The most likely diagnosis is celiac disease. This patient has the combination of loose stools, unintentional weight loss, elevated liver chemistry test results, and iron deficiency anemia without an obvious source of blood loss. Minor elevations of serum aminotransferase levels are found in up to 50% of patients with celiac disease and may be the only presenting feature in approximately 9% of patients. Serum aspartate aminotransferase (AST) is usually less than 80 U/L, and alanine aminotransferase (ALT) is usually less than 130 U/L. Patients with celiac disease can also have coexisting liver disease such as autoimmune hepatitis, primary sclerosing cholangitis, or primary biliary cirrhosis. Improvements in liver chemistry studies that are related to celiac disease occur when such patients are treated with a gluten-free diet.

Alcohol-induced liver disease (ALD) is diagnosed by a history of significant alcohol intake and clinical evidence of liver disease. Serum AST level is typically elevated two to six times the upper limit of normal in severe alcoholic hepatitis. AST levels above 500 U/L and ALT levels above 200 U/L are uncommon. An AST/ALT ratio above 2 to 3 is typical for ALD. This patient does not have significant alcohol intake or an aminotransferase profile typical of alcohol-induced liver disease.

Chronic viral hepatitis can result in modest elevations in the aminotransferase levels; typically ALT is greater than AST. The alkaline phosphatase level may be normal or slightly elevated. A diagnosis of chronic hepatitis would not account for the patient's diarrhea or iron deficiency anemia.

Primary biliary cirrhosis (PBC) is a chronic cholestatic liver disease of unknown cause. PBC is diagnosed by serum alkaline phosphatase levels greater than 1.5 times the upper limit of normal and typically normal serum total bilirubin levels. Increases in serum AST and ALT levels less than five times the upper limit of normal may be seen as well. PBC is not associated with diarrhea or iron deficiency anemia.