PowerUpR (version 1.1.0)

cra2: Two-level Cluster-randomized Trials to Detect Main, Moderation and Mediation Effects

Description

For main treatment effects, use mdes.cra2() to calculate the minimum detectable effect size, power.cra2() to calculate the statistical power, mrss.cra2() to calculate the minimum required sample size (number of clusters).

For moderator at level 1, use mdesd.mod221() to calculate the minimum detectable effect size, power.mod221() to calculate the statistical power, mrss.mod221() to calculate the minimum required sample size (number of clusters).

For moderator at level 2, use mdesd.mod222() to calculate the minimum detectable effect size, power.mod222() to calculate the statistical power, mrss.mod222() to calculate the minimum required sample size (number of clusters).

For mediator at level 1 and level 2, use power.med211() to calculate the statistical power for the 2-1-1 mediation, and power.med221() for the 2-2-1 mediation.

For cluster-randomized block designs (treatment at level 2, with fixed effects across level 3 blocks), use mdes.bcra3f2() to calculate the minimum detectable effect size, power.bcra3f2() to calculate the statistical power, and mrss.bcra3f2() to calculate the minimum required sample size (number of clusters per block).

For partially nested cluster randomized trials (interventions clusters in treatment groups) use mdes.cra2_pn() to calculate the minimum detectable effect size, power.cra2_pn() to calculate the statistical power, and mrss.cra2_pn() to calculate the minimum required sample size (number of schools).

Usage

mdes.cra2(power=.80, alpha=.05, two.tailed=TRUE,
          rel1 = 1, rho2, p=.50, g2=0, r21=0, r22=0,
          n, J)
mdesd.mod221(power=.80, alpha=.05, two.tailed=TRUE,
            rho2, omegam2, r21=0, r2m2=0,
            p=.50, q=NULL, n, J)
mdesd.mod222(power=.80, alpha=.05, two.tailed=TRUE,
            rho2, r22=0,
            p=.50, q=NULL, n, J)
power.cra2(es=.25, alpha=.05, two.tailed=TRUE,
           rel1 = 1, rho2, g2=0, p=.50, r21=0, r22=0,
           n, J)
power.mod221(es=.25, alpha=.05, two.tailed=TRUE,
             rho2, omegam2, r21=0, r2m2=0,
             p=.50, q=NULL, n, J)
power.mod222(es=.25, alpha=.05, two.tailed=TRUE,
             rho2, r22=0,
             p=.50, q=NULL, n, J)
power.med211(esa, esb1, esB, escp, two.tailed = TRUE, alpha = .05,
             mc = FALSE, nsims = 1000, ndraws = 1000,
             rhom2, rho2, r21, r22, r2m1, r2m2,
             p, n, J)
power.med221(esa, esb, escp, two.tailed = TRUE, alpha = .05,
             mc = FALSE, nsims = 1000, ndraws = 1000,
             rho2, r22, r21, r2m2,
             p = .50, n, J)
mrss.cra2(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
          n, J0=10, tol=.10, rel1 = 1,
          rho2, g2=0, p=.50, r21=0, r22=0)
mrss.mod221(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
            n, J0=10, tol=.10, rho2, omegam2, r21=0, r2m2=0,
            p=.50, q=NULL)
mrss.mod222(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
            n, J0=10, tol=.10, rho2, r22=0,
            p=.50, q=NULL)
mdes.bcra3f2(power=.80, alpha=.05, two.tailed=TRUE,
             rho2, p=.50, g2=0, r21=0, r22=0,
             n, J, K)
power.bcra3f2(es=.25, alpha=.05, two.tailed=TRUE,
              rho2, p=.50, g2=0, r21=0, r22=0,
              n, J, K)
mrss.bcra3f2(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
             n, K, J0=10, tol=.10,
             rho2, p=.50, g2=0, r21=0, r22=0)
mdes.cra2_pn(power=.80, alpha=.05, two.tailed=TRUE, df=NULL,
             rho2_trt=.20, rho_ic=0, p=.50,
             r21=0, n, J, ic_size=1)
power.cra2_pn(es=.25,alpha=.05, two.tailed=TRUE, df=NULL,
              rho2_trt=.20, rho_ic=0, p=.50,
              r21=0, n, J, ic_size=1)
mrss.cra2_pn(es=.25, power=.80, alpha=.05, two.tailed=TRUE,
             z.test=FALSE, rho2_trt=.20, rho_ic=0, p=.50,
             r21=0, n, ic_size=1, J0=10, tol=.10)

Arguments

power

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

es, esa, esb, esb1, esB, escp

effect size for main/moderator effects, or for path coefficients a (treatment - mediator), b (level 2 mediator - outcome), b1 (level 1 mediator - outcome), B (overall mediator - outcome) or cp (direct treatment - outcome) in the mediation model.

alpha

probability of type I error.

two.tailed

logical; FALSE for one-tailed hypothesis testing.

df

degrees of freedom.

rho_ic

proportion of variance in the outcome that is between intervention clusters.

rho2_trt

proportion of variance in the outcome (for treatment group) that is between level 2 units.

rel1

level 1 outcome reliability coefficient (Cox \& Kelcey, 2019b).

rho2

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

rhom2

proportion of variance in the mediator between level 2 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.

p

proportion of level 2 units randomly assigned to treatment.

q

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

g2

number of covariates at level 2.

r21

proportion of level 1 variance in the outcome explained by level 1 covariates (applies to all levels in partially nested designs).

r22

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

r2m1

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

r2m2

proportion of variance in the moderator effect that is explained by level 2 predictors. For the mediation model, proportion of variance in the mediator explained by level 2 predictors.

n

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

J

level 2 sample size.

