We investigate data hiding techniques that attempt to defeat steganalysis by restoring the statistics of the composite image to resemble that of the cover. The approach is to reserve a number of host symbols for statistical restoration: host statistics perturbed by data embedding are restored by suitably modifying the symbols from the reserved set. While statistical restoration has broad applicability to a variety of hiding methods, we illustrate our ideas here for quantization index modulation (QIM) based hiding. We propose a method for significantly reducing the detectability of QIM, while preserving its robustness to attacks. We next use the framework of statistical restoration to develop a method to combat steganalysis techniques which detect block-DCT embedding by evaluating the increase in blockiness of the image due to hiding. Numerical results demonstrating the efficacy of these techniques are provided.