Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
We conduct power analysis using optimal design software aiming for a power of 80% and a false discovery rate (alpha) of 5%.
For H1, we compare the pooled T1 & T2 (N = 8000) to the control group (N=4000). This sample gives us a MDES of 0.06 standard deviations after correcting for the imbalanced group size.
For H2, we compare the pooled T3 (N = 13 000) to the control group (N=4000). This sample gives us a MDES of 0.05 standard deviations after correcting for the imbalanced group size.
For H3, we compare the T1 (N = 4 000) to the control T2 (N=4000). This sample gives us a MDES of 0.064 standard deviations.
For H4, we compare the T3a (N = 7 000) to the control T3b (N=6000). This sample gives us a MDES of 0.05 standard deviations.
For H5, we compare the pooled T1 & T2 (N = 8000) to the pooled T3 (N=13 000). This sample gives us a MDES of 0.04 standard deviations after correcting for the imbalanced group size.
For the survey outcomes, we, conservatively, expect a response rate of 10%. Adjusting the sample size accordingly, assuming equal participation accross treatments leads to the following MDES using the comparisons outlined above.
For H1, we compare the pooled T1 & T2 (N = 800) to the control group (N=400). This sample gives us a MDES of 0.19 standard deviations after correcting for the imbalanced group size.
For H2, we compare the pooled T3 (N = 1 300) to the control group (N=400). This sample gives us a MDES of 0.16 standard deviations after correcting for the imbalanced group size.
For H3, we compare the T1 (N = 400) to the control T2 (N=400). This sample gives us a MDES of 0.20 standard deviations.
For H4, we compare the T3a (N = 700) to the control T3b (N=600). This sample gives us a MDES of 0.16 standard deviations.
For H5, we compare the pooled T1 & T2 (N = 800) to the pooled T3 (N=1300). This sample gives us a MDES of 0.13 standard deviations after correcting for the imbalanced group size.
We expect to improve power with the addition of control variables
Revision June 2026:
Outcomes registry data:
H4 is removed. Sample size numbers and MDE in H2 and H5 are updated to account for the lower sample size in T3. And power calculations for H6 and H7 are added.
For H2, we compare T3 (N = 8 000) to the control group (N = 5797). This sample gives us a MDES of 0.05 standard deviations after correcting for the imbalanced group size.
For H5, we compare the pooled T1 & T2 (N = 8000) to T3 (N = 8 000). This sample gives us a MDES of 0.05 standard deviations after correcting for the imbalanced group size.
For H6, we compare T4 (N = 5 000) to the control (N = 5797). This sample gives us a MDES of 0.06 standard deviations.
For H7, we compare T4 (N = 5 000) to T2 (N = 4 000). This sample gives us a MDES of 0.06 standard deviations.
Survey outcomes:
For H2, we compare T3 (N = 800) to the control group (N = 579). This sample gives us a MDES of 0.15 standard deviations after correcting for the imbalanced group size.
For H5, we compare the pooled T1 & T2 (N = 800) to T3 (N = 900). This sample gives us a MDES of 0.14 standard deviations after correcting for the imbalanced group size.
For H6, we compare T4 (N = 500) to the control (N = 579). This sample gives us a MDES of 0.17 standard deviations.
For H7, we compare T4 (N = 500) to T2 (N = 400). This sample gives us a MDES of 0.19 standard deviations.