What values of p1 and p2 should I use if I do not have prior data?

When prior data are unavailable, use evidence from systematic reviews, similar populations, or disease registries for p2 (unexposed group rate). For p1 (exposed group rate), calculate it from the expected relative risk: p1 = p2 × RR. If you have no data at all, use a conservative estimate of p2 = 0.10 and set the minimum detectable risk ratio based on clinical significance. Another conservative approach is to set p2 = 0.50, which maximises variance and gives the largest (most conservative) sample size — appropriate only when truly no data exist. Document all assumptions transparently in your methods section.

How to Calculate Sample Size for Cohort Study in Medical Research

Q: What is the formula for sample size in a cohort study?

The standard formula for sample size in a cohort study comparing two proportions is: n per group = (z_α/2 + z_β)² × [p1(1−p1) + p2(1−p2)] / (p1−p2)², where p1 is the event rate in the exposed group, p2 is the event rate in the unexposed group, z_α/2 is the z-score for the chosen significance level (1.96 for α=0.05), and z_β is the z-score for the desired power (0.842 for 80% power, 1.282 for 90% power). This gives the per-group sample size, so multiply by 2 for the total. Always inflate by expected loss-to-follow-up rate by dividing by (1 minus attrition proportion).

Q: What is a cohort study in medical research?

A cohort study is an observational study design in which a group of people (a cohort) who share a common characteristic or exposure is followed over time to see what outcomes develop. In a prospective cohort study, participants are enrolled before outcomes occur and followed forward in time. In a retrospective cohort study, both exposure data and outcome data are already recorded (usually in medical records) and the researcher looks backward. Cohort studies are particularly valuable for calculating incidence rates, relative risk, and attributable risk because both exposed and unexposed groups are observed.

Q: How does loss to follow-up affect cohort study sample size?

Loss to follow-up reduces the effective sample size, which weakens statistical power. To maintain the target power despite attrition, inflate the calculated sample size by dividing by (1 minus the expected attrition rate). For example, if the formula gives n=200 per group and you expect 20% loss to follow-up, the adjusted sample size is 200/(1−0.20) = 250 per group. Higher attrition rates require larger initial recruitment. Long prospective cohort studies typically plan for 10–30% loss to follow-up depending on study duration and population mobility.

Q: What is the difference between prospective and retrospective cohort studies?

In a prospective cohort study, participants are recruited at a point in time and followed forward into the future. Exposure is measured at baseline; outcomes develop during follow-up. This design is expensive and slow but avoids recall bias for exposure data. In a retrospective cohort study, both the exposure and the outcome have already occurred; the researcher uses existing records (hospital records, employment records, disease registries) to reconstruct the cohort and outcomes. Retrospective cohorts are faster and cheaper but depend entirely on the quality of historical records. Both designs use the same sample size formula.

Q: How does statistical power affect cohort study sample size?

Statistical power is the probability of correctly detecting a true effect when one exists (avoiding a Type II error). Higher power requires larger sample sizes. At 80% power (z_β = 0.842), you accept a 20% chance of missing a real effect. At 90% power (z_β = 1.282), you accept only a 10% chance. Increasing power from 80% to 90% increases required sample size by approximately 35–40% for the same parameters. Most medical studies target 80% power as a minimum, with 90% recommended for primary endpoints with major clinical or regulatory implications.

Q: What happens to sample size when the expected risk difference is small?

Sample size increases sharply as the expected risk difference (p1 − p2) decreases, because the formula uses (p1 − p2)² in the denominator. Halving the risk difference from 10% to 5% quadruples the required sample size. This is why detecting small effects — even if they are real and clinically meaningful — requires very large cohort studies. For example, detecting a 2% absolute risk difference with 80% power and α=0.05 may require thousands of participants per group, making community or multi-site recruitment essential.

Q: Can I use a 1:2 or 1:3 exposed to unexposed ratio in a cohort study?

Yes. If the exposed group is rare or expensive to recruit, you can enrol more unexposed participants to increase statistical efficiency. The formula adjusts to: n_exposed = (z_α/2 + z_β)² × [p1(1−p1) + p2(1−p2)/r] / (p1−p2)² and n_unexposed = r × n_exposed, where r is the ratio of unexposed to exposed participants. Going from 1:1 to 1:2 increases total sample size but reduces the number of harder-to-find exposed participants. Efficiency gains diminish beyond a 1:4 ratio — adding more controls beyond that does not meaningfully reduce total sample size requirements.

Q: How do I report sample size justification for a cohort study?

Report the following in the methods section: (1) the sample size formula or software used, (2) the assumed event rate in the unexposed group (p2) and its source, (3) the assumed event rate in the exposed group (p1) or the minimum detectable relative risk, (4) the significance level (α) and whether one-tailed or two-tailed, (5) the target statistical power, (6) the calculated per-group and total sample size, and (7) any inflation for loss to follow-up with the assumed attrition rate and final recruitment target. Many journals require all seven elements for peer review.

