Abstract Elastography provides significant information on staging of fibrosis in patients with liver disease and may be of some value in assessing steatosis. However, there remain questions as to the role of steatosis and fibrosis as cofactors influencing the viscoelastic measurements of liver tissues, particularly shear wave speed (SWS) and shear wave attenuation (SWA). In this study, by employing the theory of composite elastic media as well as two independent experimental measurements on oil-in-gelatin phantoms and also finite element simulations, it is consistently shown that fat and fibrosis jointly influence the SWS and SWA measurements. At a constant level of fat, fibrosis stages can influence the SWA by factors of 2–4. Moreover, the rate of increase in SWA with increasing fat is strongly influenced by the stages of fibrosis; softer background cases (low fibrosis stages) have higher rate of SWA increase with fat than those with stiffer moduli (higher fibrosis stages). Meanwhile, SWS results are influenced by the presence of fat, however the degree of variability is more subtle. The results indicate the importance of jointly considering fat and fibrosis as contributors to SWS and SWA measurements in complex liver tissues and in the design and interpretation of clinical trials.