PowerUpR (version 1.1.0)

cra3: Three-level Cluster-randomized Trials to Detect Main, Moderation, and Mediation Effects

Description

For main treatment effects, use mdes.cra3() to calculate the minimum detectable effect size, power.cra3() to calculate the statistical power, mrss.cra3() to calculate the minimum required sample size (number of clusters).

For moderator at level 1, use mdesd.mod331() to calculate the minimum detectable effect size, power.mod331() to calculate the statistical power, mrss.mod331() to calculate the minimum required sample size (number of clusters).

For moderator at level 2, use mdesd.mod332() to calculate the minimum detectable effect size, power.mod332() to calculate the statistical power, mrss.mod332() to calculate the minimum required sample size (number of clusters).

For moderator at level 3, use mdesd.mod333() to calculate the minimum detectable effect size, power.mod333() to calculate the statistical power, mrss.mod333() to calculate the minimum required sample size (number of clusters).

For mediator at level 3, use power.med331(), for mediator at level 2, use power.med321(), for mediator at level 1, use power.med311() to calculate the statistical power.

For cluster-randomized block designs (treatment at level 3, with fixed effects across level 4 blocks), use mdes.bcra4f3() to calculate the minimum detectable effect size, power.bcra4f3() to calculate the statistical power, and mrss.bcra4f3() to calculate the minimum required sample size (number of clusters per block).

Usage

mdes.cra3(power=.80, alpha=.05, two.tailed=TRUE,
          rho2, rho3, p=.50, g3=0, r21=0, r22=0, r23=0,
          n, J, K)
mdesd.mod331(power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, omegam2, omegam3,
            r21=0, r2m3=0,
            p=.50, q=NULL, n, J, K)
mdesd.mod332(power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, omegam3, r21=0, r22=0, r2m3=0,
            p=.50, q=NULL, n, J, K)
mdesd.mod333(power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, r21=0, r22=0, r23=0,
            p=.50, q=NULL, n, J, K)
power.cra3(es=.25, alpha=.05, two.tailed=TRUE,
           rho2, rho3, g3=0, r21=0, r22=0, r23=0,
           p=.50, n, J, K)
power.mod331(es=.25, alpha=.05, two.tailed=TRUE,
             rho2, rho3, omegam2, omegam3,
             r21=0, r2m3=0,
             p=.50, q=NULL, n, J, K)
power.mod332(es=.25, alpha=.05, two.tailed=TRUE,
             rho2, rho3, omegam3, r21=0, r22=0, r2m3=0,
             p=.50, q=NULL, n, J, K)
power.mod333(es=.25, alpha=.05, two.tailed=TRUE,
             rho2, rho3, r21=0, r22=0, r23=0,
             p=.50, q=NULL, n, J, K)
power.med331(esa, esB, two.tailed=TRUE, alpha=.05,
             mc=TRUE, nsims=1000, ndraws=1000,
             rho2, rho3, gm3=4, r2m3=0, r21=0, r22=0,
             g3=5, r23=0, p=.50, n, J, K)
power.med321(esa, esB, two.tailed=TRUE, alpha=.05,
             mc=TRUE, nsims=1000, ndraws=1000,
             rhom3, rho2, rho3, r2m2=0,
             gm3=4, r2m3=0, r21=0, r22=0, g3=5, r23=0,
             p=.50, n, J, K)
power.med311(esa, esB, two.tailed=TRUE, alpha=.05,
             mc=TRUE, nsims=1000, ndraws=1000,
             rhom2, rhom3, rho2, rho3,
             r2m1=0, r2m2=0, gm3=4, r2m3=0,
             r21=0, r22=0, g3=5, r23=0,
             p=.50, n, J, K)
mrss.cra3(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
          n, J, K0=10, tol=.10,
          rho2, rho3, p=.50, g3=0, r21=0, r22=0, r23=0)
mrss.mod331(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, omegam2, omegam3,
            r21=0, r2m3=0,
            p=.50, q=NULL, n, J, K0=10, tol=.10)
mrss.mod332(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, omegam3, r21=0, r22=0, r2m3=0,
            p=.50, q=NULL, n, J, K0=10, tol=.10)
mrss.mod333(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
            rho2, rho3, r21=0, r22=0, r23=0,
            p=.50, q=NULL, n, J, K0=10, tol=.10)
mdes.bcra4f3(power=.80, alpha=.05, two.tailed=TRUE,
             rho2, rho3, p=.50, r21=0, r22=0, r23=0, g3=0,
             n, J, K, L)
power.bcra4f3(es=.25, alpha=.05, two.tailed=TRUE,
              rho2, rho3, p=.50, r21=0, r22=0, r23=0, g3=0,
              n, J, K, L)
mrss.bcra4f3(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
             n, J, L, K0=10, tol=.10,
             rho2, rho3, p=.50, g3=0, r21=0, r22=0, r23=0)

