Moderating effects of (a, b) parent restriction of food, (c, d) concern over parent weight, and (e, f) child reported parental acceptance on zBMI changes by treatment group. Mixed effects regression models showed that baseline levels of restriction (P = 0.011), concern over parent weight (P = 0.009), and child reported parent acceptance (P = 0.045) interacted with treatment to influence outcome. In each interaction, children randomized to the increase healthy food group showed better long-term change than children randomized to the reduce high energy-dense food group for children with high levels of the moderating variable.