The minter package provides tools for calculating effect sizes used in meta-analyses, with specialized support for factorial experimental designs and repeated measures studies. The package offers a consistent interface for computing various effect size metrics.
minter implements two primary categories of effect size calculations:
For experimental designs with two factors (2×2 designs), supporting three types of comparisons:
For longitudinal studies examining treatment × time interactions in before-after or time-series designs.
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.
Recent developments by Shinichi Nakagawa and Daniel Noble have expanded these methods to include lnVR and lnCVR effect sizes, providing researchers with comprehensive tools for meta-analyzing factorial experiments.
We thank Shinichi Nakagawa and Daniel Noble for their theoretical contributions to factorial meta-analysis methodology and for generously sharing unpublished formulas.
The Standardized Mean Difference measures the magnitude of treatment effects in standard deviation units, making it comparable across studies with different measurement scales.
SMD_ind()Computes the individual or simple effect of Factor A compared to
Control. This is equivalent to the classic SMD that can be computed
using metafor’s escalc() function with
measure = "SMD".
Formula: \[d_{ind} = \frac{\bar{X}_A - \bar{X}_{Ctrl}}{S_{pooled}} \cdot J(m)\]
where the pooled standard deviation is: \[S_{pooled} = \sqrt{\frac{(n_A-1)sd_A^2 + (n_{Ctrl}-1)sd_{Ctrl}^2}{n_A + n_{Ctrl} - 2}}\]
Small-sample bias correction: The \(J(m)\) correction (Hedges correction) is applied by default but can be disabled: \[J(m) = 1 - \frac{3}{4m-1}\] where \(m = n_A + n_{Ctrl} - 2\)
Sampling variance: \[var(d_{ind}) = \frac{1}{n_A} + \frac{1}{n_{Ctrl}} + \frac{d^2}{2(n_A + n_{Ctrl})}\]
SMD_main()Computes the main effect of Factor A averaged across levels of Factor B, analogous to main effects in factorial ANOVA.
Formula: \[d_{main} = \frac{(\bar{X}_A + \bar{X}_{AB}) - (\bar{X}_{B} + \bar{X}_{Ctrl})}{2S_{pooled}} \cdot J(m)\]
Pooled standard deviation (four groups): \[S_{pooled} = \sqrt{\frac{(n_A-1)sd_A^2 + (n_B-1)sd_B^2 + (n_{AB}-1)sd_{AB}^2 + (n_{Ctrl}-1)sd_{Ctrl}^2}{n_A + n_B + n_{AB} + n_{Ctrl} - 4}}\]
Degrees of freedom: \(m = n_A + n_B + n_{AB} + n_{Ctrl} - 4\)
Sampling variance: \[var(d_{main}) = \frac{1}{4} \left(\frac{1}{n_A} + \frac{1}{n_B} + \frac{1}{n_{AB}} + \frac{1}{n_{Ctrl}} + \frac{d_{main}^2}{2(n_A + n_B + n_{AB} + n_{Ctrl})}\right)\]
SMD_inter()Computes the interaction between factors A and B, measuring how the effect of A depends on the level of B.
Formula: \[d_{inter} = \frac{(\bar{X}_{AB} - \bar{X}_B) - (\bar{X}_A - \bar{X}_{Ctrl})}{S_{pooled}} \cdot J(m)\]
Uses the same four-group pooled standard deviation as the main effect.
Sampling variance: \[var(d_{inter}) = \frac{1}{n_A} + \frac{1}{n_B} + \frac{1}{n_{AB}} + \frac{1}{n_{Ctrl}} + \frac{d_{inter}^2}{2(n_A + n_B + n_{AB} + n_{Ctrl})}\]
The Log Response Ratio measures proportional changes between treatments, providing intuitive interpretation.
