Age (average ± standard deviation = 60.3 ± 9.9 years), time between stroke and assessment of FMii (2.5 ± 2.2 days), lesion location, FMii (28.9 ± 23.1), ΔFM (14.9 ± 13.8), and acute dynamometry (12.3 ± 13.3 kg) for the individual patients of the imaged sample are presented in . Patients in the imaged sample were assessed for FMii and scanned with fMRI at 2.5 ± 2.2 days poststroke for 2 minutes 52 seconds per hand during the same MRI session as their clinical exams. Patients had been instructed just prior to scanning to attempt hand closure (for a specified hand) at an auditorily cued 1-Hz pace (alternating with rest epochs with matched auditory stimuli). Only fMRI data from the affected hand were used in this analysis. Follow-up for assessment of ΔFM in the imaged patient sample was at 93 ± 17 days poststroke.
Basic information for imaged sample
Twelve out of 30 patients of the imaged sample had an acute dynamometry score of 0 kg () and thus are likely to have been unable to execute the instructions of the motor task. Formally, however, the experimental condition was not the performance of the task but was rather the instruction to perform the task: behavior is never directly under experimental control and so cannot be properly thought of as an experimental variable. Behavior was not measured, but acute dynamometry score was used as a covariate in the estimation of CV fMRI recovery pattern expression (Z), and so any task-related fMRI activation linearly related to dynamometry could not contribute to prediction of recovery through Z. For example, if task-related activation was simply a reflection of dynamometry score, then Z would be pure noise with respect to ΔFM, even though dynamometry and ΔFM are correlated (R = 0.56, 2-tailed P = 0.001).
Z was computed in a CV fashion by essentially taking the inner product of a given patient's task-related fMRI activation data with an estimated fMRI recovery pattern whose computation involved neither that patient's fMRI data nor their ΔFM. Z was significantly correlated with ΔFM in the net imaged sample (R = 0.56, 1-tailed P < 0.001; it is not a typo that this is the same R value as immediately previous). While this result is more robust than a correlation between non-CV recovery pattern expression and ΔFM, it still does not directly indicate the accuracy of prediction. Thus, we also assessed SPE (squared prediction error, an explicit measure of prediction accuracy) in the imaged sample for posterior means of ΔFM given FMii and/or Z. To facilitate appreciation of the magnitude of SPE relative to the range of the FM (0–66), below we will express SPE in the form of “x2,” where x is thus in FM units.
The posterior mean of ΔFM given FMii
required an estimated fΔFM|FMii
(which was allowed to be a mixture of K
proportional recovery models for each severity category) and fZ|ΔFM
. Based on fitting using the nonimaged sample, the minimum AIC fΔFM|FMii
for the patients with nonsevere FMii
= 3 (
) and the minimum AIC model for the patients with severe FMii
= 1 (
). We were surprised that a 1-component model fit best for the severe FMii
group given our previous findings suggesting a mixture of 2 proportional recovery models (one with β ~ 0.7 and one with β ~ 0) (Prabhakaran et al. 2008
). Visual inspection of the nonimaged sample data confirmed, however, that there was a more even distribution of ΔFM in the patients with severe FMii
from the University of Freiburg (who were not represented in that study) than from Columbia University (who were); we proceeded regardless. The sole parameter of fZ|ΔFM
. Its maximum likelihood estimator was obtained from Z
and ΔFM in the imaged sample:
A preliminary remark is that it is not a mathematical necessity that SPEs decrease (just as the AIC need not decrease) as more information is added to the prediction algorithm; this can be contrasted with the necessary increase of R2 for a linear regression model as the rank of the design matrix increases. We now present the (average) SPE in the net imaged sample for the various posterior means of ΔFM: The SPE of the posterior mean of ΔFM given dichotomous stroke severity was 162. The SPE given FMii was 102, while the SPE given FMii and Z was 82. The decrease in SPE in the net imaged sample from using FMii to using FMii and Z was not significant (t(29) = 1.14, 1-tailed P = 0.13).
A more meaningful understanding of the effect of conditioning is provided by looking at SPE separately in the nonsevere (N = 23) and severe (N = 7) FMii patient subgroups of the imaged sample (; ). SPE tends to be much greater in all models in the patients with severe FMii. Relative to conditioning on dichotomous stroke severity alone, conditioning on FMii improves SPE only in the patients with nonsevere FMii (nonsevere FMii: ΔSPE = 132, t(22) = 5.89, 1-tailed P < 0.0001; severe FMii: ΔSPE = −22, t(6) = −0.66, 1-tailed P = 0.73). Relative to conditioning on FMii, conditioning on FMii and Z improves SPE meaningfully, although not significantly, only in the patients with severe FMii (nonsevere FMii: ΔSPE = 22, t(22) = 0.98, 1-tailed P = 0.17; severe FMii: ΔSPE = 122, t(6) = 1.03, 1-tailed P = 0.17).
Mean SPE of ΔFM for various prediction models (columns)
Figure 3. Predicted (posterior mean) versus observed ΔFM when using either 1) FMii (patients with nonsevere FMii: white squares; patients with severe FMii: black squares) or 2) FMii and Z (patients with nonsevere FMii: red circles; patients with severe (more ...)
A way of reexpressing SPE for a given predictive model is in terms of the percentage of the total sum of squares of ΔFM explained by that model, which equals 100·(1 − [mean SPE for that model]/[mean ΔFM2]). Unlike R2, this value need not be positive. For patients with nonsevere FMii, this value was 96% and 97% when conditioning on FMii and [FMii, Z], respectively. For patients with severe FMii, this value was 16% and 47% when conditioning on FMii and [FMii, Z], respectively.