Recovery after stroke: not so proportional after all?

Abstract

The proportional recovery rule asserts that most stroke survivors recover a fixed proportion of lost function. To the extent that this is true, recovery from stroke can be predicted accurately from baseline measures of acute post-stroke impairment alone. Reports that baseline scores explain more than 80%, and sometimes more than 90%, of the variance in the patients' recoveries, are rapidly accumulating. Here, we show that these headline effect sizes are likely inflated. The key effects in this literature are typically expressed as, or reducible to, correlation coefficients between baseline scores and recovery (outcome scores minus baseline scores). Using formal analyses and simulations, we show that these correlations will be extreme when outcomes are significantly less variable than baselines, which they often will be in practice regardless of the real relationship between outcomes and baselines. We show that these effect sizes are likely to be over-optimistic in every empirical study that we found that reported enough information for us to make the judgement, and argue that the same is likely to be true in other studies as well. The implication is that recovery after stroke may not be as proportional as recent studies suggest.

Figure 1

A canonical example of spurious r(X, Δ). Baselines scores are uncorrelated with outcomes (A), but baseline scores appear to be strongly correlated with recovery (B). That correlation can be used to derive predicted recovery, which is strongly correlated with empirical recovery (C), but predicted outcomes, derived from that predicted recovery, are still uncorrelated with empirical outcomes (D).

Figure 2

The relationship between r(X,Y), r(X,Δ) and σ_Y/σ_X. Note that the x-axis is log-transformed to ensure symmetry around 1; when X and Y are equally variable, log(σ_Y/σ_X) = 0. Supplementary material, proposition 7 in Appendix A, provides a justification for unambiguously using a ratio of standard deviations in this figure, rather than σ_Y and σ_X as separate axes. The two major regimes of Equation 1 are also marked in red. In Regime 1, Y is more variable than X, so contributes more variance to Δ, and r(X,Δ) ≈ r(X,Y). In Regime 2, X is more variable than Y, so X contributes more variance to Δ, and r(X,Δ) ≈ r(X,−X) (i.e. −1). The transition between the two regimes, when the variability ratio is not dramatically skewed either way, also allows for spurious r(X,Δ). For the purposes of illustration, the figure also highlights six points of interest on the surface, marked A–F; examples of simulated recovery data corresponding to these points are provided in Fig. 3.

Figure 3

Exemplar points on the surface in Fig. 2 . Simulated recovery data, corresponding to the points A–F marked on the surface in Fig. 1. (A) Baselines and outcomes are entirely independent [r(X,Y) = 0], yet r(X,Δ) is relatively strong; this is the canonical example of mathematical coupling, first introduced by Oldham (1962). (B) Recovery is constant with minimal noise, so baselines and outcomes are equally variable (σ_Y/σ_X ≈ 1) and recovery is unrelated to baseline scores (r(X, Δ) ≈ 0). (C and D) Outcomes are more variable than baselines (σ_Y/σ_X ≈ 5), and r(X,Δ) converges to r(X,Y). (E) Recovery is 70% of lost function, so outcomes are less variable than baselines (σ_Y/σ_X ≈ 0.3); even with shuffled outcomes data (F) baselines and recovery still appear to be strongly correlated.

Figure 4

r(X,Y) and r(X,Δ) have the same residuals. Left: Least squares linear fits for analyses relating baselines to (top) outcomes and (bottom) recovery, using the fitters’ data reported by Zarahn et al. (2011). Middle: Plots of residuals relative to each least squares line, against the fitted values in each case. Right: A scatter plot of the residuals from the model relating baselines to change, against the residuals from the model relating baselines to outcomes: the two sets of residuals are the same.