Planning a cluster randomized trial with unequal cluster sizes: practical issues involving continuous outcomes

A cluster randomized trial involves randomizing social units or clusters of individuals rather than the individuals themselves. This design, which is increasingly being used for evaluating healthcare, screening and educational interventions presents specific constraints that must be considered during planning and analysis [ 1, 2]. Indeed, the responses of individuals within a cluster tend to be more similar than those of individuals of different clusters, and we thus define the clustering effect as 1 + ( m - 1) ρ, where m is the average number of subjects per cluster and ρ the intraclass correlation coefficient (ICC). This clustering effect is used during the planning of cluster randomized trials as an inflation factor to increase the sample size required by an individual randomization trial. However, such an approach does not take into account variations in cluster size, which might differ greatly. Indeed, as illustrated by Kerry et al [ 3], cluster size may depend on, for example, (i) the potential of recruitment of the cluster (i.e., the number of subjects belonging to each cluster), (ii) the eligible fraction of subjects, which may vary among clusters, or (iii) the ability of physicians to recruit subjects within each cluster. Such an imbalance in cluster size reduces the power of the trial and has to be taken into account in the sample size calculation.

Kerry et al [ 3] assessed the theoretical efficacy of 3 weightings of the inflation factor but in the context of cluster level analysis, so summary statistics are estimated at the cluster level and the unit of analysis remains the cluster. Manatunga et al [ 4], however, assessed a correction on the basis of the assumed distribution of cluster sizes in the context of marginal models, but the authors' simulations covered a range of ICCs larger than those usually observed in cluster randomized trials.

Our aim was therefore to assess these proposed corrections in the framework of cluster randomized trials in which the unit of analysis remains the subject, embedded in the cluster. We first describe the random effects model used to simulate clustered data; then display the simulation design used to evaluate the loss of power due to imbalance in cluster size and the findings. Corrections of the variance inflation factor to allow for cluster size inequality evaluated by simulation and robustness of these corrections to misspecification of the ICC is assessed. practical guidelines for the planning stage of cluster randomized trials are drawn and perspectives for future research.

Results are expressed as absolute bias and mean square error on the one hand, and empirical' type I error and power on the other. Table 1 displays the results associated with an a priori postulated effect size of 0.25, while Table 2 displays the results associated with a 0.50 effect size. In 7 situations, data sets could not be generated for the following combinations ES/ICC/g: 0.25/0.020/5, 0.25/0.050/5, 0.25/0.050/10, 0.25/0.100/5, 0.25/0.100/10, 0.25/0.100/20 and 0.50/0.100/5. Indeed, when the number of clusters is small and/or the ICC high, even an infinite cluster size may not allow for achieving 80% power [ 20].

No significant bias was induced by inequality in cluster size (since the relative bias was no more than about 1.5%, in absolute value), while the mean square error was barely increased in cases of severe imbalance (Pareto imbalance).

Thus, while moderate imbalances (based on an equiprobability hypothesis) and Poisson's imbalances can be neglected at the planning stage, a more severe imbalance (such as the Pareto's imbalance) should be taken into account, thus leading to an adjustment in sample size calculations.

Results are displayed in Figures 1 and 2 for an effect size of 0.25 and a priori postulated ICC values of 0.005 and 0.020, respectively. As expected [ 20], in any situation, the power decreases as the ICC increases, and this result is all the more important when the number of clusters is low. In the planning situations explored, minimum variance weights and cluster size weights curves are very close, except when 20 clusters per intervention arm are randomized and the ICC is a priori fixed at 0.020, but this latter situation is extreme, as discussed previously. Otherwise, the power associated with equal weights remains greater than that associated with minimum variance weights in any situation. However, this finding probably just reflects that the use of this weighting system leads to higher required sample sizes than the use of a minimum variance weights system (cf Tables 3 and 4) and therefore higher power. In any case, imbalance in cluster size is associated with a higher sensitivity to the a priori-specified ICC than constant cluster size. For example, let us consider the case of 20 clusters per intervention arm: if the ICC is a priori postulated at 0.005, but in reality equals 0.015, the power associated with constant cluster size decreases from 0.80 to 0.75 only, whereas the power associated with Pareto repartition decreases from 0.80 to 0.68 (with the minimum variance weighting system). However, all weighting systems show great sensitivity to the actual value of the ICC. Consider the former example (ES = 0.25, g = 20 and Pareto repartition, increase in ICC from 0.005 to 0.015), the power associated with equal weights will decrease from 0.98 to 0.90, and the power associated with cluster size weights from 0.80 to 0.68. Thus, if little prior knowledge is available concerning the value of the ICC, the sensitivity analysis involving several values of ICC is of major importance, particularly when imbalance in cluster size is expected.

