Additional examples of Missingness Mechanisms –
Follow up to SON Brown Bag Presentation – 3/20/13 (C Thompson) – Missing Data part 1
From Baraldi/Enders 2009 reference pp7-9:
Theoretical background: Rubin's missing data mechanisms
Before we can begin discussing different missing data handling options, it is important
to have a solid understanding of so-called “missing data mechanisms”. Rubin (1976) and
colleagues (Little & Rubin, 2002) came up with the classification system that is in use
today: missing completely at random (MCAR), missing at random (MAR), and missing not
at random (MNAR). These mechanisms describe relationships between measured variables
and the probability of missing data. While these terms have a precise probabilistic and
mathematical meaning, they are essentially three different explanations for why the data are
missing. From a practical perspective, the mechanisms are assumptions that dictate the
performance of different missing data techniques. We give a conceptual description of each
mechanism in this section, and supplementary resources are available to readers who want
additional details on the missing data mechanisms (Allison, 2002; Enders, 2010; Little &
Rubin, 2002; Rubin, 1976; Schafer & Graham, 2002).
To begin, data are MCAR when the probability of missing data on a variable X is
unrelated to other measured variables and to the values of X itself. In other words,
missingness is completely unsystematic and the observed data can be thought of as a
random subsample of the hypothetically complete data. As an example, consider a child in
an educational study that moves to another district midway through the study. The missing
values are MCAR if the reason for the move is unrelated to other variables in the data set
(e.g., socioeconomic status, disciplinary problems, or other study-related variables). Other
examples of MCAR occur when a participant misses a survey administration due to
scheduling difficulties or other unrelated reasons (such as a doctor's appointment), a
computer randomly misreads grid-in sheets, or an administrative blunder causes several test
results to be misplaced prior to data entry. MCAR data may also be a purposeful byproduct
of the research design. For example, suppose that a researcher collects self-report data from
the entire sample but limits time-consuming behavioral observations to a random subset of
participants. We describe a number of these so-called planned missing data designs at the
end of the paper. Because MCAR requires missingness to be unrelated to study variables,
methodologists often argue that it is a very strict assumption that is unlikely to be satisfied
in practice (Raghunathan, 2004; Muthen, Kaplan, & Hollis, 1987).
The MAR mechanism requires a less stringent assumption about the reason for missing
data. Data are MAR if missingness is related to other measured variables in the analysis
model, but not to the underlying values of the incomplete variable (i.e., the hypothetical
values that would have resulted had the data been complete). This terminology is often
confusing and misleading because of the use of the word “random.” In fact, an MAR
mechanism is not random at all and describes systematic missingness where the propensity
for missing data is correlated with other study-related variables in an analysis. As an
example of an MAR mechanism, consider a study that is interested in assessing the
relationship between substance use and self-esteem in high school students. Frequent
substance abuse may be associated with chronic absenteeism, leading to a higher
probability of missing data on the self-esteem measure (e.g., because students tend to be
absent on the days that the researchers administered the self-esteem questionnaires). This
example qualifies as MAR if the propensity for missing data on the self-esteem measure is
completely determined by a student's substance use score (i.e., there is no residual
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relationship between the probability of missing data and self-esteem after controlling for
substance use). As a second example, suppose that a school district administers a math
aptitude exam, and students that score above a certain cut-off participate in an advanced
math course. The math course grades are MAR because missingness is completely
determined by scores on the aptitude test (e.g., students that score below the cut-off do not
have a grade for the advanced math course).
Finally, data are MNAR if the probability of missing data is systematically related to the
hypothetical values that are missing. In other words, the MNAR mechanism describes data
course grades). Although the magnitude of the bias depends on the correlation between the
omitted aptitude variable and the course grades (bias increases as the correlation increases),
the analysis is nevertheless consistent with an MNAR mechanism. Later in the manuscript,
we describe methods for incorporating so-called auxiliary variables that are related to
missingness into a statistical analysis. Doing so can mitigate bias (i.e., by making the MAR
mechanism more plausible) and can improve power (i.e., by recapturing some of the
missing information).
