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Letter

0021-9681.82 010073.05103 0080 0 1982 Pergamon Press Ltd

to the Editor

DETERMINING THE REQUIRED ACCRUAL RATE FOR FIXED-DURATION CLINICAL TRIALS

(Rewired

in remedform

THE PROBLEM

10 Norrmher 1980)

of sample-size determination for clinical trials and epidemiological studies continues to attract considerable attention, as evidenced by a row of publications in this journal during the past decade. Sample-size determination in the situation of gradual accrual has been viewed from two different angles: determining the required duration of the trial given a certain accrual rate or determining the necessary accrual rate given a fixed duration. Although the former approach is more applicable in many cases, when the number of patients available is quite limited, the latter approach seems more natural in others. This is particularly true when the intention of the trial is to obtain a L-year survival rate after (L + 1) years of study. In view of the continued popularity of 5-yr survival rates. especially in the evaluation of cancer survival, this situation is of considerable importance. The duration problem has been treated by Pasternack & Gilbert [l] and by George & Desu [2], whereas the latter problem of fixed duration has been investigated by Pasternack [3]. The approach of Pasternack utilizes life-table analysis, and assumes constant accrual and a constant force of mortality from interval to interval. It is attractive in that the method easily can be generalized to incorporate other assumptions, and does not depend on a particular underlying parametric model. The purpose of this letter is to point out and briefly investigate certain concerns arising from the tables given in [3]. (a) It is claimed (footnote to Table 3) that the detection of an increase in the 5-yr survival rate from P to (P + S) at a given significance level with a certain power, would require the same accrual rate as an increase from (1 - P - 6) to (1 - P). Tables are therefore given only for control group survival less than or equal to 0.50. Whereas such symmetry certainly holds for a trial where each patient is followed for 5 yr, it is not the case here. This is due to the fact that more patients are available for estimation in later life table intervals in cases where the survival rates are higher. The required accrual rates given in the tables are therefore generally too conservative when survival in the control group is .50”)$or higher. For example, in the sample calculation (Table 3), where control group survival is 0.80, c( = 0.05, fi = 0.20, 6 = 0.10 and 0.15, the accrual rates should be 69 and 22fyear rather than 95 and 36/year. New Tables (Tables 1 and 2) have been generated using the formula given in the appendix of Pasternack [3]. These new tables again assume 5 yr of patient accrual with one additional year of follow-up. They differ from Pasternack’s tables only where at least one of the survival probabilities exceeds 0.50 (one or two other entries differ by 1. probably due to computer roundoff). Quite large differences may be noted in certain cases. For example, in order to detect an increase from 0.05 to 0.10 survival with 80”” power at a significance level of 0.05 would require 222 patients/year, whereas an increase from 0.90 to 0.95 survival would require only 148 patients/year. (b) Many of the accrual rates given in the tables are quite small. When the 5-yr survival rate in the control group is low, very few or no patients may be left in the later 73

Letter to the Editor

TABLE 1. REQUIRED SIZE OF ANNUAL COHORT ENTERED PER GROUP (CONTROL AND EXPERIMENTAL) TO DETERMINE A STATISTICALLY SIGNIFICANT DIFFERENCE IN ~-YEAR SURVIVAL RATES (ONE-TAILED TEST)(The upper. middle and lower figures refer to the cohort

size required

for a Type

I error

= 0.05 and Type respectively)

Anticipated Anticipated _5-yr survival rate of the control group

(‘:“I

5

10

5

222 307 388 333 461 582 422 584 738 493 683 862 549 759 959 589 816 1031 617 854 1079 631 874 1104 633 877 1108 623 863 1091 602 834 1054 570 789 997 526 729 921 472 654 827 408 565 714 333 461 583 247 342 433 148 207 263

69 95 120 95 131 165 115 159 200 131 181 228 143 197 249 151 209 264 156 216 273 158 219 277 157 218 275 153 212 268 146 203 256 137 189 239 124 172 217 109 151 191 91 126 159 69 97 123 44 62 79

