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BCG vaccine against tuberculosis: random-effects meta-analysis of 13 trials (metafor's published example)

ToolMeta-Analysis
SourceViechtbauer, W. (2010). "Conducting Meta-Analyses in R with the metafor Package." Journal of Statistical Software, Vol. 36, No. 3, pp. 1–48. Pages 14 (random-effects model), 19 (allocation subgroups), 24 (leave-one-out), 30 (regression test), 31–32 (trim and fill); data of Colditz et al. (1994).
LevelGraduate

Problem

Thirteen trials of the BCG vaccine report tuberculosis cases among vaccinated and unvaccinated participants (357,347 in all). Pool the risk ratios under a fixed-effect and a random-effects model, measure the heterogeneity, compare the three methods of treatment allocation and look for small-study effects.

Inputs to Enter

  • Effect measure: risk ratio from events and totals; 13 lines such as Aronson 1948: 4 of 123 vaccinated, 11 of 139 controls; subgroup column allocation (random, alternate, systematic), sort column year
  • Fixed-effect method Mantel–Haenszel; τ² by restricted maximum likelihood; Wald interval, 95 %; prediction interval with t on k − 1 degrees of freedom
  • Subgroups with one pooled τ²; cumulative order by year; trim and fill with the L0 estimator, automatic side, fixed-effect centre
Open the Meta-Analysis tool (loaded with this example)

Answers

  • Random effects: ln RR = −0.7145, SE 0.1798, z = −3.974 (paper −0.7145, 0.1798, −3.9744); RR = 0.4894 (0.3441 to 0.6962) (paper 0.49, 0.34 to 0.70)
  • τ² = 0.3132 with Q-profile interval 0.1197 to 1.111 (paper 0.3132, 0.1197 to 1.1115); Q = 152.23 on 12 df; I² = 92.22 %, H² = 12.86 (paper 92.22 %, 12.86)
  • Fixed effect: Mantel–Haenszel RR 0.6353 (0.5881 to 0.6862) and inverse variance 0.6503 (0.6007 to 0.7040), the values Stata's metan prints; 95 % prediction interval 0.1360 to 1.762
  • Allocation subgroups (pooled τ²): ln RR −0.9658 random, −0.5180 alternate, −0.4289 systematic; Q between = 1.7675 on 2 df, p = 0.4132 (paper the same)
  • Egger's regression t = −1.401 on 11 df, p = 0.1887 (paper −1.4013, 0.1887); Begg's τ = 0.026, p = 0.95
  • Trim and fill: 4 studies missing on the right side; fixed-effect ln RR after filling −0.2910, SE 0.0383, Q = 262.73 (paper 4, −0.2910, 0.0383, 262.7316), RR 0.7475
  • Leave-one-out: the random-effects RR stays between 0.452 (without TPT Madras) and 0.533 (without Hart and Sutherland), as in the paper's table

Notes

The tool's default inputs are this data set; press Submit. The paper fits the fixed-effect model by inverse variance; the page shows that estimate beside the Mantel–Haenszel one.