scoreci: Score confidence intervals and tests for a single binomial or...

In ratesci: Confidence Intervals and Tests for Comparisons of Binomial Proportions or Poisson Rates

Score confidence intervals and tests for a single binomial or Poisson rate, or for comparisons of independent rates, with or without stratification.

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 variance 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 adjustment (for strictly conservative coverage).

Also includes score intervals for a single binomial proportion or Poisson rate ("p"). These are based on the Wilson score interval, and when corrected for skewness, coverage is almost identical to the mid-p method, or to Clopper-Pearson when also continuity-adjusted.

Hypothesis tests for association or non-inferiority are provided using the same score, to ensure consistency between test and CI. 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 random effects "TDAS" method, which incorporates stratum variability using a t-distribution score (inspired by Hartung-Knapp-Sidik-Jonkman). For fixed-effects analysis of stratified datasets, with weighting = "MH" for RD or RR, or weighting = "INV" for OR, omitting the skewness correction produces the CMH test, together with a coherent confidence interval for the required contrast. Alternatively, weighting = "INV" for any contrast gives intervals consistent with the efficient score test.

Usage

scoreci(
  x1,
  n1,
  x2 = 0,
  n2 = 0,
  distrib = "bin",
  contrast = "RD",
  level = 0.95,
  skew = TRUE,
  simpleskew = FALSE,
  or_bias = TRUE,
  ORbias = NULL,
  rr_tang = NULL,
  RRtang = NULL,
  bcf = ifelse(contrast != "p", TRUE, FALSE),
  cc = FALSE,
  theta0 = NULL,
  precis = 6,
  plot = FALSE,
  plotmax = 100,
  hetplot = FALSE,
  xlim = NULL,
  ylim = NULL,
  stratified = FALSE,
  weighting = NULL,
  mn_tol = 1e-08,
  MNtol = NULL,
  wt = NULL,
  sda = NULL,
  fda = NULL,
  dropzeros = FALSE,
  random = FALSE,
  prediction = 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; "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 or Gart-Nam method) or not (for the Miettinen-Nurminen method).
simpleskew	Logical (default FALSE) indicating whether to use the "simplified" skewness correction instead of the quadratic solution. See Laud 2021 for details. NOTE: this version of the score is only suitable for obtaining confidence limits, not p-values.
or_bias	Logical (default is TRUE for contrast = "OR", otherwise NULL) indicating whether to apply additional bias correction for OR derived from Gart 1985. (Laud 2018). Only applies if contrast is "OR".
ORbias	(deprecated: argument renamed to or_bias.)
rr_tang	Logical indicating whether to use Tang's score for RR: Stheta = (p1hat - p2hat * theta) / p2d (see Tang 2020). Default TRUE for stratified = TRUE, with weighting = "IVS" or "INV". Forced to FALSE for stratified = TRUE with other weightings. Has no effect when stratified = FALSE, as p2d terms cancel out. Experimental for distrib = "poi".
RRtang	(deprecated: argument renamed to rr_tang.)
bcf	Logical (default TRUE) indicating whether to apply 'N-1' variance correction in the score denominator. Applicable to distrib = "bin" only. NOTE: bcf = FALSE option is really only included for legacy validation against previous published methods (i.e. Gart & Nam, Mee, or standard Chi-squared test) and for contrast = "p".
cc	Number or logical (default FALSE) specifying (amount of) continuity adjustment. Numeric value between 0 and 0.5 is taken as the gamma parameter in Laud 2017, Appendix S2 (cc = TRUE translates to 0.5 for 'conventional' Yates adjustment). IMPORTANT NOTES: This adjustment (conventionally but controversially termed 'continuity correction') is aimed at approximating strictly conservative coverage, NOT for dealing with zero cell counts. Such 'sparse data adjustments' are not needed in the score method, except to deal with double-zero cells for stratified RD (& double-100% cells for binomial RD & RR) with IVS/INV weights. The continuity adjustments provided here have not been fully tested for stratified methods, but are found to match the continuity-adjusted version of the Mantel-Haenszel test, when cc = 0.5 for any of the binomial contrasts. Flexibility is included for a less conservative adjustment, such as cc = 0.25 suggested in Laud 2017 (see Appendix S3.4), or cc = 3/16 = 0.1875 in Mehrotra & Railkar (2000).
