How to calculate the sample size for an unmatched case-control study

Many case-control proposals size the study with the single-proportion formula (n = z² p (1 − p) / d²). That formula estimates a prevalence; it says nothing about whether you can detect an association. A case-control study compares exposure between two groups, so its sample size comes from the odds ratio you want to detect, the power to detect it, and how common the exposure is.

1. Five numbers to decide

• Exposure among controls (p₀). How common the exposure is in people without the disease. Take it from a study in a similar population, or from a survey of the source population.
• The odds ratio to detect. The smallest odds ratio that would matter, not the largest one a previous study happened to find (more on this below).
• Controls per case (r). 1 when cases and controls are equally easy to find; 2 to 4 when cases are scarce.
• Confidence. Usually 95%, two-sided (z = 1.96).
• Power. 80% (z = 0.84) or 90% (z = 1.28): the chance of finding the association if it is really there.

2. From the odds ratio to exposure in cases

The formula works with two proportions, so first turn the odds ratio into the exposure you expect among cases (p₁):

p₁ = OR × p₀ / (1 − p₀ + OR × p₀)

With exposure in 20% of controls and an odds ratio of 2.5: p₁ = 2.5 × 0.20 / (0.80 + 0.50) = 0.385. Cases would need to be exposed 38.5% of the time for the odds ratio to be 2.5.

3. The formula

For an unmatched design with r controls per case (Fleiss; normal approximation, two-sided):

n(cases) = [ z_α/2 √((1 + 1/r) p̄ q̄) + z_β √(p₁q₁ + p₀q₀ / r) ]² / (p₁ − p₀)²

where p̄ = (p₁ + r p₀) / (1 + r), q = 1 − p, and the number of controls is r × n(cases).

4. A worked example

A case-control study of risk factors for neonatal sepsis. Exposure among controls 20%, odds ratio to detect 2.5, two controls per case, 95% confidence, 80% power:

1. p₁ = 0.385 (step 2) and p̄ = (0.385 + 2 × 0.20) / 3 = 0.2615
2. First term: 1.96 × √(1.5 × 0.2615 × 0.7385) = 1.055
3. Second term: 0.84 × √(0.385 × 0.615 + 0.20 × 0.80 / 2) = 0.474
4. n(cases) = (1.055 + 0.474)² / (0.385 − 0.20)² = 2.337 / 0.0341 = 68.6, so 69 cases and 138 controls (207 in all)
5. Allowing 10% non-response in each group: 69 / 0.9 = 76.7 and 138 / 0.9 = 153.3, so 77 cases and 154 controls (231)

Round up at every step: 68.6 cases means 69.

5. The odds ratio you plan for drives everything

Same study, same exposure (20%) and two controls per case; only the odds ratio changes:

Going from an odds ratio of 2.5 to 2 nearly doubles the study; going to 1.5 multiplies it by six. This is why the odds ratio should be the smallest one that would change practice, not the one from the previous study. Published odds ratios, especially from small studies, tend to be larger than the truth; planning on them gives an underpowered study.

6. More controls per case: useful, up to a point

When cases are scarce (a rare disease, one hospital), adding controls lets you need fewer cases. At an odds ratio of 2:

The second control saves 46 cases; the fourth saves only 8, and beyond four almost nothing. Four controls per case is the usual ceiling.

7. Power, continuity correction and confounders

• 90% power instead of 80% raises the example from 69 to 93 cases (207 to 279 in all). Use it when a missed association would be costly.
• Continuity correction. Software differs: OpenEpi, for example, shows Kelsey, Fleiss, and Fleiss with continuity correction side by side, and they do not agree. With the correction, the example needs 77 cases instead of 69. Either is defensible; say which you used.
• Confounders. The formula is for a crude comparison. If you will adjust for other factors in logistic regression, and the exposure is associated with them, divide by (1 − R²), where R² is how much of the exposure the other factors explain (Hsieh et al., 1998). With R² = 0.2, 69 cases become 87.
• Matched designs (each case paired with a control of the same age, say) need a different calculation, based on discordant pairs (Dupont, 1988). This formula is for unmatched studies.

8. What to write in the methods section

“The sample size was calculated for an unmatched case-control design using the two-proportions formula (Fleiss), assuming 20% exposure among controls, an odds ratio of 2.5 to detect, two controls per case, 95% confidence and 80% power. This gave 69 cases and 138 controls; allowing for 10% non-response, 77 cases and 154 controls (231 participants) were recruited.”

9. Common mistakes

• The single-proportion formula for a case-control study.
• Disease prevalence in place of exposure among controls. The formula needs how common the exposure is, not the disease.
• The largest odds ratio from the literature as the one to detect, which gives a study too small to find a realistic effect.
• Sizing for one exposure and analysing ten. Size for the main exposure, and say the others are secondary.
• Ignoring clustering when cases and controls come from a few facilities or villages: multiply by the design effect.

References

1. Dupont WD. Power calculations for matched case-control studies. Biometrics. 1988;44(4):1157–68.
2. Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions. 3rd ed. Hoboken: John Wiley & Sons; 2003.
3. Hsieh FY, Bloch DA, Larsen MD. A simple method of sample size calculation for linear and logistic regression. Stat Med. 1998;17(14):1623–34.
4. Kelsey JL, Whittemore AS, Evans AS, Thompson WD. Methods in Observational Epidemiology. 2nd ed. New York: Oxford University Press; 1996.
5. Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York: Oxford University Press; 1982.

Tesfahun Mulatu

Epidemiologist and biostatistician with more than 10 years in academia and research, including multi-country and mixed-methods studies with local and international research institutions, and monitoring, evaluation and learning across Ethiopia’s development and humanitarian sectors on projects funded by UN agencies and international donors. Tesfahun’s work spans maternal and child health, sexual and reproductive health, gender-based violence and mental health, with over 19 publications.

Epidemiologist and biostatistician, founder of Researchite · LinkedIn