A moderate inequality in cluster sizes has little effect on power and can thus be neglected at the planning stage. However, a major imbalance in cluster sizes, like the "Pareto" imbalance, (i.e. 80% of the subjects belong to only 20% of the clusters) is associated with a loss in power, and the phenomenon is all the more important when the number of clusters is low and/or the ICC is high. In these situations, the minimum variance weights correction has good properties and allows for achieving the nominal power. This result, obtained in the extreme situation of a Pareto imbalance, suggests that this correction can be used to derive sample size or power in any situation where, in each group, cluster sizes can be separated in two strata, the small cluster stratum and the big cluster stratum. The higher sensitivity of severely unbalanced trials to the a priori-postulated value of the ICC compared to that of balanced trials emphasized the necessity of a sensitivity analysis on this parameter. We derived an adaptation of the VIF, which should be used when the imbalance is a priori hypothesized to be "a proportion of γ clusters will actually recruit a proportion τ of the subjects to be included".

A limit to this approach remains the degree of imbalance being usually difficult to foresee at the planning stage, except when, for instance, families or practices are randomized and clusters as a whole are included in the trial. In these latter situations, one may a priori know precisely the cluster size repartition and therefore use the minimum variance weights correction as initially specified by Kerry et al [ 3]. However, if, within each cluster, the physician has to recruit patients to be included in the trial, cluster size distribution may then be difficult to hypothesize. It is all the more difficult since cluster sizes are usually not reported in published clustered randomized trials. We therefore proposed to consider that cluster sizes distribution can be divided in each arm in two strata: a stratum of small clusters, and another of large clusters. This hypothesis may be debatable. However, since a moderate inequality of cluster size is of minor effect, it seems a rather useful and simple way to consider the risk of cluster size inequality at the planning stage, particularly since no precise data on cluster size inequality are available. Another limitation is that our work focused on normally distributed continuous outcomes. More work is needed to extend our results to non-normal distributions, especially with binary variables. Finally, we restricted our work to cases of no differential recruitment between arms, thus considering that imbalance is the same in the two arms. Such a hypothesis may be questionable in cluster randomized trials: since inclusion is posterior to randomization, this may indeed induce differential recruitment and imbalance in patient characteristics, which may lead to questioning the results of the study [ 24].

In conclusion, our study demonstrates that severely imbalanced trials with continuous outcomes may be highly underpowered. If such imbalance in cluster size can be anticipated at the design stage, minimum variance weights correction should be used to inflate the required sample size. A priori estimation of the expectable imbalance would be facilitated if more details on cluster sizes were given in published cluster randomized trials, as was recently advised in the extension of the CONSORT statement for cluster randomized trials [ 25]. Moreover, such publication of cluster sizes would be of particular interest to assess the real power of the trial conducted.

Four corrections have been proposed for adjusting sample size in cases of imbalance in cluster size. Considering the specific situation of a Pareto imbalance, the general form of these corrections can be simplified.

