Background Because responses to interventions can be heterogeneous, interest may lie in the extent to which any given individual’s response to an intervention relates to their baseline status (an intervention differential effect; IDE). In interventions designed to prevent excess childhood weight gain, researchers may want to investigate whether children with higher or lower body weights at baseline respond differently to the intervention. However, when investigating the potential of an IDE, it is necessary to avoid the issue of mathematical coupling (MC), where change in weight is analysed with respect to initial weight using correlation or regression. The problem of MC, and methods used to overcome it (Oldham’s method and multilevel modelling) have been described previously for outcomes that are homoscedastic. However, the literature does not explore how these methods perform in identifying IDEs in outcomes that are inherently heteroscedastic (such as growth in childhood body weight). We hypothesised that methods for detecting IDEs in heteroscedastic outcomes are only robust when analyses in the intervention group are compared with analyses in control group data.

Methods We explored the performance of Oldham’s method and multilevel modelling in overcoming MC within heteroscedastic data. We simulated longitudinal data derived from child weight growth statistics, designed to be heterogeneous to reflect real-world growth data. To emulate weight-management programmes, an intervention group was simulated with an IDE, and a control group was simulated without. Methods for detecting IDEs were evaluated: first in the intervention group only, then with analyses of the intervention group contrasted with the control group. Simulations were performed in R and MLwiN.

Results We demonstrated that Oldham’s method and multilevel modelling were biased when used to estimate an IDE within inherently heteroscedastic data. However, we showed that introducing a control group comparison enabled both methods to robustly detect an IDE in heteroscedastic data, providing that parametric assumptions of growth were justified and modelled explicitly (e.g. as linear, quadratic, etc.).

Conclusion Oldham’s method and multilevel models can robustly detect an IDE in growth data that are inherently heteroscedastic if analyses include a control group and underlying growth patterns can be parameterised appropriately. For study designs that do not collect control group data (as with most weight management programmes amongst children), identification of an IDE currently remains intractable.