Q: What is the minimum detectable relative risk in a cohort study?

The minimum detectable relative risk (MDRR) is the smallest true relative risk that your study has enough power to detect at a specified α and β. It is not calculated directly from the formula — rather, you rearrange the sample size formula to solve for RR given your fixed n. MDRR is useful when you have a fixed available sample (e.g., an existing patient cohort) and want to know what effect size you are powered to detect, rather than starting from a desired effect and calculating n. Online calculators and software such as OpenEpi can solve for MDRR directly.

Cohort studies generate some of the most powerful evidence in epidemiology — incidence rates, relative risks, and attributable fractions that no other non-experimental design can produce. But they are also among the most expensive and time-intensive research investments, making an accurate sample size calculation essential before a single participant is enrolled. Too few participants and the study cannot detect the true effect; too many and years of funding are wasted on participants beyond what the hypothesis requires. This guide walks through the complete sample size formula for cohort studies, explains every variable, demonstrates four worked examples across different disease areas, shows you how to correct for loss to follow-up, and covers the mistakes that repeatedly trigger reviewer rejection or fail ethics board review.

What Is a Cohort Study?

A cohort study is an observational epidemiological design in which a defined group of people — the cohort — is assembled based on their exposure status and followed over time to observe the development of outcomes. The word "cohort" comes from the Latin for a unit of Roman soldiers: a group with a shared characteristic who move through time together.

The defining feature of cohort studies is that exposure is measured before the outcome occurs. This temporal sequence — exposure → follow-up → outcome — is what gives cohort studies their strength in establishing the direction of association. Unlike case-control studies, cohort studies allow direct calculation of incidence rates, relative risk, attributable risk, and absolute risk differences in the exposed and unexposed groups.

Relative risk: uniquely calculable from cohort studies

Incidence rate: event density over person-time in each group

Attributable risk: absolute excess risk from exposure

Prospective vs Retrospective Cohort Studies

Cohort studies divide into two broad types based on their temporal relationship to the researcher at the time of study initiation.

Feature	Prospective Cohort	Retrospective Cohort
Timeline at enrolment	Outcome has NOT yet occurred; participants are followed forward	Both exposure and outcome have already occurred; researcher looks back
Data source	Primary data collected during follow-up (questionnaires, clinical tests)	Existing records (hospital databases, employment records, disease registries)
Exposure measurement	Measured carefully at baseline and during follow-up; minimal recall bias	Depends on historical record quality; potential for incomplete data
Outcome ascertainment	Standardised prospectively by study protocol	Based on existing diagnostic codes or records — may vary by era
Time required	Years to decades for chronic disease outcomes	Months to years — data already exist
Cost	High — participant follow-up, repeated measurements	Lower — no active follow-up required
Loss to follow-up	Real risk — must plan correction in sample size	Often lower — records are fixed (though records may be incomplete)
Classic example	Framingham Heart Study (1948–present)	Nurses' Health Study reanalysis of historical questionnaire data
Sample size formula	Same formula — parameters differ	Same formula — parameters differ

Key Point Both prospective and retrospective cohort studies use the same sample size formula. The difference lies in the parameters you plug in: a prospective study of a slowly developing disease needs to plan for substantial follow-up loss (10–30%); a retrospective study of well-maintained records may need minimal attrition adjustment but must grapple with incomplete historical data.

Why Sample Size Matters in Cohort Studies

Sample size is not a bureaucratic hurdle. It directly controls two statistical quantities that determine whether your study will produce actionable, publishable, and scientifically credible results: Type I error (α) and Type II error (β).

Type I error (α; false positive): Concluding there is an association when there is none. Controlled by the significance threshold (typically α = 0.05).
Type II error (β; false negative): Failing to detect a real association that truly exists. Controlled by study power = 1 − β. A study with 80% power accepts a 20% chance of missing a true effect.

An underpowered cohort study is not merely less informative — it is potentially actively misleading. A null result from an underpowered study (p > 0.05) cannot be interpreted as evidence of no association. It means only that the study was too small to see the association even if it exists. Systematic reviews routinely exclude underpowered studies because their confidence intervals are so wide they add no information about effect size.

Regulatory and Ethical Dimension Ethics committees and institutional review boards now routinely require a formal sample size justification as a condition of approval. A study that enrols participants without adequate power is considered ethically problematic because it exposes participants to risk (time, blood draws, follow-up inconvenience) without a realistic prospect of generating usable knowledge. Equally, a study that enrols far more participants than needed wastes participant time and research funding. The target: the smallest sample that gives adequate power to answer the research question.

