THE NUMBER OF CENTERS IN A MULTICENTER CLINICAL STUDY: EFFECTS ON STATISTICAL POWER

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1 Drug Information Journal, Vol. 34, pp , /2000 Printed in the USA. All rights reserved. Copyright 2000 Drug Information Association Inc. THE NUMBER OF CENTERS IN A MULTICENTER CLINICAL STUDY: EFFECTS ON STATISTICAL POWER ZHENGNING LIN, PHD Senior Biostatistician, Medical Information Processing and Statistics, Novartis Pharmaceuticals Corporation, East Hanover, New Jersey In the pharmaceutical industry it is often important to ensure fast patient enrollment. A common practice to speed up patient enrollment is to use more clinical centers. One of the concerns regarding this practice is that increasing the total number of centers to a certain degree may decrease the statistical efficiency of treatment comparisons. This is mainly because a study with too many clinical centers usually includes quite a few small centers and very often they do not have enough patients to represent all treatment groups. These small centers often carry little information on treatment differences. This paper utilizes a statistical model to quantify the relationship between statistical efficiency and the number of centers under typical clinical trial settings. Results in this paper provide useful statistical knowledge in choosing the number of clinical centers when planning multicenter studies. Key Words: Clinical trial; Multicenter; Number of clinical centers; Center size; Statistical power INTRODUCTION trial conduct at all study centers which may affect the study quality as more and more IN THE PHARMACEUTICAL industry it centers join the study. The approach in this is often important to ensure fast patient enpaper focuses on the following statistical perrollment. A common practice to speed up spective: How much, if any, statistical effipatient enrollment is to use more clinical cenciency is affected by the increase of study ters. One concern about this practice is centers, while the study conduct and the whether there is any statistical penalty assooverall quality of study centers remains unciated with the increase of the total number changed? I use a statistical model to quantify of study centers. What needs to be answered the relationship between the number of cenis how many study centers we can use in a ters and the statistical efficiency under typiclinical study without losing statistical effical clinical trial settings. Statistical efficiency ciency. Actually, there is no simple and genis discussed in terms of actual statistical eral answer that is appropriate for all studies, power. Results in this paper provide useful because it depends on many aspects of the statistical knowledge regarding how many centers we may use when planning a multicenter clinical study. For simplicity of expla- Reprint address: Zhengning Lin, Senior Biostatistician, nation, the discussion focuses on multicenter Medical Information Processing and Statistics, Novartis studies with the objective of comparing two Pharmaceuticals Corporation, 59 Route 10, East Hantreatments, while the conclusions are appliover, NJ zhengning.lin@pharma.novartis.com. cable to clinical studies with multiple parallel 379

2 380 Zhengning Lin treatment groups. This paper assumes that where n A and n B are the total number of patients both treatment and center effects are controlled for A and B, respectively, n Ai (or n Bi ) using a two-way (treatment-by-cen- is the number of patients in Group A (or B) at ter) analysis of variance (ANOVA) model. center i, and c is the total number of centers. Following recommendations in recent publi- Suppose that the treatment effect δ is one. cations, including those by Lin (1), Senn For 80% power we need a total of 34 patients (2,3), and Källén (4), this paper just considers based on standard (or commonly used) sample the weighted analysis of the ANOVA model size calculation. The maximum power to assess treatment differences. based on 34 patients is 80.7% when n A = n B It is well known that with the same number = 17 in a one-center study. Suppose that the of total patients, treatment comparison is block size for randomization is two per treat- most efficient when the number of patients ment group, and only one center is used. The is balanced between the two treatment imbalance of treatment allocation is controlled groups. For multicenter clinical studies this by the block size and the minimum balance is usually made at the center level, power is 80.6% when the larger group has because based on the weighted analysis of 18 patients and the smaller group has 16 pa- the ANOVA model, treatment main effect is tients. The maximum power loss due to the assessed by a weighted average of the treatment imbalance is only 0.1% in this one-center comparisons at individual centers. Al- case. If, however, we use, say, 7 centers in- though blocked randomization is usually stead, and the distribution of the 34 patients applied at study center level to balance treat- among the 7 centers is as shown in Table 1. ment allocations, it does not guarantee the Then the power is only 68.7%, a 12.0% balance in small centers when each of them drop from the maximum of 80.7% with one has few or even less than one block of patients. center. Among the 12.0% power loss, only As a result, studies using too many 1.5% is due to decreased degrees of freedom centers can lose statistical efficiency, or statistical and the other 10.5% is due to the imbalance power, due to this imbalance in a large of treatment allocations. number of small centers. The following example If we increase the number of centers, or illustrates this potential loss of statisti- the block size of the randomization in the cal power. example in Table 1, the potential power loss Consider a multicenter randomized clini- can be dramatically increased. A practical cal study to compare two treatments, A and question is: how often can we see an imbal- B, with response variable X quantitative and ance similar to this and what is the actual approximately normally distributed. Let u A risk of power loss when a study is designed and u B be the mean of X and σ be the standard to include a large number of clinical centers? deviation. Let us also define δ *u A u B */σ In the following, I use a statistical model to as the treatment effect. The statistical power of detecting a statistically significant treatment effect, with two-tailed 0.05 level test, TABLE 1 Distribution of 34 can be estimated as: Patients Among 7 Centers Center i n Ai n Bi Power = Pr t na +n B 2c > t 0.025,(nA +n B 2c) δ n Ai n Bi c i=1 n Ai + n Bi

