Figure 4.3 Histogram of FEV1: frequency scale (data from Physiology practical class, St George’s Hospital Medical School).
In Section 20.6 we were concerned with the diagnosis of particular diseases. In this section we look at it the other way round and ask what values measurements on normal, healthy people are likely to have. There are difficulties in doing this. Who is ‘normal’ anyway? In the UK population almost everyone has hard fatty deposits in their coronary arteries, which result in death for many of them. Very few rural Africans have this; they die from other causes. So it is normal in the UK to have an abnormality. We usually say that normal people are the apparently healthy members of the local population. We can draw a sample of these as described in Chapter 3 and make the measurement on them.
The next problem is to estimate the set of values. If we use the range of the observations, the difference between the two most extreme values, we can be fairly confident that if we carry on sampling we will eventually find observations outside it, and the range will get bigger and bigger (Section 4.7). To avoid this we use a range between two quantiles (Section 4.7), usually the 2.5 centile and the 97.5 centile, which is called the normal range, 95% reference range, or 95% reference interval. This leaves 5% of normals outside the ‘normal range’, which is the set of values within which 95% of measurements from apparently healthy individuals will lie.
A third difficulty comes from confusion between ‘normal’ as used in medicine and ‘Normal distribution’ as used in statistics. This has led some people to develop approaches which say that all data which do not fit under a Normal curve are abnormal! Such methods are simply absurd, there is no reason to suppose that all variables follow a Normal distribution (Sections 7.4, 7.5). The term ‘reference interval’, which is becoming widely used, has the advantage of avoiding this confusion. However, the most frequently used method of calculation rests on the assumption that the variable follows a Normal distribution.
We have already seen that in general most observations fall within two
standard deviations of the mean, and that for a Normal distribution 95%
are within these limits with 2.5% below and 2.5% above. If we estimate
the mean, m, and standard deviation, s, of data from a Normal
population we can estimate the reference interval as m - 2s
to m + 2s.
Consider the FEV1 data of Table 4.4.
Table 4.4 FEV1 (litres) of 57 male medical students (data from Physiology
practical class, St George’s Hospital Medical School)
2.85 3.19 3.50 3.69 3.90 4.14 4.32 4.50 4.80 5.20
2.85 3.20 3.54 3.70 3.96 4.16 4.44 4.56 4.80 5.30
2.98 3.30 3.54 3.70 4.05 4.20 4.47 4.68 4.90 5.43
3.04 3.39 3.57 3.75 4.08 4.20 4.47 4.70 5.00
3.10 3.42 3.60 3.78 4.10 4.30 4.47 4.71 5.10
3.10 3.48 3.60 3.83 4.14 4.30 4.50 4.78 5.10
We will estimate the reference interval for FEV1 in male medical students.
The data seem to follow a Normal distribution reasonably well
(Figure 4.3)
Figure 4.3 Histogram of FEV1: frequency scale (data from
Physiology practical class, St George’s Hospital Medical School).
We have 57 observations, mean 4.06 and standard deviation 0.67 litres. The reference interval is thus 4.06 − 1.96×0.67 = 2.7 to 4.06 + 1.96×0.67 = 5.4 litres. From Table 4.4 we see that in fact only one student (2%) is outside these limits, although the sample is rather small.
As the observations are assumed to be from a Normal distribution, standard
errors and confidence intervals for these limits are easy to find. The
estimates m and s are independent (Appendix 7A) with standard errors
√(s2/n)
and √(s2/2(n − 1)) (Section 8.2, 8.8). The
sample mean m follows a Normal distribution and
s a distribution
which is approximately Normal. Hence m − 2s is from a Normal
distribution with variance:
VAR(m − 2s) = VAR(m) + VAR(2s)
= VAR(m) + 4VAR(s)
= s2/n + 4s2/2(n
− 1)
= s2(1/n + 2/(n − 1))
Hence, provided Normal assumptions hold, the standard error of the
limit of the reference interval is
√ (s2(1/n + 2/(n − 1)))
If n is large, this is approximately
√ (3s2/n).
For the FEV1 data, the standard error is √(3 × 0.67 2/57) = 0.15. Hence the 95% confidence intervals for these limits are 2.7 +/− 1.96 × 0.15 and 5.4 +/− 1.96 × 0.15, i.e. from 2.4 to 3.0 and 5.1 to 5.7 litres.
Compare the serum triglyceride measurements of
Table 4.8.
