Type:	Package
Title:	Modified Poisson Regression for Binary Outcome and Related Methods
Version:	3.2-1
Date:	2025-11-24
Maintainer:	Hisashi Noma <noma@ism.ac.jp>
Description:	Modified Poisson, logistic and least-squares regression analyses for binary outcomes of Zou (2004) <doi:10.1093/aje/kwh090>, Noma (2025)<Forthcoming>, and Cheung (2007) <doi:10.1093/aje/kwm223> have been standard multivariate analysis methods to estimate risk ratio and risk difference in clinical and epidemiological studies. This R package involves an easy-to-handle function to implement these analyses by simple commands. Missing data analysis tools (multiple imputation) are also involved. In addition, recent studies have shown the ordinary robust variance estimator possibly has serious bias under small or moderate sample size situations for these methods. This package also provides computational tools to calculate alternative accurate confidence intervals.
Depends:	R (≥ 3.5.0)
Imports:	stats, MASS, sandwich, mice, lme4
License:	GPL-3
Encoding:	UTF-8
LazyData:	true
NeedsCompilation:	no
Packaged:	2025-11-24 07:47:59 UTC; nomah
Author:	Hisashi Noma [aut, cre]
Repository:	CRAN
Date/Publication:	2025-11-24 08:50:02 UTC

The 'rqlm' package.

Description

Modified Poisson, logistic and least-squares regression analyses for binary outcomes have been standard multivariate analysis methods to estimate risk ratio and risk difference in clinical and epidemiological studies. This R package involves an easy-to-handle function to implement these analyses by simple commands. Missing data analysis tools (multiple imputation) are also involved. In addition, recent studies have shown the ordinary robust variance estimator possibly has serious bias under small or moderate sample size situations for these methods. This package also provides computational tools to calculate accurate confidence intervals.

References

Cheung, Y. B. (2007). A modified least-squares regression approach to the estimation of risk difference. American Journal of Epidemiology 166, 1337-1344.

Noma, H. (2025). Robust variance estimators for risk ratio estimators from logistic regression in cohort and case-cohort studies. Forthcoming.

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Noma, H. and Gosho, M. (2025). Logistic mixed-effects model analysis with pseudo-observations for estimating risk ratios in clustered binary data analysis. Statistics in Medicine 44, e70280.

Noma, H., Sunada, H., and Gosho, M. (2025). Quasi-likelihood ratio tests and the Bartlett-type correction for improved inferences of the modified Poisson and least-squares regressions for binary outcomes. Statistica Neerlandica 79, e70012.

Shiiba, H. and Noma, H. (2025). Confidence intervals of risk ratios for the augmented logistic regression with pseudo-observations. Stats 8, 83.

Zou, G. (2004). A modified poisson regression approach to prospective studies with binary data. American Journal of Epidemiology 159, 702-706.

Calculating bootstrap confidence interval for modified least-squares regression based on the quasi-score statistic

Description

Recent studies revealed the robust standard error estimates of the modified least-squares regression analysis are generally biased under small or moderate sample settings. To adjust the bias and to provide more accurate confidence intervals, confidence interval and P-value of the test for risk difference by modified least-squares regression are calculated based on the bootstrap approach of Noma and Gosho (2024).

Usage

bsci.ls(formula, data, x.name=NULL, B=1000, cl=0.95, C0=10^-5,
 digits=4, seed=527916)

Arguments

Value

Results of the modified least-squares analyses are presented. Three objects are provided: Results of the modified least-squares regression with the Wald-type approximation by rqlm, the bootstrap-based confidence interval for the corresponding covariate, and P-value for the bootstrap test of RD=0.

References

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Examples

data(exdata01)

bsci.ls(y ~ x1 + x2 + x3 + x4, data=exdata01, "x3", B=10)
# For illustration. B should be >= 1000 (the number of bootstrap resampling).

Calculating bootstrap confidence interval for modified Poisson regression based on the quasi-score statistic

Description

Recent studies revealed the risk ratio estimates and robust standard error estimates of the modified Poisson regression analysis are generally biased under small or moderate sample settings. To adjust the bias and to provide more accurate confidence intervals, confidence interval and P-value of the test for risk ratio by modified Poisson regression are calculated based on the bootstrap approach of Noma and Gosho (2024).