Arguments

power

statistical power \((1-\beta)\).

es, esa, esB

effect size for main/moderator effects, or for path coefficients a (treatment - mediator), or B (overall mediator - outcome) in the mediation model.

alpha

probability of type I error.

two.tailed

logical; TRUE for two-tailed hypothesis testing, FALSE for one-tailed hypothesis testing.

rho2

proportion of variance in the outcome between level 2 units (unconditional ICC2).

rho3

proportion of variance in the outcome between level 3 units (unconditional ICC3).

rhom2

proportion of variance in the mediator between level 2 units.

rhom3

proportion of variance in the mediator between level 3 units.

omegam2

ratio of the unconditional variance in the moderator effect that is between level 2 units to the residual variance between level 2 units in the null model.

omegam3

ratio of the unconditional variance in the moderator effect that is between level 3 units to the residual variance between level 3 units in the null model.

p

proportion of level 3 units randomly assigned to treatment.

q

proportion of level 1, level 2, or level 3 units in the moderator subgroup.

g3

number of covariates at level 3.

gm3

number of covariates at level 3 for the mediation model.

r21

proportion of level 1 variance in the outcome explained by level 1 covariates.

r22

proportion of level 2 variance in the outcome explained by level 2 covariates.

r23

proportion of level 3 variance in the outcome explained by level 3 covariates.

r2m1

proportion of mediator variance at level 1 explained by level 1 predictors.

r2m2

proportion of variance in the mediator explained by level 2 predictors.

r2m3

proportion of variance in the moderator effect that is explained by level 3 predictors. For the mediation model, proportion of variance in the mediator explained by level 3 predictors.

n

harmonic mean of level 1 units across level 2 units (or simple average).

J

harmonic mean of level 2 units across level 3 units (or simple average).

K

level 3 sample size.

L

level 4 sample size (blocks).

K0

starting value for K.

tol

tolerance to end iterative process for finding K.

mc

logical; TRUE for monte carlo simulation based power.

nsims

number of replications, if mc = TRUE.

ndraws

number of draws from the distribution of the path coefficients for each replication, if mc = TRUE.

Value

fun

function name.

parms

list of parameters used in power calculation.

df

degrees of freedom.

ncp

noncentrality parameter.

power

statistical power \((1-\beta)\).

mdes

minimum detectable effect size.

K

number of level 3 units.

References

Spybrook, J., Kelcey, B., \& Dong, N. (2016). Power for detecting treatment by moderator effects in two-and three-level cluster randomized trials. Journal of Educational and Behavioral Statistics, 41(6), 605-627. 10.3102/1076998616655442

Dong, N., Kelcey, B., \& Spybrook, J. (2018). Power analyses for moderator effects in three-level cluster randomized trials. The Journal of Experimental Education, 86(3), 489-514. 10.1080/00220973.2017.1315714

Dong, N., \& Maynard, R. (2013). PowerUp!: A tool for calculating minimum detectable effect sizes and minimum required sample sizes for experimental and quasi-experimental design studies. Journal of Research on Educational Effectiveness, 6(1), 24-67. 10.1080/19345747.2012.673143

Kelcey, B., Xie, Y., Spybrook, J., \& Dong, N. (2020). Power and sample size determination for multilevel mediation in three-Level cluster-randomized trials. Multivariate Behavioral Research. Advance online publication. 10.1080/00273171.2020.1738910

Kelcey, B., Spybrook, J., Dong, N., \& Bai, F. (2020). Cross-level mediation in school-randomized studies of teacher development: Experimental design and power. Journal of Research on Educational Effectiveness. Advance online publication. 10.1080/19345747.2020.1726540

Examples

# NOT RUN {
# cross-checks for the main effect
mdes.cra3(rho3=.06, rho2=.17, n=15, J=3, K=60)
power.cra3(es=.269, rho3=.06, rho2=.17, n=15, J=3, K=60)
mrss.cra3(es=.269, rho3=.06, rho2=.17, n=15, J=3)

# cross-checks for the randomly varying cont. L1 moderator effect
mdesd.mod331(power=.80, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=NULL, n=15, J=3, K=60)
power.mod331(es=0.1248, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=NULL, n=15, J=3, K=60)
mrss.mod331(es=0.1248, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=NULL, n=15, J=3)

