Table 10.5 shows levels of zidovudine (AZT) in the blood of AIDS patients at several times after administration of the drug, for patients with normal fat absorption or fat malabsorption.

Malabsorption patients: Time since administration of zidovudine (min) 0 15 30 45 60 90 120 150 180 240 300 360 0.08 13.15 5.70 3.22 2.69 1.91 1.72 1.22 1.15 0.71 0.43 0.32 0.08 0.08 0.14 2.10 6.37 4.89 2.11 1.40 1.42 0.72 0.39 0.28 0.08 0.08 3.29 3.47 1.42 1.61 1.41 1.09 0.49 0.20 0.17 0.11 0.08 0.08 1.33 1.71 3.30 1.81 1.16 0.69 0.63 0.36 0.22 0.12 0.08 6.69 8.27 5.02 3.98 1.90 1.24 1.01 0.78 0.52 0.41 0.42 0.08 4.28 4.92 1.22 1.17 0.88 0.34 0.24 0.37 0.09 0.08 0.08 0.08 0.13 9.29 6.03 3.65 2.32 1.25 1.02 0.70 0.43 0.21 0.18 0.08 0.64 1.19 1.65 2.37 2.07 2.54 1.34 0.93 0.64 0.30 0.20 0.08 2.39 3.53 6.28 2.61 2.29 2.23 1.97 0.73 0.41 0.15 0.08 Normal absorption patients: Time since administration of zidovudine (min) 0 15 30 45 60 90 120 150 180 240 300 360 0.08 3.72 16.02 8.17 5.21 4.84 2.12 1.50 1.18 0.72 0.41 0.29 0.08 6.72 5.48 4.84 2.30 1.95 1.46 1.49 1.34 0.77 0.50 0.28 0.08 9.98 7.28 3.46 2.42 1.69 0.70 0.76 0.47 0.18 0.08 0.08 0.08 1.12 7.27 3.77 2.97 1.78 1.27 0.99 0.83 0.57 0.38 0.25 0.08 13.37 17.61 3.90 5.53 7.17 5.16 3.84 2.51 1.31 0.70 0.37A line graph of these data was shown in Figure 5.11.

One common approach to such data is to carry out a two sample t test at each time separately, and researchers often ask at what time the difference becomes significant. This is a misleading question, as significance is a property of the sample rather than the population. The difference at 15 minutes may not be significant because the sample is small and the difference to be detected is small, not because there is no difference in the population. Further, if we do this for each time point we are carrying out multiple significance tests (Section 9.10) and each test only uses a small part of the data so we are losing power (Section 9.9). It is better to ask whether there is any evidence of a difference between the response of normal and malabsorption subjects over the whole period of observation.

The simplest approach is to reduce the data for a subject to one number.
We can use the highest value attained by the subject, the time at which
this peak value was reached, or the area under the curve. The first two
are self-explanatory. The **area under the curve **or **AUC **is
found by drawing a line through all the points and finding the area between
it and the horizontal axis. The ‘curve’ is ususally formed by a series
of straight lines found by joining all the points for the subject,
and Figure 10.10 shows this
for the first participant in Table 10.5.

The area under the curve can be calculated by taking each straight line segment and calculating the area under this. This is the base multiplied by the average of the two vertical heights. We calculate this for each line segment, i.e. between each pair of adjacent time points, and add. Thus for the first subject we get

(15 − 0) × (0.08 + 13.15)/2 + (30 − 15 ) × ( 13.15 + 5.70 )/2 + ... + (360 − 300) × (0.43 + 0.32)/2 = 667.425.

This can be done fairly easily by most statistical computer packages. The area for each subject is shown in Table 10.6.

**Table 10.6** Area under the curve for data of Table 10.5 (data
from Kapembwa *et al.* 1996)

Malabsorption Normal patients patients ------------------ -------- 667.425 256.275 919.875 569.625 527.475 599.850 306.000 388.800 499.500 298.200 505.875 472.875 617.850 1377.975

We can now compare the mean area by the two sample t method. As Figures 10.11 and 10.12 show, the log area gives a better fit to the Normal distribution than does the area itself.

**Figure 10.11** Normal plot for area under the curve for the
data of Table 10.5 (data supplied by Moses Kapembwa,
personal communication).

Using the log area we get n_{1} = 9, mean_{1} = 2.639541,
s_{1} = 0.153376 for malabsorption subjects and n_{2}=5
, mean_{2} = 2.850859, s_{2} = 0.197120 for the normal
subjects. The common variance is s^{2} = 0.028635, standard error
of the difference between the means is square root of 0.028635 × (1/9+1/5),
which gives 0.094385, and the t statistic is t = (2.639541 − 2.850859)/0.094385
= −2.24 which has 12 degrees of freedom, P = 0.04. The 95% confidence interval
for the difference is 2.639541 − 2.850859 +/− 2.18 times 0.094385, giving
−0.417078 to −0.005558, and if we antilog this we get 0.38 to 0.99. Thus
the area under the curve for malabsorption subjects is between 0.38 and
0.99 of that for normal AIDS patients, and we conclude that malabsorption
inhibits uptake of the drug by this route. A fuller discussion of the analysis
of serial data is given by Matthews *et al.* (1990).

Matthews, J.N.S., Altman, D.G., Campbell, M.J., and Royston, P. (1990)
Analysis of serial measurements in medical research. *British Medical
Journal ***300 **230-35.

Adapted from pages 142–144 of
*An Introduction to Medical Statistics* by Martin Bland, 2015,
reproduced by permission of
Oxford University Press.

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