Dear gR's

(This is perhaps more g than gR, but anyway...)

An new (to me at least) and interesting (ditto) model selection algorithm, well-suited high-dimensional (sparse) problems, implemented in MIM 3.2.0.1. It is based on tests of conditional independence in low-dimensional margins. See the text below (from the online documentation) or see www.hypergraph.dk to try it out.

Best regards
David


> -------------------------------------------------------------------------------------------------------
> CISelect 
> The syntax of the command is
> CISelect <k> <options>
> where k is an integer 0,1,... and options are one or more letters from the set: O, S, E, M, Q, H, V, W, A, R . 
> The command invokes a model selection algorithm that is based on conditional independence tests in low-dimensional margins. The approach is particularly useful for high-dimensional problems since it is efficient (when k is small). The model selected should be regarded as preliminary. 
> The algorithm tests each pair of variables for conditional independence given each k-combination of the remaining variables, that is to say, each combination of k of the remaining variables. If one or more of these tests are non-significant, the edge is omitted from the selected model: otherwise it is included. For example, when k=0, the algorithm tests all pairs of variables for marginal independence. Then the edges present in the selected graph represent marginal dependences, and the missing edges marginal independences. (When all variables are continuous, such graphs are called covariance graphs: see §7.4 in Introduction to Graphical Modelling <introduc.htm> ). 
> By default k=1. In this case, for each variable pair, tests of the conditional independence of these variables given each of the remaining variables are performed, until one such test is non-significant, in which case the edge is omitted from the graph.
> The rationale for the algorithm is as follows. Testing each variable pair for conditional independence given all the remaining variables has two disadvantages: it is computationally intensive (particularly with high-dimensional discrete models) and has suboptimal power when the variables are in fact separated by a smaller variable set. Instead, the algorithm is based on testing hypotheses of the form: u is conditionally independent of v given s, where u and v are variables and s is a small set of variables. This is equivalent to the hypothesis that in the true graph, u and v are separated by s. If this is not true, then it might be because u and v are in fact adjacent, but it might also be that they are non-adjacent but not separated by s. So the edge [u,v] should not necessarily be included. But if the hypothesis is true, [u,v] should certainly be omitted. So it makes sense to omit an edge when an associated test result is non-significant, but not to include an edge when an associated test result is significant. 
> The algorithm controls the edgewise type I error rate (in other words, the false edge inclusion rate), at least for some edges. More precisely, if the true graph u and v are non-adjacent and separated by a set with k or fewer variables, the probability that the selected model includes the edge [u,v] is less than the critical level used in the individual tests. If u and v are non-adjacent but are not separated by a set with k or fewer variables, the edge [u,v] will tend to be included. The edgewise type II error rate (in other words, the false edge omission rate) depends, as usual, on the sample size and on the strength of the conditional association. The algorithm is most reliable when the true graph has the property that all non-adjacent variables have a minimal separating set with k or fewer vertices. (For example, the linkage graphs described in §2.2.5 of Introduction to Graphical Modelling>  <introduc.htm> , and the gene regulation networks described in Wille et al. (2004) may well have this property for small k). The model selected should be regarded as preliminary: it is advisable to check it using the closely related command CICheck <CICheck.htm> . 
> The option H examines also conditional independences involving all lower-order combinations, i.e, all 0-,1-,2-...k-combinations.
> The option O lists the test results in text form, and the option R does the same in R code. Otherwise no output is given.
> Option S indicates that F-tests should be used when available, rather than the (default) deviance statistics. Options E,M,Q specify methods of computation for exact tests: for more details see the TestDelete <TestDelete.htm>  command. Since these are computationally intensive, they are not usually useful in the present context. Option W specifies that ordinal-sensitive tests are to be used when available: for details see the Ordinal <Ordinal.htm>  command. The option V selects a variance-heterogeneous graphical model, and the marginal models underlying the conditional independence tests are also variance-heterogeneous: the default is to assume homogeneity throughout. 
> The CISelect command skips edges fixed using the Fix <Fix.htm>  (if both edge endpoints are fixed) and FixEdges <FixEdges.htm>  commands. The command is available in both undirected and block mode. In block mode, only conditioning sets in prior or concurrent blocks are considered.
> The critical level used by the tests may be specified using the CritLevel <CritLevel.htm>  command. Per default, as soon as a non-significant test result is found, the algorithm proceeds to the next edge, but if the A option is used, all tests are performed. 
> The algorithm generalizes the algorithm proposed (for k=1) by Wille et al (2004), Sparse graphical gaussian modeling for genetic regulatory network inference, Genome Biology. See also Kreiner, S. (1987), Analysis of Multidimensional Contingency Tables by Exact Conditional Tests: Techniques and Strategies, Scand J Statist, 14, 97-112. This paper outlines a related diagnostic method and gives a helpful discussion in the contingency table context. See also the CICheck <CICheck.htm>  command.
> 
> CICheck 
> This command examines whether the current model can be simplified or extended, based on conditional independence tests in appropriate margins. 
> The syntax is
> CICheck <options>
> where options are one or more letters from the set: O, S, E, M, Q, R, F, W, A, U . 
> By default, the algorithm operates backwards mode: that is to say, it examines whether edges present in the current model can be removed, on the basis of tests for conditional independence of the respective variables given appropriate conditioning sets. For an edge [u,v], the conditioning sets are given as the minimal u,v-separating sets, that is, the smallest sets of vertices that separate u and v, in the graph formed by omitting the edge [u,v] from the current graph. When the option F is specified, the algorithm operates in forwards mode: that is to say, it examines whether missing edges can be included, on the basis of tests for conditional independence of the respective variables given their minimal separating sets in the current graph. 
> The option O lists the test results in text form, and the option R does the same in R code. Otherwise no output is given.
> If the U option is used, the current model is updated after the command, otherwise it is not changed. In backwards mode, edges are removed if one or more of the corresponding tests are non-significant. In forwards mode, edges are added if all the corresponding tests are significant. The critical level used by the tests may be specified using the CritLevel <CritLevel.htm>  command. Per default, as soon as a non-significant test result is found, the algorithm proceeds to the next edge, but if the A option is used, all tests are performed. > 
> Note that the removal or addition of edges will influence which sets are the minimal separating sets for other edges. Also, note that in forwards mode, if a test rejects the conditional independence of two non-adjacent variables u and v given a set s, this may either be because u and v are in fact adjacent, or because s does not in fact separate them. So the edge [u,v] should not necessarily be added. Instead of using the U option, a more defensible approach is to add or remove edges thoughtfully one at a time: this is particularly true in forwards mode. 
> Options S uses F-tests, when available, rather than the (default) deviance statistics. Options E,M,Q specify methods of computation for exact tests: for details see the TestDelete <TestDelete.htm>  command. Option W specifies that ordinal-sensitive tests are to be used when available: for details see the Ordinal <Ordinal.htm>  command. If the current model is variance-homogeneous, the tests assume homogeneity, otherwise heterogeneity. 
> The CICheck command skips edges fixed using the Fix <Fix.htm>  (if both edge endpoints are fixed) and FixEdges <FixEdges.htm>  commands. It is available in both undirected and block mode. In block mode, the undirected procedure is carried out for each of the component models in turn. 
> See Kreiner, S. (1987), Analysis of Multidimensional Contingency Tables by Exact Conditional Tests: Techniques and Strategies, Scand J Statist, 14, 97-112 for a related diagnostic method and helpful discussion in the contingency table context. See also the CISelect <CISelect.htm>  command. 
> 
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