Accounting for expected attrition in the planning of community intervention trials

STATISTICS IN MEDICINEStatist. Med. 2007; 26:2615–2628Published online 27 October 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sim.2733

Accounting for expected attrition in the planningof community intervention trials

Monica Taljaard1,∗,†, Allan Donner2 and Neil Klar3

1Ottawa Health Research Institute, Clinical Epidemiology Program, The Ottawa Hospital, Ottawa, Canada2Department of Epidemiology and Biostatistics, Robarts Clinical Trials, Robarts Research Institute, Schulich

School of Medicine, University of Western Ontario, London, Canada3Department of Epidemiology and Biostatistics, Schulich School of Medicine, University of Western

Ontario, London, Canada

SUMMARY

Trials in which intact communities are the units of randomization are increasingly being used to evaluateinterventions which are more naturally administered at the community level, or when there is a substantialrisk of treatment contamination. In this article we focus on the planning of community intervention trials inwhich k communities (for example, medical practices, worksites, or villages) are to be randomly allocated toeach of an intervention and a control group, and fixed cohorts of m individuals enrolled in each communityprior to randomization. Formulas to determine k or m may be obtained by adjusting standard sample sizeformulas to account for the intracluster correlation coefficient �. In the presence of individual-level attritionhowever, observed cohort sizes are likely to vary. We show that conventional approaches of accountingfor potential attrition, such as dividing standard sample size formulas by the anticipated follow-up rate� or using the average anticipated cohort size m�, may, respectively, overestimate or underestimate therequired sample size when cluster follow-up rates are highly variable, and m or � are large. We present newsample size estimation formulas for the comparison of two means or two proportions, which appropriatelyaccount for variation among cluster follow-up rates. These formulas are derived by specifying a modelfor the binary missingness indicators under the population-averaged approach, assuming an exchangeableintracluster correlation coefficient, denoted by �. To aid in the planning of future trials, we recommendthat estimates for � be reported in published community intervention trials. Copyright q 2006 John Wiley& Sons, Ltd.

KEY WORDS: cluster randomized trial; loss to follow-up; missing data mechanism; intraclustercorrelation; sample size calculation; follow-up rates

∗Correspondence to: Monica Taljaard, Ottawa Health Research Institute, Clinical Epidemiology Program, The OttawaHospital, Civic Campus, 1053 Carling Avenue, F-wing F650b, Ottawa, Canada K1Y 4E9.

†E-mail: [email protected]

Contract/grant sponsor: Natural Sciences and Engineering Research Council of Canada

Received 29 March 2006Copyright q 2006 John Wiley & Sons, Ltd. Accepted 18 September 2006

2616 M. TALJAARD, A. DONNER AND N. KLAR

1. INTRODUCTION

In cluster randomized trials, naturally occurring groups of individuals such as inhabitants ofthe same village or patients attending the same medical practice, are collectively randomized tointervention or control conditions, and the outcome of interest is then measured on the individualcluster members. Randomization by cluster is a natural choice when the experimental interventioncan only be administered at cluster-level, or when there is a risk of treatment contamination due tocompeting interventions being allocated to different members of the same cluster. This design istherefore frequently used in studies of public health interventions [1], school-based health educationprogrammes [2], and health care professional behaviour change interventions [3]. In this article,we focus primarily on community intervention trials, which are characterized by a relatively smallnumber of relatively large units, such as entire villages, medical practices, schools, or worksites,being randomized. It has now become widely recognized that the correlation among responsesfrom the same cluster needs to be accounted for in the analysis of data from such trials (e.g. [4]).The within-cluster correlation is typically represented by a parameter known as the intraclustercorrelation coefficient, denoted by �.

An essential part of the planning of any randomized trial is the calculation of the sample size toprovide adequate levels of significance and power for detecting a clinically relevant interventioneffect. It is convenient to distinguish between two cases in the planning of cluster randomizedtrials. In the first case, the number of individuals to be enrolled in each cluster is not directly underthe control of the investigator, due to limited availability or eligibility of subjects. For example,a cluster may consist of all patients presenting at a medical practice with a certain disease ina specified time period. As a result, there will be natural variation among cluster sizes in thetrial. Sample size calculations are then typically carried out to determine the required numberof clusters to enrol, say k, given the average (anticipated) cluster size m [4]. Formulas whichexplicitly account for variable cluster sizes were proposed by Manatunga and Hudgens [5] andKerry and Bland [6]; these formulas require advance estimates of the distribution of cluster sizesin the trial.

