Effect Size Rules of Thumb for One-Dimensional Functional Data With an Application to Gait Analysis

When analyzing gait and other human movements, often one-dimensional (1D) functional data are considered, where a variable like joint angle changes smoothly over time from movement start to end. Interpretation of effect sizes in functional data like these generally follows the widely-cited Cohen/Sawilowsky rules of thumb. A key problem is that a given effect size occurs with greater probability for functional data than for the simple scalar (0D) case of Cohen/Sawilowsky. Here, we propose both (i) functional effect size rules of thumb that are probabilistically consistent with the Cohen/Sawilowsky guidelines for a benchmark, two-sample scenario, and (ii) a framework for adapting the benchmark interpretations to arbitrary experimental scenarios with probabilistic consistency. Analysis of an open total hip arthroplasty gait dataset showed that post-surgery effect sizes would be interpreted as 'medium' and 'less than very small' for the Cohen/Sawilowsky and proposed functional rules of thumb, respectively. Adapting the benchmark case to the actual experimental case (paired design with $n$ = 52 and highly smooth functional residuals) contrastingly yielded an effect size interpretation of 'very large'. These stark interpretation contrasts suggest that a single set of interpretation guidelines should not be applied to arbitrary experimental scenarios. We recommend using the Cohen/Sawilowsky and proposed functional rules of thumb only for a priori power analysis, and only in the absence of information regarding population variance and smoothness. For all other cases, especially post hoc effect size interpretation, we recommend using the proposed framework to yield probabilistically consistent results, and thus more meaningful cross-study interpretations. Code replicating all results is available at https://github.com/0todd0000/esrot1d.