Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
The total sample is composed of those who (i) visited a food market and/or (ii) enrolled in the FS program at one of the three schools over the past semester. This represents a sample of 4,153 students. The target assignment is 692 participants to the $10 incentive arm, 692 participants to the $20 incentive arm, and 2,769 to control. Thus, the expected sample sizes are:
Nc = 2,769
N10 = 692
N20 = 692
N$ = 1,384
The primary outcome is an indicator for survey completion. Power calculations are based on two-sided tests with significance level alpha = 0.05 and 80% power. Unless otherwise noted, calculations use a normal approximation for differences in proportions. We assume a completion rate of 25% in the control group, 45% in the $10 incentive group, and 55% in the $20 incentive group. These assumptions imply that the pooled incentive group has an expected completion rate of approximately 50%.
1. Main effect of any financial incentive on survey completion
The first power calculation concerns the pooled effect of receiving any financial incentive, combining the $10 and $20 incentive arms. The primary estimating equation for this contrast is:
Yi =s+T$i+i
where Yi is an indicator for survey completion, T$i equals one for participants assigned to either incentive arm, and s are randomization-stratum fixed effects.
Under the assumed completion rates of 25% in the control group and 50% in the pooled incentive group, the approximate standard error for the difference in completion rates is:
SE=sqrt(0.5(1-0.5)/1,384) + sqrt(0.25(1-0.25)/2,769)
This implies an approximate minimum detectable effect of 4.4 percentage points.
2. Difference between the $10 and $20 incentive arms
The second power calculation concerns whether the $20 incentive has a larger effect on survey completion than the $10 incentive. The relevant test compares completion rates between the two incentive arms:
H0: B10=B20
The corresponding estimating equation is:
Yi = as + B10*T10i + B20*T20i + ei
where T10i and T20i indicate assignment to the $10 and $20 incentive arms, respectively.
For this comparison, both incentive arms have approximately 700 participants. Assuming completion rates of 45% in the $10 arm and 55% in the $20 arm, the approximate standard error is:
SE=sqrt(0.45(1-0.45)/692) + sqrt(0.55(1-0.55)/692)
This implies an approximate minimum detectable difference of 7.5 percentage points.
3. Heterogeneity of the incentive effect across groups
The third power calculation concerns whether the effect of receiving any financial incentive on completion differs across observed characteristics. We will make many such comparisons, but an important test will be to compare whether individuals of different baseline income levels are more or less responsive to survey incentives. For analysis of this sort, the relevant estimating equation is:
Yi = as + B1 T$i + B2(T$i*HighInci) + B3 (T$i*MissInci) + ei
where low income is the omitted category. The coefficient B2 measures whether the effect of any incentive on completion differs between individuals with high vs. low income. The coefficient B3 captures the corresponding difference for participants with missing income, but this comparison is secondary.
Categorizing participants into high and low income groups using the sample mean of $300, the corresponding sample sizes and MDEs (comparing one group to all other groups) are presented in the following table:
Income Group Breakdown (N | Incentive N | Control N | MDE):
Low Income: 1,821 total (607 Inc. / 1,214 Ctrl) | MDE: 6.7 pp
High Income: 1,822 total (607 Inc. / 1,215 Ctrl) | MDE: 6.7 pp
Missing Income: 510 total (170 Inc. / 340 Ctrl) | MDE: 12.6 pp
When comparing the high income group to the low group, the MDE is 9.4 percentage points.