# scoreci: Score confidence intervals for comparisons of independent... In ratesci: Confidence Intervals for Comparisons of Binomial or Poisson Rates

## Description

Score-based confidence intervals for the rate (or risk) difference ("RD") or ratio ("RR") for independent binomial or Poisson rates, or for odds ratio ("OR", binomial only). Including options for bias correction (from Miettinen & Nurminen), skewness correction ("GNbc" method from Laud & Dane, developed from Gart & Nam, and generalised as "SCAS" in Laud 2017) and continuity correction (for strictly conservative coverage). Also includes score intervals for a single binomial proportion or Poisson rate. Based on the Wilson score interval, when corrected for skewness, coverage is almost identical to the mid-p method, or Clopper-Pearson when also continuity-corrected. This function is vectorised in x1, x2, n1, and n2. Vector inputs may also be combined into a single stratified analysis (e.g. meta-analysis), either using fixed effects, or the more general "TDAS" method, which incorporates stratum variability using a t-distribution score (inspired by Hartung-Knapp-Sidik-Jonkman).

## Usage

 ```1 2 3 4 5``` ```scoreci(x1, n1, x2 = NULL, n2 = NULL, distrib = "bin", contrast = "RD", level = 0.95, skew = TRUE, ORbias = NULL, bcf = TRUE, cc = FALSE, theta0 = NULL, precis = 6, plot = FALSE, plotmax = 100, stratified = FALSE, weighting = "IVS", wt = NULL, tdas = FALSE, warn = TRUE, ...) ```

## Arguments

 `x1, x2` Numeric vectors of numbers of events in group 1 & group 2 respectively. `n1, n2` Numeric vectors of sample sizes (for binomial rates) or exposure times (for Poisson rates) in each group. `distrib` Character string indicating distribution assumed for the input data: "bin" = binomial (default), "poi" = Poisson. `contrast` Character string indicating the contrast of interest: "RD" = rate difference (default), "RR" = rate ratio, "OR" = odds ratio. contrast="p" gives an interval for the single proportion or rate x1/n1. `level` Number specifying confidence level (between 0 and 1, default 0.95). `skew` Logical (default TRUE) indicating whether to apply skewness correction (for the SCAS method recommended in Laud 2017) or not (for the Miettinen-Nurminen method). `ORbias` Logical (default is value of skew for contrast="OR", otherwise NULL) indicating whether to apply additional bias correction for OR derived from Gart 1985. (Corrigendum to Laud 2017 in press). Only applies if contrast is "OR". `bcf` Logical (default TRUE) indicating whether to apply bias correction in the score denominator. Applicable to distrib = "bin" only. (NB: bcf = FALSE option is really only included for legacy validation against previous published methods (i.e. Gart & Nam, Mee, or standard Chi-squared test). `cc` Number or logical (default FALSE) specifying (amount of) continuity correction. Numeric value is taken as the gamma parameter in Laud 2017, Appendix S2 (default 0.5 if cc=TRUE). IMPORTANT NOTES: 1) This is a 'continuity correction' aimed at approximating strictly conservative coverage, NOT for dealing with zero cell counts. Such 'sparse data adjustments' are not needed in the score method. 2) The continuity corrections provided here have not been fully tested for stratified methods. `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% CI excludes theta0. NB: can also be used for a superiority test by setting theta0 = 0 (RD) or 1 (RR/OR). By default, a two-sided test against theta0 = 0 (or 1 as appropriate) is also output: if bcf=F and skew=F this is the same as Pearson's Chi-squared test. `precis` Number (default 6) specifying precision (i.e. number of decimal places) to be used in optimisation subroutine for the confidence interval. `plot` Logical (default FALSE) indicating whether to output plot of the score function `plotmax` Numeric value indicating