1. A Robust Procedure for Gaussian Graphical Model Search from Microarray Data with p Larger Than n
Robert Castelo and Alberto Roverato
In JMLR 7, December 2006
Presented by Kuan-ming Lin
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Gaussian Graphical Model Search From Microarray

2. Gaussian Graphical Model Search From Microarray
Outline
Motivation and challenge when p (#genes) > n (#microarrays)
Theory on undirected Gaussian graphic model (aka Markov random field, Markov network)
Full-order graph vs. partial-order (q-partial) graph approximations
qp-Procedure algorithm to build partial graphs
Simulations and experiment
Discussions
Can you please give a little explanations (words) about p>n? I believe most in our group are not clear about the relationship between gene & microarray. Also, what’s q, n?
11/30/2018
Gaussian Graphical Model Search From Microarray

3. Motivation and Challenge
A microarray could be modeled as a p-variate random variable XV ~ PV, where V={1,…, p} and PV is some distribution over the p genes defined by biological function.
A p n microarray table could be seen as n draws of XV.
We can construct a graph G(V,E) to describe gene interactions by computing cor(Xi,Xj|XQ), the conditional correlation of the ith and jth genes given Q, for all (Xi,Xj)E.
If Q=V\{i,j}, cor(Xi,Xj|XQ) can only be obtained from joint probabilities distribution PV, which is unknown and hard to estimate from n samples when p>n.
This paper proposes q-order partial correlation cor(Xi,Xj|XQ), |Q| ≤ q < n, as an approximation to the full-order one.
Could you please give a simple example, or in equation, relationship of microarray & p? Maybe equation is easier to understand. Then, you can express “joint and fully –marginal prob distributions” in mathematical form.
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Gaussian Graphical Model Search From Microarray

4. Gaussian Graphical Model and q-partial Graph
Gaussian graphical model: assume XV ~ PV = Gauss(μV, ΣVV)
If p>n, MLE of (μV, ΣVV) does not exist.[1]
q-partial graph G(q) is defined as G(q) = (V,E(q)), where (Xi,Xj)Ē(q) iff cor(Xi,Xj|XQ)=0 for some |Q| ≤ q.
q = 0: Ē(q) are zero entries of covariance matrix ΣVV.
q = p-2: Ē(q) are zero entries of concentration matrix K=(ΣVV )-1.[2]
In covariance graph, edges present whenever there exist indirect associations. Thus, covariance graph
is usually much denser than concentration graph
might be inadequate for describing gene networks
What’s n in terms of Xv & Pv?
So, from this slide, Pv=Gasussian(\mu_v, \Sigma_vv)?
I guess most ppl in our group are not familiar with Q-partial graph, could you please give a definition? Or maybe add 1 figure to show that? And I think maybe easy for you to explain.
So, seems V are nodes in the q-partial graph?
G^(q) is q-partial graph?
[1] Theorem 5.1 in Lauritzen (1996): “Graphical models.” [2] Dykstra (1970): “Establishing the positive definiteness of the sample covariance matrix.” Ann. Math. Statist. 41(6),pp. 2153–4.
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Gaussian Graphical Model Search From Microarray

5. Graph Theory Tool: Outer Connectivity
[Definition 3: outer connectivity]
Here S is a collection of S  V that separates i from j.
Examples
I don’t think someone will read the second graph here. If it’s the core of the method, then, you might explain in your word, maybe with the help of figures. If it’s not the key point, then just kick it off
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Gaussian Graphical Model Search From Microarray

6. Use Outer Connectivity to Prove Properties of q-partial Graphs
Assume G, the concentration graph, is a “perfect map,” which means
Markov: cor(XI,XJ|XU)=0 whenever U separates I and J.
faithful: cor(XI,XJ|XU)=0 implies U separates I and J.
Then this paper shows:
[preposition 1]
Preposition 2
[corollary 3 (hierarchy)]
If we define max d(Ē|G)=a,
Preposition 2 tells us that G = G(p-2) = G(p-1) = … = G(a).
Corollary 3 implies a hierarchy G = G(p-2) = G(p-1) = … = G(a)  G(a-1)  …  G(0)  complete graph.
Here at least less words. I don’t understand since I am totally lost in the last slide
If you can explain what those propositions mean, that will be great.
The 5th bullet is very good, it helps me understanding some.
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Gaussian Graphical Model Search From Microarray

