Quick Navigation
Factorial Design Effect Sizes
These functions compute effect sizes for standard 2×2 factorial designs where two factors (A and B) are manipulated, creating four treatment groups: Control, A alone, B alone, and AB combined.
Effect sizes for factorial meta-analysis were first introduced by Gurevitch et al. (2000), who presented methods for estimating main and interaction effects analogous to factorial ANOVA. Originally developed for Standardized Mean Difference, these approaches have been extended to other effect size families including response ratios and variability measures.
Log Response Ratio (lnRR)
lnRR_inter()
Formula: \[\ln RR_{inter} = \ln\left(\frac{\bar{X}_{AB}}{\bar{X}_B}\right) - \ln\left(\frac{\bar{X}_A}{\bar{X}_{Ctrl}}\right)\]
Sampling variance: \[var(\ln RR_{inter}) = \frac{sd_{AB}^2}{n_{AB}\bar{X}_{AB}^2} + \frac{sd_A^2}{n_A\bar{X}_A^2} + \frac{sd_B^2}{n_B\bar{X}_B^2} + \frac{sd_{Ctrl}^2}{n_{Ctrl}\bar{X}_{Ctrl}^2}\]
Log Variation Ratio (lnVR)
lnVR_ind()
Formula: \[\ln VR_{ind} = \ln\left(\frac{sd_{A}}{sd_{Ctrl}}\right) + \frac{1}{2(n_{A} - 1)} - \frac{1}{2(n_{Ctrl} - 1)}\]
Sampling variance: \[var(\ln VR_{ind}) = \frac{1}{2(n_{A} - 1)} + \frac{1}{2(n_{Ctrl} - 1)}\]
lnVR_main()
Formula: \[\ln VR_{main} = \frac{1}{2} \ln\left( \frac{sd_{AB} \cdot sd_{A}}{sd_{B} \cdot sd_{Ctrl}} \right) + \frac{1}{2} \left( \frac{1}{2(n_{AB} - 1)} + \frac{1}{2(n_{A} - 1)} - \frac{1}{2(n_{B} - 1)} - \frac{1}{2(n_{Ctrl} - 1)} \right)\]
Sampling variance: \[var(\ln VR_{main}) = \frac{1}{4} \left( \frac{1}{2(n_{AB} - 1)} + \frac{1}{2(n_{A} - 1)} + \frac{1}{2(n_{B} - 1)} + \frac{1}{2(n_{Ctrl} - 1)} \right)\]
lnVR_inter()
Formula: \[\ln VR_{inter} = \ln\left( \frac{sd_{AB} / sd_{B}}{sd_{A} / sd_{Ctrl}} \right) + \frac{1}{2(n_{AB} - 1)} - \frac{1}{2(n_{A} - 1)} - \frac{1}{2(n_{B} - 1)} + \frac{1}{2(n_{Ctrl} - 1)}\]
Sampling variance: \[var(\ln VR_{inter}) = \frac{1}{2(n_{AB} - 1)} + \frac{1}{2(n_{A} - 1)} + \frac{1}{2(n_{B} - 1)} + \frac{1}{2(n_{Ctrl} - 1)}\]
Log Coefficient of Variation Ratio (lnCVR)
The Log Coefficient of Variation Ratio combines information about both mean responses and variability by comparing coefficients of variation (\(CV = sd / \bar{x}\)).
lnCVR_ind()
Formula: \[\ln CVR_{ind} = \ln\left( \frac{CV_{A}}{CV_{Ctrl}} \right) + \frac{1}{2(n_{A} - 1)} - \frac{1}{2(n_{Ctrl} - 1)}\]
Sampling variance: \(var(\ln CVR_{ind}) = \frac{sd_{Ctrl}^2}{n_{Ctrl}\bar{X}_{Ctrl}^2} + \frac{1}{2(n_{Ctrl} - 1)} + \frac{sd_A^2}{n_A\bar{X}_A^2} + \frac{1}{2(n_A - 1)}\)
This assumes no correlation between mean and variance (Nakagawa et al. 2015) and is computed as the sum of lnRR and lnVR variances.
lnCVR_main()
Formula: \[\ln CVR_{main} = \frac{1}{2} \ln\left( \frac{CV_{AB} \cdot CV_{A}}{CV_{B} \cdot CV_{Ctrl}} \right) + \frac{1}{2} \left( \frac{1}{2(n_{AB} - 1)} + \frac{1}{2(n_{A} - 1)} - \frac{1}{2(n_{B} - 1)} - \frac{1}{2(n_{Ctrl} - 1)} \right)\]
Sampling variance: \(var(\ln CVR_{main}) = var(\ln RR_{main}) + var(\ln VR_{main})\)
This follows the assumption of no correlation between mean and variance.
