Missing Data

The complete case approach (aka listwise deletion or uniform nonresponse) assumes that data are missing completely at random (MCAR) and omits participants with missing data. When data is MCAR, the mechanism of the missing data depends neither on the observed values nor the unobserved values. In other words, the probability of an observation being missing depend neither on the observed nor unobserved values, Prob(R|Y_Observed, Y_Missing) = Prob(R). An example would be people with no prior DUI arrest history are less likely to anwer questions pertaining to DUI arrest. This is a fairly strong assumption but if that's the case, limiting the analyssi to observations with non-missing values can give valid inferences. In other words, the observation with missing value can take any random value from another observation.

Data is missing at random (MAR) when the mechanism of the missing data depends only on the observed values and not the unobserved values. In other wordds, the probability of an observation being missing does depend on the observed values and not the unobserved values, Prob(R|Y_Observed, Y_Missing) = Prob(R|Y_Observed). Put it differently, the probabiltiy of an observation being missing on that variable does not depend on that variable after controlling for other relevant variable(s). For instance, depressed individuals may be less likely to interact with other individuals. Thus, reported frequency of social interaction will be related to mental health well-being. As such, valid inrerences can be obtained using information observed from other variables in the dataset.

Data is missing not at random (NMAR) if the mechanism of the missing data depends only on the unobserved values. In other wordds, the probability of an observation being missing does depend on the unobserved values, Prob(R|Y_Observed, Y_Missing) = Prob(R|Y_Missing). For example, people with multiple prior DUI arrests may be less likely to report alcohol use. In this situation, even if we take all the information we can observed into account, the reason for why an observation is missing on certain variables still depends on the unobserved values. As such, in order to obtain valid inferences, the pattern / mechanism of missingness should be taken into account and we need to jointly model the outcome of interest (Y_i) and the missing data pattern / mechanism (R_i).