maximum value to be displayed on x-axis of plots (useful for ratio contrasts which can be infinite). `stratified` Logical (default FALSE) indicating whether to combine vector inputs into a single stratified analysis. IMPORTANT NOTE: The mechanism for stratified calculations is enabled for contrast = "p", but the performance of the resulting intervals has not been fully evaluated. `weighting` String indicating which weighting method to use if stratified = "TRUE": "IVS" = Inverse Variance of Score (default, see Laud 2017 for details), "MH" = Mantel-Haenszel, "MN" = Miettinen-Nurminen iterative weights. `wt` Numeric vector containing (optional) user-specified weights. `tdas` Logical (default FALSE) indicating whether to use t-distribution method for stratified data (defined in Laud 2017). `warn` Logical (default TRUE) giving the option to suppress warnings. `...` Other arguments.

## Value

A list containing the following components:

estimates

a matrix containing estimates of the rates in each group and of the requested contrast, with its confidence interval

pval

a matrix containing details of 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

call

details of the function call

If stratified = TRUE, the following outputs are added:

Qtest

a vector of values describing and testing heterogeneity

weighting

a string indicating the selected weighting method

stratdata

a matrix containing stratum estimates and weights

## Author(s)

Pete Laud, [email protected]

## References

Laud PJ. Equal-tailed confidence intervals for comparison of rates. Pharmaceutical Statistics 2017; 16:334-348.

Laud PJ, Dane A. Confidence intervals for the difference between independent binomial proportions: comparison using a graphical approach and moving averages. Pharmaceutical Statistics 2014; 13(5):294–308.

Miettinen OS, Nurminen M. Comparative analysis of two rates. Statistics in Medicine 1985; 4:213-226.

Farrington CP, Manning G. Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine 1990; 9(12):1447-1454.

Gart JJ. Analysis of the common odds ratio: corrections for bias and skewness. Bulletin of the International Statistical Institute 1985, 45th session, book 1, 175-176.

Gart JJ, Nam Jm. Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness. Biometrics 1988; 44(2):323-338.

Gart JJ, Nam Jm. Approximate interval estimation of the difference in binomial parameters: correction for skewness and extension to multiple tables. Biometrics 1990; 46(3):637-643.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34``` ``` #Binomial RD, SCAS method: scoreci(x1 = c(12,19,5), n1 = c(16,29,56), x2 = c(1,22,0), n2 = c(16,30,29)) #Binomial RD, MN method: scoreci(x1 = c(12,19,5), n1 = c(16,29,56), x2 = c(1,22,0), n2 = c(16,30,29), skew = FALSE) #Poisson RR, SCAS method: scoreci(x1 = 5, n1 = 56, x2 = 0, n2 = 29, distrib = "poi", contrast = "RR") #Poisson RR, MN method: scoreci(x1 = 5, n1 = 56, x2 = 0, n2 = 29, distrib = "poi", contrast = "RR", skew = FALSE) #Binomial rate, SCAS method: scoreci(x1 = c(5,0), n1 = c(56,29), contrast = "p") #Binomial rate, Wilson score method: scoreci(x1 = c(5,0), n1 = c(56,29), contrast = "p", skew = FALSE) #Poisson rate, SCAS method: scoreci(x1 = c(5,0), n1 = c(56,29), distrib = "poi", contrast = "p") #Stratified example, using data from Hartung & Knapp: scoreci(x1 = c(15,12,29,42,14,44,14,29,10,17,38,19,21), x2 = c(9,1,18,31,6,17,7,23,3,6,12,22,19), n1 = c(16,16,34,56,22,54,17,58,14,26,44,29,38), n2 = c(16,16,34,56,22,55,15,58,15,27,45,30,38), stratified = TRUE) #TDAS example, using data from Hartung & Knapp: scoreci(x1 = c(15,12,29,42,14,44,14,29,10,17,38,19,21), x2 = c(9,1,18,31,6,17,7,23,3,6,12,22,19), n1 = c(16,16,34,56,22,54,17,58,14,26,44,29,38), n2 = c(16,16,34,56,22,55,15,58,15,27,45,30,38), stratified = TRUE, tdas = TRUE) ```

ratesci documentation built on Feb. 15, 2018, 5 p.m.