Epistasis is an important and poorly understood aspect of mutations and strongly influences the evolutionary impact of genetic variation on adaptation and fitness. Although recent studies have begun to characterize the distribution of epistatic effects between mutations affecting fitness, there is currently a lack of empirical information on the underlying biological causes of these epistatic interactions. What are the functional constraints that determine the effectiveness of a compensatory mutation at restoring fitness? We have measured the effect-sizes of 52 compensatory mutations affecting nine different deleterious mutations in the major capsid and spike proteins of the DNA bacteriophage phiX174. On average, an experimentally detectable compensatory mutation recovers about two-thirds of the fitness cost of the preceding deleterious mutation. Variation in fitness effect-sizes is only weakly associated with measures of the distance separating the deleterious and compensatory mutations in the amino acid sequence or the folded protein structure. However, there is a strong association of fitness effect-size with the correlation in the effects of the mutations on the biochemical properties of amino acids. A compensatory mutation has the largest effect-size, on average, when both the compensatory and deleterious mutations have radical effects on the overall biochemical make-up of the amino acids. By examining the relative contributions of specific biochemical properties to variation in fitness effect-size, we find that the area and charge of amino acids have a major influence, which suggests that the complexity of the amino acid phenotype is simplified by selection into a reduced number of phenotypic components.