Characteristics of the Pareto imbalance

g refers to the number of clusters within each arm and m ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeaaaa@2E27@ is the average cluster size

Equal weights ( denoted w ₁) [ 3]

With an equal weights correction, the VIF is expressed as:

where m ¯ = 1 g ∑ j = 1 g m j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeaiabg2da9maalaaabaGaeGymaedabaGaem4zaCgaamaaqahabaGaemyBa02aaSbaaSqaaiabdQgaQbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaem4zaCganiabggHiLdaaaa@3B51@

With a Pareto imbalance, this equation is expressed as:

V I F w 1 = m ¯ w 1 g ( 0.8 g 0.25 m ¯ w 1 + 0.2 g 4 m ¯ w 1 ) ( 1 − ρ ) + m ¯ w 1 ρ = 3 . 2 5 + ( m ¯ w 1 − 3.25 ) ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6FAB@

where m ¯ w 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeGymaedabeaaaSqabaaaaa@30F2@ (the average cluster size for which an equal weights correction is used) is defined as:

m ¯ w 1 g = 2 T 2 E S 2 [ 3.25 + ( m ¯ w 1 − 3.25 ) ρ ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeGymaedabeaaaSqabaGccqWGNbWzcqGH9aqpdaWcaaqaaiabikdaYiabdsfaunaaCaaaleqabaGaeGOmaidaaaGcbaGaemyrauKaem4uam1aaWbaaSqabeaacqaIYaGmaaaaaOWaamWaaeaacqaIZaWmcqGGUaGlcqaIYaGmcqaI1aqncqGHRaWkdaqadaqaaiqbd2gaTzaaraWaaSbaaSqaaiabdEha3naaBaaameaacqaIXaqmaeqaaaWcbeaakiabgkHiTiabiodaZiabc6caUiabikdaYiabiwda1aGaayjkaiaawMcaaGGaciab=f8aYbGaay5waiaaw2faaaaa@4D05@

which leads to :

m ¯ w 1 = 6.5 ( 1 − ρ ) T 2 g E S 2 − 2 ρ T 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeGymaedabeaaaSqabaGccqGH9aqpdaWcaaqaaiabiAda2iabc6caUiabiwda1maabmaabaGaeGymaeJaeyOeI0ccciGae8xWdihacaGLOaGaayzkaaGaemivaq1aaWbaaSqabeaacqaIYaGmaaaakeaacqWGNbWzcqWGfbqrcqWGtbWudaahaaWcbeqaaiabikdaYaaakiabgkHiTiabikdaYiab=f8aYjabdsfaunaaCaaaleqabaGaeGOmaidaaaaakiabcYcaSaaa@4805@

where ES refers to the effect size and T = t _{(1 - α/2),2(g- 1)} + t _{(1 - β),2(g - 1)}

Cluster size weights ( denoted w ₂) [ 3]

where m A ¯ = ∑ j = 1 g m j 2 ∑ j = 1 g m j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabd2gaTnaaBaaaleaacqWGbbqqaeqaaaaakiabg2da9maalaaabaWaaabCaeaacqWGTbqBdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdEgaNbqdcqGHris5aaGcbaWaaabCaeaacqWGTbqBdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGNbWza0GaeyyeIuoaaaaaaa@450E@

With a Pareto imbalance, we can write the equation as:

m ¯ A = 0.8 g ( 0.25 m ¯ w 2 ) 2 + 0.2 g ( 4 m ¯ w 2 ) 2 m ¯ w 2 g = 3.25 m ¯ w 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5B15@

So the VIF is reduced to:

V I F w 2 = 1 + ( 3.25 m ¯ w 2 − 1 ) ρ ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqWGjbqscqWGgbGrdaWgaaWcbaGaem4DaC3aaSbaaWqaaiabikdaYaqabaaaleqaaOGaeyypa0JaeGymaeJaey4kaSYaaeWaaeaacqaIZaWmcqGGUaGlcqaIYaGmcqaI1aqncuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGHsislcqaIXaqmaiaawIcacaGLPaaaiiGacqWFbpGCcaWLjaGaaCzcamaabmaabaGaeGOnaydacaGLOaGaayzkaaaaaa@46C8@

and

m ¯ w 2 = 2 ( 1 − ρ ) T 2 g E S 2 − 6.5 ρ T 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGH9aqpdaWcaaqaaiabikdaYmaabmaabaGaeGymaeJaeyOeI0ccciGae8xWdihacaGLOaGaayzkaaGaemivaq1aaWbaaSqabeaacqaIYaGmaaaakeaacqWGNbWzcqWGfbqrcqWGtbWudaahaaWcbeqaaiabikdaYaaakiabgkHiTiabiAda2iabc6caUiabiwda1iab=f8aYjabdsfaunaaCaaaleqabaGaeGOmaidaaaaaaaa@471D@