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. (If bcf = FALSE and skew = FALSE this gives a Farrington-Manning test.) By default, a two-sided test for association against theta0 = 0 (for RD) or 1 (for RR/OR) is also output: If bcf = FALSE and skew = FALSE this is the same as K. Pearson's Chi-squared test in the single stratum case. bcf = TRUE gives E. Pearson's 'N-1' Chi-squared test for a single stratum, (Recommended by Campbell 2007: https://doi.org/10.1002/sim.2832) and (with default weighting and random = FALSE) the CMH test for stratified tables. Default bcf = TRUE and 'skew = TRUE produces a skewness-corrected version of the 'N-1' Chi-squared test or CMH. This correction will only change the p-value if group sizes are unequal.
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).
hetplot	Logical (default FALSE) indicating whether to output plots for evaluating heterogeneity of stratified datasets.
xlim	pair of values indicating range of values to be plotted.
ylim	pair of values indicating range of values to be plotted.
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 (see Laud 2017 for details); "INV" = Inverse Variance (bcf omitted, default for contrast = "OR" giving CMH test); "MH" = Mantel-Haenszel (n1j * n2j) / (n1j + n2j) (default for contrast = "RD" or "RR" giving CMH test); (= sample size for contrast = "p"); "MN" = Miettinen-Nurminen weights. (similar to MH for contrast = "RD" or "RR", similar to INV for contrast = "OR"); "Tang" = (n1j * n2j) / (n1j + n2j) / (1 - pj) from Tang 2020, for an optimal test of RD if RRs are constant across strata. (Included only for validation purposes. In general, such a test would more logically use contrast = "RR" with weighting = "INV") For CI consistent with a CMH test, select skew = FALSE, random = FALSE, and use default MH weighting for RD/RR and INV for OR. Weighting = "MN" also matches the CMH test. For the Radhakrishna optimal (most powerful) test, select INV weighting. Note: Alternative user-specified weighting may also be applied, via the 'wt' argument.
mn_tol	Numeric value indicating convergence tolerance to be used in iteration with weighting = "MN".
MNtol	(deprecated: argument renamed to mn_tol)
wt	Numeric vector containing (optional) user-specified weights. Overrides weighting if non-empty.
sda	Sparse data adjustment to avoid zero variance when x1 + x2 = 0: Only applied when stratified = TRUE. Default 0.5 for RD with IVS/INV weights. Not required for RR/OR, default is to remove double-zero strata instead.
fda	Full data adjustment to avoid zero variance when x1 + x2 = n1 + n2: Only applied when stratified = TRUE. Default 0.5 for RD & RR with IVS/INV weights. Not required for OR, default is to remove affected strata.
dropzeros	Logical (default FALSE) indicating whether to drop uninformative strata for RR/OR (i.e. strata with x1 + x2 = 0), even when the choice of weights would allow them to be retained for a fixed effects analysis. Has no effect on estimates, just the heterogeneity test.
random	Logical (default FALSE) indicating whether to perform random effects meta-analysis for stratified data, using the t-distribution (TDAS) method for stratified data (defined in Laud 2017). NOTE: If random = TRUE, then skew = TRUE only affects the per-stratum estimates.
prediction	Logical (default FALSE) indicating whether to produce a prediction interval (work in progress).
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 requested contrast and its confidence interval, and the estimated rates in each group: (p1hat, p2hat) are (r1, r0) from Miettinen-Nurminen, or (r1*, r0*) when stratified; (p1mle, p2mle) are (R1, R0), or (R1*, R0*) when stratified, evaluated at the MLE for the contrast parameter, incorporating any specified skewness/bias corrections.

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, including a score-based version of a Q statistic and p-value, I^2 and tau^2 to quantify heterogeneity, and a test for qualitative interaction analogous to the Gail and Simon test.

weighting

a string indicating the selected weighting method.

stratdata

a matrix containing stratum estimates and weights.

Author(s)

Pete Laud, [e-mail]

References

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

Laud PJ. Corrigendum: Equal-tailed confidence intervals for comparison of rates. Pharmaceutical Statistics 2018; 17:290-293.

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.

Tang Y. Score confidence intervals and sample sizes for stratified comparisons of binomial proportions. Statistics in Medicine 2020; 39:3427-3457.

Examples

# 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
)

# "Random effects" 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, random = TRUE
)

# Stratified example, with extremely rare instance of non-calculable skewness
# correction seen on plot of score function:

scoreci(
  x1 = c(1, 16), n1 = c(20, 40), x2 = c(0, 139), n2 = c(80, 160),
  contrast = "RD", skew = TRUE, simpleskew = FALSE,
  distrib = "bin", stratified = TRUE, plot = TRUE, weighting = "IVS"
)

ratesci documentation built on June 21, 2025, 1:07 a.m.