The second case, which is of interest in this article, applies in many community interventiontrials where the size of each community necessitates enrolment of a smaller subsample from eachcommunity. In such trials, sample size calculations need to be carried out to determine the numberof clusters to enrol (k) and/or the subsample size (m). Note that, although the actual size of eachcommunity may vary, m is most often selected to be constant—at least for planning purposes. Forexample, the Community Intervention Trial for Smoking Cessation involved 22 communities in theUS and Canada, and cohorts of approximately 550 smokers were identified in each community tobe followed prospectively [7]. In the Healthy Worker Project [8], a work-site intervention trial forweight control and smoking cessation, 32 worksites with between 400 and 900 employees wereenrolled, and random samples of 200 employees were selected for participation at each site. In acluster randomized trial of an intervention to promote the uptake of recommendations in The WHOReproductive Health Library [9], the investigators wished to estimate the prevalence of obstetricpractices in 40 hospitals based on a minimum of 100 eligible women in each hospital; to reachthe desired sample size, they planned to enrol women until the required number has been reachedor for six months, whichever comes first.

Standard formulas to determine k or m in such settings are presented by Donner and Klar [4];however, these formulas assume complete follow-up. In reality, trials are subject to individual-level and/or cluster-level attrition, and as a consequence of individual-level attrition, observed

Copyright q 2006 John Wiley & Sons, Ltd. Statist. Med. 2007; 26:2615–2628DOI: 10.1002/sim

EXPECTED ATTRITION IN THE PLANNING OF COMMUNITY INTERVENTION TRIALS 2617

subsample sizes are likely to vary from cluster to cluster. The primary objective of this article isto propose sample size formulas which account for potential individual-level and/or cluster-levelattrition, and to compare the proposed formulas with conventional approaches of accounting forattrition which do not explicitly take variable cluster follow-up rates into account. We focus ontrials in which k communities are to be randomly allocated to each of an intervention and a controlgroup, with fixed cohorts of m individuals enrolled in each community prior to randomization,where k and/or m need to be determined. In Section 2, standard tests and sample size formulas forthe comparison of two means or proportions in such trials are reviewed. In Section 3, the assumedmechanism generating attrition is defined. As is customary for the purpose of estimating samplesize, the mechanism is assumed to be missing completely at random (MCAR) in the well-knownterminology of Rubin [10]: under MCAR, the probability that a study subject is missing does notdepend on any missing or non-missing data in the study. In Section 4, the unconditional varianceof the estimated intervention effect under this mechanism is derived, and then used in Section 5to derive adjusted sample size formulas for the comparison of two means or two proportions. InSection 6, the proposed sample size formulas are evaluated using a simulation study; and Section 7concludes with a discussion.

2. SAMPLE SIZE CALCULATION ASSUMING NO ATTRITION

Let Yi jl denote the outcome for the lth individual (l = 1, . . . ,m) in the j th cluster ( j = 1, . . . , k)in treatment group i = 1, 2. We assume that Yi jl ∼ N(�i , �

2) with �2 = �2A+�2W, where �2A and �2Ware between-cluster and within-cluster variance components, respectively, and � = �2A/�2. Supposethe aim of the trial is to test H0 : �2 = �1 based on the difference between the group means Y 2−Y 1,where Y i =∑k

j=1∑m

l=1Yi jl/km. The variance of the difference is

Var(Y 2 − Y 1) = 2�2

km[1 + (m − 1)�]

where 1 + (m − 1)� is referred to as the variance inflation factor or design effect [4]. Then,assuming that �2 and � are known, an asymptotic test for H0 is based on

Z = Y 2 − Y 1√(2�2/km)[1 + (m − 1)�] ∼ N(0, 1)

The total number of subjects required per group can then be calculated as

km = (z�/2 + z�)22�2

�2[1 + (m − 1)�] (1)

where z�/2 and z� are the critical values of the standard normal distribution corresponding to thedesired two-sided 100� per cent significance level and power 1 − �, respectively, and � is theclinically meaningful difference to be detected. In practice, �2 and � need to be estimated, and k isusually small. The test is then based on critical values from the t-distribution with 2(k−1) degreesof freedom. To account for using standard normal critical values in formula (1), one cluster pergroup may be added for tests at the 5 per cent level of significance; and two clusters per group for



tests at the 1 per cent level of significance [11]. For tests at the 5 per cent level of significance, thenumber of clusters required per group, given a fixed subsample size m, can then be calculated as

k = 1 + (z�/2 + z�)22�2

m�2[1 + (m − 1)�] (2)

while the required subsample size, given a fixed number of clusters per group k, can be calculated as

m = (1 − �)(z�/2 + z�)22�2/�2

k − 1 − �(z�/2 + z�)22�2/�2

(3)

