A simple form of non-ignorable missing data mechanisms based on two parameters is used to characterize the amount of missing data and the severity of non-randomness in clinical trials. Based on the formulation, the effect of non-randomly missing data on simple analyses which ignore the missing data is studied for binary and normally distributed response variables. In general, the effect of the non-randomly missing data on the bias and the power increases with the severity of non-randomness. The bias can be positive or negative and the power can be less than or greater than when the data are missing at random. The results of the analysis, ignoring the missing data, can be seriously flawed if the non-randomness is severe, even when only a small proportion of the sample is missing. The problem is more pronounced in the case of normally distributed response variables with unequal variances.