10

15

20

25

30

35

40

45

50

55

60

65

70

15

80

85

90

II errors

of 0.20, 0.10 and

increase in the 5-yr survival of the experimental group (Do)

0.05,

rate

15

20

25

30

35

40

36 50 63 47 64 81 55 76 95 61 84 106 66 91 114 69 95 119 70 97 122 70 97 122 69 95 120 66 92 116 63 86 109 57 79 100 51 71 89 43 60 76 34 47 60 22 31 40

23 32 40 29 39 49 33 45 57 36 49 62 38 52 66 39 54 68 40 54 69 39 54 68 38 52 66 36 50 63 33 46 58 30 41 52 26 35 45 20 28 36 13 19 25

17 23 28 20 27 34 22 30 38 24 32 41 25 34 43 25 35 43 25 35 43 25 34 42 24 32 41 22 30 38 20 27 35 17 24 30 14 19 24 9 13 17

13 17 21 15 20 25 16 22 27 17 23 29 17 24 30 18 24 30 17 24 30 17 23 29 16 21 27 14 20 25 12 17 22 10 14 18 7 10 13

10 14 17 11 15 19 12 16 20 13 17 21 13 17 22 13 17 22 12 17 21 12 16 20 11 15 18 9 13 16 8 11 13

8 11 13 9 12 15 10 13 16 10 13 16 10 13 16 10 13 16 9 12 15 8 11 14 7 10 13 6 8 11 4 6 8

8’ 10

Letter to the Editor

75

TABLE 2. REQUIRED SIZE OF ANNUAL COHORT ENTERED PER GROUP (CONTROL AND EXPERIMENTAL) TO DETERMINE A STATISTICALLY SIGNIFICANT DIFFERENCE IN 5-YEAR SURVIVAL RATES (ONE-TAILED TEST)(The upper, middle and lower figures refer to the cohort

size required

for a Type

I error

=O.Ol and Type respectively)

Anticipated Anticipated 5-yr survival rate of the control group

11 errors

of 0.20, 0.10 and 0.05,

increase in the 5-yr survival of the experimental group (%)

rate

(‘I”)

5

10

15

20

25

30

35

40

5

360 466 565 540 700 848 685 888 1075 801 1038 1257 891 1154 1398 957 1240 1502 1001 1298 1572 1024 1328 1609 1028 1333 1614 1012 1312 1589 977 1267 1535 925 1199 1453 854 1108 1342 166 994 1205 661 858 1040 539 700 849 399 519 630 238 311 379

112 I45 175 154 199 241 187 242 292 212 275 333 232 300 363 246 318 385 254 329 398 257 333 403 255 331 400 249 322 390 237 308 373 221 287 348 201 261 316 176 229 277 146 190 231 111 145 177 69 92 113

59 76 92 76 98 118 89 115 139 99 128 I55 107 138 166 111 144 174 114 147 178 114 147 178 II2 145 I75 108 139 169 IO1 I31 159 93 120 146 82 107 129 69 90 110 54 70 86 34 46 56

38 49 59 47 60 72 53 69 83 58 75 90 62 79 96 64 82 99 64 83 100 63 82 99 61 19 96 58 75 91 54 70 84 48 62 76 41 53 65 32 42 51 21 28 35

27 35 42 32 41 49 36 46 55 39 49 59 40 52 62 41 53 63 41 53 63 40 51 62 38 49 59 35 46 55 32 41 50 27 35 43 22 28 35 14 I9 24