Consider a dichotomous outcome Yi jl such that Yi jl = 1 if the intervention is a success, and 0otherwise, and suppose the aim of the study is to test H0 : P1 = P2 where P1 and P2 are thepopulation success rates in the control and experimental groups, respectively. In the absence ofattrition, the intervention effect is estimated as P2 − P1 where Pi = Y i , and the variance is

Var(P2 − P1) =2∑

i=1

Pi (1 − Pi )

km[1 + (m − 1)�]

where � is the intracluster correlation coefficient for the binary outcome. A test for H0 is basedon the adjusted Pearson chi-square statistic [12]

�2A =2∑

i=1

km(Pi − P)2

P(1 − P)[1 + (m − 1)�] ∼ �2(1)

where P = (∑2

i=1∑k

j=1∑m

l=1 Yi jl)/2km. The required sample size per group is then given by

equation (1) with 2�2 = P1(1 − P1) + P2(1 − P2).

3. MISSING DATA MECHANISM FOR CLUSTERED DATA

Define a binary indicator variable Ri jl , such that Ri jl = 1 if Yi jl is observed, and Ri jl = 0 if Yi jlis missing. Under MCAR, we assume for all i = 1, 2, j = 1, . . . , k and l = 1, . . . ,m, that

Pr(Ri jl = 1|Yi j ) =Pr(Ri jl = 1)= � (4)

where Yi j is an (m × 1) vector of outcomes in the ijth cluster, and that

Corr(Ri jl , Ri j ′l ′) =

⎧⎪⎪⎨⎪⎪⎩0 if j �= j ′

� if j = j ′, l �= l ′

1 if j = j ′, l = l ′(5)

where −1/(m − 1)��1 denotes the intracluster correlation coefficient for the missing datamechanism. This is an example of the population-averaged approach to modelling clustered data,as described by Neuhaus [13].



We now consider the consequences of intracluster correlation in the missing data mechanism.Let Ri j =∑m

l=1Ri jl/m denote the follow-up rate in the i j th cluster. The model in (4) and (5)implies that E(Ri j ) = � and

Var(Ri j ) = �(1 − �)

m[1 + (m − 1)�]

Note that the minimum correlation �= −1/(m−1) implies that Var(Ri j ) = 0 (i.e. all clusters haveidentical follow-up rates), while the maximum correlation � = 1 implies that all the Ri jl in a clusterare identical (i.e. entire clusters are either completely observed or completely missing).

4. UNCONDITIONAL VARIANCE OF THE ESTIMATED INTERVENTION EFFECT

We now derive the unconditional variance of the estimated intervention effect assuming the MCARmodel defined in Section 3. Under this model, the mean (or proportion) in group i is estimated as

Y i =∑k

j=1∑m

l=1Ri jlYi jl∑kj=1∑m

l=1Ri jl(6)

and the conditional variance of the mean difference, as shown in Appendix A, is

Var(Y 2 − Y 1|R) = �22∑

i=1

Di∑kj=1∑m

l=1Ri jl(7)

where R is a (2km × 1) vector of missing data indicators, and

Di = 1 +[∑k

j=1(∑m

l=1Ri jl)2∑k

j=1∑m

l=1Ri jl− 1

]� (8)

is the conditional design effect.The unconditional variance can be found by averaging over the missing data mechanism

Var(Y 2 − Y 1) = ER[Var(Y 2 − Y 1|R)] + VarR[E(Y 2 − Y 1|R)] (9)

Since this expression involves the expectation of a ratio of two random variables, an exact expressionfor the unconditional variance is not available, but an approximation is derived in Appendix B as

Var(Y 2 − Y 1) ≈ 2�2G

km�(10)

where

G = 1 + (m� − 1)� + (1 − �)[1 + (m − 1)�]� (11)

The factor G can be regarded as the unconditional variance inflation factor, representing theeffects of clustering in both the outcome and the missing data mechanism. As shown inAppendix B, this approximation is satisfactory provided that CV/

√k is small (say, <0.20),

where CV= √(1 − �)[1 + (m − 1)�]/m� is the coefficient of variation of the cluster follow-up

rates Ri j .



