Confidence intervals for the rate (or risk) difference ("RD"), rate ratio
("RR") or conditional odds ratio ("OR"), for paired binomial data. (For
paired Poisson rates, suggest use the tdasci function with distrib = "poi",
and weighting = "MH", with pairs as strata.)
This function applies the score-based Tango and Tang methods for RD and
RR respectively, with iterative and closed-form versions, and an added
skewness correction for improved one-sided coverage.
Also includes MOVER options using the Method of Variance Estimates Recovery
for paired RD and RR, incorporating Newcombe's correlation correction, and
some simpler methods by Bonett & Price for RD and RR.
For OR, intervals are produced based on transforming various intervals for
the single proportion, including SCASp, mid-p and Jeffreys.
All methods have options for continuity adjustment, and the magnitude of
adjustment can be customised.
Usage
pairbinci(
x,
level = 0.95,
contrast = "RD",
method = ifelse(contrast == "OR", "SCASp", "Score"),
moverbase = ifelse(method %in% c("MOVER", "MOVER_newc", "BP"), "jeff", NULL),
bcf = TRUE,
skew = TRUE,
cc = FALSE,
theta0 = NULL,
precis = 6,
warn = TRUE,
method_RD = NULL,
method_RR = NULL,
method_OR = NULL,
cctype = NULL,
...
)Arguments
- x
A numeric vector object specified as c(a, b, c, d) where:
a is the number of pairs with the event (e.g. success) under both conditions (e.g. treated/untreated, or case/control)
b is the count of the number with the event on condition 1 only (= x12)
c is the count of the number with the event on condition 2 only (= x21)
d is the number of pairs with no event under both conditions
(Note the order of a and d is only important for contrast="RR".)- level
Number specifying confidence level (between 0 and 1, default 0.95).
- contrast
Character string indicating the contrast of interest:
"RD" = rate difference (default);
"RR" = rate ratio;
"OR" = conditional odds ratio.- method
Character string indicating the confidence interval method to be used. The following are available for
contrast = "RD"or"RR":
"Score" = (default) asymptotic score class of methods including Tango (for RD) / Tang (for RR), by iterative calculations, with optional skewness correction;
"Score_closed" = closed form solution for Tango/Tang intervals (without skewness correction);
"MOVER" = hybrid MOVER method (as per "method 8" in Newcombe, but with a choice of input methods - see moverbase);
"MOVER_newc" = hybrid MOVER methods with correction to correlation estimate (Newcombe's "method 10");
"TDAS" = t-distribution asymptotic score (experimental method, now deprecated);
"BP" = Wald with Bonett-Price adjustment for RD, or Hybrid Bonett-Price method for RR.
Forcontrast = "OR", one of the following methods may be selected, all of which are based on transformation of an interval for a single proportionb/(b+c):
"SCASp" = transformed skewness-corrected score (default);
"jeff" = transformed Jeffreys;
"midp" = transformed mid-p;
"wilson" = transformed Wilson score - included for reference only, not recommended.- moverbase
Character string indicating the base method used as input for the MOVER methods for RD or RR (when method = "MOVER" or "MOVER_newc"), and for the Hybrid BP method for RR: "jeff" = Jeffreys equal-tailed interval (default), "SCASp" = skewness-corrected score, "midp" = mid-p, "wilson" = Wilson score (not recommended, known to be skewed).
- bcf
Logical (default FALSE) indicating whether to apply variance bias correction in the score denominator. (Under evaluation, manuscript under review.)
- skew
Logical (default TRUE) indicating whether to apply skewness correction or not. (Under evaluation, manuscript under review.)
Only applies for the iterative
method = "Score".
- cc
Number or logical (default FALSE) specifying (amount of) continuity adjustment. When a score-based method is used, cc = 0.5 corresponds to the continuity-corrected McNemar test.
- theta0
Number to be used in a one-sided significance test (e.g. non-inferiority margin). 1-sided p-value will be < 0.025 iff 2-sided 95\ excludes theta0. NB: can also be used for a superiority test by setting theta0 = 0.
- precis
Number (default 6) specifying precision (i.e. number of decimal places) to be used in optimisation subroutine for the confidence interval.
