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We estimated a network via the Ising model procedure. The network contains 11 variables, and therefore 55 pairwise associations (these are called edges).

We estimated this network in 6 different samples, leading to six networks.

Our question is now whether a given edge, for instance the edge between variable 1 and 2 (one of the 55 pairwise associations in each network), varies substantially across the six networks.

For each edge, we have 6 values (the edge weight of this edge in each network), and we can compute a variance over these 6 values. We now want to test whether this variance is substantial (i.e. significant from zero).

We can also derive confidence intervals around each edge, if that is helpful. So for the edge between variable 1 and 2, we could have the values (first: edge weight; then confidence interval):

network 1: 4 (CI 3-5)
network 2: 5 (CI 4-6)
network 3: 2 (CI 1-3)
network 4: 3 (CI 2-4)
network 5: 4 (CI 3-5)
network 6: 5 (CI 4-6)

How do we test whether the variance of this particular edge is significantly different from 0? (or, if that is not possible, "substantial" [I know this is open to interpretation]).

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    $\begingroup$ Assuming no measurement error, if the observed variance is nonzero, then it is significantly above zero (at any possible level of significance), because it definitively establishes the variance is nonzero. If there is a possibility of measurement error, then what quantitative information can you provide about its variance? This is such a strange situation that it makes one wonder whether you are asking the question you need to. Could you clarify what you really mean by "significant variability across the ... networks in [an] edge"? $\endgroup$
    – whuber ♦
    Commented Jun 24, 2015 at 16:04
  • $\begingroup$ Thank you. In a minor analysis of a paper, we want to establish whether the 6 networks estimated via Ising Models differ (same variables, but different samples). This is extremely demanding and worth at least a PhD project, so not feasible for this paper. So we look for simpler ways. One idea we had was determine the variability of each edge across the 6 networks, and if this variability is substantial, that provides some information. I could bootstrap confidence interval around each edge in each network. That would provide information about error. Would that help? $\endgroup$
    – Torvon
    Commented Jun 24, 2015 at 20:57
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    $\begingroup$ That's quite a different description of the problem! Please edit your post so that readers understand what you're looking for. $\endgroup$
    – whuber ♦
    Commented Jun 24, 2015 at 20:59

1 Answer 1

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Assuming you have large set of data , the sample variance chi square distribution of data will tend to be normal , then you can use t-test to reject the null hypothesis of your variance to be zero

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    $\begingroup$ This recommendation is of doubtful validity. Exactly how would you apply a t-test? Why, indeed, could it be applicable at all to a null hypothesis of zero variance? In such a case the sample variance does not have either a chi-square distribution or a normal distribution: it's constantly zero. $\endgroup$
    – whuber ♦
    Commented Jun 24, 2015 at 16:07
  • $\begingroup$ I thought of this as well originally, but I also believe that the assumptions would not be fulfilled. $\endgroup$
    – Torvon
    Commented Jun 24, 2015 at 20:57
  • $\begingroup$ t-test is used to test whe ~ther the mean of RV differs significantly from some value hence considering sample variance as RV ~ Normal within large sample , i thought he can use it $\endgroup$
    – Mohamed Baddar
    Commented Jun 26, 2015 at 15:18

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