Usage

bsci.pois(formula, data, x.name=NULL, B=1000, eform=FALSE, cl=0.95, C0=10^-5,
 digits=4, seed=527916)

Arguments

Value

Results of the modified Poisson analyses are presented. Three objects are provided: Results of the modified Poisson regression with the Wald-type approximation by rqlm, the bootstrap confidence interval for the corresponding covariate, and P-value for the bootstrap test of RR=1.

References

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Examples

data(exdata01)

bsci.pois(y ~ x1 + x2 + x3 + x4, data=exdata01, "x3", B=10, eform=TRUE)
# For illustration. B should be >= 1000 (the number of bootstrap resampling).

Computation of the ordinary confidence intervals and P-values using the model variance estimator

Description

Confidence intervals and P-values for the generalized linear model and generalized linear-mixed-effects model can be calculated using the ordinary model variance estimators. Through simply entering the output objects of lm, glm, lmer, or glmer, the inference results are fastly computed. For the linear regression model, the exact confidence intervals and P-values based on the t-distribution are calculated. Also, for the generalized linear model, the Wald-type confidence intervals and P-values based on the asymptotic normal approximation are computed. The resultant coefficients and confidence limits can be transformed to exponential scales by specifying eform.

Usage

coeff(gm, eform=FALSE, cl=0.95, digits=4)

Arguments

Value

Results of inferences of the regression coefficients using the ordinary model variance estimators.

• coef: Coefficient estimates; transformed to the exponential scale if eform=TRUE.

• SE: Robust standard error estimates for coef.

• CL: Lower limits of confidence intervals.

• CU: Upper limits of confidence intervals.

• P-value: P-values for the coefficient tests.

Examples

data(exdata02)

gm1 <- glm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=binomial)
coeff(gm1,eform=TRUE)
# Logistic regression analysis
# Coefficient estimates are translated to odds ratio scales

lm1 <- lm(x1 ~ x2 + x3 + x4, data=exdata02)
coeff(lm1)
# Linear regression analysis

data(mch)

if(requireNamespace("lme4", quietly = TRUE)) {
    lmr1 <- lme4::lmer(Y ~ x + (1|SOUM), data=mch)
    coeff(lmr1)
}
# Linear mixed-effects model analysis

if(requireNamespace("lme4", quietly = TRUE)) {
	gmr1 <- lme4::glmer(y ~ x + (1|SOUM), nAGQ=25, family=binomial, data=mch)
	coeff(gmr1, eform=TRUE)
}
# Logistic mixed-effects model analysis
# Coefficient estimates are translated to odds ratio scales

A simulated example dataset

Description

A simulated cohort data with binomial outcome.

• y: Dichotomous outcome variable.

• x1: Continuous covariate.

• x2: Binary covariate.

• x3: Binary covariate.

• x4: Binary covariate.

Usage

data(exdata01)

Format

A simulated cohort data with binomial outcome (n=40).

A simulated example dataset

Description

A simulated cohort data with binomial outcome.

• y: Dichotomous outcome variable.

• x1: Continuous covariate.

• x2: Binary covariate.

• x3: Binary covariate.

• x4: Binary covariate.

Usage

data(exdata02)

Format

A simulated cohort data with binomial outcome (n=1200).

A simulated example dataset with missing covariates

Description

A simulated cohort data with binomial outcome. Some covariates involve missing data.

• y: Dichotomous outcome variable.

• x1: Continuous covariate.

• x2: Binary covariate.

• x3: Binary covariate.

• x4: Binary covariate.

Usage

data(exdata03)

Format

A simulated cohort data with binomial outcome (n=1200). Some covariates involve missing data.

A cluster-randomised trial dataset for the maternal and child health handbook

Description

A cluster-randomised trial dataset with binomial outcome.

• ID: ID variable of participants.

• SOUM: ID variable of soums (involving 18 soums).

• x: Binary variable specifying intervention groups (1=Intervention, 0=Control).

• mage: Mother's age.

• medu: Mother's education (1=uneducated, 2=elementary, 3=incomplete secondary, 4=complete secondary, 5=incomplete high, 6=high (completed collage or university)).

• mmarry: Mother's marital status (1=single, 2=married/cohabitating, 3=separated/divorce, 4=windowed/other).

