SOME RECENT RESULTS IN MODEL SELECTION - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 38

problems. Model selection is to choose a statistical model that is based on
a subset of variables, such that the chosen model has optimal predictive
power; while variable selection is to determine the subset of variables that
have predictive eect. Conceptually, the model selection may take a subset
of the variables from the variable selection to create a model.
It is interesting to read the model-free variable selection approach in Li,
et al. 30 . They adopt the framework that is the same as the one for \central
subspace." Their proposed procedure is like back elimination, where step-
wise statistical hypothesis testings are used to guide the variable selection.
We believe that more results along this line will come out in a near future.
Some potential research problems include: what is the statistical properties
of these methods?
5.6. Beyond Model Selection
The model selection considered here is just one stage of statistical inference.
Other researchers have considered the statistical properties of the outcomes
from these model selection methods. As an example, Shen et al. 44;45 con-
sider the bias of model selection, and suggest methods to correct it. Efron 13
studied the relation between the outcomes and prediction power via their
covariance structure. These works require mathematical formulations that
are very dierent from the one that is considered here. We choose not to
explore further in this direction.
5.7. Beyond Ordinary Linear Regression
In the contemporary statistics, ordinary linear regression is a classical how-
ever small fraction. Many other models have been created and studied in
statistical practice. We notice some recent works on model selection in lon-
gitudinal data analysis 3;17 and survival analysis 16 .
5.8. Bayesian Approach
Due to the diculty in solving the model selection problem | as mentioned
earlier, they are NP-hard in general | researchers have explored random
sampling approaches. Some computational experiments are described in
[21], and later on, more thorough Bayesian approaches are developed in [7,
20]. An interesting Monte Carlo strategy is introduced in [31] too.
Although interesting results are obtained in experiments, a major prob-
lem associating with this approach is the lack of theoretical justication.