Graphical Modelling is a approach to data analysis based on statistical models that can be displayed as graphs.
In these graphs, nodes represent variables, and edges drawn between nodes represent conditional dependences. That is to say, a line or arrow is drawn between two nodes unless the two variables are conditionally independent given some or all of the remaining variables.
In this way, the graphs supply precise representations of the interrelationships between the variables in the model. Being able to work directly with the graphs promotes an understanding of the dependence structure of the data.

Types of Graphs
MIM supports four types of graph:

Undirected Graphs

These have undirected edges which are drawn as lines, not arrows. These graphs are suitable for cross-sectional data, or situations in which there is no causal or other orderings between the variables. For example
mathmark.jpg.jpg
represents the dependence structure between five continuous variables (mathematics marks data).

Directed Graphs

These graphs use directed edges, i.e. drawn as arrows. These express direction of influence or, sometimes, causal direction.
The following graph is taken from a clinical study on depression. After initial classification of the severity of their depression, patients were randomized to one of two treatments. Their response was recorded after one, two and three weeks.
depress.jpg
After one week, treatment has no effect on the response, but does after two and three weeks. The edges between treatment and response represent causal influence.
The variables here are discrete. By convention, discrete variables are drawn as filled circles (dots), and continuous variables are drawn as hollow circles.

Chain Graphs

These combine the undirected and directed graphs. Prior to analysis the variables are grouped into blocks. Variables in the same block may be linked by lines (not arrows), whereas variables in different blocks are linked by arrows (not lines).
An example of a chain graph is:
ksl.gif
There are four blocks. Variables within a block are connected by lines, those between blocks by arrows.
From this graph we can see, for example, that the occurrence of hypertension and FEV (forced expiration volume - a measure of lung function) are influenced by the same factors, but that for given values of these factors, they are determined independently.

Factor Graphs

Factor graphs (sometimes called interaction graphs) display the interaction terms in a model, rather than its conditional independence structure. These graphs are useful when working with hierarchical models. For example, the factor graph
breast.jpg
displays a model with four discrete variables (A, B, C and G) and six continuous ones (U, V, W, X, Y and Z). The model contains a discrete term (A.B.G), two linear terms (A.B.V and A.C.G.Z), and three quadratic terms (W.Y, B.U.Y and X.Z).


File translated from TEX by TTH, version 3.81.
On 19 Nov 2008, 14:53.