Strategy for power calculation for interactions: Application to a trial of interventions to improve uptake of bowel cancer screening

Poorer postcodes within 5 regions in England have a lower response to bowel-cancer screening invitations than do richer postcodes. An extension of the sample-size formula for two proportions is used to determine that needed to detect an increase in response rate that varies by deprivation quintile. The proportions plugged into the formula are weighted averages based on the relationship between response and deprivation; the response rate is adjusted to be constant across deprivation quintiles. From a baseline period between October 2006 and January 2009, detection of an absolute or relative increase of at least 1,2,3,4 and 5% in response rate is required for the richest to poorest quintiles respectively because the interventions were chosen as those most likely to have an effect in the lower socioeconomic groups. A computer simulation experiment shows that the approach is more conservative than a likelihood-ratio calculation, and it appears sensible when compared with repeated application of a two-sample calculation at each quintile.