Using Power Analysis to Choose the Unit of Randomization, Outcome, and Approach for Subgroup Analysis for a Multilevel Randomized Controlled Clinical Trial to Reduce Disparities in Cardiovascular Health

We give examples of three features in the design of randomized controlled clinical trials which can increase power and thus decrease sample size and costs. We consider an example multilevel trial with several levels of clustering. For a fixed number of independent sampling units, we show that power can vary widely with the choice of the level of randomization. We demonstrate that power and interpretability can improve by testing a multivariate outcome rather than an unweighted composite outcome. Finally, we show that using a pooled analytic approach, which analyzes data for all subgroups in a single model, improves power for testing the intervention effect compared to a stratified analysis, which analyzes data for each subgroup in a separate model. The power results are computed for a proposed prevention research study. The trial plans to randomize adults to either telehealth (intervention) or in-person treatment (control) to reduce cardiovascular risk factors. The trial outcomes will be measures of the Essential Eight, a set of scores for cardiovascular health developed by the American Heart Association which can be combined into a single composite score. The proposed trial is a multilevel study, with outcomes measured on participants, participants treated by the same provider, providers nested within clinics, and clinics nested within hospitals. Investigators suspect that the intervention effect will be greater in rural participants, who live farther from clinics than urban participants. The results use published, exact analytic methods for power calculations with continuous outcomes. We provide example code for power analyses using validated software. Supplementary Information The online version contains supplementary material available at 10.1007/s11121-024-01673-y.

The following statements of the theory for the general linear mixed and multivariate models uses a unified notation system that avoids notational conflicts.An explicit process is given to transform any multivariate model to a mixed model.For each design question considered, inputs for power analysis are given for a multivariate model and then for a mixed model.A method for defining simpler models that are power equivalent to the original model of interest is outlined.
The (left) direct product of two matrices is written A ⊗ B = {a ij B}.More generally, for a set of square matrices, {A i }, the direct sum, written D = N i=1 A i , is the block diagonal matrix with the {A i } on the diagonal.
It is common to define the general linear mixed model as a statement about only the observations for independent sampling unit i ∈ {1, . . ., N } (see Muller and Stewart (2006), Section 6.7).The subscripts m and M distinguish mixed from multivariate model properties.The mixed model for p i × 1 vector of responses y i includes fixedeffect predictors X mi (p i × q m , fixed and known) and random-effect predictors Z i (p i × r, fixed and known): (1) Here (q m × 1) matrix β m contains unknown fixed-effect parameters, and X mi includes both between-and within-hospital design information.The distributions are For independent sampling unit i and e mi = Z i d i + e mi ∼ N (0, Σ mi ), the "population-average" form of a mixed model is (3) Muller and Stewart (2006), Section 5.2, provided explicit notation to write the model equation for all data considered together as Here and A general linear mixed model hypothesis about fixed effects can be written in terms of The multivariate general linear model, and its special case the univariate general linear model, is customarily written to describe all observations in the sample: with N rows corresponding to independent sampling units, p columns in Y , B and E and q columns in X M .Here row i (E) ∼ N p (0, Σ M ).Fixed, conforming and known constant contrast matrices C M and U define the secondary parameter matrix Θ M = C M B M U .For fixed, conforming and known constant matrix Θ 0M , the multivariate general linear hypothesis is Assuming X M is fixed and known without appreciable error and the type I error rate is α, six matrices suffice to fully specify a multivariate power analysis: X M , Σ M , C M , B M , U and Θ 0M .All X M are full rank of q, which implies the univariate error degrees of freedom are ν e = N −rank(X M ) = N − q.
It is convenient to consider the essence matrix, Es(X M ), which is created by deleting any duplicate rows in X M .With N g the replication factor, the number of independent sampling units for each row of Es(X M ), without loss of generality, for a balanced design X M =Es(X M ) ⊗ 1 Ng .With G rows in Es(X M ), a balanced design has a total sample size of N = G • N g .An unbalanced designs has N g varying across rows of Es(X M ) and N = G g=1 N g specifies the total sample size (total number of independent sampling units).

Features Common to All Examples
All examples share the following assumptions.1) The number of independent sampling units is the same for all levels of any effect that varies between independent sampling units.2) All clusters are of equal size.3) There are equal numbers of sub-clusters at each level of nesting.4) The only predictors are fixed and known indicator variables.5) Reference cell coding (Muller & Fetterman, 2003) is used.6) Every independent sampling unit has the same covariance pattern for all observations.
All power values were calculated with the software POWERLIB (Johnson et al., 2009).Code is freely available at https://www.SampleSizeShop.org).POWERLIB is open source code that runs in SAS/IML c .Each program defined a "BETASCAL" vector, b = 0 1 2 • • • d /d.POWERLIB computes a power value for each choice of B • b k .Choosing d ≈ 100 provided enough resolution to give smooth plots.In the mathematics in Online Resource 1, δ reflects BETASCAL.
The cluster dimensions are, from innermost to outermost, k = k 1 k 2 k 3 = 6 6 8 , corresponding to participants nested in providers, providers nested in clinics, and clinics nested in hospitals.Corresponding intraclass correlation parameters for the variance component model (Longford, 1987) all observations from a single hospital is The examples share many, but not all, dimensions.All examples assume 8 outcomes are measured on k * = k 1 • k 2 • k 3 clustered participants for each of 20 hospitals (the independent sampling unit).The subgroup model for question 3 assumes 10 hospitals per subgroup with 5 in each treatment, which gives a total of N = 10 independent sampling units.All other models have N = 20 independent sampling units.The total number of observations is n Power varies nonlinearly with every dimension, as well as the correlation and variance parameters.

