Studies to detect rare events are usually carried out after the introduction of a new drug or treatment into general use, to exclude the occurrence of unusual adverse events. These are sometimes called post marketing studies, or Phase IV Trials.

The idea is to require all users to report side effects or complications, so that rare adverse outcomes are recorded and dealt with.

The sample size represents a count of the number of negative cases that accompany a nominated number of positive cases, assuming the data has a negative binomial distribution, which is very similar to the Poisson distribution. It is calculated with the following parameters.

• Power (1-β), where β is the probability of Type II Error, so that power is the likelihood that the rare event will be detected if it exists. The table in the next panel provides sample size for powers of 0.8, 0.9, 0.95, and 0.99. Although power=0.08 is commonly used in clinical research, the importance of detecting unusual and unexpected adverse outcome is such that the power used is commonly set at 0.95, or even 0.99 for those adverse outcomes that are life threatening.
• The Mean Event Rate λ, (number of events / number of observations). This is the critical level of probability, above which is considered unacceptable and requiring remedy. The table in the next panel provides sample size for λ=0.0001 (1 in 10,000) to λ=0.01 (1%) with 0.0001 intervals.
• The Critical Event Limit r is the pre-determined number of events that, if occurred within the sample size observed, will led to the conclusion that λ is exceeded. Although r is usually set to 1, the table in the next panel provides for up to r=3.
• The algorithm calculates sample size required using iterative approximation until all the parameters are satisfied. Clinically, the sample size is usually rounded upwards to the next thousand.

An example : Using the very first row of the exmple data :

• If we wish to detect an event as rare as 1 in 10,000 (λ=0.0001)
• and if the power of detection is set at 0.95 (95% sure we will detect it if it exists as frequently as 1 in 10,000)
• then we can conclude that the event is more frequent than 1 in 10,000 if we observe 1 incident in 30,000 cases (rounded up from 29958), or two incidents in 48,000 cases (rounded up from 47439), or 3 incidences in 63,000 cases (rounded up from 62960).

The sample size table is constructed using the algorithm described in the reference

