![](https://csdnimg.cn/release/download_crawler_static/9369266/bg4.jpg)
the assumption that missingness stand for wrong answers hardly
holds in real applications and, as it will be seen in the section
Missing-as-Wrong (MAW) Transformations and the BLIM, it may
lead to biased estimates of the BLIM’s parameters.
Two Extensions of the BLIM to Missing Data
In this section, two different extensions of the BLIM are pro-
posed that are adequate when missing answers are allowed. The
first extension satisfies the parameter distinctiveness condition and
is based on the assumption that the data generating process is
MCAR. For this reason, it is called IMBLIM. In the second
extension, called MissBLIM, a conditional distribution of the
missing data, given the knowledge states is assumed. This last
model is appropriate when the data generating process is MNAR.
In both extensions, as in the BLIM, we denote with R Q the
collection of all problems in Q that a single student correctly
solved. Additionally, we indicate with M Q \R the set of prob-
lems for which neither a correct nor a wrong response was re-
corded. Moreover, W denotes the collection of all problems for
which an incorrect response was observed. Thus, the data in a
single observation are a triple 具R, M, W典 of pairwise disjoint sub-
sets of Q. Because the set W is fully deducible from the other two
(W ⫽ Q \ 共R M兲), in the sequel it will be dropped, and the
notation 具R, M典 will be used for representing the observed data.
The set R will be referred to as the partially observed response
pattern, and M as the missing response pattern of a student.
Indicating with P(R, M) the probability of observing the pair
具R, M典 in a randomly sampled student, a rather straightforward
extension of Equation (1) of the BLIM to observed and missing
data is
P(R, M) ⫽
兺
KK
P(R, M
ⱍ
K)
K
, (5)
where P(R, M
|
K) is the conditional probability of observing
具R, M典, given knowledge state K. This equation constitutes the
starting point for both models derived in the sequel. The likelihood
of the models, for a finite sample D of size N has general form
L⬘(⬘
ⱍ
D) ⫽
N!
兿
具R, M典O
F(R, M)!
兿
具R, M典O
P(R, M)
F(R, M)
, (6)
where ⬘ is the vector of the model parameters, O ⫽
兵具R, M典 2
Q
⫻2
Q
: R M ⫽ O其 is the collection of all theoreti-
cally observable pairs 具R, M典, and F(R, M) is the observed fre-
quency of the pair 具R, M典 in the data set.
Ignorable Missingness: The IMBLIM Model
A typical approach for postulating a probability model for
missing data is to separate a “complete data probability model”
from a “decimation mechanism.” The former specifies the proba-
bilities of the complete data, whereas the latter specifies the
probability of a pattern of missingness, given a particular pattern of
complete data. The process that generates an observable pair
具R, M典 is then regarded as a process where a complete response
pattern R
*
Q is generated using the complete data probability
model, and then a “decimation” of the complete response pattern
R
ⴱ
takes place, according to the decimation mechanism. The result
is the observable pair 具R, M典, where M Q is a collection of
problems selected by the decimation mechanism, and R R
ⴱ
\M.
It seems reasonable to assume that the process outlined above is
governed by a probability P(R
ⴱ
, M
|
K) of obtaining the complete
response pattern R
ⴱ
and the pattern M of missingness, given
knowledge state K K. Additionally—and this is the key assump-
tion—it is postulated that the “decimation mechanism” is indepen-
dent of the process that generates the complete data. More pre-
cisely the following assumptions are posited:
[A2] The conditional probability P(R
ⴱ
, M
|
K) decomposes as
P(R
*
, M
ⱍ
K) ⫽ P(R
*
ⱍ
K)P(M),
where P(R
ⴱ
|
K) is the conditional probability of R
ⴱ
given K, obtained
by an application of Equation (3) of the BLIM, and P(M) is the
probability of a missing answer for the items in M.
[A3] The probability P(M) of the missing response pattern M Q is
functionally independent of the error parameters
q
and
q
of the
items.
By the two assumptions [A2] and [A3] just introduced, the following
model is obtained, for the joint probabilities of R
ⴱ
and M:
P(R
*
, M) ⫽ P(M)
兺
KK
P(R
*
ⱍ
K)
K
.
This model is suitable for making predictions about the unob-
served complete pairs 具R
*
, M典. However, one will be especially
interested in making predictions about the observed, though in-
complete, pairs 具R, M典. To this aim, a marginal probability for the
single incomplete pair 具R, M典 can be obtained by considering the
collection
具R, M典
*
⫽ {R C : C M}
of all possible complete response patterns that are compatible with
the incomplete pair 具R, M典. This collection contains all possible
ways of “completing” the partial response pattern R, by taking
unions of this last with subsets of the missing response pattern M.
Then, the probability of 具R, M典 corresponds to the probability of
observing any of the possible “completions” of R that can be found
in the collection 具R, M典
*
:
P(R, M) ⫽
兺
R
*
具R, M典
*
P(R
*
, M)
⫽P(M)
兺
KK
兺
R
*
具R, M典
*
P(R
*
ⱍ
K)
K
.
Now, defining the marginal probability
P(具R, M典
*
ⱍ
K) ⫽
兺
R
*
具R, M典
*
P(R
*
ⱍ
K), (7)
it can be shown that
P(具R, M典
*
ⱍ
K
兲
⫽
关
兿
qK \
共
RM
兲

q
兴关
兿
q
共
KR
兲
\ M
共
1 ⫺
q
兲
兴
⫻
关
兿
qR\
共
KM
兲
q
兴关
兿
qQ\
共
RMK
兲
共
1 ⫺
q
兲
兴
(8)
whenever R M ⫽ O, and P共具R, M典
*
ⱍ
K兲 ⫽ 0 otherwise.
To give an example of the equality between the right-hand
term of Equation (7) and that in Equation (8), consider the set
Q {a, b, c, d}. Suppose that the complete response pattern
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MODELING MISSING DATA IN KNOWLEDGE SPACE THEORY