5. SAMPLE SIZE CALCULATION ACCOUNTING FOR ATTRITION

We now use the unconditional variance derived in the previous section to derive adjusted samplesize formulas, accounting for the missing data mechanism. For the comparison of two means, thisyields

km = (z�/2 + z�)22�2

�2× G

�(12)

and for the comparison of two proportions

km = (z�/2 + z�)2[P1(1 − P1) + P2(1 − P2)](P1 − P2)2

× G

�(13)

A conventional approach accounting for attrition in a randomized trial is to divide the requiredsample size in the absence of attrition by the anticipated follow-up rate �. In the case of a clusterrandomized trial, note that dividing the standard formula (1) by � is equivalent to assuming that� = 1 in our proposed formulas. An alternative approach in a cluster randomized trial is to usethe expected (average) cohort size m� in place of m in the standard formula (1). Note that this isequivalent to assuming that � =−1/(m − 1) in our proposed formulas.

6. SIMULATION STUDY

We conducted a simulation study to compare the proposed sample size formulas when explicitlyaccounting for variable cluster follow-up rates, with the two conventional approaches discussedin Section 5. The simulation study was designed to resemble the Healthy Worker Project [8](previously referred to in Section 1); conceptually, it consisted of three parts. First, we computedthe required sample size under the three approaches; we then generated data for the calculatedsample size in each case, and assigned individuals to be missing according to the same mechanism,as defined in Section 3. Because our focus was on the effect of misspecifying �, we assumed thatthe values for �, � and �2 used to calculate sample size, were appropriate. A test of H0 was thenconducted and the proportion of rejections of H0 over 1000 simulation runs (the empirical power)was computed.

6.1. Sample size calculation

We computed the number of worksites per group (k) given m = (300, 150, 100, 50, 25) employ-ees per site, as required to detect a reduction of �/� = − 0.2 in body mass index (BMI), anda reduction in the prevalence of smoking from 25 to 18 per cent, in each case using a two-sided test at the 5 per cent level of significance with desired 90 per cent power, assuming� = (0.001, 0.005, 0.01, 0.03, 0.1). We repeated the calculations to determine the required num-ber of employees (m) per worksite, given k = (8, 12, 20, 30, 50) available worksites per group.We carried out these calculations assuming � = 0.7 and �= − 1/(m − 1), 0.3 and 1—alwaysrounding up to the nearest integer. For BMI, one cluster per group was added as described inSection 2, to account for using standard normal critical values in the sample size formulas.



6.2. Data simulation

BMI was generated using the model

Yi jl = � + �Xi +Ui j + ei jl

where �= 25 and �= − 0.2�, Xi is the indicator for the intervention and control group, Ui j ∼N(0, �2A) represents random cluster-effects and ei jl ∼ N(0, �2W) individual error terms, and �2 = 16.Correlated binary outcomes representing a reduction from 25 to 18 per cent in smoking prevalencewere generated using the method described by Oman and Zucker [14].

Missing data indicators were then generated independently for each simulation run, such that� = 0.7 and � = 0.3, using the method of Oman and Zucker [14]. Note that these values for � and� imply a standard deviation among cluster follow-up rates of approximately 25 per cent.

6.3. Hypothesis testing

Let mi j denote the number of non-missing observations in the ijth cluster, and let Mi =∑kj=1mi j ,

and M = M1 + M2. We tested H0 : �1 = �2 using the standard two-sample t-test adjusted forclustering [4]

T = Y 2 − Y 1√S2P(C1/M1 + C2/M2)

∼ t2(k−1) (14)

where Y i was calculated as in equation (6), and

Ci =k∑j=1

mi j1 + (mi j − 1)�

Mi(15)

The pooled variance was estimated as S2P = S2A + S2W, and the intracluster correlation coefficientas �= S2A/(S2A + S2W), where

S2W =MSW=∑2

i=1∑k

j=1∑mi j

l=1(Yi jl − Y i j )2

M − 2k

is the variance within clusters, and

S2A = 1

m0[MSC − MSW]

the variance between clusters, where

MSC=∑2

i=1∑k

j=1mi j (Y i j − Y i )2

2(k − 1)

and

m0 = M −∑2i=1∑k

j=1m2i j/Mi

2(k − 1)

Note that when cluster follow-up rates are identical, this test is identical to the standard two-samplet-test based on cluster means; it is also identical to the test of the regression coefficient of the



Table I. Empirical power (nominal level= 90 per cent) of the adjusted two-sample t-test to detect areduction of 0.8 kg/m2 in BMI in the presence of 30 per cent attrition, when the sample size (k given m

or m given k) was calculated using three different values for � (true value= 0.3).