The Sample Size Formula for Cohort Studies

The standard formula for comparing two proportions in a cohort study — the event rate in the exposed group (p₁) versus the event rate in the unexposed group (p₂) — is derived from the two-sample z-test for proportions. The formula gives the sample size required per group, assuming equal group sizes:

Sample Size Per Group — Standard Two-Proportion Formula for Cohort Studies

n = (zα/2 + zβ)² × [p₁(1−p₁) + p₂(1−p₂)] / (p₁−p₂)²

n = required participants per group

p₁ = expected event rate in the exposed group

p₂ = expected event rate in the unexposed group

(p₁−p₂) = expected risk difference (absolute)

zα/2 = z-score for significance level (1.96 for α=0.05, two-tailed)

zβ = z-score for power (0.842 for 80%, 1.282 for 90%)

The formula has an intuitive structure: the numerator grows when the variance is large (events near 50% frequency) or when a higher z-score is demanded (stricter α, higher power). The denominator shrinks when the expected difference between groups is small, driving n upward — detecting a subtle effect requires a larger study than detecting a large one. Total sample size is simply 2n for equal allocation between exposed and unexposed groups.

Loss-to-Follow-Up Adjustment

Prospective cohort studies must account for participants who withdraw, are lost to follow-up, die of unrelated causes, or otherwise fail to contribute outcome data. The adjustment is straightforward:

Attrition-Adjusted Sample Size

nₙᵈᵍ = n / (1 − L)

nₙᵈᵍ = adjusted sample size per group after accounting for dropout

n = sample size per group from the main formula

L = expected proportion lost to follow-up (e.g., 0.20 for 20%)

Example: if n = 200 and L = 20%, then nₙᵈᵍ = 200 / 0.80 = 250 per group

All Variables Explained in Depth

p₂

Unexposed Group Event Rate (Baseline Risk)

The incidence or cumulative risk of the outcome in the unexposed (control) group over the study follow-up period. This is your reference rate and must come from prior literature, disease registries, or pilot data for the specific population, outcome definition, and follow-up duration you plan to use. Errors in p₂ directly distort the entire calculation because it anchors everything else.

p₁

Exposed Group Event Rate

The incidence or cumulative risk of the outcome in the exposed group. In practice, you rarely know p₁ directly — you estimate it from an expected relative risk: p₁ = p₂ × RR. If you expect a relative risk of 2.5 and p₂ = 0.10, then p₁ = 0.10 × 2.5 = 0.25. Alternatively, you can specify the minimum clinically meaningful risk difference (p₁ − p₂) you want the study to detect.

Significance Level (Type I Error Rate)

The probability of falsely rejecting the null hypothesis when it is actually true. The conventional threshold is α = 0.05 (5%). For studies with multiple primary endpoints, apply Bonferroni correction. Two-tailed tests (zα/2 = 1.96) are standard; one-tailed tests (zα = 1.645 for α = 0.05) are rarely justified in medical research because they assume in advance which direction the effect must go.

1−β

Statistical Power

The probability of correctly detecting the effect if it truly exists. Standard minimum is 80% (β = 0.20; zβ = 0.842). Regulatory and high-stakes clinical studies typically target 90% (β = 0.10; zβ = 1.282) or even 95%. Power of 80% means 1 in 5 true effects will be missed. Increasing from 80% to 90% requires approximately 35–40% more participants, depending on the other parameters.

Z-Score Reference Table

Significance Level (α)	zα/2 (Two-Tailed)	Power (1−β)	zβ	Combined (zα/2 + zβ)
0.10	1.645	80%	0.842	2.487
0.05	1.960	80%	0.842	2.802 ← most common
0.05	1.960	85%	1.036	2.996
0.05	1.960	90%	1.282	3.242
0.05	1.960	95%	1.645	3.605
0.01	2.576	80%	0.842	3.418
0.01	2.576	90%	1.282	3.858

Step-by-Step Calculation Guide

Define the primary outcome

State a single, clearly measured binary outcome (e.g., incident type 2 diabetes, first myocardial infarction, 30-day mortality). Sample size is calculated for one primary endpoint. Secondary endpoints use the same sample size but are acknowledged to be underpowered for individual analysis.

Determine p₂ from literature

Search existing cohort studies or disease registries for the cumulative incidence of your outcome over your planned follow-up period in a population similar to your unexposed group. Document the source. If multiple estimates exist, use the most conservative (closest to 0.50) or conduct sensitivity analyses.

Determine p₁ from expected RR or minimum detectable difference

Either obtain expected RR from prior evidence (p₁ = p₂ × RR) or define the minimum clinically important absolute risk difference you want to detect (p₁ = p₂ + ARD). The latter approach frames the study by clinical relevance rather than statistical convention.

Set α and power

Use α = 0.05 (two-tailed) and power ≥ 80% for most medical cohort studies. Use α = 0.01 or power = 90% for studies with regulatory implications, rare high-consequence outcomes, or where a false positive or negative would have major policy impact. Set these before calculation — never adjust them after seeing a sample size you dislike.