# cross-checks for the non-randomly varying cont. L1 moderator effect
mdesd.mod331(power=.80, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=NULL, n=15, J=3, K=60)
power.mod331(es=.0946, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=NULL, n=15, J=3, K=60)
mrss.mod331(es=.0946, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=NULL, n=15, J=3)

# cross-checks for the randomly varying bin. L1 moderator effect
mdesd.mod331(power=.80, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=.50, n=15, J=3, K=60)
power.mod331(es=.2082, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=.50, n=15, J=3, K=60)
mrss.mod331(es=.2082, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=.10, omegam3=.10,
            q=.50, n=15, J=3)

# cross-checks for the non-randomly varying bin. L1 moderator effect
mdesd.mod331(power=.80, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=.50, n=15, J=3, K=60)
power.mod331(es=.1893, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=.50, n=15, J=3, K=60)
mrss.mod331(es=.1893, alpha=.05, two.tailed=TRUE,
            rho2=.17, rho3=.06, omegam2=0, omegam3=0,
            q=.50, n=15, J=3)

# cross-checks for the randomly varying bin. L2 moderator effect
mdesd.mod332(rho3=.1, rho2=.1, omegam3=.05,
             q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
power.mod332(es=.2244, rho3=.1, rho2=.1, omegam3=.05,
             q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
mrss.mod332(es=.2244, rho3=.1, rho2=.1, omegam3=.05,
            q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4)

# cross-checks for the randomly varying cont. L2 moderator effect
mdesd.mod332(rho3=.1, rho2=.1, omegam3=.05,
            r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
power.mod332(es=.1209, rho3=.1, rho2=.1, omegam3=.05,
            r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
mrss.mod332(es=.1209, rho3=.1, rho2=.1, omegam3=.05,
            r21=.30, r22=.4, r2m3=0, n=20, J=4)

# cross-checks for the non-randomly varying bin. L2 moderator effect
mdesd.mod332(rho3=.1, rho2=.1, omegam3=0,
            q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
power.mod332(es=.2157, rho3=.1, rho2=.1, omegam3=0,
            q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
mrss.mod332(es=.2157, rho3=.1, rho2=.1, omegam3=0,
            q=.5, r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)

# cross-checks for the non-randomly varying cont. L2 moderator effect
mdesd.mod332(rho3=.1, rho2=.1, omegam3=0,
            r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
power.mod332(es=.1079, rho3=.1, rho2=.1, omegam3=0,
             r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)
mrss.mod332(es=.1079, rho3=.1, rho2=.1, omegam3=0,
             r21=.30, r22=.4, r2m3=0, n=20, J=4, K=60)

# cross-checks for the randomly varying bin. L3 moderator effect
mdesd.mod333(rho3=.1, rho2=.1, q=.5,
            r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)
power.mod333(es=.4128, rho3=.1, rho2=.1, q=.5,
             r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)
mrss.mod333(es=.4128, rho3=.1, rho2=.1, q=.5,
             r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)

# cross-checks for the randomly varying cont. L3 moderator effect
mdesd.mod333(rho3=.1, rho2=.1,
             r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)
power.mod333(es=.2064, rho3=.1, rho2=.1,
             r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)
mrss.mod333(es=.2064, rho3=.1, rho2=.1,
             r21=.3, r22=.4, r23=.5, n=20, J=4, K=60)

# 3-3-1 mediation
power.med331(esa= .50, esB = .30, rho2 = .15, rho3 = .15,
             r21 = .20, r22 = .20, g3 = 4,
             n = 20, J = 4, K = 80, p = .5)

# 3-2-1 mediation
power.med321(esa= .51, esB = .30, rhom3 = 0.27, rho2 = .15, rho3 = .19,
             r2m2 = .07, gm3 = 4, r2m3 = .16,
             r21 = .02, r22 = .41, g3 = 5, r23 = .38,
             p = .50, n = 20, J = 4, K = 60)

# 3-1-1 mediation
power.med311(esa= .49 , esB = .30,
             rhom2 = .05, rhom3 = .26, rho2 = .15, rho3 = .20,
             r2m1 = .10, r2m2 = .07, r2m3 = .17,
             r21 = .02, r22 = .41, r23 = .38,
             p = .50, n = 20, J = 4, K = 30)

# cross-checks for cluster-randomized block design
# treatment at level 3, with fixed effects across level 4 blocks
mdes.bcra4f3(rho3=.15, rho2=.15,
             n=10, J=4, K=23, L=15)
power.bcra4f3(es=0.137, rho3=.15, rho2=.15,
              n=10, J=4, K=33, L=15)
mrss.bcra4f3(es=0.137, rho3=.15, rho2=.15,
             n=10, J=4, L=15)
# }