Empirical power and calculated Empirical power and calculatedsample size [k given m] sample size [m given k]

Assumed � Assumed �

� m −1/(m − 1)∗ 0.3† 1‡ � k −1/(m − 1)∗ 0.3† 1‡

0.001 300 92.5 [5] 92.5 [5] 92.5 [5] 0.001 8 89.2 [116] 90.1 [118] 93.3 [129]0.005 91.7 [7] 91.7 [7] 95.6 [8] 0.005 86.6 [171] 88.4 [185] 93.8 [265]0.01 88.3 [9] 91.5 [10] 95.5 [12] 0.01 87.3 [424] 90.1 [692] —0.03 87.8 [20] 91.0 [22] 95.9 [27] 0.03 — — —0.10 85.8 [56] 89.1 [63] 96.0 [79] 0.10 — — —

0.001 150 92.0 [7] 92.0 [7] 93.6 [8] 0.001 12 90.1 [72] 90.1 [72] 91.3 [77]0.005 89.5 [9] 89.5 [9] 96.1 [11] 0.005 87.7 [90] 89.8 [93] 91.8 [110]0.01 88.7 [12] 88.7 [12] 93.7 [14] 0.01 88.0 [130] 90.9 [147] 95.2 [242]0.03 84.7 [22] 91.1 [24] 93.8 [29] 0.03 — — —0.10 87.9 [58] 90.2 [65] 94.7 [81] 0.10 — — —

0.001 100 91.4 [10] 91.4 [10] 91.4 [10] 0.001 20 88.9 [41] 88.9 [41] 91.8 [43]0.005 90.0 [12] 90.0 [12] 92.7 [13] 0.005 87.6 [46] 91.1 [47] 91.2 [51]0.01 89.1 [14] 91.4 [15] 93.6 [17] 0.01 87.7 [54] 90.9 [57] 93.9 [68]0.03 87.3 [25] 90.6 [27] 95.0 [32] 0.03 87.8 [224] 88.2 [596] —0.10 87.4 [61] 91.8 [68] 95.0 [84] 0.10 — — —

0.001 50 91.5 [17] 91.5 [17] 91.8 [18] 0.001 30 90.0 [27] 90.0 [27] 90.0 [27]0.005 89.7 [19] 89.7 [19] 92.9 [21] 0.005 88.3 [29] 88.3 [29] 91.0 [31]0.01 90.3 [22] 90.3 [22] 93.1 [24] 0.01 89.8 [32] 91.3 [33] 94.2 [36]0.03 88.4 [32] 89.0 [34] 94.3 [39] 0.03 89.2 [55] 89.8 [66] 95.3 [120]0.10 85.2 [67] 90.6 [75] 94.9 [90] 0.10 — — —

0.001 25 89.9 [32] 89.9 [32] 91.4 [33] 0.001 50 90.0 [16] 90.0 [16] 90.0 [16]0.005 90.0 [34] 90.0 [34] 91.6 [36] 0.005 90.9 [17] 90.9 [17] 90.9 [17]0.01 88.2 [36] 91.0 [37] 91.7 [39] 0.01 89.9 [17] 91.6 [18] 91.9 [19]0.03 88.4 [46] 90.5 [49] 91.6 [54] 0.03 88.2 [22] 90.8 [24] 93.8 [28]0.10 86.8 [81] 91.4 [88] 94.4 [104] 0.10 — — —

∗Equivalent to using the average anticipated cohort size in the standard formula.†Correct value for �.‡Equivalent to dividing the standard formula by the anticipated follow-up rate, �.

intervention indicator in a mixed-effects regression model when there are no other covariates inthe model [15].