From Howell ref (Missing):
1.1 The nature of missing data
Missing completely at random
There are several reasons why data may be missing. They may be missing because equipment malfunctioned,
the weather was terrible, people got sick, or the data were not entered correctly. Here the data are missing
completely at random (MCAR). When we say that data are missing completely at random, we mean that the
probability that an observation (Xi) is missing is unrelated to the value of Xi or to the value of any other
variables. Thus data on family income would not be considered MCAR if people with low incomes were less
likely to report their family income than people with higher incomes. Similarly, if Whites were more likely to
omit reporting income than African Americans, we again would not have data that were MCAR because
missingness would be correlated with ethnicity. However if a participant's data were missing because he was
stopped for a traffic violation and missed the data collection session, his data would presumably be missing
completely at random. Another way to think of MCAR is to note that in that case any piece of data is just as
likely to be missing as any other piece of data.
Notice that it is the value of the observation, and not its "missingness," that is important. If people who refused
to report personal income were also likely to refuse to report family income, the data could still be considered
MCAR, so long as neither of these had any relation to the income value itself. This is an important
consideration, because when a data set consists of responses to several survey instruments, someone who did
not complete the Beck Depression Inventory would be missing all BDI subscores, but that would not affect
whether the data can be classed as MCAR.
This nice feature of data that are MCAR is that the analysis remains unbiased. We may lose power for our
design, but the estimated parameters are not biased by the absence of data.
Missing at random
Often data are not missing completely at random, but they may be classifiable as missing at random (MAR).
(MAR is not really a good name for this condition because most people would take it to be synonymous with
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MCAR, which it is not. However, the label has stuck.) Let's back up one step. For data to be missing completely
at random, the probability that Xi is missing is unrelated to the value of Xi or other variables in the analysis.
But the data can be considered as missing at random if the data meet the requirement that missingness does not
depend on the value of Xi after controlling for another variable. For example, people who are depressed might
be less inclined to report their income, and thus reported income will be related to depression. Depressed people
might also have a lower income in general, and thus when we have a high rate of missing data among depressed
individuals, the existing mean income might be lower than it would be without missing data. However, if,
within depressed patients the probability of reported income was unrelated to income level, then the data would
be considered MAR, though not MCAR. Another way of saying this is to say that to the extent that missingness
is correlated with other variables that are included in the analysis, the data are MAR.
The phraseology is a bit awkward here because we tend to think of randomness as not producing bias, and thus
might well think that Missing at Random is not a problem. Unfortunately it is a problem, although in this case
we have ways of dealing with the issue so as to produce meaningful and relatively unbiased estimates. But just
because a variable is MAR does not mean that you can just forget about the problem. But nor does it mean that
You have to throw up your handes and declare that there is nothing to be done
The situation in which the data are at least MAR is sometimes referred to as ignorable missingness. This name
comes about because for those data we can still produce unbiased parameter estimates without needing to
provide a model to explain missingness. Cases of MNAR, to be considered next, could be labeled cases of
nonignorable missingness.
Missing Not at Random
If data are not MCAR or MAR then they are classed as Missing Not at Random (MNAR). For example, if we
are studying mental health and people who have been diagnosed as depressed are less likely than others to
report their mental status, the data are not missing at random. Clearly the mean mental status score for the
available data will not be an unbiased estimate of the mean that we would have obtained with complete data.
The same thing happens when people with low income are less likely to report their income on a data collection
form.
When we have data that are MNAR we have a problem. The only way to obtain an unbiased estimate of
parameters is to model missingness. In other words we would need to write a model that accounts for the
missing data. That model could then be incorporated into a more complex model for estimating missing values.
This is not a task anyone would take on lightly. See Dunning and Freedman (2008) for an example. However
even if the data are MNAR, all is not lost. Our estimators may be biased, but the bias may be small.
See missingdata.org.uk reference  Introduction to missing data Missingness mechanisms
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