Missing data, part 2. Missing data mechanisms: Missing completely at random, missing at random, missing not at random, and why they matter

In the first article,1Pham T.M. Pandis N. White I.R. Missing data, Part 1. Why missing data are a problem.Am J Orthod Dentofac Orthop. 2022; 161: 888-889Abstract Full Text Full Text PDF Scopus (2) Google Scholar we demonstrated that any analysis with missing data makes untestable assumptions about the missing values. The use of different statistical methods rests on different missing data assumptions, and it is important to be transparent about which assumption we are making when implementing a given method. As we will see in article 5, multiple imputation,2Rubin D.B. Multiple Imputation for Nonresponse in Surveys. Wiley, New York1987Crossref Google Scholar a popular approach for handling missing data, is typically performed assuming data are missing at random (described below). Rubin3Rubin D.B. Inference and missing data.Biometrika. 1976; 63: 581-592Crossref Scopus (5722) Google Scholar formally introduced the concept of the missingness mechanism, which describes how the chance of data being missing is associated with the values of the variables included in our analysis. Missingness is commonly categorized into missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). We will now illustrate these 3 missingness mechanisms in the example described in article 1. The example was created using data from a randomized controlled trial comparing how probing depth on the mandibular anterior teeth evolves between 2 types of lingual retainers.4Węgrodzka E. Kornatowska K. Pandis N. Fudalej P.S. A comparative assessment of failures and periodontal health between 2 mandibular lingual retainers in orthodontic patients. A 2-year follow-up, single practice-based randomized trial.Am J Orthod Dentofacial Orthop. 2021; 160: 494-502.e1Abstract Full Text Full Text PDF PubMed Scopus (6) Google Scholar Our data set contains data on individuals age at baseline (age25), which is fully observed with no missing values, and mean probing depth across 6 teeth at time point 1 (mean_pd1), which has some missing values. Here, the missingness mechanism refers to how the chance of mean_pd1 being missing depends on age25 (aged <25/≥25 years, fully observed) and the value (partially observed) of mean_pd1. These relationships can be presented graphically using directed acyclic graphs (DAGs), as illustrated in the Figure. A DAG displays assumptions about the relationships between variables. The assumptions are represented by lines that connect one variable to another. These lines are directed, with a single arrowhead indicating the direction of their effects. DAGs are acyclic, meaning they cannot contain any loops in which a variable causes itself.5Greenland S. Pearl J. Robins J.M. Causal diagrams for epidemiologic research.Epidemiology. 1999; 10: 37-48Crossref PubMed Scopus (2333) Google Scholar We create a variable mean_pd1_miss, a binary indicator of missingness in mean_pd1 (ie, mean_pd1_miss takes value 1 if mean_pd1 is observed and 0 if mean_pd1 is missing). We can now describe the 3 missingness mechanisms.1.MCAR: mean_pd1 is MCAR if the chance of mean_pd1 being missing is independent of age25 and the (possibly missing) value of mean_pd1. This assumption is illustrated in the Figure, A, in which no arrows are pointing to mean_pd1_miss from either age25 or mean_pd1. This assumption means that the missing data are fully comparable to the observed data.2.MAR: mean_pd1 is MAR conditional on age25 if the chance of mean_pd1 being missing is independent of the (possibly missing) value of mean_pd1 after controlling for age25 (Fig, B). This means the chance of mean_pd1 being missing can vary with age25, but within each age group, the chance of mean_pd1 being missing is the same for all individuals. In the Figure, B, both mean_pd1 and mean_pd1_miss are influenced by age25 (represented by 2 arrows going from age25), implying mean_pd1_miss is associated with mean_pd1 if we do not control for age25. Controlling for age25 (the common cause of mean_pd1 and mean_pd1_miss) removes the association between mean_pd1_miss and mean_pd1 (ie, mean_pd1_miss is independent of mean_pd1, conditional on age25). This assumption means that the missing data in a given age group are fully comparable to the observed data in the same age group.3.MNAR: mean_pd1 is MNAR given age25 if the chance of mean_pd1 being missing still depends on the (possibly missing) value of mean_pd1, even after we have controlled for age25 (Fig, C). This means that within each age group, the chance of mean_pd1 being missing may still vary with the values of mean_pd1 (eg, mean_pd1 might be missing more frequently for individuals with greater probing depth values compared with those with lower probing depth values who are in the same age group). As seen in the Figure, C, in addition to the arrow from age25 to mean_pd1_miss, another arrow goes directly from mean_pd1 to mean_pd1_miss. Hence, mean_pd1_miss is not independent of mean_pd1, even after controlling for age25. This assumption means that the missing data in a given age group are not comparable to the observed data, even in the same age group. Of these categorizations, MCAR is the most restrictive assumption, under which the missingness of the data does not relate to any values in the data set, whether observed or missing. This assumption is seldom plausible in practice because there is likely other information collected in the data set that explains how values in a variable have become missing. MNAR is the least restrictive assumption and the trickiest to deal with, as in practice, we rarely know what the appropriate model for the missingness mechanism looks like. MAR is the assumption used by the standard implementation of multiple imputation (as will be seen in article 5). In the above example, we described MAR using just a single variable, age25, but the definition extends to multiple variables. MAR can therefore be made more plausible by collecting data on additional explanatory variables that may explain the missing values. The next article in this series will explain how we can explore the missing data in order to decide which assumption is reasonable (MCAR, MAR, or MNAR) and plan analysis. The authors would like to thank Jadbinder Seehra for reviewing and commenting on draft versions of these articles.