Minimum variance weights ( denoted w ₃) [ 3]

V I F w 3 = m ¯ g ∑ j = 1 g m j 1 + ( m j − 1 ) ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqWGjbqscqWGgbGrdaWgaaWcbaGaem4DaC3aaSbaaWqaaiabiodaZaqabaaaleqaaOGaeyypa0ZaaSaaaeaacuWGTbqBgaqeaiabdEgaNbqaamaaqahabaWaaSaaaeaacqWGTbqBdaWgaaWcbaGaemOAaOgabeaaaOqaaiabigdaXiabgUcaRmaabmaabaGaemyBa02aaSbaaSqaaiabdQgaQbqabaGccqGHsislcqaIXaqmaiaawIcacaGLPaaaiiGacqWFbpGCaaaaleaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGNbWza0GaeyyeIuoaaaaaaa@4AB9@

With a Pareto imbalance, the equation can be written as:

V I F w 3 = m ¯ w 3 g 0.8 g 0.25 m ¯ w 3 1 + ( 0.25 m ¯ w 3 − 1 ) ρ + 0.2 g 4 m ¯ w 3 1 + ( 4 m ¯ w 3 − 1 ) ρ = ( 1 + ( 4 m ¯ w 3 − 1 ) ρ ) ( 1 + ( 0.25 m ¯ w 3 − 1 ) ρ ) 1 + ( m ¯ w 3 − 1 ) ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f8aYbaaaaaabaGaeyypa0ZaaSaaaeaadaqadaqaaiabigdaXiabgUcaRmaabmaabaGaeGinaqJafmyBa0MbaebadaWgaaWcbaGaem4DaC3aaSbaaWqaaiabiodaZaqabaaaleqaaOGaeyOeI0IaeGymaedacaGLOaGaayzkaaGae8xWdihacaGLOaGaayzkaaWaaeWaaeaacqaIXaqmcqGHRaWkdaqadaqaaiabicdaWiabc6caUiabikdaYiabiwda1iqbd2gaTzaaraWaaSbaaSqaaiabdEha3naaBaaameaacqaIZaWmaeqaaaWcbeaakiabgkHiTiabigdaXaGaayjkaiaawMcaaiab=f8aYbGaayjkaiaawMcaaaqaaiabigdaXiabgUcaRmaabmaabaGafmyBa0MbaebadaWgaaWcbaGaem4DaC3aaSbaaWqaaiabiodaZaqabaaaleqaaOGaeyOeI0IaeGymaedacaGLOaGaayzkaaGae8xWdihaaaaaaa@964E@

with

m ¯ w 3 g = 2 T 2 E S 2 [ ( 1 + ( 4 m ¯ w 3 − 1 ) ρ ) ( 1 + ( 0.25 m ¯ w 3 − 1 ) ρ ) 1 + ( m ¯ w 3 − 1 ) ρ ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@65CB@

which leads to m ¯ w 3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWG3bWDdaWgaaadbaGaeG4mamdabeaaaSqabaaaaa@30F6@ being the positive solution of the following equation:

