## Box 118 Calculating an odds ratio hypothetical example

Suppose there are four trials comparing 'experimental' treatment versus control, and that the numbers of deaths on each arm of these trials are as below:

'Experimental' |
Control |
p -value | |

Trial A |
29/67 |
39/72 |
0.20 |

Trial B |
3/23 |
10/30 |
0.09 |

Trial C |
7/39 |
11/28 |
0.05 |

Trial D |
45/145 |
64/172 |
0.30 |

Individually, none of these trials show a statistically significant difference between the number of deaths on the 'experimental' and control arm. For trial A, there are sixty-seven patients on the treatment arm, of whom twenty-nine have died. The expected number of deaths on the 'experimental' arm (E), O-E value (O-E), variance (V) and odds ratio (OR) can be calculated:

'Experimental' |
Control |
Total | |

Dead |
29 |
39 |
68(D) |

Alive |
38 |
33 |
71 |

Total |
67 (nt) |
72 |
139 (N) |

OR = exp[(O-E)/V] V =[E(1 - nt/N)(N - D)]/(N - 1) = exp (-3.78/8.74) = [32.78(1 - 67/139)(139 - 68)]/138 = 0.65 = 8.74

Similar values can be calculated for trials B, C and D:

O-E |
Variance |
Odds ratio | |

Trial A |
-3.78 |
8.74 |
0.65 |

Trial B |
-2.64 |
2.46 |
0.34 |

Trial C |
-3.48 |
3.25 |
0.34 |

Trial D |
-4.86 |
17.81 |
0.76 |

Combined or pooled OR can now be calculated as follows: OR = exp [£(O-E)/£V] 95%CI = exp (E(O-E)/EV) ± (1.96/

The combined OR is conventionally significant in favour of treatment with a 36 per cent reduction in the odds of death for the treatment group patients. Adapted and reproduced with permission from [50].

Section 9.3.4) on individual trials and then using the stratified log rank O-Es and their variances to calculate a hazard ratio (HR). Rather than summarizing the overall number of events as in an odds ratio, the HR makes use of the time (from randomization) until each individual event takes place, and also uses information from patients who have not yet experienced the event (censored patients).

In this way, the HR summarizes the entire 'survival' experience. The calculations follow through in exactly the same way as for calculation of an OR, simply by substituting the log rank O-E and variance for those calculated from the crude number of events. These hazard ratios represent the instantaneous risk of failure on treatment as opposed to control. An HR of less than 1.0 favours treatment whereas an HR of more than 1.0 favours control. For example, when measuring survival, an HR of 0.8 indicates a 20 per cent reduction in the overall risk of death when receiving treatment as compared to control. Such time-to-event analyses can be extremely important in chronic illness where a prolongation of survival, time without evidence of disease, or time without symptoms is important.

While not all systematic reviews of survival-type endpoints have time and resource available to collect and analyse IPD, they should make the most efficient use of the summary statistical data available. Where meta-analyses are based on published information, provided that trials are sufficiently well reported, it maybe possible to estimate HRs from a variety of statistical summary measures rather than calculating odds ratios [25] at fixed timepoints, as is usually done for this type of meta-analysis. Where log HRs, HRs or log rank O-Es plus variances are presented, these can be used directly in the calculation of an overall HR. Even when these are not presented, manipulation of the chi-square value, p-value or variance can be used to calculate an HR indirectly. Some of the methods of doing these calculations are shown in Box 11.9 opposite. The particular method used will depend on what information is presented for particular trials, and different methods can be used for different trials as appropriate. If there is not sufficient information to obtain an estimated HR from summary statistics then the HR can be estimated by splitting the published survival curves into discrete time intervals, calculating an OR for each of these periods and then combining the ORs over time. This is often the only practical way to proceed as few papers present sufficient information to permit manipulation of summary statistics by either the direct or indirect method. For example, a study using information from fourteen IPD meta-analysis comparisons, involving a number of cancers and a total of 138 RCTs found that only 60 per cent of published trial reports had estimatable HRs [49].

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