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## Missing at Random (MAR)

### Missing Not at Random (MNAR)

Missing data mechanisms **concern the relationship between missing data and the values of variables in the data matrix**. Given this focus, missing data mechanisms can be categorized into three classes: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR).

**Missing Completely at Random (MCAR)**

Missing completely at random (MCAR) is the only missing data mechanism that can actually be verified. Missing data are MCAR when the probability of missing data on a variable is unrelated to any other measured variable and is unrelated to the variable with missing values itself. In other words, the missingness on the variable is completely unsystematic. For example when data are missing for respondents whose questionnaire was lost in the mail. This assumption can be tested by separating the missing and the complete cases and examining the group characteristics. If characteristics are not equal for both groups, the MCAR assumption does not hold.

Missing data are missing at random (MAR) when the probability of missing data on a variable is related to some other measured variable in the model, but not to the value of the variable with missing values itself. For example, only older people have missing values for IQ. In that case, the probability of missing data on IQ is related to age. The assumption that the mechanism is MAR, can unfortunately not be confirmed, because it cannot be tested if the probability of missing data on a variable is solely a function of other measured variables. It is recommended to incorporate correlates of missingness into the missing data handling procedure to diminish bias and improve the chances of satisfying the MAR assumption.

Data are missing not at random (MNAR) when the missing values on a variable are related to the values of that variable itself, even after controlling for other variables. For example, when data are missing on IQ and only the people with low IQ values have missing observations for this variable. A problem with the MNAR mechanism is that it is impossible to verify that scores are MNAR without knowing the missing values.

Example of MNAR data