Degrees of freedom calculation for cluster bias correction when there is clustering in one treatment group only

This function calculates the degrees of freedom for studies with clustering in one treatment group only, using Equation (7) from Hedges & Citkowicz (2015).

Usage

df_h_1armcluster(N_total, ICC, N_grp, avg_grp_size = NULL, n_clusters = NULL)

Arguments

N_total: Numerical value indicating the total sample size of the study.
ICC: Numerical value indicating the intra-class correlation (ICC) value.
N_grp: Numerical value indicating the sample size of the arm/group containing clustering.
avg_grp_size: Numerical value indicating the average cluster size.
n_clusters: Numerical value indicating the number of clusters in the treatment group.

Value

Returns a numerical value indicating the cluster adjusted degrees of freedom.

Details

When clustering is present the $N-2$ degrees of freedom ($df$) will be a rather liberal choice, partly overestimating the small sample corrector $J$ and partly underestimating the true variance of (Hedges') $g_T$. The impact of the calculated $df$ will be most consequential for small (sample) studies. To overcome these issues, Hedges & Citkowicz (2015) suggest obtaining the degrees of freedom from

$$ h = \dfrac{[(N-2)(1-\rho) + (N^T-n)\rho]^2} {(N-2)(1-\rho)^2 + (N^T-n)n\rho^2 + 2(N^T-n)(1-\rho)\rho}$$

where $N$ is the total sample size, $N^T$ is the sample size of the treatment group, containg clustering, $n$ is average cluster size and $\rho$ is the (imputed) intraclass correlation.

Note

Read Taylor et al. (2020) to understand why we use the $g_T$ notation. Find suggestions for how and which ICC values to impute when these are unknown (Hedges & Hedberg, 2007, 2013).

References

Hedges, L. V., & Citkowicz, M (2015). Estimating effect size when there is clustering in one treatment groups. Behavior Research Methods, 47(4), 1295-1308. doi:10.3758/s13428-014-0538-z

Hedges, L. V., & Hedberg, E. C. (2007). Intraclass correlation values for planning group-randomized trials in education. Educational Evaluation and Policy Analysis, 29(1), 60–87. doi:10.3102/0162373707299706

Hedges, L. V., & Hedberg, E. C. (2013). Intraclass correlations and covariate outcome correlations for planning two- and three-Level cluster-randomized experiments in education. Evaluation Review, 37(6), 445–489. doi:10.1177/0193841X14529126

Taylor, J.A., Pigott, T.D., & Williams, R. (2020) Promoting knowledge accumulation about intervention effects: Exploring strategies for standardizing statistical approaches and effect size reporting. Educational Researcher, 51(1), 72-80. doi:10.3102/0013189X211051319

Examples

df <- df_h_1armcluster(N_total = 100, ICC = 0.1, N_grp = 60, avg_grp_size = 5)
df
#> [1] 95.4


# Testing function
N <- 100
rho <- 0.1
NT <- 60
n <- 5

df_raw <- ((N-2)*(1-rho) + (NT-n)*rho)^2 /
          ( (N-2)*(1-rho)^2 + (NT-n)*n*rho^2 + 2*(NT-n)*(1-rho)*rho )

round(df_raw, 2)
#> [1] 95.4