Reference

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 144-145

	Power=0.8			Power=0.9			Power=0.95			Power=0.99				Power=0.8			Power=0.9			Power=0.95			Power=0.99
λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3
0.0001	16095	29943	42791	23026	38897	53225	29958	47439	62960	46052	66389	84062	0.0002	8048	14972	21396	11513	19450	26612	14979	23721	31479	23026	33187	42032
0.0003	5365	9982	14264	7676	12966	17741	9986	15813	20987	15351	22126	28022	0.0004	4024	7486	10698	5757	9725	13306	7490	11860	15740	11513	16595	21017
0.0005	3219	5989	8559	4606	7780	10645	5992	9489	12592	9211	13276	16814	0.0006	2683	4991	7132	3838	6483	8871	4993	7907	10493	7676	11064	14012
0.0007	2300	4278	6113	3290	5557	7604	4280	6778	8995	6579	9484	12007	0.0008	2012	3743	5349	2879	4863	6653	3745	5930	7871	5757	8299	10507
0.0009	1789	3328	4755	2559	4322	5914	3329	5271	6996	5117	7377	9340	0.0010	1610	2995	4280	2303	3890	5323	2996	4744	6296	4606	6639	8406
0.0011	1464	2723	3890	2094	3537	4839	2724	4313	5724	4187	6036	7642	0.0012	1342	2496	3566	1919	3242	4436	2497	3954	5247	3838	5533	7005
0.0013	1239	2304	3292	1772	2993	4094	2305	3650	4843	3543	5106	6467	0.0014	1150	2139	3057	1645	2779	3802	2140	3389	4498	3290	4742	6005
0.0015	1073	1997	2853	1536	2594	3549	1998	3163	4198	3071	4426	5605	0.0016	1006	1872	2675	1440	2432	3327	1873	2966	3935	2879	4149	5255
0.0017	947	1762	2518	1355	2289	3131	1763	2791	3704	2709	3906	4946	0.0018	895	1664	2378	1280	2161	2957	1665	2636	3498	2559	3689	4671
0.0019	848	1576	2253	1212	2048	2802	1577	2497	3314	2424	3495	4425	0.0020	805	1498	2140	1152	1945	2662	1498	2372	3148	2303	3319	4203
0.0021	767	1426	2038	1097	1853	2535	1427	2259	2999	2193	3161	4003	0.0022	732	1362	1946	1047	1769	2420	1362	2157	2862	2094	3018	3821
0.0023	700	1302	1861	1002	1692	2314	1303	2063	2738	2003	2887	3655	0.0024	671	1248	1783	960	1621	2218	1249	1977	2624	1919	2766	3503
0.0025	644	1198	1712	922	1556	2129	1199	1898	2519	1843	2656	3363	0.0026	620	1152	1646	886	1497	2048	1153	1825	2422	1772	2554	3234
0.0027	597	1110	1585	853	1441	1972	1110	1757	2332	1706	2459	3114	0.0028	575	1070	1529	823	1390	1901	1070	1695	2249	1645	2372	3003
0.0029	555	1033	1476	794	1342	1836	1034	1636	2172	1588	2290	2899	0.0030	537	999	1427	768	1297	1775	999	1582	2099	1536	2214	2803
0.0031	520	966	1381	743	1255	1717	967	1531	2031	1486	2142	2712	0.0032	503	936	1338	720	1216	1664	937	1483	1968	1440	2075	2628
0.0033	488	908	1297	698	1179	1613	908	1438	1908	1396	2012	2548	0.0034	474	881	1259	678	1145	1566	882	1396	1852	1355	1953	2473
0.0035	460	856	1223	658	1112	1521	856	1356	1799	1316	1897	2402	0.0036	448	832	1189	640	1081	1479	833	1318	1749	1280	1845	2335
0.0037	435	810	1157	623	1052	1439	810	1283	1702	1245	1794	2272	0.0038	424	788	1127	606	1024	1401	789	1249	1657	1212	1747	2213
0.0039	413	768	1098	591	998	1365	769	1217	1615	1181	1702	2156	0.0040	403	749	1070	576	973	1331	749	1186	1574	1152	1660	2102
	Power=0.8			Power=0.9			Power=0.95			Power=0.99				Power=0.8			Power=0.9			Power=0.95			Power=0.99
λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3
0.0041	393	731	1044	562	949	1299	731	1158	1536	1124	1620	2051	0.0042	384	713	1019	549	927	1268	714	1130	1500	1097	1581	2002
0.0043	375	697	996	536	905	1238	697	1104	1465	1071	1544	1956	0.0044	366	681	973	524	885	1210	681	1079	1431	1047	1509	1911
0.0045	358	666	951	512	865	1183	666	1055	1400	1024	1476	1869	0.0046	350	651	931	501	846	1158	652	1032	1369	1002	1444	1828
0.0047	343	638	911	490	828	1133	638	1010	1340	980	1413	1789	0.0048	336	624	892	480	811	1109	625	989	1312	960	1384	1752
0.0049	329	612	874	470	794	1087	612	969	1285	940	1355	1716	0.0050	322	599	856	461	778	1065	600	949	1260	922	1328	1682
0.0051	316	588	840	452	763	1044	588	931	1235	903	1302	1649	0.0052	310	576	823	443	749	1024	577	913	1211	886	1277	1617
0.0053	304	565	808	435	734	1005	566	896	1188	869	1253	1587	0.0054	299	555	793	427	721	986	555	879	1166	853	1230	1557
0.0055	293	545	779	419	708	968	545	863	1145	838	1207	1529	0.0056	288	535	765	412	695	951	535	848	1125	823	1186	1502
0.0057	283	526	751	404	683	934	526	833	1105	808	1165	1475	0.0058	278	517	738	397	671	918	517	818	1086	794	1145	1450
0.0059	273	508	726	391	660	903	508	805	1068	781	1126	1425	0.0060	269	500	714	384	649	888	500	791	1050	768	1107	1401
0.0061	264	491	702	378	638	873	492	778	1033	755	1089	1378	0.0062	260	483	691	372	628	859	484	766	1016	743	1071	1356
0.0063	256	476	680	366	618	845	476	754	1000	731	1054	1335	0.0064	252	468	669	360	608	832	469	742	984	720	1038	1314
0.0065	248	461	659	355	599	819	461	730	969	709	1022	1294	0.0066	244	454	649	349	590	807	454	719	954	698	1006	1274
0.0067	241	447	639	344	581	795	448	709	940	688	991	1255	0.0068	237	441	630	339	573	783	441	698	926	678	977	1237
0.0069	234	434	621	334	564	772	435	688	913	668	963	1219	0.0070	230	428	612	329	556	761	428	678	900	658	949	1201
0.0071	227	422	603	325	548	750	422	669	887	649	936	1185	0.0072	224	416	595	320	541	740	417	659	875	640	923	1168
0.0073	221	411	587	316	533	730	411	650	863	631	910	1152	0.0074	218	405	579	312	526	720	405	642	851	623	897	1136
0.0075	215	400	571	308	519	710	400	633	840	615	886	1121	0.0076	212	394	564	303	512	701	395	625	829	606	874	1106
0.0077	210	389	556	300	506	692	390	617	818	599	863	1092	0.0078	207	384	549	296	499	683	385	609	808	591	852	1078
0.0079	204	380	542	292	493	674	380	601	797	583	841	1065	0.0080	202	375	535	288	487	666	375	593	787	576	830	1051
	Power=0.8			Power=0.9			Power=0.95			Power=0.99				Power=0.8			Power=0.9			Power=0.95			Power=0.99
λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	λ	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3	r=1	r=2	r=3
0.0081	199	370	529	285	481	658	370	586	778	569	820	1038	0.0082	197	366	522	281	475	650	366	579	768	562	810	1026
0.0083	194	361	516	278	469	642	361	572	759	555	800	1013	0.0084	192	357	510	275	464	634	357	565	750	549	791	1001
0.0085	190	353	504	271	458	627	353	559	741	542	781	990	0.0086	188	349	498	268	453	619	349	552	733	536	772	978
0.0087	185	345	492	265	448	612	345	546	724	530	764	967	0.0088	183	341	487	262	443	605	341	540	716	524	755	956
0.0089	181	337	481	259	438	599	337	533	708	518	746	945	0.0090	179	333	476	256	433	592	333	528	700	512	738	934
0.0091	177	330	471	254	428	585	330	522	692	507	730	924	0.0092	175	326	466	251	423	579	326	516	685	501	722	914
0.0093	174	322	461	248	419	573	323	511	677	496	714	904	0.0094	172	319	456	245	414	567	319	505	670	490	707	895
0.0095	170	316	451	243	410	561	316	500	663	485	699	885	0.0096	168	312	446	240	406	555	313	495	656	480	692	876
0.0097	166	309	442	238	402	549	309	490	650	475	685	867	0.0098	165	306	437	235	397	544	306	485	643	470	678	858
0.0099	163	303	433	233	393	538	303	480	636	466	671	850	0.0100	161	300	428	231	389	533	300	475	630	461	664	841