Apply the formula to get n per group

Substitute all values into the formula. Round up to the nearest whole number (never round down — this would reduce power below the target). Multiply by 2 for total sample size with 1:1 exposed:unexposed allocation.

Adjust for expected loss to follow-up

For prospective studies, inflate each group size: nₙᵈᵍ = n / (1 − L). Plan a conservative L based on study duration, population mobility, and disease severity. A 2-year study in a stable outpatient population might use L = 10–15%; a 10-year community study might use L = 20–30%.

Consider unequal group sizes if indicated

If exposed individuals are rare or costly to recruit, enrol more unexposed controls. The formula adjusts to: n₁ = (zα/2 + zβ)² × [p₁(1−p₁) + p₂(1−p₂)/r] / (p₁−p₂)² where r = n₂/n₁. Maximum efficiency gain occurs at r = 2 (1:2 ratio). Beyond r = 4, gains are negligible.

Document and justify all assumptions

Write out every assumption in your research protocol: the values of p₁, p₂, α, power, L, and the source for each estimate. Ethics committees and journal reviewers will ask for the basis of each assumption. An unjustified assumption is a protocol weakness that invites revision requests.

Worked Epidemiology Examples

The following four examples cover a range of disease areas, follow-up durations, and parameter choices. Each shows the complete calculation with all working visible.

Smoking and Chronic Obstructive Pulmonary Disease — Classic Prospective Cohort

10-year follow-up • Binary outcome: incident COPD diagnosis • 1:1 allocation • α = 0.05, Power = 80%

A pulmonologist plans a 10-year prospective cohort study comparing current smokers (exposed) with lifelong non-smokers (unexposed) for incident COPD. Literature estimates the 10-year COPD incidence in non-smokers at 6% and the expected relative risk for smokers at 3.0, giving an expected incidence of 18% in smokers.

Given Parameters p₂ (unexposed, non-smokers) = 0.06 (6%)
p₁ (exposed, smokers) = p₂ × RR = 0.06 × 3.0 = 0.18 (18%)
α = 0.05 (two-tailed) → zα/2 = 1.960
Power = 80% → zβ = 0.842
Expected attrition over 10 years: L = 20%
Variance Term p₁(1−p₁) = 0.18 × 0.82 = 0.1476
p₂(1−p₂) = 0.06 × 0.94 = 0.0564
Sum = 0.1476 + 0.0564 = 0.2040
Risk Difference Squared (p₁−p₂)² = (0.18−0.06)² = (0.12)² = 0.0144
Sample Size Per Group n = (1.960 + 0.842)² × 0.2040 / 0.0144
n = (2.802)² × 0.2040 / 0.0144
n = 7.851 × 0.2040 / 0.0144
n = 1.602 / 0.0144 = 111.3 → 112 per group
Attrition Adjustment (L = 20%) nₙᵈᵍ = 112 / (1−0.20) = 112 / 0.80 = 140 per group
Total Sample Total = 140 × 2 = 280 participants

Interpretation: Recruit 140 smokers and 140 non-smokers (280 total). After expected 20% attrition (28 lost per group), 112 complete participants per group will give 80% power to detect the difference between 6% and 18% COPD incidence at α = 0.05. Had we not planned for attrition, we would have enrolled only 112 per group and likely finished with ~89 completers — reducing power to approximately 68%, well below acceptable levels.

Physical Activity and Type 2 Diabetes — Small Risk Difference, High Power Target

5-year follow-up • Outcome: new T2DM diagnosis • Power = 90% • Shows impact of small absolute difference

Researchers plan a 5-year prospective cohort study examining whether regular vigorous exercise (at least 150 minutes/week) reduces type 2 diabetes incidence in adults aged 40–65. The 5-year incidence in sedentary adults (unexposed) is estimated at 12% from national registry data. The research team considers a 4 percentage-point absolute reduction (from 12% to 8%) clinically meaningful. The study targets 90% power.

Given Parameters p₂ (sedentary, unexposed) = 0.12 (12%)
p₁ (active, exposed) = 0.08 (8%) → RR = 0.67 (33% risk reduction)
α = 0.05 (two-tailed) → zα/2 = 1.960
Power = 90% → zβ = 1.282
Attrition over 5 years: L = 15%
Variance Term p₁(1−p₁) = 0.08 × 0.92 = 0.0736
p₂(1−p₂) = 0.12 × 0.88 = 0.1056
Sum = 0.0736 + 0.1056 = 0.1792
Risk Difference Squared (p₁−p₂)² = (0.08−0.12)² = (−0.04)² = 0.0016
Sample Size Per Group n = (1.960 + 1.282)² × 0.1792 / 0.0016
n = (3.242)² × 0.1792 / 0.0016
n = 10.511 × 0.1792 / 0.0016
n = 1.884 / 0.0016 = 1,177.0 → 1,177 per group
Attrition Adjustment (L = 15%) nₙᵈᵍ = 1,177 / (1−0.15) = 1,177 / 0.85 = 1,385 per group
Total Sample Total = 1,385 × 2 = 2,770 participants

Why Is This So Large? The small absolute risk difference (8% vs 12% = only 4 percentage points) drives the (p₁−p₂)² denominator to 0.0016 — a tiny number that forces n into the thousands. Detecting a 33% relative risk reduction with 90% power requires a large study because even though the relative effect sounds substantial, the absolute numbers are small. If the team reduced the power target to 80% (zβ = 0.842), n drops to approximately 900 per group — still large, but 35% more feasible.

