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An implementation of the unified framework for assessing Parrtial Association between ordinal variables after adjusting for a set of covariates (Dungang Liu, Shaobo Li, Yan Yu and Irini Moustaki (2020), accepted by the Journal of the American Statistical Association). This package provides a set of tools to quantify, visualize, and test partial associations between multiple ordinal variables. It can produce a number of ϕ measures, partial regression plots, 3-D plots, and p-values for testing H0 : ϕ = 0 or H0 : ϕ ≤ δ


The PAsso package is currently available on PAsso CRAN.

# Install the development version from GitHub
if (!requireNamespace("devtools")) install.packages("devtools")

Install PAsso from the CRAN

# Install from CRAN

# For macOS, if you have "error: 'math.h' file not found" for installing PAsso v0.1.9,
# the solution could be:
                  repos = NULL, type = "source")
# If error "there is no package called 'gsl'" comes, try:
install.packages("gsl", type = "mac.binary")


The following example shows the R code for evaluating the partial association between a binary variable PreVote.num and a ordinal variable PID, while adjusting for age, education, and income. Specifically, PreVote.num is the respondent’s voting preference between Donald Trump and Hilary Clinton. And PID is the respondent’s party identification with 7 ordinal levels from strong democrat (=1) to strong republication (=7). The data set is drawn from the 2016 American National Election Study.


PAsso_1 <- PAsso(responses = c("PreVote.num", "PID"),
                 adjustments = c("income.num", "age", "edu.year"),
                 data = ANES2016,
                 uni.model = "probit",
                 method = c("kendall"))

# Print the partial association matrix only
print(PAsso_1, 5)

# Provide:
# 1. partial association matrix;
# 2. marginal association matrix for comparison purpose;
# 3. summary of models' coefficients for model diagnostics and interpretation
summary(PAsso_1, 4)

# Plot partial association regression plot: residuals

# Retrieve residuals that are used as ingredients for partial assocaition analyses
test_resids <- residuals(PAsso_1, draw = 1)

# test function: Conduct inference based on object of "PAsso.test" class ----------------------------

system.time(Pcor_SR_test1 <- test(object = PAsso_1, bootstrap_rep = 100, H0 = 0, parallel = F))
print(Pcor_SR_test1, digits=6)

# diagnostic.plot function -----------------------------------------------------
check_qq <- diagnostic.plot(object = PAsso_1, output = "qq")

check_fitted <- diagnostic.plot(object = PAsso_1, output = "fitted")

check_covar <- diagnostic.plot(object = PAsso_1, output = "covariate")

# Or more specific, draw residual-vs-covariate plot for the second model with
# response "PID" and covariate "income.num" 
diagnostic.plot(object = PAsso_1, output = "covariate", x_name = "income.num", model_id = 2)

# general association measure and 3-D plot for VOTE and PID ------------------
library("copula"); library("plotly")

# Draw all pairs
testPlots <- plot3D(PAsso_1)
testPlots$`PreVote.num v.s. PID`

# Draw just one pair
testPlots2 <- plot3D(object = PAsso_1, y1 = "PreVote.num", y2 = "PID")

# "PAsso" advanced using of the function: Input a few models directly ------------------------------
fit_vote <- glm(PreVote.num ~ income.num + age + edu.year, data = ANES2016,
                family = binomial(link = "probit"))

fit_PID <- polr(as.factor(PID) ~ income.num + age + edu.year, data = ANES2016,
                method = "probit", Hess = TRUE)


system.time(PAsso_adv1 <- PAsso(fitted.models=list(fit_vote, fit_PID),
                                association = c("partial"),
                                method = c("kendall"),
                                resids.type = "surrogate")

# Partial association coefficients 
print(PAsso_adv1, digits = 3)
summary(PAsso_adv1, digits = 3)


Li et al. (2021). PAsso: an R Package for Assessing Partial Association between Ordinal Variables. The R Journal, 13(2), 239–252, <10.32614/RJ-2021-088>

Liu, D., Li, S., Yu, Y., & Moustaki, I. (2020). Assessing partial association between ordinal variables: quantification, visualization, and hypothesis testing. Journal of the American Statistical Association, 1-14. <10.1080/01621459.2020.1796394>

Liu, D., & Zhang, H. (2018). Residuals and diagnostics for ordinal regression models: A surrogate approach. Journal of the American Statistical Association, 113(522), 845-854. <10.1080/01621459.2017.1292915>

Greenwell, B.M., McCarthy, A.J., Boehmke, B.C. & Liu, D. (2018) Residuals and diagnostics for binary and ordinal regression models: An introduction to the sure package. The R Journal. <10.32614/RJ-2018-004>