7. Sufficient Conditions and Relation to Graph Sparseness
[Theorem 4]
Corollary 5
[Theorem 6: sparseness]
Corollary 5 gives us a way to ensure G = G(q) given G unknown but G(q) known.
Theorem 6 implies that to yield sparser G(q), q has to be increased.
This one is very good.
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Gaussian Graphical Model Search From Microarray

8. Tests for conditional independence in qp-Procedure: Non-Rejection Rate
Here the expectation is taken over all possible Q’s, and T is the indicator variable of whether the null hypothesis H0: ij.Q = cor(Xi,Xj|XQ) = 0 (against HA: cor(Xi,Xj|XQ)  0) is not rejected via the following regression coefficient test.
The usual t-test for zero:
Here A=Q{i,j}.
This regression coefficient test is optimal in the sense that it is uniformly most powerful unbiased (UMPU).[1]
The paragraph is long
Cox and Wermuth (1996): “Multivariate dependencies: Models, analysis and interpretation.” [1] Page 397 in S.L. Lehmann (1986): “Testing statistical hypotheses, 2nd ed.”
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Gaussian Graphical Model Search From Microarray

9. The qp-Procedure Algorithm
[5.2 on p.2632]
β* (type-II error) indicates the power of the tests. Here we see the reason for choosing the t-test on regression coefficients: it is optimized for power.
In practice, not every Q is considered; Monte Carlo method is applied.
Very good
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Gaussian Graphical Model Search From Microarray

10. Explain qp-Procedure via Toy Example
p=164 variables, 20 non-zero blocks in concentration matrix K
each 1010
each overlaps 44 with neighboring blocks
1206 (9%) present edges
G(3) is the complete graph
G(20) = G
Use this K to generate n=40 samples
In qp-procedure, 500 Q’s are sampled and applied t-tests for each of the 13,366 (i,j) pairs.
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Gaussian Graphical Model Search From Microarray

11. Gaussian Graphical Model Search From Microarray
Toy Example continued
Figure 3 shows that larger q leads to higher separation of non-rejection rates between present and absent edges.
Figure 4 gives the size of the largest cliques; every circle below the dotted line (sample size) corresponds to a model whose dimension is small enough for standard techniques like MLE.
q=20, β*=.975: only 34 (2.8%) of 1206 edges are wrongly removed.
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Gaussian Graphical Model Search From Microarray

12. Experiment on Simulated Data: p=150, n=20, 150(G1) or 50(G2)
d(E1|G1)=5, |E1|=375 (3.4%)
d(E2|G2)=20, |E2|=1499 (13.4%)
Figure 6 shows that for G1, sparse graph can be obtained by increasing q.
Figure 9 shows that for G2, sparse graph cannot be obtained for various values of n and q.
The results imply that the intrinsic graph (G2) may be too dense for q-partial graph heuristic to work.
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Gaussian Graphical Model Search From Microarray

13. Experiment on Breast Tumor Microarrays: p=150, n=49
The 150 genes are associated with estrogen receptor pathway
q=20, β*=0.975, |E|=7240 (64.8%), largest clique size=24
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Gaussian Graphical Model Search From Microarray

14. Gaussian Graphical Model Search From Microarray
Discussions
This paper has two main contributions (both unrelated to actual microarray analysis):
Theory related to q-partial graphs
qp-procedure
qp-procedure is robust w.r.t. faithfulness assumption.
Violations for a few conditional distributions won’t hurt much because statistical tests are averaged from many Q’s.
But when p becomes large, as usual cases in microarrays:
For efficiency, the procedure has to perform fewer tests for each pair. The robustness therefore cannot be guaranteed.
qp-procedure can be generalized to non-Gaussian.
Different assumptions on microarray distributions may apply.
11/30/2018
Gaussian Graphical Model Search From Microarray