• mprig1: First pregnancy (1=Yes, 2=No).

• height: Mother's height.

• weight: Mother's weight.

• time: Travel time from mother's home to antenatal care clinic.

• Y: Outcome variable: Number of antenatal visits.

• y: Outcome variable: Whether the number of antenatal visits is >= 6 (0 or 1).

• ses: Quintile groups by the social-economic index (= 1, 2, 3, 4, 5).

Usage

data(mch)

Format

A data frame with 500 participants with 18 soums.

References

Mori, R., Yonemoto, N., Noma, H., et al. (2015). The Maternal and Child Health (MCH) handbook in Mongolia: a cluster-randomized, controlled trial. PloS One 10: e0119772.

Multiple imputation analysis for the generalized linear model

Description

Multiple imputation analysis for the generalized linear model is performed for the imputed datasets generated by mice function in mice package. For computing covariance matrix estimate, the ordinary Rubin's rule is adapted to the model variance estimates.

Usage

mi_glm(ice, formula, family=gaussian, offset=NULL, eform=FALSE, cl=0.95, digits=4)

Arguments

Value

Results of the multiple imputation analysis for the generalized linear model. For computing covariance matrix estimate, the ordinary Rubin's rule is adapted to the model variance estimates.

• coef: Coefficient estimates; transformed to the exponential scale if eform=TRUE.

• SE: Standard error estimates for coef.

• CL: Lower limits of confidence intervals.

• CU: Upper limits of confidence intervals.

• df: Degree of freedom for the t-approximation.

• P-value: P-values for the coefficient tests.

References

Little, R. J., and Rubin, D. B. (2019). Statistical Analysis with Missing Data, 3rd edition. New York: Wiley.

Examples

library("mice")

data(exdata03)

exdata03$x2 <- factor(exdata03$x2)
exdata03$x3 <- factor(exdata03$x3)
exdata03$x4 <- factor(exdata03$x4)

ice5 <- mice(exdata03,m=5)
# For illustration. m should be >=100.

mi_glm(ice5, y ~ x1 + x2 + x3 + x4, family=binomial, eform=TRUE)
# Logistic regression analysis
# Coefficient estimates are translated to odds ratio scales

mi_glm(ice5, x1 ~ x2 + x3 + x4, family=gaussian)
# Ordinary least-squares regression analysis with the model variance estimator

Multiple imputation analysis for modified Poisson and least-squares regressions

Description

Multiple imputation analysis for modified Poisson and least-squares regressions is performed for the imputed datasets generated by mice function in mice package. For computing covariance matrix estimate, the ordinary Rubin's rule is adapted to the sandwich variance estimates. Its validity is checked by several simulation studies for general GEE applications by Beunckens et al. (2008), Birhanu et al. (2011) and Yoo (2010).

Usage

mi_rqlm(ice, formula, family=poisson, eform=FALSE, cl=0.95, digits=4)

Arguments

Value

Results of the multiple imputation analysis for modified Poisson and least-squares regressions. For computing covariance matrix estimate, the ordinary Rubin's rule is adapted to the sandwich variance estimates.

• coef: Coefficient estimates; transformed to the exponential scale if eform=TRUE.

• SE: Robust standard error estimates for coef.

• CL: Lower limits of confidence intervals.

• CU: Upper limits of confidence intervals.

• df: Degree of freedom for the t-approximation.

• P-value: P-values for the coefficient tests.

References

Aloisio, K. M., Swanson, S. A., Micali, N., Field, A., and Horton, N. J. (2014). Analysis of partially observed clustered data using generalized estimating equations and multiple imputation. Stata Journal, 14, 863-883.

Beunckens, C., Sotto, C., and Molenberghs., G. (2008). A simulation study comparing weighted estimating equations with multiple imputation based estimating equations for longitudinal binary data. Computational Statistics and Data Analysis, 52, 1533-1548.

Birhanu, T., Molenberghs, G., Sotto, C., and Kenward, M. G. (2011). Doubly robust and multiple-imputation-based generalized estimating equations. Journal of Biopharmaceutical Statistics, 21, 202-225.

Little, R. J., and Rubin, D. B. (2019). Statistical Analysis with Missing Data, 3rd edition. New York: Wiley.

