Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
To simulate minimum detectable effect sizes, we use the pilot of the lab-in-the-field experiment with a sample of 33 parents who have similar characteristics to our target sample. Given the limited size of this sample, as well as the initial wording of questions and procedures used in the pilot (which was later improved and refined), this data is expected to be noisier than our target sample.
To estimate the minimum detectable effect size, we simulate effect sizes for the main specification (Equation 4) aiming for 80% power with our target sample of 2,400 respondents. Our approach follows Campos-Mercade (2024). To do so, we use the mean and standard deviation of log earnings in the pilot data as the dependent variable.
According to these simulations, for βp, the productivity of time investments, we are positioned to detect parameter sizes bigger than 0.045 SD (0.059 SD when clustering at the individual level) of the outcome variable (log earnings). With the pilot data variation, this estimate corresponds to a face value of 0.07 (0.09) for βp, corresponding to a minimum detectable increase of 0.07% (0.09%) in log earnings for a 1% increase in time investments.
For γp, the productivity of child initial skills, we are positioned to detect parameters bigger than 0.105 SD (0.135 SD when clustering at the individual level), with a face value of 0.17 (0.21), corresponding to a minimum detectable increase of 0.17% (0.21%) in log earnings for a 1% increase in initial skills.