Table 1.
The 33 studies reporting a positive (white) or negative (grey) statistical significance of a prognostic threshold of peak VO2.
Figure 1.
Simulated population characterized by gradually increasing risk and effectiveness of a series of potential prognostic thresholds by Kaplan-Meier and log-rank analysis.
In 1500 notional patients, with a wide spread of annual mortality (evenly distributed from 0.01 to 15.00%), we run survival simulation and use Kaplan-Meier and log-rank analysis to examine the prognostic power of many potential threshold values of the risk factor. For three examples amongst the many thresholds tested, the upper panels show the resulting Kaplan-Meier curves. In the lower panels, the results of the full range of tested thresholds are shown. The threshold that gave the highest chi-squared value (equivalent to the smallest p value) was taken as the “optimal” threshold.
Table 2.
Apparent loss of prognostic power of Peak VO2 threshold over time and likelihood of different prognostic thresholds giving positive results.
Figure 2.
Relationships between the threshold tested and the individual studies' mean: examples from peak VO2 (panel a), LVEF (panel b) and BNP (panel c).
In the studies testing a threshold and finding it to be significant (open circles), the threshold reported may be either slightly higher than the mean of the study or slightly lower, but in all cases it is not far from the mean; in contrast it is often far from the mean in the studies testing a threshold and finding it to be non significant (black dots). Dotted lines in each panel represent the line of equivalence.
Table 3.
Comparison of the main features of studies testing a threshold and finding it to be significant or non significant.
Table 4.
The 35 studies reporting a positive (white) or negative (grey) statistical significance of a prognostic threshold of ejection fraction.
Table 5.
The 20 studies reporting a positive (white) or negative (grey) statistical significance of a prognostic threshold of brain natriuretic peptide.
Figure 3.
Mathematical simulation of sample selection from the general population: correlations between the sample mean and the apparently-optimal prognostic threshold.
Sub-populations with different ranges of risk simulating a shift in the mean peak VO2 were created and strong correlations between population mean and optimal thresholds by Kaplan-Meier and ROC analysis were found.
Figure 4.
Apparently-optimal prognostic thresholds in twelve different types of relationship between the risk factor and mortality.
For each type of relationship, 10 simulations were conducted, and the 10 apparently-optimal thresholds derived from Kaplan Mayer analysis were found. They are shown by vertical arrows (where multiple arrows would have been superimposed, they have been placed one above another).
Figure 5.
Apparent optimal prognostic threshold, by Kaplan-Meier and ROC method, arising from a mathematically simulated population with known, smooth gradation of risk.
The position of the apparently optimal threshold is almost completely determined by the risk factor mean. Several overlapping samples are taken from a single population of smoothly varying risk.
Figure 6.
Two different types of threshold: apparently-optimal versus decision-making thresholds.
Cartoon illustrating two distinct, unrelated, values that are both called “threshold”. The statistically optimal threshold value of a continuous risk factor for subdividing the population (left panel) has no relevance to the question of what value of a risk factor should be used to decide whether to intervene or not (right panel). The former, the “observed prognostic threshold”, will generally be the middle of whatever population happens to be studied, if mortality varies roughly linearly with the risk factor. The latter, the “ideal clinical decision-making threshold”, will critically depend also on the outcomes with intervention, and will move as the success of the package of medical therapy (and of transplantation) changes with time. There is no sense in using one as a proxy for the other.
Figure 7.
Example of use of flexible non-linear function to describe the relationships between age (left) and peak VO2 (right) and log odds of death using 208 patients.
The shaded areas represent the 95% confidence intervals for this function. Flexible non-linear functions have numerous benefits over categorization, including improved precision, avoidance of assumption of a discontinuous relationship, maximisation of applicability to the individual and importantly avoidance of giving other variables or interactions artificially high weights. Inspection of the resulting plots above can make obvious the lack of a discontinuity in risk.