For smoking status, H0 : P1 = P2 was tested using

�2A =2∑

i=1

Mi (Pi − P)2

P(1 − P)Ci∼ �2(1) (16)



Table II. Empirical power (nominal level= 90 per cent) of the adjusted Pearson �2-test to detect a reductionof 7 per cent in smoking prevalence in the presence of 30 per cent attrition, when the sample size (k

given m or m given k) was calculated using three different values for � (true value= 0.3).

Empirical power and calculated Empirical power and calculatedsample size [k given m] sample size [m given k]

Assumed � Assumed �

� m −1/(m − 1)∗ 0.3† 1‡ � k −1/(m − 1)∗ 0.3† 1‡

0.001 300 91.6 [5] 91.6 [5] 91.6 [5] 0.001 8 89.7 [141] 90.7 [143] 88.5 [147]0.005 86.0 [7] 90.1 [8] 93.4 [9] 0.005 87.8 [232] 89.4 [259] 91.3 [356]0.01 87.4 [11] 91.2 [12] 94.1 [14] 0.01 — — —0.03 86.5 [25] 89.7 [28] 95.8 [35] 0.03 — — —0.10 84.9 [75] 91.2 [85] 94.7 [106] 0.10 — — —

0.001 150 89.1 [8] 89.1 [8] 89.1 [8] 0.001 12 89.7 [91] 89.7 [92] 91.2 [94]0.005 88.8 [11] 88.8 [11] 91.1 [12] 0.005 86.8 [122] 89.5 [129] 92.4 [149]0.01 88.5 [14] 89.4 [15] 94.0 [18] 0.01 88.2 [211] 89.9 [262] 94.2 [583]0.03 88.6 [29] 89.6 [31] 95.1 [38] 0.03 — — —0.10 85.8 [78] 90.7 [88] 94.9 [109] 0.10 — — —

0.001 100 87.4 [11] 91.6 [12] 91.6 [12] 0.001 20 89.3 [54] 89.3 [54] 91.4 [55]0.005 86.5 [14] 89.1 [15] 92.4 [16] 0.005 91.4 [63] 88.0 [65] 87.8 [69]0.01 90.6 [18] 90.0 [19] 93.6 [21] 0.01 86.9 [80] 87.7 [86] 92.6 [105]0.03 87.6 [32] 90.2 [35] 93.1 [41] 0.03 — — —0.10 87.0 [82] 90.4 [91] 95.6 [112] 0.10 — — —

0.001 50 88.7 [22] 88.7 [22] 88.7 [22] 0.001 30 89.8 [35] 90.6 [36] 90.8 [36]0.005 88.4 [24] 90.6 [25] 91.8 [26] 0.005 90.0 [39] 89.1 [40] 91.5 [42]0.01 89.5 [28] 89.5 [29] 90.9 [31] 0.01 87.5 [45] 89.3 [47] 90.1 [52]0.03 88.6 [42] 88.6 [45] 92.7 [51] 0.03 85.8 [118] 91.8 [176] —0.10 86.3 [91] 90.6 [100] 95.1 [122] 0.10 — — —

0.001 25 89.4 [42] 89.4 [42] 90.3 [43] 0.001 50 90.7 [21] 90.7 [21] 90.7 [21]0.005 89.0 [45] 89.0 [45] 90.3 [46] 0.005 89.9 [22] 89.4 [23] 88.2 [23]0.01 89.5 [48] 90.2 [49] 90.8 [51] 0.01 90.5 [24] 90.8 [25] 92.4 [26]0.03 85.9 [62] 89.7 [65] 92.9 [71] 0.03 88.7 [35] 89.6 [39] 93.0 [52]0.10 88.2 [109] 88.9 [119] 94.2 [140] 0.10 — — —

∗Equivalent to using the average anticipated cohort size in the standard formula.†Correct value for �.‡Equivalent to dividing the standard formula by the anticipated follow-up rate, �.

where P = (∑2

i=1∑k

j=1∑mi j

l=1Yi jl)/M is the overall proportion of smokers and Ci is defined inequation (15).

6.4. Results

Empirical power values for BMI are reported in Table I and for smoking status in Table II, withthe calculated sample sizes (k or m) given in parentheses. Note that a few table cells for m givenk are left blank; in these cases, the sample size formula yielded negative estimates for m, so thatno data could be generated. Negative estimates for m, which may occur even in the absence of



attrition, signify that there is no m large enough for the available number of clusters to yield thedesired power: as pointed out by Donner and Klar [4] (Section 5.5.2), power may be increasedindefinitely by increasing k; however, when k is fixed, power may be improved by increasing monly to an upper bound which is limited by �. When using the proposed sample size formulas,negative estimates for m are naturally more likely to occur when assuming that � = 1; in practice,erroneously assuming that � = 1 may therefore result in abandoning a study when a possible mmay have been found if � were known.

