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iop: Inflated Ordered Probit and Logit Models

Documentation site: https://bagozzib.github.io/iop/.

iop estimates ordered probit and ordered logit models whose outcome contains an inflated category – a single ordered category (bottom, middle, top, or any other) that mixes observations generated by the ordered process with observations generated by a distinct split-population process that places them in that category regardless of the ordered mechanism. It fits the zero-inflated ordered probit of Harris and Zhao (2007) and its middle- and top-inflated extensions (Bagozzi and Mukherjee 2012; Bagozzi, Hill, Moore and Mukherjee 2015; Bagozzi, Joo and Mukherjee 2024), generalized to any inflated category and to the logit link, alongside the plain ordered models on the same engine. It provides:

  • oprobit(), ologit() – the standard ordered models, with survey weights, offsets, partial proportional-odds (non-parallel) effects (parallel =, parallel_test(), parallel = "auto"), and analytic, robust, or cluster-robust standard errors;
  • iop() – the inflated ordered probit for any single inflated category (inflate = "bottom" | "middle" | "top" or a category label), with optional correlated errors (correlated = TRUE: ZiOPC / MiOPC / TiOPC) and optional category-specific split equations (split = "category": the generalised GZiOP / GMiOP of Brown, Harris and Spencer 2020, tested against the common split by split_test());
  • iol() – the inflated ordered logit counterpart;
  • panel tools on every model: unit random intercepts (re =, adaptive Gauss–Hermite quadrature, optionally in both equations), unit fixed effects (fe =, with the split-panel jackknife fe_correction = "jackknife"), the mundlak() correlated-random-effects device, and cluster-robust standard errors;
  • one set of methods for all four: summary(), predict() by regime and by type of zero (with delta-method se.fit), first_difference() and ame() (average marginal effects, by stage – outcome equation, inflation equation, or both – and with the two-types-of-zeros decomposition) and their plot() methods, the diord()/piord()/qiord()/riord() distribution family of the response, vuong(), lr_test(), inflation_test() (Vuong and a parametric-bootstrap likelihood-ratio test), split_test(), classification() (confusion table, Brier and ranked probability scores), compare_models(), simulate() (DHARMa-ready), ranef(), analytic / robust / cluster-robust / bootstrap standard errors, and broom / modelsummary / texreg integration.

The likelihood, its analytic gradient, and the bivariate-normal probabilities are implemented in C++.

Installation 

# development version
# install.packages("remotes")
remotes::install_github("bagozzib/iop", build_vignettes = TRUE)

Documentation

  • vignette("iop") – getting started: the four estimators on the bundled political-violence (zero-inflated), trade-agreement (top-inflated), and simulated survey (middle-inflated) data.
  • vignette("quantities") – predicted probabilities with standard errors, first differences, average marginal effects, plots, regression tables, and simulated-residual diagnostics.
  • vignette("panels") – random intercepts, fixed effects and the jackknife, the Mundlak device, cluster-robust standard errors, and the package’s Monte Carlo on short panels.
  • vignette("model") – the likelihood and sign conventions, identification, multi-start estimation and boundary cases, which test for which comparison, and validation.
  • ?iop for the estimators’ full argument list; ?"iop-package" for an index.

Quick start 

library(iop)

## political violence (none / repression / civil war): zero-inflated ordered probit
data(bp)
m_op   <- oprobit(violence ~ loggdppc + parliament + disaster + major_oil + major_primary, data = bp)
m_ziop <- iop(violence ~ loggdppc + parliament + disaster + major_oil + major_primary |
                loggdppc + parliament + disaster + major_oil + major_primary,
              data = bp, inflate = "bottom")
summary(m_ziop)
inflation_test(m_ziop)                     # Vuong (raw / AIC / BIC) vs the plain ordered probit
compare_models(op = m_op, ziop = m_ziop)

## regime-specific quantities
head(predict(m_ziop, type = "inflated"))   # P(structurally peaceful)
predict(m_ziop, newdata = data.frame(loggdppc = 8, parliament = 1, disaster = 0,
                                     major_oil = 0, major_primary = 0), se.fit = TRUE)
first_difference(m_ziop, "loggdppc", from = 7, to = 9)           # total effect
first_difference(m_ziop, "loggdppc", from = 7, to = 9, stage = "inflation")
plot(ame(m_ziop, vars = c("loggdppc", "disaster")))              # average marginal effects

## correlated errors (ZiOPC)
m_ziopc <- iop(violence ~ loggdppc + parliament + disaster + major_oil + major_primary |
                 loggdppc + parliament + disaster + major_oil + major_primary,
               data = bp, inflate = "bottom", correlated = TRUE)
lr_test(m_ziop, m_ziopc)                   # rho = 0?

