Annex 6a: Flexible Functional Forms for the Acquisition of Life Potential
This annex provides a technical discussion of issues raised by incorporating late fetal deaths (stillbirths) into the global burden of disease, as measured within the disability-adjusted life year (DALY) framework. One approach is simply to take the DALY loss at birth and discount back to the time of the stillbirth, indicating that there are no life years to lose before birth, but that there are still all the postpartum life years. Essentially this is the standard DALY, but with an age-weighting function equal to 0 before birth. This is feasible, but has several potential drawbacks, in particular, any reasonable discount rate (for example, 3 percent) would thence count all late fetal losses almost the same as a loss at birth. This approach yields the DALYsSB(3,0,1) measure as described in the main text, and table 6B.6 presents global burden of disease estimates using DALYsSB(3,0,1) because these are the simplest extension of DALYs(3,0).
However, as with traditional DALYs, DALYsSB(3,0,1) assume instantaneous acquisition of life potential (ALP), as illustrated in figure 6.2 and discussed in the main text. Whether or not one wishes to include stillbirths in the global burden of disease, this discontinuity (at some given age) is troublesome. The purpose of this annex is to provide a flexible, yet tractable, explicit function that allows for gradual ALP.
One natural approach is to weight the YLL from outside the integral instead of from the inside (as with age weighting), that is, to create a multiplier function (the ALP function), which takes on values between 0 and 1 as a function of age, and use it to ratchet down the YLL function, potentially starting before birth. For convenience and with some regard to the known physiological underpinnings, we take this starting point in time to be the beginning of the third trimester of pregnancy. Roughly speaking, the rate of natural fetal loss becomes noticeable after the beginning of some level of consciousness during the second half of the second trimester. One could force this function to equal 1 at birth, recovering the standard DALYs from that point onward, and this will be a special case of our formulation. However, we have no definitive reason to think that ALP is necessarily complete at birth. Indeed, quite a bit of evidence suggests that in many (if not all) societies worldwide, infants are not given full status, for instance, they are not always named immediately. Thus we wish to allow for continued gradual acquisition after birth and up to some time T that signifies full standing or full ALP. Likewise, starting the acquisition only at birth but proceeding gradually afterward is perfectly possible.
Turning to the specifics, denote the ALP multiplier function by f(t), where t is measured in years and ranges from -0.25 (that is, 13 weeks before birth, the beginning of the third trimester) to T. The function is meaningfully defined for any finite value of T, though it is natural to assume that full life potential is achieved by puberty at the latest. Thus f(-0.25) = 0 and f(T) = 1. We let f0 = f(0) be the value at 0. Of course, starting times other than -0.25 are perfectly legitimate as well, but -0.25 is the natural choice given the standard definitions of stillbirth and the gathering and reporting of data using that definition.
We need a functional form that smoothly begins at 0 and rises to f0, which is at least weakly convex (following the intuition that life potential is acquired increasingly rapidly as birth is approached), and whose curvature is parametrizable. The natural choice is xGamma with Gamma ≥ 1. This has canonical endpoints of 0 and 1, where xGamma takes on the values 0 and 1, respectively, for any Gamma, so that as we change the curvature (or skewness), the endpoints remain fixed. Fitting this to our specific domain, we get x = 4t + 1 for -0.25 ≤ t < 0. Finally, if we wish the skewness parameter to lie between 0 and 1 as well (for clarity), we can define g so that g = 1/(1 - g) for 0 ≤ g < 1. This yields f_(t) = f0(4t + 1)1/(1-g) for -0.25 ≤ t< 0. Thus g = 0 produces a straight line (zero curvature), while g = 1 (defined by fiat) is infinitely skewed: 0 until birth and then jumping to f0.
For t ≥ 0, we consider the symmetric version of the same polynomial, that is, 1 - (1 - x)Beta. Again we fit this to our domain, namely, from t = 0 to t = T, and define b so that b = 1/(1 - b) for the skewness. This yields f+ (t) = 1 - (1 - f0)[(T - t)/T]1/(1-b) for 0 ≤ t ≤ T. We check that indeed f+(0) = f0 and f+(T) = 1 according to this formula for any 0 ≤ Beta ≤ 1. If T = 1, the formula simplifies to f+(t) = 1 - (1 - f0)(1 - t)1/(1-b). This leaves four parameters: f0, T, g, and b. We can additionally impose g = b if we wish, but this is unnecessary.
