**1.a Calculating the effective number of trials**

Consider the situation where three treatments, A, B and C, have been compared head to head in randomized clinical trials. For any one trial, assume that the estimated treatment effect has variance

*v.* For a meta-analysis of

*2k* trials, using the inverse variance approach would produce an estimated variance of the pooled treatment effect of

*σ*
^{
2
}
*/2k.* By the expected variance of an indirect comparison, if we have two comparison including

*k* trials, we would expect an indirect variance estimate of

*σ*
^{
2
}
*/k + σ*
^{
2
}
*/k = 2σ*
^{
2
}
*/k.* Now letting

*R* denote a ratio describing the relationship between the precision of indirect and direct evidence; we can derive

*R* as follows

That is, in the scenario where the number of trials are equal in the two comparisons informing the indirect comparison (and the other above assumptions are met), it would require four trials in the indirect evidence to produce the same precision as that corresponding to a single head-to-head trial. We can generalize this ratio to the situation where the number of trials is not equal in the two comparisons informing the indirect evidence. Let

*k*
_{
AC
} and

*k*
_{
BC
} be the number of trials informing the comparison of A vs. C and B vs. C, respectively. For a single meta-analysis, with

*k*
_{
AC
}
*+ k*
_{
BC
} trials we would expect a variance of the pooled effect of

*σ*
^{2}
*/(k*
_{
AC
}
*+ k*
_{
BC
}
*).* Moreover, we would expect a variance from the indirect comparison of

*σ*
^{2}
*/k*
_{
AC
}
*+ v/k*
_{
BC
}
*.* Proceeding as above we then have

This formula creates the basis for the results presented in Table
1.

**1.b Calculating the effective number of patients**

Consider the situation where three treatments, A, B and C, have been compared head to head in randomized clinical trials. Assume that the population variance of comparative treatment effects is the same for A vs. B, A vs. C and B vs. C, and assume the population variance produced by a fixed-effect pairwise meta-analysis can be regarded as a large well-designed clinical trial. Let *n*
_{
AB
}, *n*
_{
AC
} and *n*
_{
BC
} denote the meta-analysis sample size (total number of patients) for the three comparisons A vs. B, A vs. C and B vs. C, respectively.

We are interested in finding the ratio between the variance of the direct meta-analysis pooled treatment effect estimate and the variance of the indirect meta-analysis pooled treatment estimate. Let

*R* denote this ratio, and let

*σ*
_{
AB
}
^{
2
},

*σ*
_{
AC
}
^{
2
} and

*σ*
_{
BC
}
^{
2
} denote the population variances for the three comparisons (where we assume

*σ*
_{
AB
}
^{
2
} =

*σ*
_{
AC
}
^{
2
} =

*σ*
_{
BC
}
^{
2
} =

*σ*
^{
2
}). Then we have

Thus, by multiplying this ratio with the total indirect sample size (n

_{AC} + n

_{BC}) we have that the formula for the effective indirect sample size is

When heterogeneity exists for one or both of the comparisons in the indirect evidence, one can penalize the sample size by multiplying by the ‘lack of homogeneity,’ much similar to what is done for a heterogeneity correction of a required meta-analysis sample size. With estimates of the percentage of variation in the meta-analysis due to between-trial heterogeneity for A vs. C,

*I*
_{
AC
}
^{
2
}
*,* and for B vs. C,

*I*
_{
BC
}
^{
2
}
*,* we can derive penalized sample sizes within each comparison

and subsequently use these penalized sample sizes in the formula for the effective indirect sample size.

**2. Information fraction and power calculations – worked example**

For low-dose NRT vs. high-dose NRT, the direct evidence includes 3,605 patients (and no heterogeneity). Indirect evidence exists with inert control as the common comparator. The comparison of low-dose NRT and inert control includes 19,929 patients, but with 63% heterogeneity, so the heterogeneity penalized sample size is 19,929×(1–0.63) = 7,373. The comparison of high-dose NRT vs. inert control includes 2,487 patients, but with 60% heterogeneity, so the heterogeneity penalized sample size is 2,487×(1–0.60) = 1,492. The effective sample size from this indirect comparison is therefore

A second indirect comparison with varenicline as the common comparator only includes 32 patients in one of the two involved comparisons. The effective sample size of this indirect comparison (

*n*
_{
indirect
} = 31) is so comparably small that we choose to ignore it. Adding the above calculated indirect sample sizes to the direct evidence sample size, we get effective total sample sizes of

This total effective sample sizes correspond to information fractions of 92% and 60% and statistical power estimates of 88% and 72%.

For low-dose NRT vs. buproprion, no direct evidence exists. Indirect evidence exists through inert control as the common comparator. As above, the sample size for low-dose NRT vs. inert control is 19,929, or 7,373 if heterogeneity penalized. The sample size for buproprion vs. inert control is 12,567, or 12,567×(1–0.39) = 7,666 when heterogeneity is penalized. Therefore, the total effective sample sizes (which are equal to the effective indirect sample sizes) are

This total effective sample sizes correspond to information fractions of >100% and 60% and statistical power estimates of 95% and 71%.

For low-dose NRT vs. varenicline, the direct evidence includes 740 patients (and no heterogeneity). As above, the sample size for low-dose NRT vs. inert control is 19,929, or 7,373 if heterogeneity is penalized. The sample size for varenicline vs. inert control is 4,331, or 4,331 × (1–0.69) = 1,343 if heterogeneity is penalized. Therefore, the total indirect sample sizes are

and so the total effective sample are

All power, and information fraction calculations above are geared to detect an assumed relative improvement in smoking cessation of 20%. All calculations are highly sensitive to the assumed relative improvement. In particular, assuming larger improvements would result in substantially larger power and information fraction estimates.