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Online Appendix for the following JACC: Cardiovascular Imaging article
TITLE: The Limit of Plausibility For Predictors of Response: Application to Biventricular
Pacing (Cardiac Resynchronization Therapy), Systematic Review and Design Steps For
Reliable Research
AUTHORS: Sukhjinder S. Nijjer, BSC, MB CHB, Punam A. Pabari, MB CHB, PHD,
Berthold Stegemann, PHD, Vittorio Palmieri, MD, PHD, Francisco Leyva, MD, Cecilia
Linde, MD, PHD, Nick Freemantle, PHD, Justin E. Davies, BSC, MBBS, PHD, Alun D.
Hughes, MB BS, PHD, Darrel P. Francis, MA, MD
___________________________________________________________________________
APPENDIX
Determining the contraction factor
Contraction factor imposed by dyssynchrony marker
If a mechanical dyssynchrony marker is measured several times in one patient, the observed
values (xobserved) will not all be identical, but rather be scattered randomly with a notional
stable true underlying value (xtrue) and an error component (xerror), i.e. xobserved = xtrue + xerror.
Here, the word “error” includes observer error, equipment error, operator error, and (most
importantly) genuine biological variability, all of which cause the observed value to differ
from the true underlying value.
When one measurement in one patient is compared with one measurement in another patient,
only part of the difference is due to true underlying difference between patients: the
remainder is due to error. Across a population, the total observed variance between
individuals is composed of two components: the true underlying variance between
individuals (Variance of xtrue) and the size of the error component (Variance of xerror). The
total variance of xobserved is:
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 of 𝑥𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 of 𝑥𝑡𝑟𝑢𝑒 + 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 of 𝑥𝑒𝑟𝑟𝑜𝑟
The component of variation arising from xerror is clinically completely uninformative because
it is random, changing each time it is measured rather than representing anything meaningful
for the patient. Therefore of the variation observed in x, only the remaining proportion is
meaningful and therefore has any hope of correlating with another variable:
Proportion of variation in 𝑥𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 that is meaningful =
=1−
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑡𝑟𝑢𝑒
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑒𝑟𝑟𝑜𝑟
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
The final formula above expresses quantitatively the notion that if a measurement has wide
test-retest irreproducibility (xerror), then only a small proportion of the observed differences
between patients are actually meaningful. It goes on to say that if test-retest reproducibility is
as wide as the observed spread in the population then all observed differences between
patients are completely meaningless.
Only the meaningful proportion of the variation can correlate with another variable, and so if
there is any underlying relationship between x and another variable y, with an underlying
association strength of R2underlying, the observed association strength R2observed will be
contracted by this factor.
Contraction factor imposed by response marker
The same applies for a marker of response: yobserved = ytrue + yerror. Here, “error” refers to all
variability between LV function measurements over time within individuals who do not
receive biventricular pacing. HSSCSs characteristically do not report this. EMRCTs
sometimes do, as the standard deviation of change in LV measurement (e.g., ΔLVEF) over
time, in the control arm.
In the control arm, there is no biventricular pacing is given, and so “error” is the only source
of variation: this means Variance of yerror = Variance of ΔLVEF in the control arm. In the
device arm, there is both “error” variation and variation due to biventricular pacing having
different-sized effects in different patients. This means that the Variance of yerror + Variance
of ytrue = Variance of ΔLVEF in the biventricular pacing arm.
Applying the same reasoning as for x, we obtain:
Proportion of Variance in 𝑦𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 that is meaningful = 1 −
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑦 𝑖𝑛 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑎𝑟𝑚
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑦 𝑖𝑛 𝐶𝑅𝑇 𝑎𝑟𝑚
Only this meaningful proportion of the variation can correlate with another variable, and so if
there is any underlying relationship between y and another variable x, with an underlying
association strength of R2underlying, the observed association strength R2observed will be
contracted by this factor.
The combined contraction factor
If there is an underlying relationship between x and y, the observed relationship strength will
be lower, because of both of the above contraction processes:
R2observed = R2underlying × [1 −
Variance of xerror
Variance of y in Control arm
] × [1 −
]
Variance of xobserved
Variance of y in CRT arm
How to screen the plausibility of a reportedly observed R2 value
If a study reports a high R2observed value, a plausibility check can quickly be carried out with
the above formula, perhaps made more convenient by rewriting it like this:
2
2
𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑅𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
= 𝑅𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
[1 −
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑒𝑟𝑟𝑜𝑟
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑦 𝑖𝑛 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑎𝑟𝑚
] × [1 −
]
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑦 𝑖𝑛 𝐶𝑅𝑇 𝑎𝑟𝑚
For example, if the study’s reported standard deviation of dyssynchrony is 50 ms, and the
blinded test-retest reproducibility standard deviation of xerror is 30 ms, then the first bracketed
term is 1 – 302/502. The chosen response marker may have credible reports of spontaneous
variation, for example from an RCT. If the SD of “response” was in the control arm 9 units,
and in the biventricular pacing arm 11 units, then the second bracketed term is 1 – 92/112.
Thus, if the reported R2 observed is 0.80, this would only be sustainable if the underlying R2
value were 0.80/[(1 – 302/502) (1 – 92/112)] = 3.8. This goes beyond “highly skilled” into
“impossible.”
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