Meta-analysis is the statistical part of a systematic review. Many researchers have\ud used selection functions to model publication bias in a meta-analysis. The main\ud problem with this approach is that it is impossible to verify that the selection function\ud truly represents the selection process, and so the use of selection functions can only\ud be seen as part of a sensitivity analysis. In this thesis we present new methods that\ud involve selection functions that aim to make as few strong assumptions about selection\ud as possible, including the use of a non-parametric permutation test, and the use of a\ud step selection function. We also investigate the use of parametric selection functions\ud and suggest how researchers could use these as part of a sensitivity analysis, by looking\ud at a range of plausible values for the overall selection probability. As part of this\ud sensitivity analysis, we assess the effectiveness of the Bounds method as presented\ud by Henmi et al. Throughout the thesis we illustrate all methods with numerical\ud examples, including a meta-analysis investigating the effects of environmental tobacco\ud smoke on the risk of lung cancer in non-smokers
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