First, W is not the specimen thickness. B is the specimen thickness. W is the width. Second, ASTM E 1290 and other fracture toughness standards (ASTM E 1820, etc.) are quite clear in specifying the relationship of W to B, so that a valid test is obtained.
So, are you referring to the specimen thickness B? If so, then thickness can have a large influence on the results, depending on the material being tested. Plane stress or plane strain conditions produce different results due to the effect of out of plane stresses, and specimen thickness is a huge part of this. You should choose the specimen size to be appropriate for your loading conditions, e.g. if your final application has a thickness of 4 mm, then this is appropriate for your test, regardless of other tests having been performed at 10 mm.