Non-regular inference for dynamic weighted ordinary least squares: understanding the impact of solid food intake in infancy on childhood weight.

A dynamic treatment regime (DTR) is a set of decision rules to be applied across multiple stages of treatments. The decisions are tailored to individuals, by inputting an individual's observed characteristics and outputting a treatment decision at each stage for that individual. Dynamic weighted ordinary least squares (dWOLS) is a theoretically robust and easily implementable method for estimating an optimal DTR. As many related DTR methods, the dWOLS treatment effects estimators can be non-regular when true treatment effects are zero or very small, which results in invalid Wald-type or standard bootstrap confidence intervals. Inspired by an analysis of the effect of diet in infancy on measures of weight and body size in later childhood-a setting where the exposure is distant in time and whose effect is likely to be small-we investigate the use of the $m$-out-of-$n$ bootstrap with dWOLS as method of analysis for valid inferences of optimal DTR. We provide an extensive simulation study to compare the performance of different choices of resample size $m$ in situations where the treatment effects are likely to be non-regular. We illustrate the methodology using data from the PROmotion of Breastfeeding Intervention Trial to study the effect of solid food intake in infancy on long-term health outcomes.