# 4: Paired contrasts

    library(ratesci)

## Confidence intervals and tests for paired comparisons of binomial proportions

The input data for paired proportions takes a different structure, compared with the data for independent proportions:

|  | Event B |  |
|  | Success | Failure | Total
Event A | Success | \(a \ (p_{11})\) | \(b \ (p_{12})\) | \(x_1 = a + b \ (p_1)\)
| Failure | \(c \ (p_{21})\) | \(d \ (p_{22})\) | \(c + d \ (1 -p_1)\)
| Total | \(x_2 = a + c \ (p_2)\) | \(b + d \ (1 - p_2)\) | \(N\)

### SCAS and other asymptotic score methods for RD and RR

To calculate a confidence interval (CI) for a paired risk difference (\(\hat \theta_{RD} = \hat p_1 - \hat p_2\), where \(\hat p_1 = (a+b)/N\), \(\hat p_2 = (a+c)/N\)), or relative risk (\(\hat \theta_{RR} = \hat p_1 / \hat p_2\)), the skewness-corrected asymptotic score (SCAS) method is recommended, as one that succeeds, on average, at containing the true parameter \(\theta\) with the appropriate nominal probability (e.g. 95%), and has evenly distributed tail probabilities (Laud 2025, under review). It is a modified version of the asymptotic score methods by (Tango 1998) for RD, and (Nam and Blackwelder 2002) and (Tang, Tang, and Chan 2003) for RR, incorporating both a skewness correction and a correction in the variance estimate.

The plots below illustrate the one-sided and two-sided interval coverage probabilities achieved by SCAS compared to some other popular methods1, when \(N=40\) and the correlation coefficient is 0.25. A selection of coverage probability plots for other sample sizes and correlations can be found in the “plots” folder of the cpplot GitHub repository.

`pairbinci()` takes input in the form of a vector of length 4, comprising the four values `c(a, b, c, d)` from the above table, which are the number of paired observations having each of the four possible pairs of outcomes.

For example, using the dataset from a study of airway reactivity in children before and after stem cell transplantation, as used in (Fagerland, Lydersen, and Laake 2014):

    out <- pairbinci(x = c(1, 1, 7, 12))
    out$estimates
    #>       lower    est   upper level  p1hat p2hat  p1mle p2mle phi_hat phi_c
    #> [1,] -0.528 -0.286 -0.0184  0.95 0.0952 0.381 0.0952 0.381  0.0795     0
    #>      psi_hat
    #> [1,]    1.71

The underlying z-statistic is used to obtain a two-sided hypothesis test against the null hypothesis of no difference (`pval2sided`). Note that this is equivalent to an ‘N-1’ adjusted version of the McNemar test. The facility is also provided for a custom one-sided test against any specified null hypothesis value \(\theta_0\), e.g. for non-inferiority testing (`pval_left` and `pval_right`).

    out$pval
    #>      chisq pval2sided theta0 scorenull pval_left pval_right
    #> [1,]  4.29     0.0384      0     -2.07    0.0192      0.981

For a confidence interval for paired RR, use:

    out <- pairbinci(x = c(1, 1, 7, 12), contrast = "RR")
    out$estimates
    #>       lower   est upper level  p1hat p2hat  p1mle p2mle phi_hat phi_c psi_hat
    #> [1,] 0.0429 0.263 0.928  0.95 0.0952 0.381 0.0994 0.379  0.0795     0    1.71
    out$pval
    #>      chisq pval2sided theta0 scorenull pval_left pval_right
    #> [1,]  4.29     0.0384      1     -2.07    0.0192      0.981

To obtain the legacy Tango and Tang intervals for RD and RR respectively, you may set the `skew` and `bcf` arguments to `FALSE`. Also switching to `method = "Score_closed"` takes advantage of closed-form calculations for these methods (whereas the SCAS method is solved by iteration).

### MOVER methods

For application of the MOVER method to paired RD or RR, an estimate of the correlation coefficient is included in the formula. A correction to the correlation estimate, introduced by Newcombe, is recommended, obtained with `method = "MOVER_newc"`. As for unpaired MOVER methods, the default base method used for the individual (marginal) proportions is the equal-tailed Jeffreys interval, rather than the Wilson Score as originally proposed by Newcombe (obtained using `moverbase = "wilson"`). The combination of the Newcombe correlation estimate and the Jeffreys intervals gives the designation “MOVER-NJ”. This method is less computationally intensive than SCAS, but coverage properties are inferior, and there is no corresponding hypothesis test.

    pairbinci(x = c(1, 1, 7, 12), contrast = "RD", method = "MOVER_newc")$estimates
    #>       lower    est   upper level  p1hat p2hat phi_hat
    #> [1,] -0.511 -0.286 -0.0324  0.95 0.0952 0.381       0

For cross-checking against published example in (Fagerland, Lydersen, and Laake 2014)

    pairbinci(x = c(1, 1, 7, 12), contrast = "RD", method = "MOVER_newc", moverbase = "wilson")$estimates
    #>       lower    est   upper level  p1hat p2hat phi_hat
    #> [1,] -0.507 -0.286 -0.0256  0.95 0.0952 0.381       0

### Conditional odds ratio

Confidence intervals for paired odds ratio are obtained conditional on the number of discordant pairs, by transforming a confidence interval for the proportion \(b / (b+c)\). Transformed SCAS (with or without a variance bias correction, to ensure consistency with the above SCAS hypothesis tests for RD and RR) or transformed mid-p intervals are recommended (Laud 2025, under review).

    out <- pairbinci(x = c(1, 1, 7, 12), contrast = "OR")
    out$estimates
    #>       lower   est upper
    #> [1,] 0.0077 0.162 0.912
    out$pval
    #>      chisq pval2sided theta0 scorenull pval_left pval_right
    #> [1,]  4.29     0.0384      1     -2.07    0.0192      0.981

To select an alternative method, for example transformed mid-p:

    pairbinci(x = c(1, 1, 7, 12), contrast = "OR", method = "midp")$estimates
    #>        lower  est upper   x     n
    #> [1,] 0.00629 0.16 0.924 Inf -1.14

## References

Fagerland, Morten W., Stian Lydersen, and Petter Laake. 2014. “Recommended Tests and Confidence Intervals for Paired Binomial Proportions.” Statistics in Medicine 33 (16): 2850–75. https://doi.org/10.1002/sim.6148.
Nam, Jun-mo, and William C. Blackwelder. 2002. “Analysis of the Ratio of Marginal Probabilities in a Matched-Pair Setting.” Statistics in Medicine 21 (5): 689–99. https://doi.org/10.1002/sim.1017.
Tang, Nian-Sheng, Man-Lai Tang, and Ivan Siu Fung Chan. 2003. “On Tests of Equivalence via Non-Unity Relative Risk for Matched-Pair Design.” Statistics in Medicine 22 (8): 1217–33. https://doi.org/10.1002/sim.1213.
Tango, Toshiro. 1998. “Equivalence Test and Confidence Interval for the Difference in Proportions for the Paired-Sample Design.” Statistics in Medicine 17 (8): 891–908. https://doi.org/10.1002/(sici)1097-0258(19980430)17:8<891::aid-sim780>3.0.co;2-b.

1. SCASu = SCAS omitting variance bias correction; AS = Tang; MOVER-NJ = Method of Variance Estimates Recovery, based on Newcombe’s correlation adjustment but using Jeffreys equal-tailed intervals instead of Wilson; MOVER-W = MOVER using Wilson intervals, and omitting Newcombe’s correlation correction; BP = Bonett-Price↩︎