m ¯ w 3 2 ρ [ g E S 2 − 2 ρ T 2 ] + m ¯ w 3 ( 1 − ρ ) [ g E S 2 − 8.5 ρ T 2 ] − 2 ( 1 − ρ ) 2 T 2 = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaDaaaleaacqWG3bWDdaWgaaadbaGaeG4mamdabeaaaSqaaiabikdaYaaaiiGakiab=f8aYnaadmaabaGaem4zaCMaemyrauKaem4uam1aaWbaaSqabeaacqaIYaGmaaGccqGHsislcqaIYaGmcqWFbpGCcqWGubavdaahaaWcbeqaaiabikdaYaaaaOGaay5waiaaw2faaiabgUcaRiqbd2gaTzaaraWaaSbaaSqaaiabdEha3naaBaaameaacqaIZaWmaeqaaaWcbeaakmaabmaabaGaeGymaeJaeyOeI0Iae8xWdihacaGLOaGaayzkaaWaamWaaeaacqWGNbWzcqWGfbqrcqWGtbWudaahaaWcbeqaaiabikdaYaaakiabgkHiTiabiIda4iabc6caUiabiwda1iab=f8aYjabdsfaunaaCaaaleqabaGaeGOmaidaaaGccaGLBbGaayzxaaGaeyOeI0IaeGOmaiZaaeWaaeaacqaIXaqmcqGHsislcqWFbpGCaiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaakiabdsfaunaaCaaaleqabaGaeGOmaidaaOGaeyypa0JaeGimaadaaa@65C0@

Distribution-based correction ( denoted d) [ 4]

V I F d = 1 + ( E ( m ) 2 + var ( m ) E ( m ) − 1 ) ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqWGjbqscqWGgbGrdaWgaaWcbaGaemizaqgabeaakiabg2da9iabigdaXiabgUcaRmaabmaabaWaaSaaaeaacqWGfbqrdaqadaqaaiabd2gaTbGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaOGaey4kaSIagiODayNaeiyyaeMaeiOCai3aaeWaaeaacqWGTbqBaiaawIcacaGLPaaaaeaacqWGfbqrdaqadaqaaiabd2gaTbGaayjkaiaawMcaaaaacqGHsislcqaIXaqmaiaawIcacaGLPaaaiiGacqWFbpGCaaa@4ACE@

So we have:

with

m ¯ d g = 2 T 2 E S 2 [ 1 + ( 3.25 m ¯ d − 1 ) ρ ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWGKbazaeqaaOGaem4zaCMaeyypa0ZaaSaaaeaacqaIYaGmcqWGubavdaahaaWcbeqaaiabikdaYaaaaOqaaiabdweafjabdofatnaaCaaaleqabaGaeGOmaidaaaaakmaadmaabaGaeGymaeJaey4kaSYaaeWaaeaacqaIZaWmcqGGUaGlcqaIYaGmcqaI1aqncuWGTbqBgaqeamaaBaaaleaacqWGKbazaeqaaOGaeyOeI0IaeGymaedacaGLOaGaayzkaaacciGae8xWdihacaGLBbGaayzxaaaaaa@4887@

that is to say:

m ¯ d = 2 ( 1 − ρ ) T 2 g E S 2 − 6.5 ρ T 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeamaaBaaaleaacqWGKbazaeqaaOGaeyypa0ZaaSaaaeaacqaIYaGmdaqadaqaaiabigdaXiabgkHiTGGaciab=f8aYbGaayjkaiaawMcaaiabdsfaunaaCaaaleqabaGaeGOmaidaaaGcbaGaem4zaCMaemyrauKaem4uam1aaWbaaSqabeaacqaIYaGmaaGccqGHsislcqaI2aGncqGGUaGlcqaI1aqncqWFbpGCcqWGubavdaahaaWcbeqaaiabikdaYaaaaaaaaa@45CD@

One then recognizes the results obtained using the cluster size weights correction.

Characteristics of the Pareto-like imbalance

g refers to the number of clusters within each arm and m ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGTbqBgaqeaaaa@2E27@ is the average cluster size

