Most analyses of randomised trials with incomplete outcomes make untestable assumptions and should therefore be subjected to sensitivity analyses. However, methods for sensitivity analyses are not widely used. We propose a mean score approach for exploring global sensitivity to departures from missing at random or other assumptions about incomplete outcome data in a randomised trial. We assume a single outcome analysed under a generalised linear model. One or more sensitivity parameters, specified by the user, measure the degree of departure from missing at random in a pattern mixture model. Advantages of our method are that its sensitivity parameters are relatively easy to interpret and so can be elicited from subject matter experts; it is fast and non-stochastic; and its point estimate, standard error and confidence interval agree perfectly with standard methods when particular values of the sensitivity parameters make those standard methods appropriate. We illustrate the method using data from a mental health trial.