To overcome this discrete threshold problem we will replace the above form of score function with a smoothed version. One appealing form of this is a continuous function for every step. Each step could be replaced by a suitably scaled multiple of a cumulative distribution function (cdf) chosen simply for convenience since any cdf ranges from 0 to 1. We can use the notion of a mean to centre the new function at or close to the old threshold value while the use of a standard deviation, based in our case on the estimate of the day-to- day variability, allows us to dictate the steepness of the; new 'smoothed' step.
In all our illustrations we have chosen to use the family of cumulative distribution functions of the logistic distribution simply out of numerical convenience but all of the work could be repeated using any family of cdfs. It would be very surprising if any substantial
differences in the results from our techniques arose from a different choice of underlying family of cdfs. No other version that we have tried has proved to have any substantial effect on our score functions.
The methods which are about to be described should not be interpreted from a probabilistic viewpoint since they are, in the first order, devices based on practical convenience to smooth out discrete boundaries, and we only exploit any probability properties en route to developing smoothed diagnostic indices.
In the single threshold case, a score will be calculated which will depend smoothly on the magnitude of the variable of interest x,
increasing from zero to K as x increases. To assess how sharply the ’score’ will increase in the vicinity of the thresholds, estimates of the repeat variability as described in 4.4 will be used to dictate an appropriate 'standard deviation’ for the probability distribution which is being used as the smooth function. In this case the repeat variation will be represented in the form of day-to-day variability since data is available to estimate this. Incorporating an estimate of the day-to-day variability in the function allows a score to be calculated which takes account of random fluctuation in the measurement jc. Thus the bulk of each point will only be allocated if we are sure that the true value
x is beyond the old threshold value (accounting for any error).
In this way, an observed value of x which lies some way from the critical value b, but is from an ECG measurement associated with quite a substantial standard deviation <jv, may be allocated a significant proportion of the score K. In another case it might be that value of x which lies numerically a little nearer to b but with much smaller recording-to-recording variation may score less.
Our chosen smooth score function therefore has the following form :
1 1 S',„(x) ~ K x F{a%g x\ b - G x} ea* = £ —---- l + ea* where a , = ^ 1 ^ 1 . °x
K - maximum no. of points to be allocated,
b = old threshold value
x - observed value of the ECG variable of interest g x = estimate of day-to-day variability associated with
the observation x .
In a somewhat arbitrary manner the distribution has been 'centred' at ( b - ctx) rather than at b in order to give a score of 0.1K
at the old threshold value b. The value ( b - a x) represents a level which is one standard deviation below the old threshold value and ensures that observations falling less than one standard deviation below b will all receive contributions of at least 0.5K to the diagnostic index which we feel is a reasonable criterion to adopt.
Figure 5.3 shows the smoothed score function «S/;i+(a) for the single threshold case while Figure 5.4 demonstrates how this can be extended to the multiple threshold situation simply by summing a series of cdfs, i.e. Sm\ x ) = K a, a , a 3 e x e x e 1 H--- + --- h 1 + *"- l + e"2* l + e where a. CF
Score Function
ECG Measurement x
(old threshold)
Figure 5 .3 The sm oothed score function (Single threshold)
Score Function
b + c
ECG Measurement x
Figure 5 .4 The sm oothed score function (Multiple thresholds)
As a specific example, Figure 5.5 illustrates the smoothed score function specific to the portion of the final LV score associated with the Lewis Index, taking into account the estimated day-to-day variability. This score function will replace the 'old' discrete rule (also represented in Figure 5.5) which was:
S(x) =
0
if Lewis Index >age and sex - dependent limit otherwise Score S„'(x) 2 0 1000 2000 3000 4000 Lewis Index
Old Threshold (at Age=30) for Lewis Index (Males)
Figure 5 .5 Sm oothed score function for Lewis Index
Similarly, the 'old' discrete rule for RV5 or RV6 which works as follows:
S(x)
has been replaced by a suitably smoothed version which takes account of the multiple steps. A comparison of the two approaches can;be seen in Figure 5.6.
'2 if RV5 or RV6 > age anc sex-dependent limit 3 if RV5 or RV6 > limit + 0.5 mV
4 if RV5 or RV6 > limit + 1.0 mV 5 if RV5 or RV6 > limit + 1.5 mV
E \ A Score 5 •• 4 - 3 • 2 - 0 --- 1--- 1--- 1--- 1--- . f---p . 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Old threshold (at Age - 40) for RV5 (Males)
Figure 5 .6 Sm oothed score function for RV5