V I F w 3 = m ¯ g ∑ j = 1 g m j 1 + ( m j − 1 ) ρ = m ¯ g ( 1 − γ ) g 1 − τ 1 − γ m ¯ 1 + ( 1 − τ 1 − γ m ¯ − 1 ) ρ + γ g τ γ m ¯ 1 + ( τ γ m ¯ − 1 ) ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaaeeqaaiabdAfawjabdMeajjabdAeagnaaBaaaleaacqWG3bWDdaWgaaadbaGaeG4mamdabeaaaSqabaGccqGH9aqpdaWcaaqaaiqbd2gaTzaaraGaem4zaCgabaWaaabCaeaadaWcaaqaaiabd2gaTnaaBaaaleaacqWGQbGAaeqaaaGcbaGaeGymaeJaey4kaSYaaeWaaeaacqWGTbqBdaWgaaWcbaGaemOAaOgabeaakiabgkHiTiabigdaXaGaayjkaiaawMcaaGGaciab=f8aYbaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdEgaNbqdcqGHris5aaaaaOqaaiabg2da9maalaaabaGafmyBa0MbaebacqWGNbWzaeaadaqadaqaaiabigdaXiabgkHiTiab=n7aNbGaayjkaiaawMcaaiabdEgaNnaalaaabaWaaSaaaeaacqaIXaqmcqGHsislcqWFepaDaeaacqaIXaqmcqGHsislcqWFZoWzaaGafmyBa0MbaebaaeaacqaIXaqmcqGHRaWkdaqadaqaamaalaaabaGaeGymaeJaeyOeI0Iae8hXdqhabaGaeGymaeJaeyOeI0Iae83SdCgaaiqbd2gaTzaaraGaeyOeI0IaeGymaedacaGLOaGaayzkaaGae8xWdihaaiabgUcaRiab=n7aNjabdEgaNnaalaaabaWaaSaaaeaacqWFepaDaeaacqWFZoWzaaGafmyBa0MbaebaaeaacqaIXaqmcqGHRaWkdaqadaqaamaalaaabaGae8hXdqhabaGae83SdCgaaiqbd2gaTzaaraGaeyOeI0IaeGymaedacaGLOaGaayzkaaGae8xWdihaaaaaaaaa@821D@

So we obtain:

V I F = [ 1 + ( 1 − τ 1 − γ m − 1 ) ρ ] [ 1 + ( τ γ m − 1 ) ρ ] τ [ 1 + ( 1 − τ 1 − γ m − 1 ) ρ ] + ( 1 − τ ) [ 1 + ( τ γ m − 1 ) ρ ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqWGjbqscqWGgbGrcqGH9aqpdaWcaaqaamaadmaabaGaeGymaeJaey4kaSYaaeWaaeaadaWcaaqaaiabigdaXiabgkHiTGGaciab=r8a0bqaaiabigdaXiabgkHiTiab=n7aNbaacqWGTbqBcqGHsislcqaIXaqmaiaawIcacaGLPaaacqWFbpGCaiaawUfacaGLDbaadaWadaqaaiabigdaXiabgUcaRmaabmaabaWaaSaaaeaacqWFepaDaeaacqWFZoWzaaGaemyBa0MaeyOeI0IaeGymaedacaGLOaGaayzkaaGae8xWdihacaGLBbGaayzxaaaabaGae8hXdq3aamWaaeaacqaIXaqmcqGHRaWkdaqadaqaamaalaaabaGaeGymaeJaeyOeI0Iae8hXdqhabaGaeGymaeJaeyOeI0Iae83SdCgaaiabd2gaTjabgkHiTiabigdaXaGaayjkaiaawMcaaiab=f8aYbGaay5waiaaw2faaiabgUcaRmaabmaabaGaeGymaeJaeyOeI0Iae8hXdqhacaGLOaGaayzkaaWaamWaaeaacqaIXaqmcqGHRaWkdaqadaqaamaalaaabaGae8hXdqhabaGae83SdCgaaiabd2gaTjabgkHiTiabigdaXaGaayjkaiaawMcaaiab=f8aYbGaay5waiaaw2faaaaaaaa@7757@

g i n i = 1 2 g 2 m ¯ ∑ i = 1 g ∑ j = 1 g | m i − m j | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGNbWzcqWGPbqAcqWGUbGBcqWGPbqAcqGH9aqpdaWcaaqaaiabigdaXaqaaiabikdaYiabdEgaNnaaCaaaleqabaGaeGOmaidaaOGafmyBa0MbaebaaaWaaabCaeaadaaeWbqaamaaemaabaGaemyBa02aaSbaaSqaaiabdMgaPbqabaGccqGHsislcqWGTbqBdaWgaaWcbaGaemOAaOgabeaaaOGaay5bSlaawIa7aaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaem4zaCganiabggHiLdaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGNbWza0GaeyyeIuoaaaa@50E0@