K

number of level 3 units (blocks).

ic_size

sample size for each intervention cluster.

J0

starting value for J.

tol

tolerance to end iterative process for finding J.

z.test

logical; TRUE for z-test.

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.

J

number of level 2 units.

References

Cox, K., \& Kelcey, B. (2019a). Optimal sample allocation in group-randomized mediation studies with a group-level mediator. The Journal of Experimental Education, 87(4), 616-640. 10.1080/00220973.2018.1496060

Cox, K., \& Kelcey, B. (2019b). Optimal design of cluster-and multisite-randomized studies using fallible outcome measures. Evaluation Review, 43(3-4), 189-225. 10.1177/0193841X19870878

Dong, N., Spybrook J., Kelcey, B., \& Bulus, M. (2021). Power analyses for moderator effects with (non)random slopes in cluster randomized trials. Methodology, 17(2), 92-110. 10.5964/meth.4003

Kelcey, B., Dong, N., Spybrook, J., \& Cox, K. (2017). Statistical power for causally defined indirect effects in group-randomized trials with individual-level mediators. Journal of Educational and Behavioral Statistics, 42(5), 499-530. 10.3102/1076998617695506

Kelcey, B., Dong, N., Spybrook, J., \& Shen, Z. (2017). Experimental power for indirect effects in group-randomized studies with group-level mediators. Multivariate behavioral research, 52(6), 699-719. 10.1080/00273171.2017.1356212

Kelcey, B., \& Shen, Z. (2020). Strategies for efficient experimental design in studies probing 2-1-1 mediation. The Journal of Experimental Education, 88(2), 311-334. 10.1080/00220973.2018.1533796

Kelcey, B., Spybrook, J., \& Dong, N. (2019). Sample size planning for cluster-randomized interventions probing multilevel mediation. Prevention Science, 20(3), 407-418. 10.1007/s11121-018-0921-6

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., \& 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

Lohr, S., Schochet, P. Z., \& Sanders, E. (2014). Partially nested randomized controlled trials in education research: A guide to design and analysis. NCER 2014-2000. National Center for Education Research. https://ies.ed.gov/ncer/pubs/20142000/pdf/20142000.pdf

Examples

# NOT RUN {
# cross-checks for the main effect
mdes.cra2(rho2=.17, n=15, J=20)
power.cra2(es=.629, rho2=.17, n=15, J=20)
mrss.cra2(es=.629, rho2=.17, n=15)

# cross-checks for the randomly varying cont. L1 moderator effect
mdesd.mod221(rho2=.17, omegam2=.10, n=15, J=20)
power.mod221(es=.3563, rho2=.17, omegam2 =.10, n=15, J=20)
mrss.mod221(es=.3563, rho2=.17, omegam2 =.10, n=15)

# cross-checks for the non-randomly varying cont. L1 moderator effect
mdesd.mod221(rho2=.17, omegam2=0, n=15, J=20)
power.mod221(es=0.2957, rho2=.17, omegam2 =0, n=15, J=20)
mrss.mod221(es=0.2957, rho2=.17, omegam2 =0, n=15)

# cross-checks for the randomly varying bin. L1 moderator effect
mdesd.mod221(rho2=.17, omegam2=.10, q=.50, n=15, J=20)
power.mod221(es=.647, rho2=.17, omegam2 =.10, q=.50, n=15, J=20)
mrss.mod221(es=.647, rho2=.17, omegam2 =.10, q=.50, n=15)

# cross-checks for the non-randomly varying bin. L1 moderator effect
mdesd.mod221(rho2=.17, omegam2=0, q=.50, n=15, J=20)
power.mod221(es=0.5915, rho2=.17, omegam2 =0, q=.50, n=15, J=20)
mrss.mod221(es=0.5915, rho2=.17, omegam2 =0, q=.50, n=15)

# cross-checks for the cont. L2 moderator effect
mdesd.mod222(rho2=.17, n=15, J=100)
power.mod222(es=0.2757, rho2=.17, n=15, J=100)
mrss.mod222(es=0.2757, rho2=.17, n=15)

# cross-checks for the bin. L2 moderator effect
mdesd.mod222(rho2=.17, q=.50, n=15, J=100)
power.mod222(es=0.5514, rho2=.17, q=.50, n=15, J=100)
mrss.mod222(es=0.5514, rho2=.17, q=.50, n=15)

# 2-2-1 mediation
power.med221(esa=0.6596, esb=0.1891, escp=.1,
             rho2=.15, r22=.52, r21=.40, r2m2=.50,
             n=100, J=40, p=.5)

# 2-1-1 mediation
power.med211(esa=0.4135, esb1=0.0670, esB=0.3595, escp=.1,
            rhom2=.3, rho2=.3, r22=.6, r21=.6, r2m2=.6, r2m1=.6,
            n=30, J=80, p=.1)

# cross-checks for cluster-randomized block design
# treatment at level 2, with fixed effects across level 3 blocks
mdes.bcra3f2(rho2=.10, n=20, J=44, K=5)
power.bcra3f2(es = .145, rho2=.10, n=20, J=44, K=5)
mrss.bcra3f2(es = .145, rho2=.10, n=20, K=5)

# cross-checks for partially nested cluster-randomized trial
mdes.cra2_pn(n=40, J=70, rho2_trt=.15, rho_ic=.10, ic_size=10)
power.cra2_pn(es=.305, n=40, J=70, rho2_trt=.15, rho_ic=.10, ic_size=10)
mrss.cra2_pn(es=.305, n=40, rho2_trt=.15, rho_ic=.10, ic_size=10)
# }