The decision to be an organ donor may seem easy for some: You sign an agreement that will let your heart, kidneys, liver, pancreas and other organs be used after your death in a way that helps the recipients lead fuller, healthier lives.

But for other people, the choice is harder. Some fear that a doctor may not work as hard to save them because he or she wants their organs for other patients, or that their organs might be removed prematurely (although there is no evidence to support either of these concerns). There may also be a psychological cost of having to think about your own death at a time when you are still relatively healthy.…