Given the characteristics of the Pareto-like imbalance presented in appendix B, considering that clusters are ordered hierarchically by increasing size, the matrix of the difference | m _i - m _j | can be written as:

M = ( 0 ( γ g , γ g ) 1 ( γ g , ( 1 − γ ) g ) | τ − γ γ ( 1 − γ ) | 1 ( ( 1 − γ ) g , γ g ) | τ − γ γ ( 1 − γ ) | 0 ( ( 1 − γ ) g , ( 1 − γ ) g ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGnbqtcqGH9aqpdaqadaqaauaabeqaciaaaeaacqaIWaamdaWgaaWcbaWaaeWaaeaaiiGacqWFZoWzcqWGNbWzcqGGSaalcqWFZoWzcqWGNbWzaiaawIcacaGLPaaaaeqaaaGcbaGaeGymaeZaaSbaaSqaamaabmaabaGae83SdCMaem4zaCMaeiilaWYaaeWaaeaacqaIXaqmcqGHsislcqWFZoWzaiaawIcacaGLPaaacqWGNbWzaiaawIcacaGLPaaaaeqaaOWaaqWaaeaadaWcaaqaaiab=r8a0jabgkHiTiab=n7aNbqaaiab=n7aNnaabmaabaGaeGymaeJaeyOeI0Iae83SdCgacaGLOaGaayzkaaaaaaGaay5bSlaawIa7aaqaaiabigdaXmaaBaaaleaadaqadaqaamaabmaabaGaeGymaeJaeyOeI0Iae83SdCgacaGLOaGaayzkaaGaem4zaCMaeiilaWIae83SdCMaem4zaCgacaGLOaGaayzkaaaabeaakmaaemaabaWaaSaaaeaacqWFepaDcqGHsislcqWFZoWzaeaacqWFZoWzdaqadaqaaiabigdaXiabgkHiTiab=n7aNbGaayjkaiaawMcaaaaaaiaawEa7caGLiWoaaeaacqaIWaamdaWgaaWcbaWaaeWaaeaadaqadaqaaiabigdaXiabgkHiTiab=n7aNbGaayjkaiaawMcaaiabdEgaNjabcYcaSmaabmaabaGaeGymaeJaeyOeI0Iae83SdCgacaGLOaGaayzkaaGaem4zaCgacaGLOaGaayzkaaaabeaaaaaakiaawIcacaGLPaaaaaa@8068@

Where 0 _(γg,γg) and 0 _{((1-γ)g,(1-γ)g)} are squared matrices of size γg and (1- γ) g respectively and 1 _{(γg,(1-γ)g)} and 1 _{((1-γ)g,γg)} are matices of size γg × (1- γ) g and (1- γ) g × γg respectively, containing only 1 s.

Thus:

g i n i = 1 2 g 2 m ¯ 2 g 2 γ ( 1 − γ ) | τ − γ γ ( 1 − γ ) | m ¯ g i n i = | τ − γ | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdEgaNjabdMgaPjabd6gaUjabdMgaPjabg2da9maalaaabaGaeGymaedabaGaeGOmaiJaem4zaC2aaWbaaSqabeaacqaIYaGmaaGccuWGTbqBgaqeaaaacqaIYaGmcqWGNbWzdaahaaWcbeqaaiabikdaYaaaiiGakiab=n7aNnaabmaabaGaeGymaeJaeyOeI0Iae83SdCgacaGLOaGaayzkaaWaaqWaaeaadaWcaaqaaiab=r8a0jabgkHiTiab=n7aNbqaaiab=n7aNnaabmaabaGaeGymaeJaeyOeI0Iae83SdCgacaGLOaGaayzkaaaaaaGaay5bSlaawIa7aiqbd2gaTzaaraaabaGaem4zaCMaemyAaKMaemOBa4MaemyAaKMaeyypa0ZaaqWaaeaacqWFepaDcqGHsislcqWFZoWzaiaawEa7caGLiWoaaaaa@60CF@