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Cost effectiveness analysis (CEA) of randomised trials are an important source of evidence for informing policy makers on how to best allocate limited resources. Missing data are a common issue in trial-based CEA, and methods such as multiple imputation are now commonly used to account for the missing values, assuming the data are 'missing at random' (MAR). This implies that the reasons for the missing data can be explained by the observed data. However, the missingness is often related to unobserved values, that is data are 'missing not at random' (MNAR, or 'informative'). For example, patients whose health status is relatively poor may be less likely to return health questionnaires, even conditional on their observed characteristics. In these settings, methodological guidance recommends assessing whether conclusions are sensitive to departures from the MAR assumption. Sensitivity analysis strategies for handling MNAR is an area of rapid development in medical statistics, but this form of uncertainty has not yet been appropriately addressed in health economics. This PhD thesis aims to develop practical, accessible sensitivity analysis strategies and software tools to handle MNAR data in trial-based CEA. The thesis critically assessed the statistical methods for handling MNAR data in CEA practice, and identified barriers to more widespread use of these methods, via a systematic review and stakeholder focus groups. The research then focused on two strategies to conduct sensitivity analysis under MNAR assumptions: pattern-mixture models, which involve imputing the data assuming MAR, then modifying the imputed values to reflect possible departures from that assumption; and reference-based imputation, where the data are imputed assuming a distribution borrowed from a 'reference group'. These approaches were illustrated in CEAs of 10TT and CoBalT trials, which evaluated weight loss and depression interventions. Software code and practical guidance are provided to facilitate implementation in practice.