Missing at Random (MAR)

Secondly, data can be Missing at Random (MAR). Here, the missingness of data depends on observed variables. A brief narrative shall provide some intuition for the MAR condition: in a national survey of small business owners about the business climate, there is a question about the local cost of energy. Chances are that the owner of a business that uses little electricity, e.g., a yoga studio, may not know of the current cost of 1 kWh of electric energy and could not answer that question, thus producing a missing value in the questionnaire. On the other hand, the owner of an energy-intensive business, e.g., an electroplating shop, would presumably be keenly aware of the electricity price and able to respond accordingly. In this story, the probability of non-response is presumably inversely proportional to the energy consumption of the business.

In the subnetwork shown below, X3_obs is the observed variable that causes the missingness, e.g., the energy consumption in our story. X2_obs would be the stated price of energy if known. X2 would represent the actual price of energy in our narrative. Indeed, from the researcher’s point of view, the actual cost of energy in each local market and for each electricity customer is hidden.

To simulate this network, we need to define its parameters, i.e., the quantitative part of the network structure:

• X2 is a continuous variable with values between 0 and 1. Here, too, we have arbitrarily defined a Normal distribution for modeling the DGP.

• MAR_X2 is a boolean variable with one parent, which specifies that the missingness probability depends directly on the fully observed variable X3_obs. The exact values are not important here, as we only need to know that the probabilities of missingness are inversely proportional to the values of X3_obs:

$$P(MA{R_{X2}} = true|X{3_{obs}}) \propto \frac{1}{{X{3_{obs}}}}$$

• X2_obs has two parents, i.e., the data-generating variable X2 and the missingness mechanism MAR_X2. The conditional probability distribution of X2_obs can be described by the following deterministic rule: IF MAR_X2 THEN X2_obs=? ELSE X2_obs=X2

Given the fully specified network, we can now simulate the impact of the missingness mechanism on the observable variable X2_obs.

As the above screenshot shows, the mean and standard deviation in the Monitor of X2_obs indicates that the distribution of the observed values of X2 differs significantly from the original distribution, leading to an overestimation of X2 in this example. We can simulate the deletion of incomplete observations by setting negative evidence on “?” in the Monitor of X2_obs (green arrow labeled “Delete”). The simulated distribution of X2_obs (right) clearly differs from the one of X2 (left).