#> Estimate Std_Error z_value p_value
#> y>=2 1.279 NA NA NA
#> y>=3 0.541 0.063 8.542 0.000
#> y>=4 -0.075 0.083 -0.898 0.369
#> y>=5 -1.047 0.107 -9.826 0.000
#> y>=6 -1.852 0.124 -14.901 0.000
#> y>=7 -2.684 0.149 -17.990 0.000
#> Dscrmn 1.016 0.096 10.569 0.000
#>
#> $jobs_guar7
#> Estimate Std_Error z_value p_value
#> y>=2 2.136 0.173 12.377 0
#> y>=3 1.352 0.153 8.860 0
#> y>=4 0.607 0.141 4.299 0
#> y>=5 -0.520 0.137 -3.797 0
#> y>=6 -1.611 0.141 -11.429 0
#> y>=7 -2.785 0.163 -17.043 0
#> Dscrmn 1.305 0.114 11.448 0
#>
#> $gov_services7
#> Estimate Std_Error z_value p_value
#> y>=2 3.950 0.222 17.760 0.000
#> y>=3 2.859 0.182 15.707 0.000
#> y>=4 1.831 0.158 11.592 0.000
#> y>=5 0.247 0.147 1.679 0.093
#> y>=6 -1.001 0.154 -6.490 0.000
#> y>=7 -2.020 0.169 -11.947 0.000
#> Dscrmn -1.363 0.116 -11.715 0.000
#>
#> $FS_poor3
#> Estimate Std_Error z_value p_value
#> y>=2 -1.180 0.179 -6.601 0
#> y>=3 -4.459 0.243 -18.357 0
#> Dscrmn 1.918 0.164 11.679 0
#>
#> $FS_childcare3
#> Estimate Std_Error z_value p_value
#> y>=2 -0.808 0.148 -5.474 0
#> y>=3 -4.051 0.192 -21.132 0
#> Dscrmn 1.608 0.128 12.535 0
#>
#> $FS_crime3
#> Estimate Std_Error z_value p_value
#> y>=2 -0.845 0.066 -12.866 0
#> y>=3 -3.150 0.108 -29.048 0
#> Dscrmn 0.516 0.059 8.823 0
#>
#> $FS_publicschools3
#> Estimate Std_Error z_value p_value
#> y>=2 -1.790 0.136 -13.197 0
#> y>=3 -4.144 0.188 -22.022 0
#> Dscrmn 1.302 0.111 11.751 0
#>
#> $FS_welfare3
#> Estimate Std_Error z_value p_value
#> y>=2 1.054 0.117 8.970 0
#> y>=3 -1.355 0.116 -11.650 0
#> Dscrmn 1.178 0.099 11.937 0
#>
#> $FS_envir3
#> Estimate Std_Error z_value p_value
#> y>=2 -0.855 0.106 -8.071 0
#> y>=3 -3.499 0.159 -22.023 0
#> Dscrmn 1.101 0.092 11.953 0
#>
#> $FS_socsec3
#> Estimate Std_Error z_value p_value
#> y>=2 -1.091 0.104 -10.535 0
#> y>=3 -4.278 0.178 -24.033 0
#> Dscrmn 1.028 NA NA NA
coef_mean(nes_m1)
#> Estimate Std_Error z_value p_value
#> x_(Intercept) -0.480 0.105 -4.570 0.000
#> x_partyindependent 0.386 0.086 4.473 0.000
#> x_partyRepublican 1.133 0.135 8.408 0.000
#> x_educ2 0.037 0.079 0.467 0.641
#> x_partyindependent:educ2 0.235 0.117 2.007 0.045
#> x_partyRepublican:educ2 0.428 0.148 2.886 0.004
coef_var(nes_m1)
#> Estimate Std_Error z_value p_value
#> z_(Intercept) -0.097 0.139 -0.697 0.486
#> z_partyindependent 0.166 0.100 1.661 0.097
#> z_partyRepublican 0.172 0.126 1.373 0.170
Latent scores
The latent_scores() function can be used to extract the Expected A Posteriori (EAP) estimates of the latent ability parameters, along with their “prior” estimates (without the random effects). In this example, the latent ability estimates can be interpreted as the estimated ideological positions of ANES respondents on economic issues.
pref <- latent_scores(nes_m1)
summary(pref)
#> post_mean post_sd prior_mean prior_sd
#> Min. :-2.082000 Min. :0.3940 Min. :-0.4800000 Min. :0.953
#> 1st Qu.:-0.751000 1st Qu.:0.4788 1st Qu.:-0.4440000 1st Qu.:0.953
#> Median :-0.104000 Median :0.5280 Median :-0.0950000 Median :1.035
#> Mean :-0.000147 Mean :0.5469 Mean :-0.0001561 Mean :1.001
#> 3rd Qu.: 0.629500 3rd Qu.:0.6090 3rd Qu.: 0.1770000 3rd Qu.:1.035
#> Max. : 3.359000 Max. :0.9780 Max. : 1.1170000 Max. :1.039
Identification constraints.
The constr parameter in the hgrm() and hltm() function can be used to specify the type of constraints used to identify the model. The default option, "latent_scale", constrains the mean of the latent ability parameters to zero and the geometric mean of their prior variance to one; Alternatively, "items" sets the mean of the item difficulty parameters to zero and the geometric mean of the discrimination parameters to one.
In practice, one may want to interpret the effects of the mean predictors (in the above example, education and party affiliation) on the standard deviation scale of the latent trait. This can be easily achieved through rescaling their point estimates and standard errors.
library(dplyr)
total_sd <- sqrt(var(pref$post_mean) + mean(pref$post_sd^2))
coef_mean_sd_scale <- coef_mean(nes_m1) %>%
mutate(`Estimate` = `Estimate`/total_sd,
`Std_Error` = `Std_Error`/total_sd)
coef_mean_sd_scale
#> Estimate Std_Error z_value p_value
#> 1 -0.42437486 0.09283200 -4.570 0.000
#> 2 0.34126812 0.07603383 4.473 0.000
#> 3 1.00170150 0.11935543 8.408 0.000
#> 4 0.03271223 0.06984503 0.467 0.641
#> 5 0.20776686 0.10344137 2.007 0.045
#> 6 0.37840092 0.13084892 2.886 0.004
hIRT with fixed item parameters
Sometimes, the researcher might want to fit the hIRT models using a set of fixed item parameters, for example, to make results comparable across different studies. The hgrm2() and hltm2() functions can be used for this purpose. They are illustrated in more detail in the package documentation.