Yoo, B. (2010). The impact of dichotomization in longitudinal data analysis: a simulation study. Pharmaceutical Statistics, 9, 298-312.

Examples

library("mice")

data(exdata03)

exdata03$x2 <- factor(exdata03$x2)
exdata03$x3 <- factor(exdata03$x3)
exdata03$x4 <- factor(exdata03$x4)

ice5 <- mice(exdata03,m=5)
# For illustration. m should be >=100.

mi_rqlm(ice5, y ~ x1 + x2 + x3 + x4, family=poisson, eform=TRUE)
# Modifed Poisson regression analysis
# Coefficient estimates are translated to risk ratio scales

mi_rqlm(ice5, y ~ x1 + x2 + x3 + x4, family=gaussian)
# Modifed least-squares regression analysis

Calculating confidence interval for modified least-squares regression based on the quasi-score test

Description

Recent studies revealed the robust standard error estimates of the modified least-squares regression analysis are generally biased under small or moderate sample settings. To adjust the bias and to provide more accurate confidence intervals, confidence interval and P-value of the test for risk difference by modified least-squares regression are calculated based on the quasi-score test of Noma and Gosho (2024).

Usage

qesci.ls(formula, data, x.name=NULL, cl=0.95, C0=10^-5, digits=4)

Arguments

Value

Results of the modified least-squares analyses are presented. Three objects are provided: Results of the modified least-squares regression with the Wald-type approximation by rqlm, quasi-score confidence interval for the corresponding covariate, and P-value for the quasi-score test of RD=0.

References

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Examples

data(exdata01)

qesci.ls(y ~ x1 + x2 + x3 + x4, data=exdata01, "x3")

Calculating confidence interval for modified Poisson regression based on the quasi-score test

Description

Recent studies revealed the risk ratio estimates and robust standard error estimates of the modified Poisson regression analysis are generally biased under small or moderate sample settings. To adjust the bias and to provide more accurate confidence intervals, confidence interval and P-value of the test for risk ratio by modified Poisson regression are calculated based on the quasi-score test of Noma and Gosho (2024).

Usage

qesci.pois(formula, data, x.name=NULL, eform=FALSE, cl=0.95, C0=10^-5, digits=4)

Arguments

Value

Results of the modified Poisson analyses are presented. Three objects are provided: Results of the modified Poisson regression with the Wald-type approximation by rqlm, quasi-score confidence interval for the corresponding covariate, and P-value for the quasi-score test of RR=1.

References

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Examples

data(exdata01)

qesci.pois(y ~ x1 + x2 + x3 + x4, data=exdata01, "x3", eform=TRUE)

Augmented (modified) logistic regression analyses for estimating risk ratio

Description

Logistic regression with augmented pseudo-observations for estimating risk ratios is performed. This function is handled by a similar way with lm or glm. Also, the resultant coefficients and confidence limits can be transformed to exponential scales by specifying eform. The Morel-Bokossa-Neerchaal-type small-sample corrected estimator is adopted for standard error estimation as the default method.

Usage

qlogist(formula, data, eform=TRUE, cl=0.95, digits=4, var.method="MBN")

Arguments

Value

Results of the augmented (modified) logistic regression analysis.

• coef: Coefficient estimates; transformed to the exponential scale if eform=TRUE.

• SE: Robust standard error estimates for coef.

• CL: Lower limits of confidence intervals.

• CU: Upper limits of confidence intervals.

• P-value: P-values for the coefficient tests.

References

Diaz-Quijano, F. A. (2012). A simple method for estimating relative risk using logistic regression. BMC Medical Research Methodology 12, 14.

Gosho, M., Sato, Y., and Takeuchi, H. (2014). Robust covariance estimator for small-sample adjustment in the generalized estimating equations: a simulation study. Science Journal of Applied Mathematics and Statistics 2, 20-25.

Morel, J. G., Bokossa, M., and Neerchal, N. (2003). Small sample correction for the variance of GEE estimators. Biometrical Journal 45, 395-409.

Noma, H. (2025). Robust variance estimators for risk ratio estimators from logistic regression in cohort and case-cohort studies. Forthcoming.

Noma, H., and Gosho, M. (2025). Logistic mixed-effects model analysis with pseudo-observations for estimating risk ratios in clustered binary data analysis. Statistics in Medicine 44, e70280.

