FYI when I worked for HP, HR implemented a similar policy and I as a union rep worked with them on the 12 months of stats following the policy. We also saw a similar effect to that observed by Pudney. However, we saw a 21% increase in incidence of stress related sickness, so in the end HR rescinded the policy, as overall sickness absence rose, especially long term sickness. They consequently introduced "duvet" days where you could just send a text to say you weren't coming in and it was subtracted from annual leave or flexi - no conditions were applied and you didn't need to give a reason. Single day absences dropped by over 15% and overall sickness rates by 8%.
Sometimes, people just need the day off.