Odds ratio — Sample Size Calculator

Determines the sample size required to detect a specified odds ratio between cases and controls, based on the expected exposure distribution and study design.

Key takeaways

What is calculated: Determine the number of cases and controls at which the odds ratio can be detected with the target power.
Quantities you must supply: Exposure prevalence in controls; Odds ratio to detect; Significance level; Power; Controls per case.
Type of calculation: Power-based: it answers how much data is needed to detect a stated effect. Power 0.80 at α = 0.05 is the usual target; confirmatory studies use 0.90.

Method used for Odds ratio

Unmatched case-control study — odds ratio (Schlesselman) Two-proportion calculation parameterised by control exposure prevalence and the odds ratio. Case-control sampling fixes the numbers of cases and controls by design, so incidence is not estimable and the odds ratio is the appropriate measure.

When to use

Use when the primary objective is to determine whether there is a meaningful association between an exposure and an outcome, expressed as an odds ratio in a case-control design.

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Sample size

Exposure prevalence in controls (p₀)

Odds ratio to detect (OR)

Significance level (α)

Type I error rate.

Power (1 − β)

Probability of detecting the specified effect.

Alternative

A two-sided alternative splits α between both tails; a one-sided alternative places all of α in one tail, needs a smaller n, and must be pre-specified.

Controls per case (k)

Set to 1 for equal group sizes.

Anticipated dropout / non-response (proportion)

Final n is inflated by 1/(1 − dropout).

Reset to defaults

Awaiting inputs

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Set the parameters and calculate.

Methodology — Unmatched case-control study — odds ratio (Schlesselman)

Formula used Unmatched case-control study — odds ratio (Schlesselman)

n = \frac{{[z_{1 - α / 2} \sqrt{(1 + 1 / k) \bar{p} \bar{q}} + z_{1 - β} \sqrt{p_{1} q_{1} + p_{0} q_{0} / k}]}^{2}}{{(p_{1} - p_{0})}^{2}}

Number of cases required; controls are k times this

p_{1} = \frac{O R \cdot p_{0}}{1 - p_{0} + O R \cdot p_{0}}

Exposure prevalence in cases

$z_{1 - α / 2} (two-sided) in place of z_{1 - α} (one-sided)$ Two-sided alternative selected: α is split between both tails, so the larger quantile is used and the requirement rises

where —

p₀: Exposure prevalence among controls, representing the source population. 0 – 1 exclusive
OR: Odds ratio to detect. > 0; must differ from 1
α: Probability of rejecting a true null hypothesis (Type I error). 0.0001 – 0.5; conventionally 0.05, or 0.025 one-sided for regulatory non-inferiority
1 − β: Probability of rejecting the null hypothesis when the specified alternative is true. 0.50 – 0.9999; conventionally 0.80 or 0.90
k: Controls per case. 0.05 – 20
ℓ: Expected proportion of enrolled units lost before analysis. 0 – 0.95

How this method works

Sampling is on outcome status, so the estimable measure is the odds ratio. The exposure prevalence among cases is derived from the control prevalence and the odds ratio, and the two-proportion formula then applies.

Sampling is on outcome, so the estimable measure is the odds ratio; exposure prevalence among cases follows from p₀ and the OR.

Power is maximised when control exposure is near 0.5 and falls sharply for rare or near-universal exposures. Recruiting more controls helps, but the gain follows 4k/(1+k)² and is negligible beyond about 1:4. Assumes unmatched analysis.

Null hypothesis. H0: OR = 1, equivalently exposure prevalence is equal in cases and controls.

Alternative hypothesis. Two-sided H1: OR != 1. One-sided H1: OR > 1 or OR < 1.

Calculation procedure

State the exposure prevalence among controls, $p_{0}$ , and the odds ratio $O R$ worth detecting.
Convert to the case exposure prevalence: $p_{1} = \frac{p_{0} O R}{1 + p_{0} (O R - 1)}$ .
Set the number of controls per case, $k$ , and compute the pooled proportion for that ratio.
Evaluate the two-proportion formula on $p_{1}$ against $p_{0}$ to obtain the number of cases; controls are $k$ times that.
Round each group up, then inflate for non-response. Gains from extra controls per case flatten beyond about $k = 4$ .

Exact or approximate

Approximate — a normal approximation to the binomial.

The formula shown always matches what was computed: when a one-sided test is selected the rendered quantile changes from z_1−α/2 to z_1−α, and where several methods exist, the formula follows the method selected in the calculator.

Advantages & limitations

Advantages

Efficient for rare outcomes.
Supports recruiting several controls per case when cases are scarce.
Requires only the control exposure prevalence and the target odds ratio.

Limitations

Power falls sharply for rare or near-universal exposures.
Gains from extra controls follow 4k/(1+k)² and are negligible beyond about four per case.
Not valid for matched designs.

Assumptions

Cases and controls are sampled independently from the same source population.
Controls represent the exposure distribution of the population that produced the cases.
Exposure is measured without differential misclassification.
The analysis will be unmatched.

Applicable adjustments

Supports the control-to-case ratio and dropout inflation.

Conclusion

Report cases, controls and the control-to-case ratio, together with the assumed control exposure prevalence. If matching is planned, use the paired calculator instead.

References

Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York: Oxford University Press; 1982.
Dupont WD. Power calculations for matched case-control studies. Biometrics. 1988;44(4):1157–1168. Link
Woodward M. Epidemiology: Study Design and Data Analysis. 3rd ed. Boca Raton: Chapman & Hall/CRC; 2014.