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
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")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
#> xpartyindependent 0.386 0.086 4.473 0.000
#> xpartyRepublican 1.133 0.135 8.408 0.000
#> xeduc2 0.037 0.079 0.467 0.641
#> xpartyindependent:educ2 0.235 0.117 2.007 0.045
#> xpartyRepublican: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
#> zpartyindependent 0.166 0.100 1.661 0.097
#> zpartyRepublican 0.172 0.126 1.373 0.170
#>
#> Log Likelihood: -16259.16The 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.
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
#> xpartyindependent 0.386 0.086 4.473 0.000
#> xpartyRepublican 1.133 0.135 8.408 0.000
#> xeduc2 0.037 0.079 0.467 0.641
#> xpartyindependent:educ2 0.235 0.117 2.007 0.045
#> xpartyRepublican: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
#> zpartyindependent 0.166 0.100 1.661 0.097
#> zpartyRepublican 0.172 0.126 1.373 0.170The 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. :1.039The 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) %>%
tibble::rownames_to_column("term") %>%
mutate(`Estimate` = `Estimate`/total_sd,
`Std_Error` = `Std_Error`/total_sd)
coef_mean_sd_scale
#> term Estimate Std_Error z_value p_value
#> 1 x(Intercept) -0.42437486 0.09283200 -4.570 0.000
#> 2 xpartyindependent 0.34126812 0.07603383 4.473 0.000
#> 3 xpartyRepublican 1.00170150 0.11935543 8.408 0.000
#> 4 xeduc2 0.03271223 0.06984503 0.467 0.641
#> 5 xpartyindependent:educ2 0.20776686 0.10344137 2.007 0.045
#> 6 xpartyRepublican:educ2 0.37840092 0.13084892 2.886 0.004Sometimes, 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.