This means that the probability of missingness is not related neither to the recorded variables of the data nor the unobserved variable $Z$ itself. In other words, the behavior of $M_{Z}$ is statistically independent of all remaining variables of the data. For example, if the participants throw a coin to decide if they will answer a question or not, the missing item is MCAR.
The independence of $M_{Z}$ from the missing values $Z$ cannot be tested because we do not have access to the values which people refused to give answers.

The second type of Missing data is the mechanism Missing at Random (MAR). We assume that the sets $X$ and $Z$ and the indicator function $M_{Z}$ are the same as before. The MAR can be defined as
$$\mathbb{P}(M_{Z}=1|X,Z)=\mathbb{P}(M_{Z}=1|X).$$
To be more specific, the data are MAR if the missing values are related to the variables $X$ which have been recorded but it is independent from the variable which is missing from the data. …show more content…
For example, if we conduct a survey for the people's weight and we have missing values in the variable "weight", we may assume that the missingness comes from the overweight people who are ashamed to report their weight. However, we cannot be sure if this assumption is true or not. The MAR assumption will be correct if the people answer to questions like, "what is your height","how often do you work out". Such variables are highly correlated to the unknown variable and this will help us to reduce the dependence with the unobserved variable itself. (PaulD.Allison) We have to note that the MAR assumption is less restrictive than the MCAR