Confidence Intervals
Proportions
One proportion
P = n / x
where n - number of outcomes; x - number of observations.
Absolute risk difference
Diff(𝛿) = P₁ - P₂ = n₁ / x₁ - n₂ / x₂
Risk Ratio
RR = P₁ / P₂ = (n₁ / x₁) - (n₂ / x₂)
Odd Ratio
OR = (n₁ / (x₁ - n₁)) - (n₂ / (x₂ - n₂))
propci
ClinicalTrialUtilities.propci
— Functionpropci(x::Int, n::Int; alpha=0.05, method = :default)::ConfInt
Confidence interval for proportion.
Computation methods:
- :wilson | :default - Wilson's confidence interval (CI) for a single proportion (wilson score);
- :wilsoncc - Wilson's CI with continuity correction (CC);
- :cp - Clopper-Pearson exact CI;
- :soc - SOC: Second-Order corrected CI;
- :blaker - Blaker exact CI for discrete distributions;
- :arcsine - Arcsine CI;
- :wald - Wald CI without CC;
- :waldcc - Wald CI with CC;
propci(tab::ConTab{2,2}; alpha::Real = 0.05, method::Symbol = :default)
Confidence interval for proportions a / (a + b) and c / (c + d)
diffpropci
ClinicalTrialUtilities.diffpropci
— Functiondiffpropci(x1::Int, n1::Int, x2::Int, n2::Int;
alpha::Real = 0.05, method::Symbol = :default)::ConfInt
Confidence interval for proportion difference.
Computation methods:
:nhs
- Newcombes Hybrid (wilson) Score interval for the difference of proportions;:nhscc
- Newcombes Hybrid Score CC;:ac
- Agresti-Caffo interval for the difference of proportions;:mn
|:default
- Method of Mee 1984 with Miettinen and Nurminen modification;:mee
|:fm
- Mee maximum likelihood method;:wald
- Wald CI without CC;:waldcc
- Wald CI with CC;
References
- nhs, nhscc - Newcombe RG (1998), Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods. Statistics in Medicine 17, 873-890.
- ac - Agresti A, Caffo B., “Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures”, American Statistician 54: 280–288 (2000)
- mn - Miettinen, O. and Nurminen, M. (1985), Comparative analysis of two rates. Statist. Med., 4: 213-226. doi:10.1002/sim.4780040211
- mee - Mee RW (1984) Confidence bounds for the difference between two probabilities, Biometrics40:1175-1176
- Brown, L.D., Cai, T.T., and DasGupta, A. Interval estimation for a binomial proportion. Statistical Science, 16(2):101–117, 2001.
- Farrington, C. P. and Manning, G. (1990), “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, 9, 1447–1454
- Li HQ, Tang ML, Wong WK. Confidence intervals for ratio of two Poisson rates using the methodof variance estimates recovery. Computational Statistics 2014; 29(3-4):869-889
- Brown, L., Cai, T., & DasGupta, A. (2003). INTERVAL ESTIMATION IN EXPONENTIAL FAMILIES. Statistica Sinica, 13(1), 19-49.
diffpropci(tab::ConTab{2,2}; alpha::Real = 0.05, method::Symbol = :default)::ConfInt
Confidence interval for proportion difference: (a / (a + b)) - (c / (c + d))
rrpropci
ClinicalTrialUtilities.rrpropci
— Functionrrpropci(x1::Int, n1::Int, x2::Int, n2::Int; alpha::Real = 0.05,
method::Symbol = :default)::ConfInt
Confidence interval for relative risk.
Computation methods:
- :mn | :default - Miettinen-Nurminen Score interval;
- :cli | :walters - Crude log interval;
- :li | :katz - Log interval for the risk ratio;
- :mover - Method of variance estimates recovery;
rrpropci(tab::ConTab{2,2}; alpha::Real = 0.05, method::Symbol = :default)::ConfInt
Confidence interval for relative risk.
orpropci
ClinicalTrialUtilities.orpropci
— Functionorpropci(x1::Int, n1::Int, x2::Int, n2::Int; alpha::Real = 0.05,
method::Symbol = :default)::ConfInt
Confidence interval for odd ratio.
Computation methods:
- :mn - Miettinen-Nurminen CI (deprecated);
- :mn2 | :default - Miettinen-Nurminen CI;
- :woolf - Woolf logit CI;
- :awoolf | :gart - Adjusted Woolf interval (Gart adjusted logit);
- :mover - Method of variance estimates recovery;
orpropci(tab::ConTab{2,2}; alpha::Real = 0.05, method::Symbol = :default)::ConfInt
Confidence interval for odd ratio.
Means
meanci
ClinicalTrialUtilities.meanci
— Functionmeanci(m::Real, σ²::Real, n::Int; alpha::Real = 0.05,
method=:default)::ConfInt
Confidence interval for mean, where:
m - mean; σ² - variance; n - observation number.
Computation methods:
- :norm - Normal distribution (default);
- :tdist - T Distribution.
diffmeanci
ClinicalTrialUtilities.diffmeanci
— Functiondiffmeanci(m1::Real, σ²1::Real, n1::Real, m2::Real, σ²2::Real, n2::Real;
alpha::Real = 0.05, method::Symbol = :default)::ConfInt
Confidence interval for mead difference.
m1, m2 - mean; σ²1, σ²2 - variance; n1, n2 - observation number.
Computation methods:
- :ev - equal variance (default);
- :uv - unequal variance with Welch-Satterthwaite df correction.
Cochran–Mantel–Haenszel confidence intervals
Table cell map:
group | outcome 1 | outcome 2 |
---|---|---|
group 1 | a | b |
group 2 | c | d |
diffcmhci
ClinicalTrialUtilities.diffcmhci
— Functiondiffcmhci(data; a = :a, b = :b, c = :c, d = :d,
alpha = 0.05, method = :default)::ConfInt
Cochran–Mantel–Haenszel confidence intervals for proportion difference.
data- data with 4 columns, each line represent 2X2 table
abcd - data table names (number of subjects in 2X2 table):
diffcmhci(a::Vector, b::Vector, c::Vector, d::Vector;
alpha = 0.05, method = :default)::ConfInt
Cochran–Mantel–Haenszel confidence intervals for proportion difference.
abcd - vector of cells in in 2X2 tables:
orcmhci
ClinicalTrialUtilities.orcmhci
— Functionorcmhci(data; a = :a, b = :b, c = :c, d = :d,
alpha = 0.05, logscale = false)::ConfInt
Cochran–Mantel–Haenszel confidence intervals for odd ratio.
rrcmhci
ClinicalTrialUtilities.rrcmhci
— Functionrrcmhci(data; a = :a, b = :b, c = :c, d = :d,
alpha = 0.05, logscale = false)::ConfInt
Cochran–Mantel–Haenszel confidence intervals for risk ratio.