21 26 31 24 30 36 26 33 40 27 35 42 28 36 43 28 36 44 28 36 43 27 34 41 25 32 39 23 29 35 20 25 31 16 20 25 10 14 I7

16 21 25 18 23 28 20 25 30 21 26 31 21 26 32 21 26 32 20 25 31 19 24 29 17 22 27 15 I9 23 I2 16 19 8 I1 13

13 17 20 15 19 22 15 20 23 16 20 24 16 20 24 15 20 23 14 I9 22 13 I7 21 12 15 18 9 12 I5 6 9 11

10

15

20

25

30

35

40

45

50

55

60

65

70

15

80

85

90

16

Letter to the Editor

life table intervals. In particular, the survival in year 5 is based on only one cohort (the one entered during the first study year). If this cohort dies before 5 yr of follow-up the 5-yr survival rate and/or its variance cannot be estimated. The probability of this occurring in the control group is (1 - P,)” where P, is the true 5-yr survival rate and n is the number of patients entered into the control group in the first study year (n = accrual rate). Using entries from Table 1 for power = 0.80 and significance level = 0.05, this probability is greater than or equal to 5”” for: P, = 0.05 with 6 > 0.15; P, = 0.10 with 6 3 0.20 6 3 0.35. For P, = 0.05 with fi = 0.15 the probability estimable or 0 survival is 31”,,. In trials with very low 5-yr survival probabilities it increase either accrual rates or the duration of the study. another possibility is to increase the number of patients than adhering to a constant accrual rate. (c) The small sample sizes also raise concerns with significance level and power. This is due to two reasons:

and P, = 0.20 with of obtaining a not

may therefore be necessary to If enough patients are available, entered in the first year, rather regard

to potential

effects

on

1. The sample size formula is based on an asymptotic variance and an asymptotically standard normal test statistic. Small sample sizes may affect both significance level and power. 2. The sample size formula is based on the expected number of patients entering each life table interval. Due to the variability in the number of patients surviving preceding life table intervals, the actual number observed to enter intervals 225 at the analysis stage will vary. This may affect power, but not significance level, since the test performed at the analysis stage is conditional on the observed number entering each interval. In order to clarify possible effects on significance level and power, a small-scale Monte Carlo study was undertaken for presumed power = 0.80 and significance level = 0.05. Control group survival probabilities were taken as P, = 0.10, 0.20.. . 0.90 and the treatment difference as (5 = 0.05,0.10, 0.15, 0.20.. ,0.35. The corresponding sample sizes were as in Table 1. Cases of greater than or equal to 5”,, probability of obtaining 0 survival in the control group were excluded, as were accrual rates greater than or equal to 200, year. The following table gives estimated power and significance level by accrual rate with 95”, confidence intervals. The results are grouped for greater precision. The number of simulation runs refers to the number under each of alternative and null hypotheses.

Accrual rate (n year) n < 20 < n 30 < n n 2

20 < 30 < 40 40

Significance level (7;) 5.7 5.7 5.3 5.0

+ * + *

0.6 0.5 0.6 0.7

Number simulation

Power (“J 80.6 80.0 81.2 80.2

+ k & +

I.0 0.9 1.1 1.3

of runs

5900 7100 5000 3700

Although this table is not conclusive because of too few simulation runs and grouping of the accrual rates, it indicates that power is satisfactory and probably slightly higher than required. The significance level seems to be between 0.05 and 0.06. The results, therefore. are satisfactory for the cases investigated with some caution in order for n < 30. In summary, it is recommended that the tables given in [3] not be used for control group survival probabilities greater than or equal to. 0.60, and that some caution be taken for accrual rates less than 30 per year. In particular, alternative statistical techniques, such as rank tests or parametric analyses, may be contemplated. and /or the size

71

Letter to the Editor

of the trial increased rates.

in order

to avoid 0 survival,

in the case of very low 5yr

MARI

University

survival

PALTA

of Iowa

REFERENCES 1. 2. 3.

Pasternack BS, Gilbert HS: Planning the duration of long-term survival time studies designed for accrual by cohorts. J Chron Dis 24: 681-700, 1971 George SL, Dew MM: Planning the size and duration of a clinical trial studying the time to some critical event. J Chron Dis 27: 15-24, 1974 Pasternack BS: Sample sizes for clinical trials designed for patient accrual by cohorts. J Chron Dis 25: 6733681. 1972