# 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.propciFunction
propci(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.diffpropciFunction
diffpropci(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.rrpropciFunction
rrpropci(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;
• :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.orpropciFunction
orpropci(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;
• :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.meanciFunction
meanci(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.diffmeanciFunction
diffmeanci(m1::Real, σ²1::Real, n1::Real, m2::Real, σ²2::Real, n2::Real;
alpha::Real = 0.05, method::Symbol = :default)::ConfInt

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:

groupoutcome 1outcome 2
group 1ab
group 2cd

### diffcmhci

ClinicalTrialUtilities.diffcmhciFunction
diffcmhci(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.orcmhciFunction
orcmhci(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.rrcmhciFunction
rrcmhci(data; a = :a, b = :b, c = :c, d = :d,
alpha = 0.05, logscale = false)::ConfInt

Cochran–Mantel–Haenszel confidence intervals for risk ratio.