The proportional recovery rule redux: Arguments for its biological and predictive relevance

The proportional recovery rule (PRR) posits that most stroke survivors can expect to reverse a fixed proportion of motor impairment. As a statistical model, the PRR explicitly relates change scores to baseline values -- an approach that has the potential to introduce artifacts and flawed conclusions. We describe approaches that can assess associations between baseline and changes from baseline while avoiding artifacts either due to mathematical coupling or regression to the mean due to measurement error. We also describe methods that can compare different biological models of recovery. Across several real datasets, we find evidence for non-artifactual associations between baseline and change, and support for the PRR compared to alternative models. We conclude that the PRR remains a biologically-relevant model of recovery, and also introduce a statistical perspective that can be used to assess future models.

Inflated Estimates of Proportional Recovery From Stroke

The proportional recovery rule states that most survivors recover a fixed proportion (≈70%) of lost function after stroke. A strong (negative) correlation between the initial score and subsequent change (outcome minus initial; ie, recovery) is interpreted as empirical support for the proportional recovery rule. However, this rule has recently been critiqued, with a central observation being that the correlation of initial scores with change over time is confounded in the situations in which it is typically assessed. This critique has prompted reassessments of patients’ behavioral trajectory following stroke in 2 prominent papers. The first of these, by van der Vliet et al presented an impressive modeling of upper limb deficits following stroke, which avoided the confounded correlation of initial scores with change. The second by Kundert et al reassessed the value of the proportional recovery rule, as classically formulated as the correlation between initial scores and change. They argued that while effective prediction of recovery trajectories of individual patients is not supported by the available evidence, group-level inferences about the existence of proportional recovery are reliable. In this article, we respond to the van der Vliet and Kundert papers by distilling the essence of the argument for why the classic assessment of proportional recovery is confounded. In this respect, we reemphasize the role of mathematical coupling and compression to ceiling in the confounded nature of the correlation of initial scores with change. We further argue that this confound will be present for both individual-level and group-level inference. We then focus on the difficulties that can arise from ceiling effects, even when initial scores are not being correlated with change/recovery. We conclude by emphasizing the need for new techniques to analyze recovery after stroke that are not confounded in the ways highlighted here.

Statistical limitations on drawing inferences about proportional recovery

BackgroundNumerous studies have found large statistical relationships between the amount of recovery and initial impairments in people with stroke. When change scores are regressed onto initial impairments, the resulting slope is approximately ≈0.7for a variety of outcomes. These findings have led to the 70% “proportional recovery rule” and the argument that proportional recovery represents a biological phenomenon. Previous studies of proportional recovery are confounded by statistical limitations that come from regressing change scores onto initial impairments in bounded scales.ObjectiveOur goal is to show that data claimed as evidence for proportional recovery are generally consistent with random patterns of recovery, once statistical limitations are taken into account.MethodsUsing a pooled dataset of N = 373 Fugl-Meyer Assessment (FMA) upper extremity scores extracted from published literature, we ran simulations to illustrate three main arguments: (1) Mathematical coupling renders the traditional null-hypothesis significance test irrelevant in proportional recovery studies; (2) Proportional recovery is one of many alternative hypotheses; (3) Current evidence claimed in favor of proportional recovery is consistent with uniform random recovery.ResultsOur simulations show that if all data were included (no exclusion of “non-fitters”) regressing change scores onto initial impairments in a bounded scale would lead to a slope of ≈ 0.5. Similarly, cluster analysis will spuriously identify groups of fitters and non-fitters, leading to a slope for the fitters of ≈ 0.7, when the underlying recovery is random.ConclusionsThese results cast doubt on the validity of “proportional recovery” as a population level-statistic and a biological phenomenon.

Does Mathematical Coupling Matter to the Acute to Chronic Workload Ratio? A Case Study From Elite Sport

Purpose: Criticisms of the acute to chronic workload ratio (ACWR) have been that the mathematical coupling inherent in the traditional calculation of the ACWR results in a spurious correlation. The purposes of this commentary are (1) to examine how mathematical coupling causes spurious correlations and (2) to use a case study from actual monitoring data to determine how mathematical coupling affects the ACWR. Methods: Training and competition workload (TL) data were obtained from international-level open-skill (basketball) and closed-skill (weightlifting) athletes before their respective qualifying tournaments for the 2016 Olympic Games. Correlations between acute TL, chronic TL, and the ACWR as coupled/uncoupled variations were examined. These variables were also compared using both rolling averages and exponentially weighted moving averages to account for any potential benefits of one calculation method over another. Results: Although there were some significant differences between coupled and uncoupled chronic TL and ACWR data, the effect sizes of these differences were almost all trivial (g = 0.04–0.21). Correlations ranged from r = .55 to .76, .17 to .53, and .88 to .99 for acute to chronic TL, acute to uncoupled chronic TL, and ACWR to uncoupled ACWR, respectively. Conclusions: There may be low risk of mathematical coupling causing spurious correlations in the TL–injury-risk relationship. Varying levels of correlation seem to exist naturally between acute and chronic TL variables regardless of coupling. The trivial to small effect sizes and large to nearly perfect correlations between coupled and uncoupled AWCRs also imply that mathematical coupling may have little effect on either calculation method, if practitioners choose to apply the ACWR for TL monitoring purposes.

Breaking Proportional Recovery After Stroke

People with hemiparesis after stroke appear to recover 70% to 80% of the difference between their baseline and the maximum upper extremity Fugl-Meyer (UEFM) score, a phenomenon called proportional recovery (PR). Two recent commentaries explained that PR should be expected because of mathematical coupling between the baseline and change score. Here we ask, If mathematical coupling encourages PR, why do a fraction of stroke patients (the “nonfitters”) not exhibit PR? At the neuroanatomical level of analysis, this question was answered by Byblow et al—nonfitters lack corticospinal tract (CST) integrity at baseline—but here we address the mathematical and behavioral causes. We first derive a new interpretation of the slope of PR: It is the average probability of scoring across remaining scale items at follow-up. PR therefore breaks when enough test items are discretely more difficult for a patient at follow-up, flattening the slope of recovery. For the UEFM, we show that nonfitters are most unlikely to recover the ability to score on the test items related to wrist/hand dexterity, shoulder flexion without bending the elbow, and finger-to-nose movement, supporting the finding that nonfitters lack CST integrity. However, we also show that a subset of nonfitters respond better to robotic movement training in the chronic phase of stroke. These persons are just able to move the arm out of the flexion synergy and pick up small blocks, both markers of CST integrity. Nonfitters therefore raise interesting questions about CST function and the basis for response to intensive movement training.