Previous studies had demonstrated that the T-g of polymer thin films is strongly dependent on the interactions, gamma (s), between the polymer and the underlying substrate. We present a study of the glass transition temperature, T-g, in thin films of polystyrene, PS, as a function of film thickness and as a function of gamma (s) by measuring the change in the thermal expansion using X-ray reflectivity. The T-g for PS on native silicon oxide was found to decrease with decreasing film thickness, consistent with results by others. Using random copolymers of styrene and methyl methacrylate anchored to the substrate, gamma (s) could be varied by changing the styrene fraction, f. With a constant PS film thickness of 330 Angstrom, the T-g was depressed by similar to 20 degreesC as f was decreased from I to 0.75. An analysis analogous to the Gibbs-Thompson model indicated that the surface energy was not a suitable parameter to use to describe the effects of interfacial interactions on the T-g of polymer thin films. An associated local fractional change in the segment density at the substrate interface is instead proposed to describe the changes in T-g observed with different gamma (s). Only a small change (< 1.4%) in the density at the interface is required to produce a 20 degreesC depression in T-g found in this experiment.