Docetaxel with prednisone or prednisolone for the treatment of ...

Using the method outlined by Parmar and
colleagues, 25 we undertook the estimation of
HRs and corresponding 95% CIs if such data were
not reported in the trial publications identified. If
the HR is not reported, it can be computed
directly if the observed and expected numbers are
presented for both treatment groups. However,
this information is rarely reported. The next
preferred method is estimating the HR from the
quoted p-value and number of observed events in
the trial (an example of which is shown below).
The final option is estimating the HR and CIs
directly from the survival curve.
Survival curves are fairly commonly reported;
however, this method is prone to further
challenges. In order to estimate the HR in this way,
a general approach is to split the time axis into T
non-overlapping time intervals. Then, using the
probabilities of survival for each group estimated
from the survival curve, the HRs for each time
interval are calculated. These are then combined
in a stratified way across time intervals to obtain
an overall HR for the trial. This technique has its
challenges, including being time consuming, and
problems can arise when attempting to read
survival probabilities from poorly drawn curves,
meaning that duplicate data extraction from the
survival curve is of paramount importance.
Example using p-values and
observed numbers of events
In order to estimate the HR and its corresponding
95% CIs, we need to extract the p-value (the logrank
and Mantel–Haenszel statistics are
considered to be equivalent here) and the total
number of observed events, which will be known
as p and O, respectively. The process of calculating
the HR and its 95% CIs is iterative, consisting of
six steps as follows:
Step 1. Calculate V:
V = O/4
Step 2. Calculate O r – E r
(O r – E r)= 1 – 2 × √ –– O Φ –1 (1 – p/2)
Appendix 5
Calculation of hazard ratios
© Queen’s Printer and Controller of HMSO 2007. All rights reserved.
Health Technology Assessment 2007; Vol. 11: No. 2
O r = observed number of events in study group,
E r = log rank expected number of events in study
group, Φ = cumulative distribution function of the
normal distribution.
Step 3. Calculate ln(HR) using the answers from
Steps 1 and 2:
ln(HR) = (O r – E r)/V
Step 4. Calculate HR:
HR = exp[ln(HR)]
Step 5. Calculate the variance of ln(HR),
var[ln(HR)]:
var[ln(HR)] = 1/V
Step 6. Calculate the 95% CI for HR:
95% CI = exp{ln(HR) ± 1.96 × √ –––––––––––
var[ln(HR)]}
Worked example – CALGB 9182
overall survival
In this example, an unadjusted p-value of 0.77 (p)
is presented with a total of 58 deaths in the
mitoxantrone group and 68 deaths in the
hydrocortisone group at the time of analysis. This
gives a total number of observed events of
58 + 68 = 126 (O). Working through the steps as
outlined above, we have:
Step 1. Calculate V.
V = O/4
V = 126/4 = 31.5
Step 2. Calculate O r – E r
(O r – E r) = 1 – 2 × √ ––––
126 Φ –1 (1 – 0.77/2)
= 1 – 2 × 11.22 × 0.2923 = 1.64095
Step 3. Calculate ln(HR) using the answers from
Steps 1 and 2:
ln(HR) = (O r – E r)/V
ln(HR) = 1.64095/31.5 = 0.052094
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