## Binary data

The statistical techniques most frequently employed in meta-analyses are based on the Mantel-Haenszel method of combining data over a series of 2 x 2 contingency tables. Basically this involves comparing the observed and expected event rate for the 'experimental' treatment and control groups within each trial and then combining the results of the individual trials. There are a number of ways of doing this. The simplest, an example of which is shown in Box 11.8, makes use of the proportion of patients event-free at a given time point. For each individual trial the overall number of events within the study is used to calculate the expected number of events that would occur if there were no difference between the two arms of the trial. The difference between the actual observed (O) and the expected (E) number of events is then calculated for the 'experimental' arm, giving the O-E value. A negative O-E value indicates that the 'experimental' treatment group has fared better than the control group, whilst a positive O-E value indicates the opposite. If the 'experimental' treatment has no effect, then each individual O-E could be either positive or negative and will differ only randomly from zero. Likewise, the summated O-E from all the studies will differ only randomly from zero if there is no difference between the 'experimental' and control groups. If, however, the treatment does have a beneficial effect, then there will be a trend for individual O-E values to be negative, and the overall summated value to be clearly so.

The O-E and its variance can then be used to calculate odds ratios (OR) for each trial. This OR gives the ratio of the odds of an event among the 'experimental' group to the corresponding odds of an event among the control group patients. An OR value of 1.0 represents equal odds or no difference between treatments, while a value of 0.5 indicates a halving of the odds of an event measured for patients in the 'experimental' arm. An OR across trials can be calculated by summing the individual O-E and variance values in such a way that the pooled OR represents a weighted average of the individual ORs, with the individual trials that contain most information (usually the largest trials) having the greatest influence. This approach is known as the fixed effect model. An alternative approach where the pooled estimate of effect takes account of both the within trial variance and the between trial variance is known as the random effect model. The pros and cons of each are discussed in Section 11.6.1.

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