Practical implication: A study of this scale requires multi-site recruitment, grant funding for 5+ years, and a dedicated data management team. A pilot study of 200 participants could validate recruitment rates and attrition assumptions before committing to full-scale enrolment. Always run a feasibility assessment before finalising the sample size for a study of this magnitude.

Occupational Solvent Exposure and Hepatocellular Carcinoma — Unequal Groups (1:3 Ratio)

Retrospective cohort • Rare exposed group (factory workers) • 1:3 allocation • Rare outcome

An occupational health researcher uses a company employment database to assemble a retrospective cohort. Factory workers with documented solvent exposure (exposed group) are rare — only about 200 workers meet the exposure criteria from 20 years of records. The researcher plans to match them with unexposed administrative staff in a 1:3 ratio (exposed:unexposed). The 20-year cumulative incidence of hepatocellular carcinoma in occupationally unexposed adults is approximately 0.8%. Expected RR with chronic solvent exposure is 4.0. α = 0.05, power = 80%.

Given Parameters p₂ (unexposed staff) = 0.008 (0.8%)
p₁ (exposed workers) = 0.008 × 4.0 = 0.032 (3.2%)
r = 3 (three unexposed per one exposed)
α = 0.05 → zα/2 = 1.960 Power = 80% → zβ = 0.842
Retrospective design: L = 5% (incomplete records)
Unequal Ratio Formula for n₁ (exposed group) n₁ = (zα/2 + zβ)² × [p₁(1−p₁) + p₂(1−p₂)/r] / (p₁−p₂)²
p₁(1−p₁) = 0.032 × 0.968 = 0.030976
p₂(1−p₂)/r = (0.008 × 0.992)/3 = 0.007936/3 = 0.002645
Sum = 0.030976 + 0.002645 = 0.033621
(p₁−p₂)² = (0.032−0.008)² = (0.024)² = 0.000576
n₁ = (2.802)² × 0.033621 / 0.000576
n₁ = 7.851 × 0.033621 / 0.000576 = 0.2640 / 0.000576 = 458.3 → 459 exposed
Unexposed Group n₂ = r × n₁ = 3 × 459 = 1,377 unexposed
Attrition Adjustment (5%) Exposed: 459 / 0.95 = 484 (but only ~200 available!)
Unexposed: 1,377 / 0.95 = 1,449 unexposed

The Feasibility Problem The formula requires 484 exposed workers but only 200 exist in the records. This reveals a fundamental feasibility barrier. The researcher has three options: (1) calculate the minimum detectable RR with n₁ = 200, accepting reduced power; (2) extend the exposure definition to include related solvents and expand the eligible factory cohort; (3) increase the unexposed:exposed ratio beyond 3:1, though efficiency gains diminish past 1:4. Using n₁ = 200, the study will have only ~55% power — insufficient for a primary confirmatory study but potentially sufficient for a hypothesis-generating analysis.

Lesson: Always check feasibility before finalising parameters. If your formula returns a number larger than your available population, redesign the study (lower power, redefine exposure, extend timeframe, expand case ascertainment) rather than proceeding with a fatally underpowered study.

Breastfeeding Duration and Childhood Asthma — Sensitivity Analysis on Power

Prospective cohort • 5-year child follow-up • Shows how power choice changes total N

A paediatrician studies whether exclusive breastfeeding for at least 6 months (exposed) versus formula feeding (unexposed) reduces childhood asthma diagnosis by age 5. The 5-year asthma incidence in formula-fed children is estimated at 22%. An absolute 6 percentage-point reduction to 16% in breastfed children is considered clinically important. The research team wants to compare sample sizes for 80%, 85%, and 90% power at α = 0.05 with L = 15% attrition.