In Tables I and II, empirical power values at least 2 per cent below the nominal level of 90per cent are highlighted. Note that—except for random variation in the outcome and the missing datamechanism—empirical power should be at least 90 per cent when the assumed value for � is correct.The results show that the proposed formulas substituting the appropriate value for � generallyyielded adequate sample size; on the other hand, the ‘average size method’ (� =−1/(m − 1))tended to underestimate the required sample size, while dividing by the anticipated follow-up rate(� = 1) tended to overestimate the required sample size. The impact of misspecificying � was mostnoticeable when � or m was large; for example, when � = 0.001, increasing � from −1/(m − 1)to 0.3 had virtually no effect on k; however, it resulted in an increase of 2 to 3 clusters per groupwhen �= 0.03, and an increase of 7 to 10 clusters per group when � = 0.1.

7. DISCUSSION

We presented sample size formulas to account for potential attrition in community interventiontrials where cohorts of size m are to be enrolled in each community prior to randomization. Theseformulas account for potential loss in power due to reduction in the total sample size, as wellas variability among cohort follow-up rates. This variability was quantified by a parameter �, theintracluster correlation coefficient for the missing data mechanism. We showed that two conven-tional approaches of dealing with attrition, namely dividing standard formulas by the anticipatedfollow-up rate, or using the average anticipated cohort size, can be expressed in terms of substitut-ing maximum, or minimum values for �, respectively. Using a simulation study, we showed thatsubstituting the appropriate value for � can avoid potential underestimation of sample size dueto the average size method; it can also avoid potentially over-powered studies due to assumingthat � = 1. We showed that the potential benefits of substituting appropriate values for � are mostnoticeable when � or m are large.

In the simulation study, we assumed that �, � and � are known; in practice, suitable estimatesfor these parameters, in addition to �, will have to be obtained in advance of the trial. Publishedestimates for �, covering a wide range of potential outcomes, are available in the literature and therange is increasing (e.g. [16–18]). In the absence of reliable estimates for �, a sensitivity analysisis recommended by substituting a range of plausible values for �. It is likely that in many plannedcommunity intervention trials, varying � will have little or no impact on the estimated samplesize: as can be seen from equation (11), the impact depends on �, m and �. Thus, when � is verysmall, as is frequently the case in community intervention trials, the three approaches accountingfor attrition may yield similar sample size estimates. When the difference between substituting theminimum and maximum values for � is minimal, we recommend using � = 1 as a conservativeestimate. However, to aid in the planning of future trials, we also recommend that estimates for �(as well as � and �) be published in reports of community intervention trials. To estimate �, thestandard approach for binary outcomes can be used (e.g. [4]).



In most cluster randomized trials, consent is obtained at both individual-level and cluster-level,and a combination of individual-level and cluster-level attrition is then possible. In the planning ofsuch trials, the formulas in Section 5 can be used to compute the required sample size, assumingfirst that no clusters will drop out; the resulting value for k can then be divided by �, where � nowrefers to the follow-up probability for decision-makers at cluster-level. However, note that whenestimating �, a distinction should be made between individual-level and cluster-level attrition: intrials where entire clusters have dropped out due to decision-makers at cluster-level withdrawingconsent, these clusters should not be included in the calculation of �.

A reviewer suggested that it may be easier to postulate plausible values for observed cohort sizesthan for �, because minimum and maximum cohort sizes (or follow-up rates) are often reportedin cluster randomized trials. This would provide a starting point to estimating the coefficient ofvariation CV of observed cohort sizes (or cluster follow-up rates). By making use of the fact that

�= (CV)2m�

(m − 1)(1 − �)− 1

(m − 1)(17)

our proposed formulas may also be expressed in terms of CV. This would be equivalent to theapproach by Manatunga and Hudgens [5], who expressed variability among cluster sizes in theabsence of attrition, using the coefficient of variation. Variability in this setting has also previouslybeen expressed using binomial, Poisson and Pareto models [19]. A key difference with the settingof interest here, is that we consider imbalance solely as a result of attrition; thus, minimum andmaximum possible cluster sizes (0 and m, respectively) are known. An advantage of expressingimbalance in terms of the intracluster correlation coefficient � is that its maximum possible valueis always 1; in contrast, the maximum possible coefficient of variation—which will occur whenall but one cluster are missing, the remaining cluster having an arbitrary non-zero number ofobservations (see [20])—depends on k and is therefore more difficult to generalize.

