The evolution of heterogametic sex chromosomes is often-but not always-accompanied by the evolution of dosage compensating mechanisms that mitigate the impact of sex-specific gene dosage on levels of gene expression. One emerging view of this process is that such mechanisms may only evolve in male-heterogametic (XY) species but not in female-heterogametic (ZW) species, which will consequently exhibit "incomplete" sex chromosome dosage compensation. However, recent results suggest that at least some Lepidoptera (moths and butterflies) may prove to be an exception to this prediction. Studies in bombycoid moths indicate the presence of a chromosome-wide epigenetic mechanism that effectively balances Z chromosome gene expression between the sexes by reducing Z-linked expression in males. In contrast, strong sex chromosome dosage effects without any reduction in male Z-linked expression were previously reported in a pyralid moth, suggesting a lack of any such dosage compensating mechanism. Here we report an analysis of sex chromosome dosage compensation in Heliconius butterflies, sampling multiple individuals for several different adult tissues (head, abdomen, leg, mouth, and antennae). Methodologically, we introduce a novel application of linear mixed-effects models to assess dosage compensation, offering a unified statistical framework that can estimate effects specific to chromosome, to sex, and their interactions (i.e., a dosage effect). Our results show substantially reduced Z-linked expression relative to autosomes in both sexes, as previously observed in bombycoid moths. This observation is consistent with an increasing body of evidence that some lepidopteran species possess an epigenetic dosage compensating mechanism that reduces Z chromosome expression in males to levels comparable with females. However, this mechanism appears to be imperfect in Heliconius, resulting in a modest dosage effect that produces an average 5-20% increase in male expression relative to females on the Z chromosome, depending on the tissue. Thus our results in Heliconius reflect a mixture of previous patterns reported for Lepidoptera. In Heliconius, a moderate pattern of incomplete dosage compensation persists apparently despite the presence of an epigenetic dosage compensating mechanism. The chromosomal distributions of sex-biased genes show an excess of male-biased and a dearth of female-biased genes on the Z chromosome relative to autosomes, consistent with predictions of sexually antagonistic evolution.