Data Entry   - The data is in 3 columns separated by white spaces   - Each row is a sepaqrate test     - Column 1 is Power (1-β)     - Column 2 is the Expected Event Rate, the probability or proportion to detect     - Column 3 is the Critical tolerance limit, the number of positives within the sample size        for a decision that the Expected Event Rate is Exceeded

R codess for sample size to detect rare events

fuctions for calculation

FindB <- function(lambda,n,a)  # Poisson coefficient
{
  b = 0
  for(y in 1:a)
  {
    x = y - 1
    b = b + (n * lambda)^x * exp(-n * lambda) / factorial(x)  
  }
  return (b)
}
# interative approximation for the correct sample size nm
SSizAInc <- function(lambda,beta,r)  
{
  if(r==1)
  {
    return (-log(beta) / lambda)
  }
  nL = 2
  nR = r * -log(beta) / lambda;   
  nM = (nL + nR) / 2
  bL = FindB(lambda, nL, r) - beta
  bR = FindB(lambda, nR, r) - beta
  bM = FindB(lambda, nM, r) - beta
  if(bL*bR>0)
  {
    return (0)
  }
  if(bL*bM>0) { nL = nM }  else { nR = nM }
  j=0;
  while(j<100 & abs(bM)>0.00001)
  {
    nM = (nL + nR) / 2
    bL = FindB(lambda,nL,r) - beta
    bM = FindB(lambda,nM,r) - beta
    if(bL*bM>0) { nL = nM } else { nR = nM }
    j = j + 1
  }
  return (nM)
}

Main program

# Data entry as a table
myDat = ("
pow   lambda r      # power, expected proportion, critical tolerance limit
0.90  0.001  1
0.90  0.001  2
0.99  0.002  1
0.99  0.002  3
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
df                                           # display input data
# Calculations
ssiz <- vector()       # vector (array) to contain sample sizes 
for(i in 1:nrow(df))
{
  beta = 1 - df$pow[i]
  lambda = df$lambda[i]
  r = df$r[i]
  ssiz <- append(ssiz,ceiling(SSizAInc(lambda,beta,r)))
}

df$SSiz <- ssiz     # add sample size column to data frame
df                  # display input data + sample size calculated

The results are

> df                  # display input data
   pow lambda r
1 0.90  0.001 1
2 0.90  0.001 2
3 0.99  0.002 1
4 0.99  0.002 3

> df                 # display input data + sample size calculated
   pow lambda r SSiz
1 0.90  0.001 1 2303
2 0.90  0.001 2 3890
3 0.99  0.002 1 2303
4 0.99  0.002 3 4204