Fixed Parameters p₂ (formula-fed) = 0.22 p₁ (breastfed) = 0.16
Variance: p₁(1−p₁) + p₂(1−p₂) = (0.16×0.84) + (0.22×0.78) = 0.1344 + 0.1716 = 0.3060
(p₁−p₂)² = (0.16−0.22)² = (−0.06)² = 0.0036
Ratio: 0.3060 / 0.0036 = 85.0
80% Power (zβ = 0.842) n = (1.960 + 0.842)² × 85.0 = (2.802)² × 85.0 = 7.851 × 85.0 = 667.3 → 668
Adjusted: 668 / 0.85 = 787 per group → 1,574 total
85% Power (zβ = 1.036) n = (1.960 + 1.036)² × 85.0 = (2.996)² × 85.0 = 8.976 × 85.0 = 763.0 → 763
Adjusted: 763 / 0.85 = 898 per group → 1,796 total
90% Power (zβ = 1.282) n = (1.960 + 1.282)² × 85.0 = (3.242)² × 85.0 = 10.511 × 85.0 = 893.4 → 894
Adjusted: 894 / 0.85 = 1,052 per group → 2,104 total

Power comparison: Moving from 80% to 85% power adds 222 participants (14% more). Moving from 80% to 90% adds 530 participants (34% more). The decision between these power targets is a trade-off between resource cost and the risk of a false negative. For a primary study seeking to establish this association for the first time, 90% power is defensible. For a confirmatory study where prior evidence is already strong, 80% may suffice.

The Attrition Correction: How Loss to Follow-Up Inflates Sample Size

Loss to follow-up is one of the most underappreciated threats to cohort study validity and the most common reason prospective studies arrive at analysis with insufficient events to answer their question. The table below shows how different attrition levels inflate an example base sample size of 300 per group:

0% Loss

300

Calculated n

300

Adjusted n

No inflation

10% Loss

300

Calculated n

334

Adjusted n

+34 extra

20% Loss

300

Calculated n

375

Adjusted n

+75 extra

30% Loss

300

Calculated n

429

Adjusted n

+129 extra

Estimating Your Attrition Rate Base your attrition assumption on: (1) published dropout rates from similar cohort studies in the same population, (2) the study duration (longer = more dropout), (3) participant burden (frequent clinic visits = higher dropout), and (4) disease severity (seriously ill populations have higher mortality-related censoring). When in doubt, use a higher attrition estimate — over-recruiting by 5–10% is far less costly than finishing with inadequate data.

Impact of Confidence Level, Power, and Risk Difference

Understanding how each parameter drives sample size helps you make informed trade-offs when full-scale recruitment is constrained by budget, time, or available population.

Effect of α Level (Confidence Level)

Tightening the significance threshold from α = 0.05 to α = 0.01 increases the zα/2 term from 1.96 to 2.576. Since sample size scales with the square of (zα/2 + zβ), this increases sample size by roughly 49% at 80% power. A α = 0.01 threshold is warranted when a false positive would have serious downstream consequences — for example, incorrectly declaring a substance carcinogenic and triggering costly regulatory action.

Effect of Power Level

Every 5-percentage-point increase in power approximately adds 14–20% more participants (the exact amount varies with the other parameters). The minimum acceptable power is 80% for most peer-reviewed medical research. Ethics committees will question studies powered below this threshold. A power of 70% or below is generally considered inadequately powered and unlikely to be published if the primary endpoint is not statistically significant.

Effect of Risk Difference

Because risk difference appears squared in the denominator, it has a disproportionately large effect on sample size. Halving the expected risk difference quadruples the required sample size. This is why large, expensive cohort studies investigating small exposures — such as the effect of low-dose environmental pollutants on cardiovascular outcomes — often require tens of thousands of participants and multi-decade follow-up periods.

Common Sample Size Calculation Mistakes

Mistake 1: Using prevalence instead of cumulative incidence for p₂

Cohort studies track new cases over time, so the correct measure is the cumulative incidence (risk) over the follow-up period — the proportion of disease-free participants at baseline who develop the outcome by end of follow-up. Plugging in the overall disease prevalence (which includes existing cases) inflates p₂ and artificially deflates the required sample size.

✓ Fix: Use incidence data specific to your follow-up duration from prospective cohort studies or disease surveillance reports, not cross-sectional prevalence data.

Mistake 2: Ignoring loss to follow-up in prospective studies

Researchers frequently calculate n from the formula and directly report that number as their recruitment target, without inflating for expected dropout. A prospective study enrolling exactly the formula-calculated n will almost certainly finish underpowered because some fraction of participants will not complete follow-up.

✓ Fix: Always apply the attrition correction nₙᵈᵍ = n / (1−L) for prospective studies. For studies longer than 5 years, consider a conservative L of 25–30% unless very strong reasons exist for a lower estimate.

Mistake 3: Over-optimistic expected risk difference

Researchers sometimes choose p₁ and p₂ that produce a conveniently small and feasible n, rather than values grounded in realistic epidemiological expectations. If the true effect is smaller than assumed, the study will be underpowered for the actual difference and may miss it entirely.

✓ Fix: Use the minimum clinically important difference — the smallest effect that would change clinical practice — as your expected difference, not the most optimistic estimate from a small pilot study. Pilot studies often overestimate effects due to selection bias and random high estimates.

Mistake 4: Forgetting that total sample size = 2n, not n

The formula gives n per group. A surprisingly common protocol error states the total sample as n rather than 2n, effectively cutting the study to half the required power without realising it. This halves the number of participants in each arm and substantially reduces power.