Reversible Models
With the assumptions, all models for the examples are reversible, as defined by Chi, Glueck, and Muller (2019).A reversible linear model can be stated as either a general linear mixed model or as a general linear multivariate model.The multivariate statement allows using power and sample size methods that give exact results for the examples (Chi et al., 2019;Muller, Lavange, Ramey, & Ramey, 1992) .For the scenarios considered, the Hotelling-Lawley multivariate test statistic is equivalent to the mixed model Wald test statistic (Gurka, Edwards, & Muller, 2011).Chi et al. (2019) provided sufficient conditions to determine whether a mixed model is reversible.
An explicit transformation process allows converting any general linear multivariate model into a general linear mixed model.Equation 12.6 and subsequent expression in Section 12.1 from Muller and Stewart (2006) define the process.Transposing the multivariate model and applying the vec operator gives the data stacked by independent sampling unit: The last form coincides with equation 4, and hence defines a population-average mixed model for all observations.For independent sampling unit i: Theorem 1.5 in Muller and Stewart (2006) gives In summary, a general linear multivariate model and associated multivariate general linear hypothesis may be expressed as a general linear mixed model and associated general linear hypothesis with the following steps.Multivariate model Y = X M B + E corresponds to the mixed model statement describing all of the data, y m = X m β m + e m .Equivalently, y mi = X mi β m + e mi states the mixed model by describing only the data for independent sampling unit i.The multivariate hypothesis H 0M : C M B M U = Θ 0M corresponds to the general linear hypothesis for the mixed model For all of the data For the observations for independent sampling unit i, Here e mi ∼ N p (0, Σ M ) and e m ∼ N N •p (0, I N ⊗ Σ M ).For either model statement 1.4 Question 1 (Figure 2): Does Power Vary with Randomization Level?
Each level of randomization requires a distinct multivariate model.In all models, δ is the intervention effect.For randomization level 1) participant, 2) provider, 3) clinic, or 4) hospital, the corresponding models are For question 1, levels 1-3 have between contrast matrix 1 and level 4 has C M 1,4 = 0 1 .The exact Hotelling-Lawley test statistic is referenced to an F distribution with degrees of freedom {1, 19} for levels 1-3 and {1, 18} for level 4. The within-independent sampling unit contrast matrix varies across levels for question 1: ) In all cases, Θ M 1,L = δ/8 and Θ 0M 1,L = 0 .The value reflects the assumption of a single composite outcome and the alternative hypothesis that only 1 of 8 of the variables in the composite respond to the intervention.
Since all multivariate models are reversible, equation 9 allows describing corresponding population average mixed models using the following expressions: For randomization level 1) participant, 2) provider, 3) clinic, or 4) hospital, The last equation uses the equivalence The mixed model contrast matrices for randomization at the level of participant, provider, clinic or hospital, are In all cases θ m1,L = δ/8 .All mixed models have 1.5 Question 2 (Figure 3): Do Composite and Multivariate Tests Differ in Power?
The example randomizes at the level of the independent sampling unit (Hospital), a group-randomized design, and has no subgroups.Answering the question compares two distinct tests within a single model.Using reference cell coding, the multivariate model is The within contrasts are for composite and for multivariate ) with degrees of freedom {1, 18} for the composite F and {1, 11} for the multivariate F .Also, Θ 2C = δ/8 and Θ 2M = δ 0 0 0 0 0 0 0 .
1.6 Question 3 (Figure 4): Do Subgroup and Pooled Analyses Differ in Power?
The multivariate models are for a subgroup and for pooled data The corresponding mixed model design matrices are The parameter matrices are The mixed model contrasts are
Power analyses for Question 2, comparing composite and multivariate tests, used the transformation T 4 in equation 59 for the composite test.The multivariate test used T 4M = I 8 ⊗ 1 k3 ⊗ 1 k2 ⊗ 1 k1 /k * .The transformed models used U = 1 for the composite test and U = I 8 for the multivariate test.The values of C M 2 and Θ 0M,2 remain the same in the transformed models.
Power analyses for Question 3, subgroup and pooled analysis, used the transformation T 4 in equation 59 for both the subgroup and pooled models.The transformed models used The values of C M 3 and Θ 0M,3 remain the same in the transformed models.
The values of C M and Θ 0M remain the same in the transformed models.Replacing U with I b defines the power equivalent hypothesis in the transformed model asH 0 : C M B H = Θ 0M .With Σ * = U Σ M U ,in the power equivalent model, row i (E H ) ∼ N p (0, Σ * ).