Quality Assessment of Total Cholesterol Measurements in the WHO MONICA Project
We will compare the variances of the uncorrected mean value X and the corrected mean value Y. From Section 5.5.2 the correction formula is
Y = 100X/(100+b),
and the variance of the corrected value is
Var(Y) = 1002Var(X/100+b),
which, using a first order Taylor approximation (see e.g. page 325 of reference 1) is approximately
Var(Y) = 1002{Var(X)/[E(100+b)]2 + [E(X)]2Var(b)/[E(100+b)]4}.
We will now find approximate estimates for each of the parameters of this formula, so that the estimates are conservative with respect to the effect of the bias correction.
To estimate Var(b), we can make use of condition (ii) of Section 5.5.2, which states that the difference between the maximum and minimum pool biases is less than 5. An intuitively extreme situation would then be that each bias is at the end-point of an interval of length 5, which would give a maximum variance of 25/4 for a pool bias (cf. Bernoulli distribution with parameter p, where the length of interval would be one, and the maximum variance would be 1/4, at p=0.5). Then if b is the mean value of more than six pools, the variance (and standard error) of the mean will be at most 1.
Alternatively, we could assume that the pool biases are distributed uniformly, and the distance between the lowest and the highest observed pool bias is 5. Then, assuming we have n pools, the minimum variance unbiased estimate for the range of the the uniform distribution is 5(n+1)/(n-1) (see e.g. page 390 of reference 2). As the variance of the uniform distribution with range r is r2/12, a good estimate of the variance of the pool bias is 25(n+1)2/12(n-1)2. This variance is less than 25/4 (cf. the first approach above) whenever n>3. Therefore, 1 from the first approximation should be a sufficiently conservative estimate for Var(b) for the situation which fulfills conditions (i)-(v) for the correction.
E(100+b) is approximately 100.
E(X) is mostly between 5 and 6 mmol/l in the MONICA populations. Therefore we will use the conservative 6 mmol/l.
Var(X), the variance of the sample mean for a sample size of 200 within a ten-year age/sex group, is of the magnitude of 0.01. (This we see from the MONICA first and second survey data books, MNM 178A and MNM 260A.)
After substituting these we get
Var(Y) = Var(X) + 36/1002 = 0.01 + 0.0036.
Taking the square roots, we get the standard errors
SE(X) = 0.1, and SE(Y) = 0.12.
The increase in the standard error after the bias correction (i.e. from SE(X) to SE(Y)) is so small, that there will be no need to complicate analyses of the cholesterol data by taking into account the variance of the correction (Var(b)). When deriving this conclusion it was assumed that the correction was based on six EQC pools. If the number of pools is higher, the maximum expected effect of the correction on the variance will be even smaller.
References of Appendix 2