Clinical Trial Safety and Dose Finding

Adverse Event Detection Tables: Probability, Sample Size and the Rule of Three

How likely a trial is to see a rare adverse event, how many patients it needs, and what zero events really rules out.

Incidence 0.01% to 20%
n = 10 to 10,000
Rule of three

Adverse Event Detection

Three tables: probability of at least one event, patients needed, and the upper confidence limit when no events are seen.

Probability of observing at least one event

Probability of observing at least one event
True incidence of the event → Patients (n) ↓	0.05%	0.1%	0.2%	0.5%	1%	2%	5%	10%
10	0.0050	0.0100	0.0198	0.0489	0.0956	0.1829	0.4013	0.6513
20	0.0100	0.0198	0.0392	0.0954	0.1821	0.3324	0.6415	0.8784
30	0.0149	0.0296	0.0583	0.1396	0.2603	0.4545	0.7854	0.9576
40	0.0198	0.0392	0.0770	0.1817	0.3310	0.5543	0.8715	0.9852
50	0.0247	0.0488	0.0953	0.2217	0.3950	0.6358	0.9231	0.9948
60	0.0296	0.0583	0.1132	0.2597	0.4528	0.7024	0.9539	0.9982
75	0.0368	0.0723	0.1394	0.3134	0.5294	0.7802	0.9787	0.9996
100	0.0488	0.0952	0.1814	0.3942	0.6340	0.8674	0.9941	1−
150	0.0723	0.1394	0.2594	0.5285	0.7785	0.9517	0.9995	1−
200	0.0952	0.1814	0.3299	0.6330	0.8660	0.9824	1−	1−
250	0.1175	0.2213	0.3938	0.7144	0.9189	0.9936	1−	1−
300	0.1393	0.2593	0.4515	0.7777	0.9510	0.9977	1−	1−
400	0.1813	0.3298	0.5510	0.8653	0.9820	0.9997	1−	1.0000
500	0.2212	0.3936	0.6325	0.9184	0.9934	1−	1−	1.0000
750	0.3128	0.5278	0.7772	0.9767	0.9995	1−	1.0000	1.0000
1,000	0.3935	0.6323	0.8649	0.9933	1−	1−	1.0000	1.0000
1,500	0.5277	0.7770	0.9504	0.9995	1−	1−	1.0000	1.0000
2,000	0.6322	0.8648	0.9818	1−	1−	1.0000	1.0000	1.0000
3,000	0.7770	0.9503	0.9975	1−	1−	1.0000	1.0000	1.0000
5,000	0.9180	0.9933	1−	1−	1.0000	1.0000	1.0000	1.0000
10,000	0.9933	1−	1−	1.0000	1.0000	1.0000	1.0000	1.0000

Patients needed to observe at least one event

Patients needed to observe at least one event
Probability of seeing ≥ 1 event → Incidence ↓	50%	80%	90%	95%	99%
0.01%	6,932	16,094	23,025	29,956	46,050
0.02%	3,466	8,047	11,512	14,978	23,024
0.05%	1,386	3,219	4,605	5,990	9,209
0.1%	693	1,609	2,302	2,995	4,603
0.2%	347	804	1,151	1,497	2,301
0.5%	139	322	460	598	919
1%	69	161	230	299	459
2%	35	80	114	149	228
5%	14	32	45	59	90
10%	7	16	22	29	44
20%	4	8	11	14	21

Upper confidence limit for the incidence when no events are observed

Upper confidence limit for the incidence when no events are observed
One-sided confidence level → Patients (n) ↓	90%	95%	99%	Rule of three (3/n)
10	20.57%	25.89%	36.90%	30.00%
20	10.87%	13.91%	20.57%	15.00%
30	7.39%	9.50%	14.23%	10.00%
40	5.59%	7.22%	10.87%	7.50%
50	4.50%	5.82%	8.80%	6.00%
60	3.76%	4.87%	7.39%	5.00%
75	3.02%	3.92%	5.96%	4.00%
100	2.28%	2.95%	4.50%	3.00%
150	1.52%	1.98%	3.02%	2.00%
200	1.14%	1.49%	2.28%	1.50%
250	0.917%	1.19%	1.83%	1.20%
300	0.765%	0.994%	1.52%	1.00%
400	0.574%	0.746%	1.14%	0.75%
500	0.459%	0.597%	0.917%	0.6%
750	0.307%	0.399%	0.612%	0.4%
1,000	0.23%	0.299%	0.459%	0.3%
1,500	0.153%	0.2%	0.307%	0.2%
2,000	0.115%	0.15%	0.23%	0.15%
3,000	0.0767%	0.0998%	0.153%	0.1%
5,000	0.046%	0.0599%	0.0921%	0.06%
10,000	0.023%	0.03%	0.046%	0.03%

Tip: click any value to highlight its row and column.

How to read this table

Probability table: find the row for the number of patients and the column for the true incidence.
Patients needed: find the row for the incidence and the column for how sure you want to be of seeing at least one event.
No events seen: the upper confidence limit is the highest incidence still consistent with zero events.

Worked example

A safety database of 300 patients has a 95% chance of containing at least one case of an adverse event with a true incidence of 1%. To be 95% sure of seeing at least one case, about 299 patients are needed.

If none of the 300 patients has the event, the one-sided 95% upper limit for its incidence is 0.994%, close to the rule-of-three value 3/300 = 1%. This is why ICH E1 asks for 300 to 600 patients treated for 6 months: enough to see events with an incidence of about 1%.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

P(≥ 1 event) = 1 − (1 − p)ⁿ · n = ⌈ln(1 − P) / ln(1 − p)⌉ · upper limit (0 events) = 1 − α^(1/n) ≈ 3/n

Probabilities are 1 − (1 − p)ⁿ, sample sizes the smallest n with 1 − (1 − p)ⁿ ≥ P, and upper limits the exact (Clopper-Pearson) one-sided bound 1 − (1 − γ)^(1/n) for zero events, all assuming independent patients with a common incidence.

Frequently asked questions

What is the rule of three?

If no events occur in n patients, 3/n is an approximate 95% upper confidence limit for the event rate. It is accurate for n above about 30.

Does this apply to events that need time to develop?

Only if every patient is followed long enough for the event to occur. For time-dependent risks, use exposure-adjusted rates and person-time instead.

Adverse Event Detection Tables: Probability, Sample Size and the Rule of Three

Adverse Event Detection

Probability of observing at least one event

Patients needed to observe at least one event

Upper confidence limit for the incidence when no events are observed

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

Related calculators and tutorials