- warn
Logical (default TRUE) giving the option to suppress warnings.
- method_RD
(deprecated: parameter renamed to method)
- method_RR
(deprecated: parameter renamed to method)
- method_OR
(deprecated: parameter renamed to method)
- cctype
(deprecated: new equivariant cc method implemented instead.)
- ...
Other arguments.
Value
A list containing the following components:
- data
the input data in 2x2 matrix form.
- estimates
the requested contrast, with its confidence interval and the specified confidence level, along with estimates of the marginal probabilities and the correlation coefficient (uncorrected and corrected).
- pval
the corresponding 2-sided significance test against the null hypothesis that p_1 = p_2, and one-sided significance tests against the null hypothesis that theta >= or <= theta0 as specified.
- call
details of the function call.
References
Tango T. Equivalence test and confidence interval for the difference in proportions for the paired-sample design. Statistics in Medicine 1998; 17:891-908
Newcombe RG. Improved confidence intervals for the difference between binomial proportions based on paired data. Statistics in Medicine 1998; 17:2635-2650
Tango T. Improved confidence intervals for the difference between binomial proportions based on paired data by Robert G. Newcombe, Statistics in Medicine, 17, 2635-2650 (1998). Statistics in Medicine 1999; 18(24):3511-3513
Nam J-M, Blackwelder WC. Analysis of the ratio of marginal probabilities in a matched-pair setting. Stat Med 2002; 21(5):689–699
Tang N-S, Tang M-L, Chan ISF. On tests of equivalence via non-unity relative risk for matched-pair design. Statistics in Medicine 2003; 22:1217-1233
Agresti A, Min Y. Simple improved confidence intervals for comparing matched proportions. Statistics in Medicine 2005; 24:729-740
Bonett DG, Price RM. Confidence intervals for a ratio of binomial proportions based on paired data. Statistics in Medicine 2006; 25:3039-3047
Tang M-L, Li H-Q, Tang N-S. Confidence interval construction for proportion ratio in paired studies based on hybrid method. Statistical Methods in Medical Research 2010; 21(4):361-378
Tang N-S et al. Asymptotic confidence interval construction for proportion difference in medical studies with bilateral data. Statistical Methods in Medical Research. 2011; 20(3):233-259
Yang Z, Sun X and Hardin JW. A non-iterative implementation of Tango's score confidence interval for a paired difference of proportions. Statistics in Medicine 2013; 32:1336-1342
Fagerland MW, Lydersen S, Laake P. Recommended tests and confidence intervals for paired binomial proportions. Statistics in Medicine 2014; 33(16):2850-2875
Laud PJ. Equal-tailed confidence intervals for comparison of rates. Pharmaceutical Statistics 2017; 16:334-348.
DelRocco N et al. New Confidence Intervals for Relative Risk of Two Correlated Proportions. Statistics in Biosciences 2023; 15:1–30
Chang P et al. Continuity corrected score confidence interval for the difference in proportions in paired data. Journal of Applied Statistics 2024; 51-1:139-152
Laud PJ. Comments on "New Confidence Intervals for Relative Risk of Two Correlated Proportions" (2023). Statistics in Biosciences 2025; https://doi.org/10.1007/s12561-025-09479-4
Laud PJ. Improved confidence intervals and tests for paired binomial proportions. (2025, Under review)
Author
Pete Laud, p.j.laud@sheffield.ac.uk
Examples
