Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
For the power calculations we consider two sets of outcome variables. The first set of outcome variables uses the administrative files and can, hence, be based on the full sample. The second set of outcome variables are based on 3 electronic surveys sent by e-mail: (1) an initial survey at the start of the unemployment spell, (2) an intermediate survey after about 3.5 months of unemployment, and (3) an exit survey one month after exit from unemployment. We plan to send initial surveys to about 50% of the retained treated sample and to about 25% of the control sample, i.e. to between 10,300 and 13,125 individuals in each treatment arm (between 30,900 and 39,375 individuals in total). By contrast, intermediate surveys are planned to be sent to the full treatment population and 50% of the control population, i.e. to between 20,600 and 26,250 individuals in each treatment arm (between 61,800 and 78,750 individuals in total). Nevertheless, these numbers should be regarded as targets. Due to budgetary and administrative restrictions we might have to reduce these targets. Based on a pilot survey we set the first survey response rate equal to 30%. Conditional on response in a prior survey, the subsequent response rate is assumed to be 70%. These response rates are taken into account when conducting the power analysis for the outcomes that are based on the survey outcomes. For outcomes based on the administrative data only, these response rates are set to 100%
Since we have no clear a priori about the means and standard deviations of the outcome variables, we conduct the power analyses for two cases: (1) for a standardized outcome variable, i.e. a variable of which the mean is normalized to zero and the standard deviation to one, and for (2) a discrete dichotomous variable with two possible outcomes and with mean equal to 0.5. We set in all power analyses the significance level to 5% and the power to 80%.
We consider for each outcome three different effects: treatment 1 (T1) versus control (C), T2 versus C and T1 versus T2. Because the control group is double as large as each of the treated groups, the power analysis for the outcome variables based on administrative files differs according to the considered treatment effect: the minimum detectable effect size (MDE) will be larger for the contrast between T1 and T2 than for the other two contrasts. By contrast, since in the surveys we set the sample size to be equal for each treatment arm, the MDE is the same for all three treatment effects.
1. Set of outcome variables based on admin files
(i) For the standardized outcome (mean equal to zero and standard deviation equal to one) the MDE ranges, depending on aforementioned sample size ranges, between 0.0212 and 0.0239 of a standard deviation when we consider the contrasts between T1 (or T2) and C. For the contrast between T1 and T2 the MDE ranges between 0.0245 and 0.0276.
(ii) For the binary discrete outcome with mean equal to 0.5, the MDE for the contrasts between T1 (or T2) and C ranges between 0.0106 and 0.0119. For the contrast between T1 and T2 it ranges between 0.0130 and 0.0146.
2. Set of outcome variables based on the intermediate and the outcome surveys. For outcomes based on these surveys, we consider that the response rate of 30% reduces the effective sample size by treatment arm to between 6,180 and 7,950. Hence the MDE of a standardized outcome ranges between 0.0444 and 0.0504, while for a binary discrete outcome it ranges between 0.0222 and 0.0252.
3. Set of outcome variables based on an interaction between variables in the initial survey on the one hand, and, on the other hand, outcomes in a) the administrative data, b) the intermediate survey and the outcome survey. This aims, amongst other, at measuring the moderating effect of initial motivation on the other outcomes.
(i) the administrative data, the effective survey size must only be reduced by the non-response rate (70%) in the initial survey. The effective sample size by treatment arm ranges therefore between 3,090 and 3,938. The MDE of a standardized interacted outcome therefore ranges between 0.0631 and 0.0713.
(ii) the intermediate survey and the outcome survey, the effective survey size must be reduced by the non-response rate in the initial survey (70%) as well as by the non-response response rate in the intermediate survey, conditional on response in the initial survey (30%). The effective sample size by treatment arm therefore ranges between 2,163 and 2,757. The MDE of a standardized interacted outcome therefore ranges between 0.0755 and 0.0852.