## top-inflated: escape-flexibility provisions in trade agreements
data(pta)
m_tiop <- iop(flexibility ~ depth * democracy + gdp + gdppc + trade + gattwto + members +
                democratization | gdp + gdppc + democracy + democratization,
              data = pta, inflate = "top")

## partial proportional odds for the plain ordered logit
m_pl <- ologit(flexibility ~ depth + democracy + gdp, data = pta, parallel = "auto")
m_pl$autofit$relaxed

## panels: cluster-robust SEs, random intercepts, the Mundlak device
m_cl <- iop(violence ~ loggdppc + parliament + disaster | loggdppc + parliament + disaster,
            data = bp, inflate = "bottom", cluster = "country")
m_re <- oprobit(violence ~ loggdppc + parliament + disaster, data = bp, re = "country")
head(ranef(m_re))
md <- mundlak(violence ~ loggdppc + disaster, data = bp, unit = "country")
oprobit(md$formula, md$data, cluster = "country")

## tables
texreg::screenreg(list(m_op, m_ziop))
modelsummary::modelsummary(list(OP = m_op, ZiOP = m_ziop))

Model

Outcome stage (ordered probit/logit with cutpoints tau): P(y <= j | s = 1) = F(tau_j - x'beta). Inflation stage: P(s = 1) = F(z'gamma) is the probability of the ordered (non-inflated) regime, so positive inflation coefficients raise the probability of the ordered regime; units in the inflated regime (s = 0) are observed in the inflated category k with certainty. Hence

P(y = j) = F(z'gamma) * pi_j(x) + 1{j = k} * [1 - F(z'gamma)].

With correlated = TRUE (probit only) the two latent errors are bivariate normal with correlation rho, and the regime-1 cell probabilities are rectangle probabilities of the bivariate normal. See vignette("model").

Validation

The package’s tests check the bivariate-normal routine against pbivnorm and mvtnorm, every analytic gradient against numerical differentiation, oprobit()/ologit() against MASS::polr, ordinal::clm, and VGAM::vglm, ordinal::clmm for the random-intercept models, parameter recovery on simulated inflated data, and – on the bundled bp data – Table 1 of Bagozzi, Hill, Moore and Mukherjee (2015): OP, ZiOP, ZiOPC, and ZiOPC2 log-likelihoods and the error correlations. Beyond the test suite, iop() reproduces Table 1 and the marginal-effects figures of Bagozzi and Mukherjee (2012), the TiOP/TiOPC log-likelihoods of Bagozzi, Joo and Mukherjee (2024), and the Python idcempy implementation where it converges; the original likelihood code of each paper evaluated at iop’s estimates returns iop’s log-likelihood to machine precision. See data-raw/oracles/README.md in the source repository, and inst/mc/ for the Monte Carlo behind the panel recommendations.

Citation

citation("iop") gives the package citation and the articles that introduced the models.

References

Harris, M.N. and Zhao, X. (2007). A zero-inflated ordered probit model, with an application to modelling tobacco consumption. Journal of Econometrics, 141, 1073-1099.

Bagozzi, B.E. and Mukherjee, B. (2012). A mixture model for middle category inflation in ordered survey responses. Political Analysis, 20, 369-386.

Bagozzi, B.E., Hill, D.W., Moore, W.H. and Mukherjee, B. (2015). Modeling two types of peace: The zero-inflated ordered probit (ZiOP) model in conflict research. Journal of Conflict Resolution, 59, 728-752.

Bagozzi, B.E., Joo, M.M. and Mukherjee, B. (2024). Top-category inflation in ordered international relations outcomes. Foreign Policy Analysis, 20, orae006.