✓ Fix: Always explicitly state in the protocol: "n = [value] per group; total sample = [2×value]." Document the per-group and total figures separately.

Mistake 5: Not conducting sensitivity analyses

A single point estimate of sample size based on one set of assumed parameters creates a brittle calculation. If any assumption is wrong — and at least some usually are — the planned sample size may be inappropriate. Ethics committees and grant reviewers increasingly expect authors to show how sample size changes across a plausible range of assumptions.

✓ Fix: Run the formula across a 2×2 grid of plausible p₂ values (low / high) and RR values (conservative / optimistic). Present all four resulting sample sizes and explain which assumption set you selected and why.

Mistake 6: Using one-tailed instead of two-tailed tests without justification

One-tailed tests require fewer participants than two-tailed tests for the same power, because they only look for an effect in one direction. This is tempting when recruits are scarce, but using a one-tailed test is only valid when there is a strong prior reason to believe the exposure can only increase (or only decrease) the outcome. In most epidemiological situations this cannot be justified, and using one-tailed p-values is likely to trigger reviewer rejection.

✓ Fix: Default to two-tailed tests (zα/2 = 1.96 for α = 0.05). If you believe a one-tailed test is genuinely justified, state the specific mechanistic reason in the protocol and acknowledge the limitation in the discussion.

Research Reporting Examples

Journal methods sections should include all the information a reader needs to independently verify the sample size calculation. The following examples show compliant and non-compliant reporting formats.

✓ Complete Reporting — STROBE-Compliant

"We estimated sample size based on a 10-year cumulative COPD incidence of 6% in non-smokers (derived from the Global Burden of Disease 2020 cohort estimates for our target age group) and an expected relative risk of 3.0 in smokers, yielding an expected incidence of 18% in the exposed group. Using the two-proportion z-test formula with a two-tailed significance level of α = 0.05 and 80% power (zβ = 0.842), we calculated a required sample size of 112 participants per group. To account for an estimated 20% loss to follow-up over 10 years, we inflated this to 140 per group, for a total enrolment target of 280 participants. A sensitivity analysis showed that if the unexposed group incidence were as high as 9%, the required n per group drops to 68, and if as low as 4%, it rises to 178; in both cases our target enrolment of 140 per group maintains at least 80% power."

✕ Incomplete Reporting — Common Protocol Weakness

"Based on previous literature, we determined that 280 participants would be sufficient for the study at the 5% level of significance and 80% power. Loss to follow-up was not expected to significantly affect the results."

✓ Reporting for Unequal Allocation (1:3 ratio)

"Because occupationally exposed workers (n = 200 maximum available) were substantially outnumbered by eligible unexposed controls, we adopted an unequal allocation ratio of 1:3 (exposed:unexposed). Using the modified two-proportion formula for unequal groups, with p₁ = 0.032, p₂ = 0.008, r = 3, α = 0.05 (two-tailed), and 80% power, we calculated a required sample of 459 exposed workers. Since only 200 exposed workers meet eligibility criteria, we present a sensitivity analysis indicating that n₁ = 200 with n₂ = 600 unexposed controls provides approximately 55% power — sufficient for this hypothesis-generating study but not for confirmatory inference. Full confirmation would require a multi-site collaboration to access the required 459 exposed workers."

✓ Reporting for Sensitivity Analysis on Power

"We calculated sample sizes for three power levels: 80% (n = 787 per group; total 1,574), 85% (n = 898 per group; total 1,796), and 90% (n = 1,052 per group; total 2,104), all with α = 0.05, expected asthma incidence 22% in formula-fed and 16% in breastfed children, and 15% attrition adjustment. Given budget constraints permitting enrolment of approximately 1,800 participants, we selected 85% power as the primary target (n = 898 per group), with pre-specified recruitment extension to 2,104 participants if interim feasibility review after 12 months confirms sufficient enrolment rates."

Frequently Asked Questions

What is the formula for sample size in a cohort study? +

The standard formula is: n = (zα/2 + zβ)² × [p₁(1−p₁) + p₂(1−p₂)] / (p₁−p₂)², where p₁ is the event rate in the exposed group, p₂ is the event rate in the unexposed group, zα/2 = 1.96 (for α = 0.05), and zβ = 0.842 (for 80% power). This gives the per-group sample size. Multiply by 2 for the total (1:1 allocation). For prospective studies, apply the attrition correction: nₙᵈᵍ = n / (1−L), where L is the expected proportion lost to follow-up.

What is a cohort study in medical research? +

A cohort study assembles a group of people defined by their exposure status at baseline and follows them over time to observe who develops the outcome of interest. Unlike case-control studies, both exposed and unexposed groups are followed prospectively (or reconstructed retrospectively from records), allowing direct calculation of incidence rates and relative risk. Famous examples include the Framingham Heart Study (cardiovascular risk factors), the Nurses' Health Study (diet and chronic disease), and the UK Biobank (genetic and environmental health determinants).