lnRR_ind()Formula: \[\ln RR_{ind} = \ln\left(\frac{\bar{X}_A}{\bar{X}_{Ctrl}}\right)\]
Sampling variance: \[var(\ln RR_{ind}) = \frac{sd_A^2}{n_A\bar{X}_A^2} + \frac{sd_{Ctrl}^2}{n_{Ctrl}\bar{X}_{Ctrl}^2}\]
lnRR_main()Formula: \[\ln RR_{main} = \ln\left(\frac{\bar{X}_A + \bar{X}_{AB}}{\bar{X}_{Ctrl} + \bar{X}_B}\right)\]
Sampling variance: \[var(\ln RR_{main}) = \left(\frac{1}{\bar{X}_A + \bar{X}_{AB}}\right)^2 \left(\frac{sd_A^2}{n_A} + \frac{sd_{AB}^2}{n_{AB}}\right) + \left(\frac{1}{\bar{X}_{Ctrl} + \bar{X}_B}\right)^2 \left(\frac{sd_{Ctrl}^2}{n_{Ctrl}} + \frac{sd_B^2}{n_B}\right)\]
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: \[var(\ln RR_{inter}) = \frac{sd_{AB}^2}{n_{AB}\bar{X}_{AB}^2} + \frac{sd_A^2}{n_A\bar{X}_A^2} + \frac{sd_B^2}{n_B\bar{X}_B^2} + \frac{sd_{Ctrl}^2}{n_{Ctrl}\bar{X}_{Ctrl}^2}\]
The Log Variation Ratio compares variability between treatments, providing insights into how treatments affect response consistency.
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: \[var(\ln VR_{inter}) = \frac{1}{2(n_{AB} - 1)} + \frac{1}{2(n_{A} - 1)} + \frac{1}{2(n_{B} - 1)} + \frac{1}{2(n_{Ctrl} - 1)}\]
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.
These functions compute effect sizes for treatment × time interactions in longitudinal studies, comparing how experimental and control groups change over time.
time_SMD()Computes the standardized mean difference for the interaction between experimental treatment and time.
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: \[var(\ln RR) = \frac{(sd_{t0,Exp}^2 \bar{X}_{t1,Exp}^2 + sd_{t1,Exp}^2 \bar{X}_{t0,Exp}^2 - 2r_{Exp} \bar{X}_{t0,Exp} \bar{X}_{t1,Exp} sd_{t0,Exp} sd_{t1,Exp})}{n_{Exp} \bar{X}_{t0,Exp}^2 \bar{X}_{t1,Exp}^2} +\] \[\frac{(sd_{t0,Ctrl}^2 \bar{X}_{t1,Ctrl}^2 + sd_{t1,Ctrl}^2 \bar{X}_{t0,Ctrl}^2 - 2r_{Ctrl} \bar{X}_{t0,Ctrl} \bar{X}_{t1,Ctrl} sd_{t0,Ctrl} sd_{t1,Ctrl})}{n_{Ctrl} \bar{X}_{t0,Ctrl}^2 \bar{X}_{t1,Ctrl}^2}\]
time_lnVR()Formula: \[\ln VR = \ln\left(\frac{sd_{t1,Exp} / sd_{t1,Ctrl}}{sd_{t0,Exp} / sd_{t0,Ctrl}}\right)\]
Sampling variance: \(var(\ln VR) = \frac{(1 - r_{Exp}^2)}{n_{Exp} - 1} + \frac{(1 - r_{Ctrl}^2)}{n_{Ctrl} - 1}\)
time_lnCVR()Formula: \[\ln CVR = \ln\left(\frac{CV_{t1,Exp} / CV_{t1,Ctrl}}{CV_{t0,Exp} / CV_{t0,Ctrl}}\right)\]
Sampling variance: \[var(\ln CVR) = var(\ln RR) + var(\ln VR)\]
All functions share common parameters:
data: Data frame containing the
variablescol_names: Vector of two strings
naming the output columns (default: c("yi", "vi"))append: Logical, whether to append
results to input data (default: TRUE)[Group]_mean,
[Group]_sd, [Group]_n for each group (Ctrl, A,
B, AB)hedges_correction: Logical, apply
small-sample bias correction (SMD only, default: TRUE)t0_[Group]_mean, t0_[Group]_sd,
t1_[Group]_mean, t1_[Group]_sd[Group]_n for each
group[Group]_cor -
correlation between time points within each groupGurevitch, 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.