Summarizing, the function we use for ALP is
If fD(t) is the standard DALY formulation (whether or not age weighting or discounting is used), then g = b = 1 (that is, discontinuous jumps around birth from 0 to 1) and fD0= 1, so that technically at age 0 the value is already 1 (so the discontinuity is on the left side of age 0 only). Given these parameters, T is immaterial, because the function achieves its maximum immediately. However, the fact that we can replicate the standard DALY means that the gradual acquisition function does indeed generalize it.
Combining these equations with the standard definition of DALYs, the total loss L(a) for a death at age a ≥ -0.25 is
where Beta is the age-weighting parameter (typically 0.04) if age weighting is used, r is the discount rate (typically 0.03), sa(x) is the survival probability for reaching age x ≥ a conditional on having reached age a, and C is the normalization parameter for the age weights (C = 0.16243, see the discussion in chapter 5).
The normalization parameter C in equation (6A.2) was chosen so that the total global burden of disease would be the same with and without age weighting. The index of age weighting referred to in the main text, K, is generated by having a weighted average—with weights of K and (1 - K), where 0 ≤ K ≤ 1—of loss functions L(a) that result from equation (6A.2) with the indicated values of Beta and C and a loss function assuming uniform age weights. That this is at least approximately the case is apparent from figure 6.4b, where the two functions cross at about age 40. Clearly this will not be true when any of the acquisition functions are used, because they reduce the YLL burden at younger ages with no corresponding increase elsewhere, leading to a reduced total burden as measured by absolute DALY levels.
Note, however, that the total burden is no longer the same even for DALYs(3,0) and DALYs(3,1), because the specific value of C was calibrated to 1990 morbidity and mortality statistics. One can readily imagine more neutral (and invariant) normalizations, such as requiring a constant integral over age of death for each of these YLL functions, or perhaps weighting this integral using an idealized survival table. Any variant along these lines would raise the total level of DALYs(3,0,.54) relative to both DALYs(3,0) and DALYs(3,1). Of course, we are for the most part interested only in the relative burden across ages or disease categories, so the absolute totals are of secondary importance.
Finally, to somewhat simplify the number of parameters in the ALP function, we introduce a notion of speed of acquisition, A. Recall that f0 can be anywhere between 0 and 1, regardless of whether the function f(t) takes on positive values before birth. If f0 = 1 (as in the original DALY), then f = 1 thereafter and the speed A is in some sense as large as possible. To generalize this idea, we look at the total area between the ALP function f(t) and the constant function 1.
Formally, this area is given by the integral of 1 - f(t), evaluated from t0 to T, where t0 is the first t s that f(t) > 0. It is thus typically either - 0.25 or 0, depending on whether we are including stillbirths. Call this integral I:
Substituting the second part of equation (6A.1), we can evaluate this integral as
Normalizing so that the speed A lies between 0 and 1 (and higher values denote faster acquisition), we define
For example, for b = 0.7 (a typical value) and t0 = 0, we obtain a simple formula for the speed parameter A, encapsulating the acquisition function in a single number: A = 1/[1 + 0.23T(1 - f0)]. There is still a trade-off between T and f0, that is, the relationship between the underlying parameters and A is not one-to-one. A single value for A could have arisen from multiple combinations parameter values, but it still serves as a useful summary statistic. Figure 6A.1 graphs (as a function of T, fixing b = 0.7 and t0 = 0) the value of f0 that yields various specified acquisition speeds A. The analogous figure 6.3 shows less variability in this ratio.
[Figure
6A.1]
We evaluate three specifications (parameter choices) for the acquisition function. These are, in order of value at birth: f1, given by (f10 = 0.25, T1 = 14, g1 = 0.5, b1 = 0.7); f2, given by (f20 = 0.3, T2 = 5, g2 = 0.4, b2 = 0.7); and f3, given by (f30 = 0.5, T3 = 2, g3 = 0.3, b3 = 0.8). The respective values for A (using t0 = -0.25) are 0.29, 0.54, and 0.84. These three acquisition functions were graphed in figure 6.4. Representative values for specific ages were listed in table 6.5, along with the corresponding values for fD(t), the traditional formulation for DALYs. Figure 6A.2 shows how the ratio of years of life lost at age 20 to age 0 for these three functions varies with A. We view f2 (with T = 5) as a reasonable intermediate choice and, with a 3 percent discount rate, have used f2 to generate what we define as DALYs(3,0,.54). Complete burden of disease calculations are reported using DALYs(3,0,.54) in table 6B.8.
[Figure
6A.2]