Table 4.8 Serum triglyceride measurements in cord blood from 282 babies (data supplied by Tessi Hanid, personal
communication)
0.15 0.29 0.34 0.38 0.41 0.46 0.52 0.56 0.64 0.80
0.16 0.30 0.34 0.38 0.41 0.46 0.52 0.56 0.64 0.80
0.20 0.30 0.34 0.38 0.41 0.46 0.52 0.56 0.65 0.82
0.20 0.30 0.34 0.39 0.42 0.46 0.52 0.57 0.66 0.82
0.20 0.30 0.34 0.39 0.42 0.47 0.52 0.57 0.66 0.82
0.20 0.30 0.34 0.39 0.42 0.47 0.52 0.58 0.66 0.82
0.21 0.30 0.34 0.39 0.42 0.47 0.52 0.58 0.66 0.83
0.22 0.30 0.35 0.39 0.42 0.47 0.53 0.58 0.66 0.84
0.24 0.30 0.35 0.40 0.42 0.47 0.54 0.58 0.67 0.84
0.25 0.30 0.35 0.40 0.44 0.48 0.54 0.59 0.67 0.84
0.26 0.31 0.35 0.40 0.44 0.48 0.54 0.59 0.68 0.86
0.26 0.31 0.35 0.40 0.44 0.48 0.54 0.59 0.70 0.87
0.26 0.32 0.35 0.40 0.44 0.48 0.54 0.59 0.70 0.88
0.27 0.32 0.36 0.40 0.44 0.48 0.54 0.60 0.70 0.88
0.27 0.32 0.36 0.40 0.44 0.48 0.55 0.60 0.70 0.95
0.27 0.32 0.36 0.40 0.44 0.48 0.55 0.60 0.72 0.96
0.28 0.32 0.36 0.40 0.44 0.48 0.55 0.60 0.72 0.96
0.28 0.32 0.36 0.40 0.44 0.48 0.55 0.60 0.74 0.99
0.28 0.32 0.36 0.40 0.45 0.48 0.55 0.60 0.75 1.01
0.28 0.32 0.36 0.40 0.45 0.48 0.55 0.60 0.75 1.02
0.28 0.33 0.36 0.40 0.45 0.48 0.55 0.60 0.76 1.02
0.28 0.33 0.36 0.40 0.45 0.49 0.55 0.61 0.76 1.04
0.28 0.33 0.37 0.40 0.45 0.49 0.56 0.62 0.78 1.08
0.28 0.33 0.37 0.40 0.45 0.49 0.56 0.62 0.78 1.11
0.29 0.33 0.37 0.41 0.46 0.50 0.56 0.63 0.78 1.20
0.29 0.33 0.37 0.41 0.46 0.50 0.56 0.64 0.78 1.28
0.29 0.33 0.38 0.41 0.46 0.50 0.56 0.64 0.78 1.64
0.29 0.33 0.38 0.41 0.46 0.50 0.56 0.64 0.78 1.66
0.29 0.34
As already noted (Sections 4.4 and 7.4), the data are highly skewed
(Figure 7.14)
Figure 7.14 Distribution of serum triglyceride (Table 4.8) in
cord blood for 282 babies, with corresponding Normal distribution curve.
and we cannot use the Normal method directly. If we did, the lower limit would be 0.07, well below any of the observations, and the upper limit would be 0.94, greater than which are 5% of the observations. It is possible for such data to give a negative lower limit.
The log transformed data give a breathtakingly symmetrical distribution
(Figure 7.15).
Figure 7.15 Distribution of log10 serum triglyceride in cord
blood for 282 babies, with corresponding Normal distribution curve.
Logs to base 10 have been used here. For the transformed data, m = −0.331, s = 0.171. The lower limit in the transformed data is −0.67, corresponding to a triglyceride level of 0.21, below which are 2.1% of observations. The upper limit is 0.01, corresponding to 1.02, above which are 2.5% of observations. The fit to the log transformed data is excellent. For the standard error of the reference limit we have √(3 × 0.1712 / 282) = 0.0176. The 95% confidence intervals are thus −0.673 +/− 1.96 × 0.0176 and 0.011 +/− 1.96 × 0.0176, i.e. −0.707 to −0.639 and −0.023 to 0.045. Transforming back to the linear scale, we get 0.196 to 0.230 and 0.948 to 1.109, found by taking the antilogs. These confidence limits can be transformed back to the original scale, unlike those in Section 10.4, because no subtraction of means has taken place.
Because of the obviously unsatisfactory nature of the Normal method for some data, some authors have advocated the estimation of the percentiles directly (Section 4.5), without any distributional assumptions. This is an attractive idea. We want to know the point below which 2.5% of values will fall. Let us simply rank the observations and find the point below which 2.5% of the observations fall. For the 282 triglycerides, the 2.5 and 97.5 centiles are found as follows. For the 2.5 centile, we find i = q(n + 1) = 0.025 × (282 +1) = 7.08. The required quantile will be between the 7th and 8th observation. The 7th is 0.21, the 8th is 0.22 so the 2.5 centile would be estimated by 0.21 + (0.22 − 0.21) × (7.08 − 7) = 0.211. Similarly the 97.5 centile is 1.039.
This approach gives an unbiased estimate whatever the distribution. The log transformed triglyceride would give exactly the same results. Note that the Normal theory limits from the log transformed data are very similar. We now look at the confidence interval. The 95% confidence interval for the q quantile, here q being 0.025 or 0.975, estimated directly from the data is found by the Binomial distribution method (Section 8.10). For the triglyceride data, n = 282 and so for the lower limit, q = 0.025, we have
j = 282 × 0.025 − 1.96 √ (282 × 0.025 × 0.975)
k = 282 × 0.025 + 1.96 √ (282 × 0.025 × 0.975)
This gives j = 1.9 and k = 12.2, which we round up to j = 2 and k = 13. In the triglyceride data the second observation, corresponding to j= 2, is 0.16 and the 13th is 0.26. Thus the 95% confidence interval for the lower reference limit is 0.16 to 0.26. The corresponding calculation for q = 0.975 gives j = 270 and k = 281. The 270th observation is 0.96 and the 281st is 1.64, giving a 95% confidence interval for the upper reference limit of 0.96 to 1.64. These are wider confidence intervals than those found by the Normal method, those for the long tail particularly so. This method of estimating percentiles in long tails is relatively imprecise.
Adapted from pages 329–331 of An Introduction to Medical Statistics by Martin Bland, 2015, reproduced by permission of Oxford University Press.
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