3 The Number of Centers in a Multicenter Clinical Study 381 quantify the relationship between the number Now let us consider the calculation of the of centers and the actual statistical power of statistical power. In clinical studies, the sta- a study. tistical power is the probability of detecting a statistically significant treatment effect. METHOD Usually the power is calculated based on fixed sample sizes. In multicenter clinical Let us describe the process of patient enrollment studies, the fixed sample sizes are at the study at Center i by a Poisson process with center level for each treatment group. This parameter λ i, i = 1,..., c, where c is the is illustrated in the power calculation formula total number of centers in the study. Then provided in the previous section, where n Ai the probability that a newly recruited patient and n Bi are both fixed values for center i. belongs to Center i is π i =λ i /Σλ j. Different At the designing stage of typical multicenter sites recruit patients with different speeds studies, however, we do not know the sample and, therefore, in general, the λ i s are not sizes at each study center and can only specify equal. Usually, the study centers are not close the total sample sizes of the treatment to each other and, therefore, we can assume groups. Therefore, the power calculation that the λ i s are statistically independent. In conditional on fixed sample sizes at the cen- the following analysis I assume that λ 1,..., ter level is often not realistic. In the following λ c are independent and identically distributed. power calculation, I define the actual statisti- While the common distribution of the cal power as the probability of detecting a λ i s may vary from study to study, one possible, significant treatment effect conditional on and maybe the simplest, distribution for the total sample sizes for the treatment λ i s is the exponential distribution. In this case groups, and the sample sizes at the study the π i s follow D(1,...,1), the Dirichlet dis- center level is not fixed. tribution with all parameters equal to one. The calculations used Monte Carlo simu- To analyze the robustness against the as- lations based on the model assumptions specified sumption of patient distribution among study earlier, with replications for each centers, the analysis also consider the case of power calculation. Statistical significance is equal λ i s. Under this assumption, all centers defined as p < 0.05 using a two-tailed test. I have the same characteristics in recruiting consider the cases when the treatment effect, patients and therefore π 1 =π 2 =...=π c = δ, is 1/1, 1/2, and 1/3, which requires a total 1/c. In either case, the mean of the λ i s does of 17, 64, and 143 patients per group, respectively, not affect the distribution of patients across to achieve 80% power using a standard the centers. It only affects the overall recruit- (or common) sample size calculation ing speeds. Table 2 shows two examples of method. As mentioned earlier, treatment 34 patients distributed across 7 centers based comparison is most efficient when the number on simulation, assuming that the πs either of patients is equal between the two comon follow D(1,..., 1), or they are all equal to paring treatments at each of all centers, and 1/c, and the randomization block size is two in this case the maximum power (with fixed per group. total sample size) is achieved. Although the Note that in (a) the total number of centers sample sizes are powered at 80%, this maxithat contributed study patients is only six, mum power sometimes can be slightly less as Center 2 failed to recruit any patient, a than 80% because of the reduced degrees of situation frequently encountered in multicen- freedom due to increased number of centers ter studies. In practice, most multicenter in the ANOVA model, which the standard studies have center imbalance similar to the sample size calculation does not take into example in (a), while some others are more account. Nevertheless, the power reduction balanced between study centers and are due to loss of degrees of freedom is negligible closer to the example in (b). for studies with large sample sizes.

4 382 Zhengning Lin TABLE 2 Two Examples of 34 Patients Distributed Across 7 Centers (a): Example of s D(1,...,1),34patients, 7 centers Center All In Group A In Group B π (%) (b): Example of i 1/7 (14.3%), 34 patients, 7 centers Center All In Group A In Group B π (%) RESULTS AND CONCLUSIONS Table 3 displays the power calculation results using a randomization block of two per treatment group. The last column, Power Loss, calculates the difference between the maximum power and the actual power, which measures the power reduction due to the imbalance of sample sizes between the two treatments at each of the study centers. From Table 3 it is evident that no matter which assumption we use for the π i s, the power losses due to the imbalance in sample sizes between the two treatments are similar and relatively small, no more than 2% when the block size is two per group. The overall difference between the actual power and the targeted power (80%) is no more than 3%. Is this conclusion true for any block size? For comparison purposes Table 4 and Table 5 list the same results based on block sizes of four and one per group, respectively. Block sizes of more than four per group are not common and are usually unnecessary. Results in Tables 4 and 5 demonstrate that the power loss can be around 4% when the block size is four per group, while the power loss with block size of one per group is around 1.5% or less. Since the power loss is no more than several percentage points, it can be controlled by increasing the sample size to make up the TABLE 3 Actual Powers with Block Size Two per Group Actual power (%) Power loss a (%) Average n A + n B size per Maximum πs πs δ c group power D(1,..,1) π i 1/c D(1,..,1) π i 1/c / / / a Power loss due to imbalance within centers = Max power - Actual power