How does loss to follow-up affect cohort study sample size? +

Every participant lost to follow-up reduces the effective analysable sample below what was planned, which reduces statistical power. To maintain target power despite dropout, inflate the initial recruitment target by dividing the formula-derived n by (1 minus the expected attrition proportion). For a 20% expected dropout: enrol n/0.80. For a 30% dropout: enrol n/0.70. Always ground your attrition estimate in similar published cohorts — an unrealistically optimistic attrition assumption is one of the most common sample size errors in cohort study protocols.

What is the difference between prospective and retrospective cohort studies? +

In a prospective cohort study, participants are enrolled at study start and followed forward in time; data are collected as events happen. In a retrospective cohort study, both the exposure and the outcome have already occurred; the researcher uses historical records to reconstruct who was exposed and who developed the outcome. Both use the same sample size formula but differ in feasibility, cost, recall bias risk, and data quality. Prospective studies are slower and more expensive but provide better-quality exposure data; retrospective studies are faster but depend entirely on record completeness.

What values of p₁ and p₂ should I use if I do not have prior data? +

When prior literature is sparse: (1) check national disease registries and surveillance systems for your country for background incidence data; (2) search for systematic reviews or meta-analyses of similar exposures in comparable populations; (3) if no data exist at all, define the clinically meaningful risk difference you want to be able to detect and back-calculate p₁ from p₂ + minimum detectable difference. As a last resort, use p₂ = 0.10–0.20 with an RR of 2.0–3.0 as a conservative starting point, then conduct a sensitivity analysis across a plausible range. Document all assumptions and their sources transparently.

How does statistical power affect cohort study sample size? +

Power = probability of detecting the true effect. The combined z-score term (zα/2 + zβ)² in the numerator grows with increasing power, proportionally increasing n. Going from 80% to 90% power increases the (zα/2 + zβ)² term from 7.85 to 10.51 — a 34% increase that carries through to n. Most medical journals and ethics committees accept 80% power as the minimum. For primary regulatory submissions, pivotal RCTs, or studies where a false negative would lead to major policy error, 90% or 95% power may be required.

What happens to sample size when the expected risk difference is small? +

Sample size is inversely proportional to the square of the risk difference. Halving the expected absolute risk difference from 10% to 5% quadruples the required n. A risk difference of 2% may require sample sizes in the tens of thousands to detect at 80% power. This is why large cohort studies of modest environmental exposures — where the true RR may be 1.1–1.3 and baseline risks are low — need enormous enrolled populations such as the UK Biobank (500,000 participants) or the US Million Veteran Program.

Can I use a 1:2 or 1:3 exposed to unexposed ratio in a cohort study? +

Yes — and this is often necessary when the exposed group is rare or hard to recruit. The formula adjusts to: n₁ = (zα/2 + zβ)² × [p₁(1−p₁) + p₂(1−p₂)/r] / (p₁−p₂)² for exposed participants, and n₂ = r × n₁ for unexposed. Going from 1:1 to 1:2 allocation meaningfully reduces n₁ (the harder-to-recruit group) by about 25%, though total sample increases slightly. Beyond 1:4, efficiency gains plateau — a 1:10 ratio does not meaningfully reduce n₁ compared with 1:4.

How do I report sample size justification for a cohort study? +

Your methods section should state: (1) the formula or software used; (2) the source and value of p₂; (3) the source and value of p₁ (or the minimum detectable RR); (4) α and whether one- or two-tailed; (5) the target power; (6) the calculated per-group and total n; (7) the assumed attrition rate and adjusted recruitment target; and ideally (8) a brief sensitivity analysis across plausible parameter values. The STROBE reporting guidelines for observational studies include sample size justification as a mandatory item (Item 10).

What is the minimum detectable relative risk in a cohort study? +

The minimum detectable relative risk (MDRR) is the smallest true RR your fixed sample size has adequate power to detect. It is calculated by rearranging the formula to solve for p₁ given a fixed n, then converting to RR = p₁/p₂. MDRR is particularly useful for retrospective cohorts or secondary analyses of existing datasets where n is fixed and you want to know what effect size you are realistically powered to detect, rather than starting from a desired effect and calculating n. OpenEpi (openepi.com) provides a free MDRR calculator for cohort studies.

Calculate your cohort study sample size instantly

StatClinic's free sample size calculator handles two-proportion cohort studies with attrition correction, unequal group ratios, and sensitivity analysis — with plain-English output ready for your thesis or grant application.

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z (α=0.05, 2-tail)	1.960
z (α=0.01, 2-tail)	2.576
zβ (80% power)	0.842
zβ (85% power)	1.036
zβ (90% power)	1.282
zβ (95% power)	1.645
Total sample	2 × n per group
Attrition adj.	n ÷ (1−L)