Costs of incorrect classification
In determining costs for incorrect decisions, we have presumed standard clinical treatment of abnormal beats, or non-treatment of normal beats according to the system's output. We have then investigated the costs, in monetary terms, of taking that course of action under each state of nature. We have endeavoured to report 'costs' as the general cost to society, rather than the cost to any particular entity. A summary of these figures is presented in Table
2. All costs have been normalised the year 2006, and are in Australian dollars except where otherwise noted.
Each application however may have different ancillary parameters, and these may affect the costs involved. The methods and figures provided in this study reflect the most general situation, as best as we could determine. Propositions we have made in calculations of costs we have stated herein, and these should be examined when applying the figures.
Proposition 1 The clinical treatment for fusions of normal and Ventricular Ectopic Beats (class ω
F
) is identical to that for Ventricular Ectopic Beats.
We make this proposition on the basis that sustained Ventricular Ectopic Beats are potentially life threatening, and must be treated. To a clinician, the fact that the polarisation of the waveform coincides with the preceding beat is merely incidental.
Proposition 2 The maximum future projection which may affect costs of misclassification is 10 years.
10 years is chosen as a reasonable period beyond which the advances in medical technology can be expected to invalidate the results of future prediction.
Datum 1 The expected loss of life of a healthy subject, projected over the next 10 years, is 1.21 years.
This datum is from the control group of Benjamin
et al. [
11]. This was a longitudinal study which investigated the mortality of subjects who had developed Atrial Fibrillation. We integrated the survival results presented by Figure A of that paper to determine the expected number of years of life lost by a healthy subject.
Datum 2 The probability that a subject with Supra-Ventricular Ectopic Beats will develop Atrial Fibrillation is 0.324.
Datum 2 comes from the results of Frost
et al. [
12].
Datum 3 The expected loss of life due to a person suffering from Atrial Fibrillation is 2.69 years (projected over 10 years).
Datum 3 is determined from Benjamin
et al. in a similar fashion to Datum 1: The calculations are presented in Table
4.
Table 4
Expected Loss of life due to Atrial Fibrillation
Women without Atrial Fibrillation | 0.87 |
Men without Atrial Fibrillation | 1.55 |
Mean Average | 1.21 |
Women with Atrial Fibrillation | 3.75 |
Men with Atrial Fibrillation | 4.05 |
Mean Average | 3.90 |
Expected Loss Due to AF | 2.69 |
Datum 4 A person's contribution to society is $44,320 per annum.
Datum 4 is the mean average wage in Australia for the year 2006[
13].
Datum 5 The cost of misdiagnosing a Supra-Ventricular Ectopic Beat as a normal beat, λ(α
N|ω
SVEB) is $38,627.
This figure is the product of Data 2, 3 and 4.
Datum 6 The probability of initially surviving Ventricular Fibrillation is .
From the text of Baum
et al.[
14], we know that, in a 3 year study of Ventricular Fibrillation cases, 146 patients out of 886 initially survived Ventricular Fibrillation.
Datum 7 The total expected loss of life by a person who initially survives Ventricular Fibrillation is 6.33 years (projected over 10 years).
To derive Datum 7 we used the results of Baum
et al.[
14]. In Figure 2 of that paper, survival curves are presented for subjects who initially survived Ventricular Fibrillation. Survival data are presented for only 24 months. We extrapolated survival data over a 10 year period by the method described above.
Datum 8 The expected loss of life, attributable to Ventricular Fibrillation, by a subject who suffers Ventricular Fibrillation is 8.19 years (projected over 10 years).
We know from Datum 3 that the expected loss of life of healthy subjects is 1.21 years. Hence we can calculate the loss due to Ventricular Fibrillation as per Table
5.
Table 5
Expected Loss of life due to Ventricular Fibrillation
Survival | 0.161
| × | 6.362
| = | 1.04 |
Non-survival | 0.84 | × | 10 | = | 8.40 |
Healthy subject | | | | | (1.21)3
|
Total loss due to VF | | | | | 8.19 |
Datum 9 The expected loss of life, attributable to Ventricular Tachycardia by a subject who suffers Ventricular Tachycardia is 2.59 years (projected over 10 years).