5 The Number of Centers in a Multicenter Clinical Study 383 TABLE 4 Actual Powers with Block Size Four per Group Actual power (%) Power loss a (%) Average n A + n B size per Maximum πs πs δ c group power D(1,..,1) π I 1/c D(1,..,1) π i 1/c b 0.1 1/ / / a Power loss due to imbalance within centers = Max power - Actual power b Negative number due to approximation is rounded to 0.0 power loss. For example, the actual power for the case of 64 patients per arm with 27 centers and a block size of four per group (Table 5) is 75.5 with πs D(1,..,1) while the sample size is based on 80% power using standard sample size calculation. To make up the 5% deficit a sample size of 73 per arm is needed to have 85% power using the standard sample size calculation. If the same average sample size per center is kept, the total number of centers becomes 31. The actual power with this D(1,..,1) assumption is 81.0%, which is sufficient for the 80% requirement. Since the power calculation re- sults are similar no matter which assumption we make for the π i s, we only need to pay attention to the average number of patients per center, and also the block size, in determining the extra number of patients needed for the study to make up the power loss. This can be done easily in practice by increasing the specified power in the standard sample size calculation using the above tables for reference. The above calculations are all based on sample sizes with 80% power. Is the power loss pattern similar for higher-powered studies? In fact, for studies with the same sample TABLE 5 Actual Powers with Block size One per Group Actual power (%) Power loss a (%) Average n A + n B size per Maximum πs πs δ c group power D(1,..,1) π i 1/c D(1,..,1) π i 1/c b 1/ / b 0.0 b 1/ a Power loss due to imbalance within centers = Max power - Actual power b Negative number due to approximation is rounded to 0.0

6 384 Zhengning Lin size, a higher power corresponds to a larger using a relatively large number of centers. δ, that is, a larger treatment effect. For example, Understanding this fact is important in deteris a study with 34 total patients (n A + n B ) mining the sample sizes for studies with powered at 80% for δ=1, and it is powered many study centers. For most Phase III stud- at 90% for δ=1.15. More calculations re- ies we may want to ensure that studies are vealed that for the same sample sizes, studies adequately powered and, therefore, the sam- with larger δs (compared to those for 80% ple sizes should not be calculated by the standard power) tend to have smaller power losses. As method without adjusting for the power an illustration, when δ=1.15, the maximum loss. Although in most cases there is no accurate power for a study with 34 total patients and way of knowing the exact statistical 7 centers is 89.0%. The actual power with power, at least this paper quantified one πs (D(1,..,1) and block size of four per source of power loss and provided a tool to group is 86.3%. Therefore, the power loss adjust the sample size. is 89.0% 86.3% = 2.7%, smaller than the Results in this paper by no means suggest power loss of 3.6% when the study is pow- that no matter how many centers we use, ered at 80% (δ=1, see Table 4). This interesting the reduction of statistical power is always finding is mathematically explained in limited by a couple of percentages. Actually, Appendix 1. even if we only increase the number of cen- From the above we can conclude the fol- ters to, say, 10, for the example of n A + n B = lowing: In general the actual power of a 34 and block size of 4 per group, the actual multicenter study can be up to 5% lower than power would reduce to 69.8% for π i 1/10. what the study is targeted for. The power In this case the maximum power is 77.3% loss caused by the imbalance of sample sizes and the power loss due to the imbalance is within study centers: already a substantial amount of 7.5%. It is not common, however, to have average center Increases with the block size, sizes less than two per group and also it Increases with the number of centers, is usually not desirable. Is closely related to the average center size, Results in this paper also indicate that, in as opposed to the total sample size, general, there is limited statistical penalty to Is around 2% or less when the block size pay if we use as many centers as logistically is no more than two per group, desirable. The advantage of speeding up the Is negligible (<1%) when the average center patient enrollment and exposing the investigroup, size is at least seven per treatment gating drug to more physicians and clinical centers is often well worth the little extra Is less than 4% when the average center cost of slightly increasing the total number size is at least 2.4 per treatment group, and of patients to make up the power loss. The Can be made up by increasing the power statistical model in this paper, however, is used in the standard sample size calculation. based on the assumption that the treatment effect (δ) at each study center is independent of the center size. Although this assumption Note that the results based on the two ex- is used in virtually all parametric statistical treme scenario, πs D(1,..,1) and π i 1/ c, are similar and therefore, these conclusions are not scenario dependent. DISCUSSION not have the opportunity to pass this learning curve. Also, clinical centers serving larger patient populations often are more experienced than those serving smaller patient pop- This paper demonstrates that standard power calculation overestimates the actual power by a few percentage points ( 5%) for studies analyses for controlled multicenter clinical studies, it often can be incorrect in practice. For studies using new treatment assessment procedures, there is often a learning curve for each clinical center and small centers do

