Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Minimum detectable effect size: Parameters will be delineated, with references from the current textile sector's achievements and other analogous studies. Using Adem et al. (2023) Summary statistics for a similar RCT in the UK distilling industry, we compute the following Minimum Detectable Effect and Implied Change
Table 1. Minimum Detectable Effect and Implied Change of the Randomized Control Trial
Treatment arm 1 Each treatment arms 2 & 3 Significance level Power Mean SD Minimum Detectable Effect Implied Change
First-Contentful-Paint 720 540 5% 80% 1.408 0.744 0.1187 8.43
First-Meaningful-Paint 720 540 5% 80% 1.554 0.833 0.1330 8.56
First-Meaningful-Paint 720 540 5% 80% 2.490 1.485 0.2370 9.52
Time to interactive 720 540 5% 80% 3.189 2.168 0.3460 10.85
Ranking Keywords 720 540 5% 80% 24.910 143.485 22.9015 91.94
Speed-Index 720 540 5% 80% 3.268 2.127 0.3395 10.39
Domain Authority 720 540 5% 80% 17.114 14.036 2.2403 13.09
Page Views 720 540 5% 80% 4368.6 46671.2 7449.1408 170.52
Below Average Count 720 540 5% 80% 2.365 1.529 0.2440 10.32
Sensitivity analysis
We evaluate the sensitivity of the results to different effect sizes and standard deviations using a representative variable, time to interactive, which takes the median value of the web measures included in Table 1. This representative variable has 1441 observations for the treatment arm 1 and 1080 observations for the treatment arms 2 and 3. Its mean value and standard deviation are 3.189 and 2.127, respectively. Thus, the effect size corresponding to testing the effect of treatment arm 1 compared to treatment arm 2 or 3 with 80% of power is 0.2399.
Table 2 shows the statistical power of the mean difference test (treatment arm 1 vs treatment arm 2 or 3) associated with different effect sizes. It can be seen that small reductions in the effect size generate big losses in power. For example, an effect size of 0.2 is only associated with a power of 65%.
Table 2. Statistical power associated with different mean differences
alpha power N N1 N2 Effect size m1 m2 SD
0.05 0.052 2521 1441 1080 0.01 3.189 3.199 2.127
0.05 0.056 2521 1441 1080 0.02 3.189 3.209 2.127
0.05 0.064 2521 1441 1080 0.03 3.189 3.219 2.127
0.05 0.075 2521 1441 1080 0.04 3.189 3.229 2.127
0.05 0.090 2521 1441 1080 0.05 3.189 3.239 2.127
0.05 0.108 2521 1441 1080 0.06 3.189 3.249 2.127
0.05 0.129 2521 1441 1080 0.07 3.189 3.259 2.127
0.05 0.154 2521 1441 1080 0.08 3.189 3.269 2.127
0.05 0.183 2521 1441 1080 0.09 3.189 3.279 2.127
0.05 0.215 2521 1441 1080 0.1 3.189 3.289 2.127
0.05 0.250 2521 1441 1080 0.11 3.189 3.299 2.127
0.05 0.289 2521 1441 1080 0.12 3.189 3.309 2.127
0.05 0.330 2521 1441 1080 0.13 3.189 3.319 2.127
0.05 0.373 2521 1441 1080 0.14 3.189 3.329 2.127
0.05 0.418 2521 1441 1080 0.15 3.189 3.339 2.127
0.05 0.464 2521 1441 1080 0.16 3.189 3.349 2.127
0.05 0.510 2521 1441 1080 0.17 3.189 3.359 2.127
0.05 0.556 2521 1441 1080 0.18 3.189 3.369 2.127
0.05 0.602 2521 1441 1080 0.19 3.189 3.379 2.127
0.05 0.646 2521 1441 1080 0.2 3.189 3.389 2.127
0.05 0.689 2521 1441 1080 0.21 3.189 3.399 2.127
0.05 0.729 2521 1441 1080 0.22 3.189 3.409 2.127
0.05 0.766 2521 1441 1080 0.23 3.189 3.419 2.127
0.05 0.800 2521 1441 1080 0.24 3.189 3.429 2.127
0.05 0.831 2521 1441 1080 0.25 3.189 3.439 2.127
0.05 0.859 2521 1441 1080 0.26 3.189 3.449 2.127
0.05 0.884 2521 1441 1080 0.27 3.189 3.459 2.127
0.05 0.905 2521 1441 1080 0.28 3.189 3.469 2.127
0.05 0.923 2521 1441 1080 0.29 3.189 3.479 2.127
0.05 0.939 2521 1441 1080 0.3 3.189 3.489 2.127
0.05 0.952 2521 1441 1080 0.31 3.189 3.499 2.127
Table 3 shows the effect of changing the standard deviation of the representative sample on the power of the test. The effect of marginal changes in SD is also asymmetric in the neighbourhood of 80% power, as reducing the SD generates only small increases in power as we approach the upper bound of 100%.
Table 3. Statistical power associated with different standard deviations
alpha power N N1 N2 Effect size m1 m2 SD
0.05 0.9793 2,521 1,441 1,080 0.2416 3.189 3.431 1.5
0.05 0.9633 2,521 1,441 1,080 0.2416 3.189 3.431 1.6
0.05 0.9418 2,521 1,441 1,080 0.2416 3.189 3.431 1.7
0.05 0.9152 2,521 1,441 1,080 0.2416 3.189 3.431 1.8
0.05 0.8846 2,521 1,441 1,080 0.2416 3.189 3.431 1.9
0.05 0.8509 2,521 1,441 1,080 0.2416 3.189 3.431 2
0.05 0.8153 2,521 1,441 1,080 0.2416 3.189 3.431 2.1
0.05 0.7786 2,521 1,441 1,080 0.2416 3.189 3.431 2.2
0.05 0.7418 2,521 1,441 1,080 0.2416 3.189 3.431 2.3
0.05 0.7055 2,521 1,441 1,080 0.2416 3.189 3.431 2.4
0.05 0.6701 2,521 1,441 1,080 0.2416 3.189 3.431 2.5
0.05 0.6361 2,521 1,441 1,080 0.2416 3.189 3.431 2.6
0.05 0.6035 2,521 1,441 1,080 0.2416 3.189 3.431 2.7
0.05 0.5727 2,521 1,441 1,080 0.2416 3.189 3.431 2.8
0.05 0.5435 2,521 1,441 1,080 0.2416 3.189 3.431 2.9
0.05 0.5161 2,521 1,441 1,080 0.2416 3.189 3.431 3
0.05 0.4904 2,521 1,441 1,080 0.2416 3.189 3.431 3.1
0.05 0.4663 2,521 1,441 1,080 0.2416 3.189 3.431 3.2
0.05 0.4438 2,521 1,441 1,080 0.2416 3.189 3.431 3.3
0.05 0.4228 2,521 1,441 1,080 0.2416 3.189 3.431 3.4
0.05 0.4031 2,521 1,441 1,080 0.2416 3.189 3.431 3.5
Reference
Anwar Adem, Richard Kneller, Cher Li (2023) Information constraints and technology efficiency: Field experiments benchmarking firms website performance. CESifo Working Paper 10457. https://www.cesifo.org/en/publications/2023/working-paper/information-constraints-and-technology-efficiency-field-experiments