This datum was obtained from Doval
et al.[
15] by the data extrapolation method described previously. In that study of 516 subjects both with and without non-sustained Ventricular Tachycardia, the projected loss over 10 years for the group with Ventricular Tachycardia was 9.24 years, whereas the projected loss for the group without Ventricular Tachycardia was 6.65 years. The difference is 2.59 years.
Datum 10 The probability that a subject who experiences one or more episodes of Ventricular Ectopic Beats, will develop Ventricular Fibrillation is 0.34.
Datum 11 The probability that a subject who experiences one or more episodes of Ventricular Ectopic Beats, will develop Ventricular Tachycardia is 0.41.
Datum 10 and Datum 11 are implied from Carrim and Khan [
16]. In that study of 44 subjects exhibiting Ventricular Ectopic Beats,
V T is the set of subjects developing Ventricular Tachycardia and
V F is the set of subjects developing Ventricular Fibrillation. From their data, we are given:
|
V T ∪
V F| = 21,
(16)
and
(17)
from which we can deduce that |V T ∩ V F| = 12, |V T| = 18 and |V F| = 15.
Thus,
(18)
and
(19)
Datum 12 The expected loss of life, when a Ventricular Ectopic Beat is misdiagnosed as a normal beat is 3.84 years.
Datum 12 is derived from Data 10, 11, 8 and 9. The derivation is shown in Table
6.
Table 6
Expected loss of life due to Ventricular Ectopic Beats
Tachycardia | 0.411
| × | 2.59 2
| = | 1.06 |
Fibrillation | 0.343
| × | 8.19 4
| = | 2.78 |
Total | | | | | 3.84 |
Datum 13 The cost of misdiagnosing a Ventricular Ectopic Beat as a normal beat, λ(α
N|ω
VEB) is $170,189.
This is the product of Datum 12 and Datum 4. By Proposition 1 we attribute the same cost to λ(α
N|ω
F).
Costs of misclassification of a normal beat as abnormal
Ventricular Ectopic Beats are of potential concern to a physician. If a system misclassifies a normal beat (ω
N), or a Supra-Ventricular Ectopic Beat (ω
SVEB) as Ventricular Ectopic Beat or a fusion beat (decisions α
VEB and α
F), the likely result is that the patient will be unnecessarily detained in observation, pending further examination. Thus, the cost of misclassification, in this case, is the cost of retaining a patient in intensive care for one day.
Datum 14
The cost of retaining a patient in intensive care for 1 day is $2146
This datum is the mean average of figures obtained from two independent Australian health insurance providers. A case study by Rechner and Lipman[
17] for the year 2003 cites the figure $2670. This study however was conducted within a teaching hospital, and we therefore expect the costs to be higher than average.
We use the cost of retaining a patient in intensive care for one day as the cost of misclassification of a normal beat as the penalty in such instances. Thus λ(α
VEB|ω
N) = λ(α
F|ω
N) = λ(α
VEB|ω
SVEB) = λ(α
F|ω
SVEB) = $2146.
From Proposition 1 we conclude that λ(λ(α
F|ω
VEB) = λ(α
VEB|ω
F) = 0.
Erroneous classification of beats as Supra-Ventricular Ectopic Beats
Unlike Ventricular Ectopic Beats, Supra-Ventricular Ectopic Beats are typically not life threatening, and are therefore not normally treated unless recurrent [
18]. Thus, the cost of misclassification as a normal beat (
λ(
α
SVEB|
ω
N)) is zero. By the same token, misclassification of Ventricular Ectopic Beats or fusion beats as Supra-Ventricular Ectopic Beats bears the same penalty as an erroneous classification as a normal beat. Thus
λ(
α
SVEB|
ω
VEB) =
λ(
α
N|
ω
VEB) and
λ(
α
SVEB|
ω
F) =
λ(
α
N|
ω
F).