lnCVR_inter()
Formula: \[\ln CVR_{inter} = \ln\left( \frac{CV_{AB} / CV_{B}}{CV_{A} / CV_{Ctrl}} \right) + \frac{1}{2} \left( \frac{1}{2(n_{AB} - 1)} - \frac{1}{2(n_{A} - 1)} + \frac{1}{2(n_{B} - 1)} - \frac{1}{2(n_{Ctrl} - 1)} \right)\]
Sampling variance: \(var(\ln CVR_{inter}) = var(\ln RR_{inter}) + var(\ln VR_{inter})\)
This follows the assumption of no correlation between mean and variance.
Repeated Measures Effect Sizes
These functions compute effect sizes for treatment × time interactions in longitudinal studies, comparing how experimental and control groups change over time.
time_SMD()
Formula: \[d = \frac{((\bar{X}_{t1,Exp} - \bar{X}_{t1,Ctrl}) - (\bar{X}_{t0,Exp} - \bar{X}_{t0,Ctrl}))}{S_{pooled}} \cdot J\]
Time-specific pooled standard deviation: \[S_{pooled} = \sqrt{\frac{((n_{Exp} - 1)(sd_{t0,Exp}^2 + sd_{t1,Exp}^2) + (n_{Ctrl} - 1)(sd_{t0,Ctrl}^2 + sd_{t1,Ctrl}^2))}{2(n_{Exp} + n_{Ctrl} - 2)}}\]
Sampling variance: \[var(d) = \frac{2(1 - r_{Exp})}{n_{Exp}} + \frac{2(1 - r_{Ctrl})}{n_{Ctrl}} + \frac{d^2}{2(n_{Exp} + n_{Ctrl})}\]
where \(r_{Exp}\) and \(r_{Ctrl}\) are the correlations between time points within each group.
time_lnRR()
Formula: \[\ln RR = \ln\left(\frac{\bar{X}_{t1,Exp} / \bar{X}_{t1,Ctrl}}{\bar{X}_{t0,Exp} / \bar{X}_{t0,Ctrl}}\right)\]
Sampling variance: \[var(\ln RR) = \frac{(sd_{t0,Exp}^2 \bar{X}_{t1,Exp}^2 + sd_{t1,Exp}^2 \bar{X}_{t0,Exp}^2 - 2r_{Exp} \bar{X}_{t0,Exp} \bar{X}_{t1,Exp} sd_{t0,Exp} sd_{t1,Exp})}{n_{Exp} \bar{X}_{t0,Exp}^2 \bar{X}_{t1,Exp}^2} +\] \[\frac{(sd_{t0,Ctrl}^2 \bar{X}_{t1,Ctrl}^2 + sd_{t1,Ctrl}^2 \bar{X}_{t0,Ctrl}^2 - 2r_{Ctrl} \bar{X}_{t0,Ctrl} \bar{X}_{t1,Ctrl} sd_{t0,Ctrl} sd_{t1,Ctrl})}{n_{Ctrl} \bar{X}_{t0,Ctrl}^2 \bar{X}_{t1,Ctrl}^2}\]
Function Parameters
All functions share common parameters:
-
data: Data frame containing the variables -
col_names: Vector of two strings naming the output columns (default:c("yi", "vi")) -
append: Logical, whether to append results to input data (default:TRUE)
References
Gurevitch, J., Morrison, J. A., & Hedges, L. V. (2000). The interaction between competition and predation: a meta-analysis of field experiments. The American Naturalist, 155(4), 435-453.
Morris, W. F., Hufbauer, R. A., Agrawal, A. A., Bever, J. D., Borowicz, V. A., Gilbert, G. S., … & Vázquez, D. P. (2007). Direct and interactive effects of enemies and mutualists on plant performance: a meta‐analysis. Ecology, 88(4), 1021-1029.
Nakagawa, S., Poulin, R., Mengersen, K., Reinhold, K., Engqvist, L., Lagisz, M., & Senior, A. M. (2015). Meta‐analysis of variation: ecological and evolutionary applications and beyond. Methods in Ecology and Evolution, 6(2), 143-152.
Lajeunesse, M. J. (2011). On the meta‐analysis of response ratios for studies with correlated and multi‐group designs. Ecology, 92(11), 2049-2055.
Personal communications: Shinichi Nakagawa and Daniel Noble for recent developments in factorial effect size calculations.