Confidence interval for a median and other quantiles
This is a section from my text book An Introduction to Medical Statistics,
Third Edition. I hope that the topic will be useful in its own right,
as well as giving a flavour of the book. It is marked "*" because it would
not be included in most undergraduate courses for healthcare students.
(Otherwise in this page, "*" means "multiply".)
* Confidence interval for a median and other quantiles
In Section 4.5 we estimated medians and other
quantiles directly from the frequency distribution. We can estimate confidence
intervals for these using the Binomial distribution. This is a large sample
method. The 95% confidence interval for the q quantile can be found
by an application of the Binomial distribution (Section 6.4, Section 6.6)
(see Conover 1980). The number of observations less than the q quantile
will be an observation from a Binomial distribution with parameters n
and
q, and hence has mean nq and standard deviation
root(nq(1q)).
We calculate j and k such that:
j = nq  1.96 root(nq(1q))
k = nq + 1.96 root(nq(1q))
We round j and k up to the next integer. Then the 95%
confidence interval is between the jth and the kth observations
in the ordered data.
For example, the following data are measurements of Forced Expiratory
Volume in one second (FEV1) for 57 male medical students:
FEV1 (litres) for 57 male medical students
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

For these FEV1 data the median is 4.1, the 29th value in the Table. For
the 95% confidence interval for the median, n = 57 and q
= 0.5, and
j = 57 * 0.5  1.96 * root(57 * 0.5 * (10.5)) = 21.10
k = 57 * 0.5 + 1.96 * root(57 * 0.5 * (10.5)) = 35.90
The 95% confidence interval is thus from the 22nd to the 36th observation,
3.75 to 4.30 litres from the Table. Compare this to the 95% confidence
interval for the mean, 3.9 to 4.2 litres, which is completely included
in the interval for the median. This method of estimating percentiles is
relatively imprecise. Another example is given in Section
15.5.
Reference
Conover, W.J. (1980) Practical Nonparametric Statistics John Wiley
and Sons, New York.
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Last updated: 17 October, 2003.
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