# Example from Fagerland et al 2014
# SCAS method for RD
pairbinci(x = c(1, 1, 7, 12), contrast = "RD", method = "Score")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat p1mle p2mle
#> [1,] -0.528113 -0.285861 -0.018422 0.95 0.095238 0.380952 0.095201 0.381062
#> phi_hat phi_c psi_hat
#> [1,] 0.079536 0 1.714286
#>
#> $pval
#> chisq pval2sided theta0 scorenull pval_left pval_right
#> [1,] 4.285714 0.03843393 0 -2.070197 0.01921697 0.980783
#>
#> $call
#> contrast method level bcf skew cc
#> "RD" "Score" "0.95" "TRUE" "TRUE" "FALSE"
#>
# Tango method
pairbinci(x = c(1, 1, 7, 12), contrast = "RD", method = "Score", skew = FALSE, bcf = FALSE)
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat p1mle p2mle
#> [1,] -0.517232 -0.285714 -0.026003 0.95 0.095238 0.380952 0.095238 0.380952
#> phi_hat phi_c psi_hat
#> [1,] 0.079536 0 1.714286
#>
#> $pval
#> chisq pval2sided theta0 scorenull pval_left pval_right
#> [1,] 4.5 0.03389485 0 -2.12132 0.01694743 0.9830526
#>
#> $call
#> contrast method level bcf skew cc
#> "RD" "Score" "0.95" "FALSE" "FALSE" "FALSE"
#>
# MOVER-NJ method
pairbinci(x = c(1, 1, 7, 12), contrast = "RD", method = "MOVER_newc", moverbase = "jeff")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat phi_hat
#> [1,] -0.510506 -0.285714 -0.032389 0.95 0.095238 0.380952 0
#>
#> $call
#> contrast method moverbase level bcf skew
#> "RD" "MOVER_newc" "jeff" "0.95" "TRUE" "TRUE"
#> cc
#> "FALSE"
#>
# SCAS for RR
pairbinci(x = c(1, 1, 7, 12), contrast = "RR", method = "Score")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat p1mle p2mle
#> [1,] 0.042901 0.262723 0.928199 0.95 0.095238 0.380952 0.099445 0.378517
#> phi_hat phi_c psi_hat
#> [1,] 0.079536 0 1.714286
#>
#> $pval
#> chisq pval2sided theta0 scorenull pval_left pval_right
#> [1,] 4.285714 0.03843393 1 -2.070197 0.01921697 0.980783
#>
#> $call
#> contrast method level bcf skew cc
#> "RR" "Score" "0.95" "TRUE" "TRUE" "FALSE"
#>
# Tang method
pairbinci(x = c(1, 1, 7, 12), contrast = "RR", method = "Score", skew = FALSE, bcf = FALSE)
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat p1mle p2mle phi_hat
#> [1,] 0.065279 0.25 0.906881 0.95 0.095238 0.380952 0.095238 0.380952 0.079536
#> phi_c psi_hat
#> [1,] 0 1.714286
#>
#> $pval
#> chisq pval2sided theta0 scorenull pval_left pval_right
#> [1,] 4.5 0.03389485 1 -2.12132 0.01694743 0.9830526
#>
#> $call
#> contrast method level bcf skew cc
#> "RR" "Score" "0.95" "FALSE" "FALSE" "FALSE"
#>
# MOVER-NJ
pairbinci(x = c(1, 1, 7, 12), contrast = "RR", method = "MOVER_newc", moverbase = "jeff")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper level p1hat p2hat phi_hat
#> [1,] 0.051297 0.25 0.873051 0.95 0.095238 0.380952 0
#>
#> $call
#> contrast method moverbase level bcf skew
#> "RR" "MOVER_newc" "jeff" "0.95" "TRUE" "TRUE"
#> cc
#> "FALSE"
#>
# Transformed SCASp method for OR
pairbinci(x = c(1, 1, 7, 12), contrast = "OR", method = "SCASp")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper
#> [1,] 0.007702 0.161863 0.912315
#>
#> $pval
#> chisq pval2sided theta0 scorenull pval_left pval_right
#> [1,] 4.285714 0.03843393 1 -2.070197 0.01921697 0.980783
#>
#> $call
#> contrast method level bcf skew cc
#> "OR" "SCASp" "0.95" "TRUE" "TRUE" "FALSE"
#>
# Transformed Wilson method
pairbinci(x = c(1, 1, 7, 12), contrast = "OR", method = "wilson")
#> $data
#> Test_2
#> Test_1 Success Failure
#> Success 1 1
#> Failure 7 12
#>
#> $estimates
#> lower est upper
#> [1,] 0.022932 0.142857 0.88996
#>
#> $call
#> contrast method level bcf skew cc
#> "OR" "wilson" "0.95" "TRUE" "TRUE" "FALSE"
#>