Investigators will perform a randomized, parallel, double-blind clinical trial of two 6-month dosing strategies of levothyroxine (LT4) in patients aged 65 years and older who are already taking a stable dose of at least 1.2 mcg/kg/day of LT4 therapy, one to maintain a target thyroid stimulating hormone (TSH) level of 0.5-2.0 mU/L (lower TSH group) and another of a lower levothyroxine dose to achieve a target TSH of 5.5-7.0 mU/L (higher TSH group). Investigators will assess the effects of levothyroxine therapy at two different TSH targets on symptoms of hypothyroidism, mood, sleep, measures of memory and executive function, weight, lipids, and a marker of bone turnover.