Graphics Reference
In-Depth Information
θ
where
f
is the probability function for a single case and
represents the parameters
of the model that yield such a particular instance of data. The main problem is that
the particular parameters' values
for the given data are very rarely known. For this
reason authors usually overcome this problem by considering distributions that are
commonly found in nature and their properties are well known as well. The three
distributions that standout among these are:
θ
1. the multivariate normal distribution in the case of only real valued parameters;
2. the multinomial model for cross-classified categorical data (including loglinear
models) when the data consists of nominal features; and
3. mixed models for combined normal and categorical features in the data [
50
,
55
].
If we call
X
obs
the observed part of
X
and we denote the missing part as
X
mis
so
that
X
, we can provide a first intuitive definition of what
missing at
random
(MAR) means. Informally talking, when the probability that an observation
is missing may depend on
X
obs
but not on
X
mis
we can state that the missing data is
missing at random.
In the case of MAR missing data mechanism, given a particular value or val-
ues for a set of features belonging to
X
obs
, the distribution of the rest of features
is the same among the observed cases as it is among the missing cases. Follow-
ing Schafer's example based on [
79
], let suppose that we dispose an
n
=
(
X
obs
,
X
mis
)
p
matrix
called
B
of variables whose values are 1 or 0 when
X
elements are observed and
missing respectively. The distribution of
B
should be related to
X
and to some
unknown parameters
×
ζ
, so we dispose a probability model for
B
described by
(
|
,ζ)
P
B
X
. Having a MAR assumption means that this distribution does not depend
on
X
mis
:
P
(
B
|
X
obs
,
X
mis
,ζ)
=
P
(
B
|
X
obs
,ζ).
(4.2)
Please be aware of MAR does not suggest that the missing data values consti-
tute just another possible sample from the probability distribution. This condition is
known as missing completely at random (MCAR). Actually MCAR is a special case
of MAR in which the distribution of an example having a MV for an attribute does
not depend on either the observed or the unobserved data. Following the previous
notation, we can say that
P
(
B
|
X
obs
,
X
mis
,ζ)
=
P
(
B
|
ζ).
(4.3)
Although there will generally be some loss of information, comparable results can be
obtained with missing data by carrying out the same analysis that would have been
performed with no MVs. In practice this means that, under MCAR, the analysis of
only those units with complete data gives valid inferences.
Please note that MCAR is more restrictive than MAR. MAR requires only that
the MVs behave like a random sample of all values in some particular subclasses
defined by observed data. In such a way, MAR allows the probability of a missing
datum to depend on the datum itself, but only indirectly through the observed values.