7 The Number of Centers in a Multicenter Clinical Study 385 Acknowledgments I would like to thank an anonymous referee for helpful comments and suggestions. REFERENCES ulations. In these cases, larger centers may provide more reliable data than smaller ones, although there are always exceptions. These issues may be too complicated to be incorporated in statistical analyses. The best way to handle them is to select and monitor all study centers carefully to assure the quality of the clinical data. For studies using well-estab- lished treatment assessment procedures, the model assumption in this paper can be appropriate when study centers are carefully selected and monitored. 1. Lin Z. An issue of statistical analysis in controlled multi-centre studies: How shall we weight the centers? Stat Med. 1999; 18: Senn S. Some controversies in planning and analyzing multi-center trials. Stat Med. 1998; 17: Senn S. Statistical Issues in Drug Development. Multi-center trails. Wiley;1997: Källén A. Treatment-by-center interaction: What is the issue? Drug Inf J. 1997; 31: APPENDIX 1 Why the Power Loss for a Higher Powered Study is Smaller The overall study power is estimated by: Power = Pr t na +n B 2c > t α/2,(na +n B 2c) δ c i=1 n Ai n Bi n Ai + n Bi = E (nji ) Pr t na +n 2c > t α/2,(na +n 2c) δ n Ai n Bi c B B i=1 n Ai + n Bi *n ji = E (nji ) 1 F t t α/2,(na +n 2c) δ n Ai n Bi c B i=1 n Ai + n Bi *n ji Power max 1 F t t α/2,(na +n 2c) δ n A + n B B 2 where Power max is the maximum power when the numbers of patients are exactly the same between the two treatments at each of all centers, and F t is the cumulative probability distribution of t na+nb 2c. Therefore, the power loss is: Loss Pwr (δ) = Power max Power = E (nji ) F t t α/2 δ c i=1 = E (nji )[G(δ)*n ji ] n Ai n Bi n Ai + n Bi F t t α/2 δ n A + n B 2 *n ji where G(x) = F t t α/2 x c i=1 n Ai n Bi n Ai + n Bi F t t α/2 x n A + n B 2 = F t (t α/2 xm) F t (t α/2 xn) where M = Σn Ai n Bi /(n Ai + n Bi ),andn = n A + n B /2. The power loss corresponding to a higher powered study (eg,

8 386 Zhengning Lin 90% power) with the same sample size is simply the power loss corresponding to a higher δ. Therefore, if Loss pwr (x) is generally a decreasing function of x then the power loss for higher powered study is less than that for a 80% powered study with the same sample size. To show that Loss pwr (x) is decreasing it is sufficient if we can prove that G(x) is decreasing. Actually, G (x) = f t (t α/2 xn)n f t (t α/2 xm)m = H(N) H(M) where f t (a) =Γ[(r + 1)/2] rπ Γ(r/2) 1 (1 + a 2 /r) (r 1)/2 is the density function of the t distribution with r degrees of freedom, and H(y) = f t (t α/2 xy)y. To show that G (x) < 0 so that G(x) is decreasing in the region of interest, we only need to prove that H(y) is decreasing for M y N while x corresponds to a test of at least 80% power. Actually, it can be shown that where H (y) = f t (xy t α/2 )r(r + (xy t α/2 ) 2 ) 1 L(y) L(y) = xy r 1 t α/2 + (r + 1)t α/2 + 1 Therefore if L(y) < 0 then H(y) has to be decreasing for M y N. Actually, L(y) < 0 xy r 1 t α/2 > (r + 1)t α/2 + 1 xy t α/2 > r + 1 t α/2 + (r + 1)t α/2 + 1 xy t α/2 > (r + 1)t α/2 (r + 1)t α/ However, (r + 1)t α/2 (r + 1)t α/2 1 1 z α/2 z α/2 = where Z α/2 is the corresponding percentile for standard normal distribution and α=0.05. However, only statistical tests with power = 66.3 can have xy t α/2 > 0.42 (when x =δ and y = M), which is usually outside the area for studies powered at 80% or more with relatively small randomization block size. Therefore, generally xy t α/2 = 0.42 and thus L(y) < 0. Therefore, the actual power loss for a higher powered study with the same total sample size is generally smaller.