Medical researchers often find that some data which they intended to collect could not be collected: for example, because participants could not be contacted or were unwilling to provide data. These missing data present problems in the analysis of the study, because including only participants who provided data may lead to incorrect results. The commonest way to handle missing data assumes that missing values are similar to observed values within subgroups: for example, for participants whose weight was observed at times 1 and 2 but missing at time 3, the missing weights at time 3 are assumed to have the same average as observed weights at time 3 in participants whose weights were similar at times 1 and 2 and observed at time 3. This approach is called "Missing at Random" and provides a good starting point for analysis but is unlikely to be entirely correct: for example, participants whose weight was unobserved at time 3 may have had a larger weight gain. It is therefore important for researchers to do sensitivity analyses in which different assumptions are made about the missing data. Our research proposes to adapt a popular method for handling missing data called Multiple Imputation by Chained Equations (MICE) to allow for a range of assumptions about the missing data. The idea of this approach is that missing values are filled in iteratively using the relationships between all the variables, and this is then done multiple times in order to express uncertainty about the missing data. However, at present the MICE method is done assuming Missing at Random. We have developed a new way to implement the MICE method which does not assume Missing at Random: instead, the researcher has to specify how big the departures from Missing at Random are, by specifying the likely average differences between missing values and observed values within subgroups. However, we have only explored the new method in idealised settings, and in particular we have not explored its use in randomised trials or in studies where outcomes are measured over time. The work will first extend the statistical theory to handle outcomes that are measured over time and see how well the method performs in randomised trials. It will then extend the methods to tackle a wide range of problems met in practice: for example different types of variables, complex analysis questions, and very large data sets. This work will be supported by writing user-friendly software to implement the new method in two widely used statistics packages. We will implement the method in practice in several data sets, including the Avon Longitudinal Study of Parents and Children where we will explore predictors of self-harm, and randomised trials in smoking cessation and weight loss. Missing self-harm, smoking cessation and weight loss data are all very unlikely to be Missing at Random: we will use our subject matter expertise to specify a range of likely average differences between missing values and observed values within subgroups and hence reach more defensible conclusions. This work is likely to raise unexpected theoretical issues which we will address. Finally, we believe that this method will be widely applicable, so we will disseminate it to researchers via tutorial articles and by running courses.