Schouten, E. G., Dekker, J. M., Kok, F. J., et al. (1993). Risk ratio and rate ratio estimation in case-cohort designs: hypertension and cardiovascular mortality. Statistics in Medicine 12, 1733-1745.

Shiiba, H., and Noma, H. (2025). Confidence intervals of risk ratios for the augmented logistic regression with pseudo-observations. Stats 8, 83.

Wang, M., and Long, Q. (2011). Modified robust variance estimator for generalized estimating equations with improved small-sample performance. Statistics in Medicine 30, 1278-1291.

Examples

data(exdata02)

qlogist(y ~ x1 + x2 + x3 + x4, data=exdata02)
# Augmented logistic regression analysis
# Coefficient estimates are translated to risk ratio scales
# MBN robust variance estimator is adopted.

qlogist(y ~ x1 + x2 + x3 + x4, data=exdata02, var.method="GST")
# GST robust variance estimator is adopted.

qlogist(y ~ x1 + x2 + x3 + x4, data=exdata02, var.method="WL")
# WL robust variance estimator is adopted.

Modified Poisson and least-squares regression analyses for binary outcomes

Description

Modified Poisson and least-squares regression analyses for binary outcomes are performed. This function is handled by a similar way with lm or glm. The model fitting to the binary data can be specified by family. Also, the resultant coefficients and confidence limits can be transformed to exponential scales by specifying eform. The Morel-Bokossa-Neerchaal-type small-sample corrected estimator is adopted for standard error estimation as the default method.

Usage

rqlm(formula, data, family=poisson, eform=FALSE, cl=0.95, digits=4, var.method="MBN")

Arguments

Value

Results of the modified Poisson and least-squares regression analyses.

• coef: Coefficient estimates; transformed to the exponential scale if eform=TRUE.

• SE: Robust standard error estimates for coef.

• CL: Lower limits of confidence intervals.

• CU: Upper limits of confidence intervals.

• P-value: P-values for the coefficient tests.

References

Cheung, Y. B. (2007). A modified least-squares regression approach to the estimation of risk difference. American Journal of Epidemiology 166, 1337-1344.

Gosho, M., Ishii, R., Noma, H., and Maruo, K. (2023). A comparison of bias-adjusted generalized estimating equations for sparse binary data in small-sample longitudinal studies. Statistics in Medicine 42, 2711-2727.

Gosho, M., Sato, Y., and Takeuchi, H. (2014). Robust covariance estimator for small-sample adjustment in the generalized estimating equations: a simulation study. Science Journal of Applied Mathematics and Statistics 2, 20-25.

Morel, J. G., Bokossa, M., and Neerchal, N. (2003). Small sample correction for the variance of GEE estimators. Biometrical Journal 45, 395-409.

Noma, H. and Gosho, M. (2025). Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions. Epidemiologic Methods 14, 20240030.

Noma, H., Sunada, H., and Gosho, M. (2025). Quasi-likelihood ratio tests and the Bartlett-type correction for improved inferences of the modified Poisson and least-squares regressions for binary outcomes. Statistica Neerlandica 79, e70012.

Wang, M., and Long, Q. (2011). Modified robust variance estimator for generalized estimating equations with improved small-sample performance. Statistics in Medicine 30, 1278-1291.

White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1-25.

Zou, G. (2004). A modified poisson regression approach to prospective studies with binary data. American Journal of Epidemiology 159, 702-706.

Examples

data(exdata02)

rqlm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=poisson, eform=TRUE)
# Modifed Poisson regression analysis
# Coefficient estimates are translated to risk ratio scales
# MBN robust variance estimator is adopted.

rqlm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=gaussian)
# Modifed least-squares regression analysis

rqlm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=gaussian, digits=3)
# Modifed least-squares regression analysis
# Number of decimal places can be changed by specifying "digits"

rqlm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=poisson, eform=TRUE, var.method="GST")
# Modifed Poisson regression analysis
# Coefficient estimates are translated to risk ratio scales
# GST robust variance estimator is adopted.

rqlm(y ~ x1 + x2 + x3 + x4, data=exdata02, family=poisson, eform=TRUE, var.method="WL")
# Modifed Poisson regression analysis
# Coefficient estimates are translated to risk ratio scales
# WL robust variance estimator is adopted.