My question centers around local independence, which is one assumption of
Item Response Theory. Local independence states that the responses to items
within the survey are statistically independent when holding constant the
underlying dimension. Specifically, I was wondering if the network nodes
created for one of the dimensions of a multidimensional survey could be
related to other nodes in other dimensions. Therefore, responses to items
about written/oral communication could be related to items about leadership,
etc.
For those unfamiliar with the Item Response Theory model it is
essentially a naive Bayes model with a single (latent) normal variable
and conditionally independent binary observations (although there are
extensions to multinomial observations). The alternative model you
are proposing has multiple latent variables or traits which are
arranged in a graphical model. There are two approaches I've seen
explored in the literature. One uses multivariate normal latent
variables (this is essentially multivariate IRT) the other uses
multinomial latent traits (this is essenntialy the Bayes nets model
Bob Mislevy and I have been working with).
There is a big independence assumption which is impossible to avoid:
all of your survey/test items are independent given the latent
variables. This tends not to be a practical limitation because
generally you can engineer the survey or test so that two items which
are not conditionally independent do not appear on the same form.
The problem you allude to in your message is not really an
independence issue but rather one of identifiability. Given that the
variables you are measuring are purely latent, your effective
definition of those variables will depend on what items are in your
survey and how you load those items onto the various factors. From a
Bayesian prespective, the issue is that you may need strong priors
about the relative importance of the various latent traits. I don't
think that this is a practical problem in this case as the pychometric
properties of the items are pretty well known.
Question 1: If one was to create a network of a multidimensional survey, do
the network nodes (and particular survey items associated with the nodes)
need to be locally independent (i.e., need to be unrelated to nodes and
items of other dimensions)?
The survey nodes should probably be independent given the latent
traits you are measuring. On the other hand, you could explicitly
model the dependence if you wanted, it would just be more work.
The higher the degree of dependence you have among the
latent nodes, the higher the computational cost of the model.
Similarly, if you have a lot of items which load onto multiple factors
this will increase the complexity. When you "moralize" the graph with
all of the items, even if the latent variables are a priori
independent, they will become dependent after measuring the item which
loads on multiple factors. [This is producing serious computational
problems in our current work.]
I'm a little bit confused about your use of the word "dimension" here,
because each latent node in your network is effectively a dimension.
If you are talking about dimensions from the original factor analysis,
aren't they already rotated so as to be independent. Therefore, even
if you rexepressed each normal factor as several multinomial variables
there would be no reason to create a priori dependencies among them
(unless you had a separate analysis of the domain). There is no
theoretical issue here, the problem is one of interpretation.
Question 2: Would anyone know of quality sources which may help someone
create adaptive multidimensional surveys using Bayesian networks?
The reference you want is:
\refrence {Almond, R.G., \& Mislevy, R.J. [1999].} ``Graphical
models and computerized adaptive testing.'' {\it Applied Psychological
Measurement.\/} {\bf 23\/} 223--238.
These might also be helpful:
\refrence {Mislevy, R.J, Almond, R.G., Yan, D. and Steinberg,
L.S. [1999]} ``Bayes Nets in Educational Assessment: Where the
numbers come from.'' In {\it Uncertainty in Artificial Intelligence
'99\/} Laskey, K.B. and Prade, H. (eds.) Morgan-Kaufmann, 437--446.
\refrence {Almond, R.G., Herskovits, E., Mislevy, R.J. and Steinberg,
L.S. [1999]} ``Transfer of Information between System and Evidence
Models.'' {\it Artificial Intelligence and Statistics 99\/}
Heckerman and Whittaker (eds.), 181--186.
\refrence {Mislevy, RJ [1994]} ``Evidence and Inference in Educational
Assessment.'' {\it Psychometrika,\/} {\bf 59\/}, 439--483.
I hope this gets you started.
--Russell Almond
Educational Testing Service
Research Statistics Group, 15-T
Princeton, NJ 08541
Phone: 609-734-1557 FAX: 609-734-5420
Email: [EMAIL PROTECTED], [EMAIL PROTECTED]
http://www.stat.washington.edu/bayes/almond/almond.html
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