Minimum detectable effect size for main outcomes (accounting for sample design and clustering)
The primary outcome variable is the Manager’s choice of feedback. Specifically, we consider whether Managers choose to increase the noise of the signal by sending a less precise message to the Worker, or to keep the signal precision as it is. Hence, the Manager’s feedback choice will be transformed into a binary outcome variable.
As a main test of our theoretical predictions, we consider pairwise comparisons of the Manager’s feedback choice between the non-instrumental and instrumental treatments for a given signal that they have received. With 6 possible signal types (ranks between 1 and 4, top half, and bottom half), 2 treatments, and 1,200 Managers, this implies that we have 100 Managers in each cell. Our power calculation is based on: (i) baseline proportions of 0.01, 0.25, 0.50, or 0.75; (ii) one-tailed z-tests of differences between two independent proportions; (iii) Type I error rate of 0.05 and power of 0.80. Given these parameters, the minimum detectable effect size is an increase in proportion of between 0.074 and 0.172 in the treated group.
The analysis of gender differences in feedback provision will primarily be conducted using parametric regressions. Using simple linear probability models, we first consider a baseline model of Manager’s feedback choice against the Worker’s gender, the Manager’s gender, the signal type received by the Manager, the treatment variable (instrumental vs. non-instrumental), as well as the Manager’s posterior and second-order beliefs about the Worker’s ability. This provides a total of 10 predictors with 1 predictor to be tested. Considering a F-test of an increase in R2 with 1,200 Managers, a Type I error rate of 0.05, and a power of 0.80, the minimum detectable effect size is 0.0066.
To allow for heterogeneity in treatment effects, we also consider interaction terms between the Worker’s gender and Manager’s signals in the regression model. This gives a total of 15 predictors of which 6 are to be tested. Using the same parameters as above, the minimum detectable effect size is 0.0114.
On the Workers’ end, the main outcome variable is a binary decision of incentive choice on their Part 3 rank (competition versus piece-rate). Our treatment comparisons will be similar to that for Managers, except that we will now consider the feedback sent by the Managers instead of the signal that they have received. Hence, the power calculations and minimum detectable effect sizes follow that of the preceding paragraphs.