A limitation of the proposed sample size formulas is the assumption that attrition is completelyrandom—in reality, attrition is likely to be related to observed or unobserved characteristics ofindividuals or clusters in the trial. Mechanisms other than MCAR are, however, much more difficultto anticipate and generalize; moreover, bias rather than power is then a more serious concern. Lastly,it should be noted that the simulation study was based on a fairly large value for � ( = 0.3). Inpractice, values for � may be substantially smaller; for example, in the Healthy Worker Project,� was approximately 0.05. When � is small, it is likely that the three approaches consideredin this article will yield similar estimates for sample size; sensitivity analyses are neverthelessrecommended, given the lack of empirical estimates for �.

APPENDIX A

In this appendix, we derive the conditional variance of the estimated intervention effect in thepresence of attrition. Let Ri++ =∑k

j=1∑m

l=1Ri jl . Assuming MCAR, the conditional variance ofthe observed mean in each group is

Var(Y i |R) =k∑j=1

Var(∑m

l=1Ri jlYi jl)

(Ri++)2



=k∑j=1

∑ml=1R

2i jl Var(Yi jl) +∑m

l=1∑m−1

l ′ �=l=1Ri jl Ri jl ′ Cov(Yi jl , Yi jl ′)

(Ri++)2

= �2

Ri++

k∑j=1

∑ml=1R

2i jl + �

∑ml=1∑m−1

l ′ �= l=1Ri jl Ri jl ′

Ri++

= �2

Ri++

k∑j=1

∑ml=1R

2i jl + �[(∑m

l=1Ri jl)2 −∑m

l=1R2i jl ]

Ri++

= �2

Ri++

[1 +

(k∑j=1

(m∑l=1

Ri jl

)2/

Ri++ − 1

)�

]

which yields the result (7).

APPENDIX B

In this appendix, we derive the unconditional variance of the estimated intervention effect in thepresence of attrition (equation (10)). Let Ri++ =∑k

j=1∑m

l=1Ri jl . Since

E(Y i |R) =∑k

j=1∑m

l=1Ri jl E(Yi jl)

Ri++= �i

is constant with respect to the missing data mechanism, the second term on the right of equation(9) is 0; the first term requires taking the expected value of the variance in equation (7), i.e.

ER Var(Y 2 − Y 1|R)= �22∑

i=1ER

(Di

Ri++

)

Provided that the coefficient of variation of Ri++ is small, this expectation can be approximatedas

ER Var(Y 2 − Y 1|R) ≈ �22∑

i=1

ER(Di )

ER(Ri++)(B1)

Following Hartley and Ross [21], proof of this assertion begins by noting that

Cov

(Di

Ri++, Ri++

)= ER(Di ) − ER(Ri++)ER

(Di

Ri++

)

so that

ER

(Di

Ri++

)= ER(Di )

ER(Ri++)− 1

ER(Ri++)Cov

(Di

Ri++, Ri++

)



The absolute value of the error in the approximation can then be written as∣∣∣∣ER

(Di

Ri++

)− ER(Di )

ER(Ri++)

∣∣∣∣ = |Corr(Di/Ri++, Ri++)√Var(Di/Ri++)

√Var(Ri++)|

ER(Ri++)

so that

|ER(Di/Ri++) − ER(Di )/ER(Ri++)|√Var(Di/Ri++)

�√Var(Ri++)

ER(Ri++)

Thus, the error in the approximation is negligible in relation to the standard error if the coefficientof variation of Ri++ is small.

A similar approach can be used to show that

ER(Di ) ≈ 1 +[Var(

∑ml=1Ri jl) + [∑m

l=1ER(Ri jl)]2ER(Ri++)

− 1

]�

Substitution into equation (B1) yields the desired result.

ACKNOWLEDGEMENTS

This work was partially supported by an Ontario Graduate Scholarship and grants from the NaturalSciences and Engineering Research Council of Canada. The authors would like to thank an anonymousreviewer for valuable suggestions on an earlier version of this manuscript.

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Accounting for expected attrition in the planning of community intervention trials