A couple dozen medical societies want Congress to kill a not-yet-launched Medicare program that aims to curb writing orders for expensive advanced imaging tests like MRIs.

The program, due to start Jan. 1, 2020, requires doctors to consult an online portal before ordering the tests if they expect to be reimbursed by Medicare. The doctor may override it, however, and order the test anyway. The first year will be a testing period with no penalties for incorrect reporting.

The goal...