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On page 378 of Raudenbush & Bryk's (2002) book, they recognize 3 possible types of correlations among two observations given two (fully or partially) crossed random effects (i.e., neighborhood & school):

(1) two observations coming from the same school and same neighborhood.

(2) two observations coming from same neighborhood but different schools.

(3) two observations coming from same school but different neighborhoods.

Question: I wonder how to justify which of the 3 types of correlations among two observations are possible in the following toy, neighborhood-school dataset given its structure (below the data)?

m = "
neighbor school
1        2
1        2
2        1
2        1
2        1
3        2
3        2
4        1
"
### DATA STRUCTURE in `R` code:
dat <- read.table(text=m,h=T)
xtabs(~neighbor+school,data = dat)

       school
neighbor 1 2
       1 0 2
       2 3 0
       3 0 2
       4 1 0
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  • $\begingroup$ You can't tell from the structure of the data what correlations will be present. This is determined by the experiment/study design. $\endgroup$
    – Robert Long
    Commented Jul 18, 2021 at 16:31
  • $\begingroup$ Also I doubt that the example of schools and neighbourhoods is fully crossed. To be fully crossed, every school has to be in every neighbourhood and every neighbourhood has to contain every school. While it's possible for a school to be in more than 1 neighbourhood, it seems impossible that they can be fully crossed. $\endgroup$
    – Robert Long
    Commented Jul 18, 2021 at 16:37
  • $\begingroup$ @RobertLong, but Robert, can't we, for example, see (from xtab() output) that there are rows that belong to the (1) same school and same neighborhood, (2) same neighborhood but different schools, and (3) same school but different neighborhoods. It seems to me that fitting such a data structure and encoding crossing in data structure enables accounting for these 3 possible correlations, if not why? $\endgroup$
    – rnorouzian
    Commented Jul 18, 2021 at 16:42
  • $\begingroup$ Sure, if the experimental design is appropriate, then these correlations might exist. But the data structure doesn't tell you that. For example there could be zero variance within a particular grouping variable, so that any correlation between obervations in that and anther grouping variable will be zero. $\endgroup$
    – Robert Long
    Commented Jul 18, 2021 at 18:28
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    $\begingroup$ @RobertLong, … experiment innumerable times then if we examine observations with in the same school at any two iterations, then they will be likely positively correlated). Is my understanding of the meaning of two observations being correlated in the context of clustered data correct? $\endgroup$
    – rnorouzian
    Commented Jul 18, 2021 at 20:18

1 Answer 1

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I'm not really sure that I understand the question:

I wonder how to justify which of the 3 types of correlations among two observations are possible in the following toy, neighborhood-school dataset given its structure (below the data)?

I don't see how you can "justify" any correlation at all. Provided that the variance components can be estimated (ie the model converges to a non-singular result) then you simply just calculate the 3 intra-class correlations (ICCs) that you mentioned.

As for the comments on the OP regarding the interpretation of these correlations, you said:

And, in fact, how come only two observations can be correlated?(I think we are talking about imaginary correlations that come about if we repeat our exact experiment innumerable times then if we examine observations with in the same school at any two iterations, then they will be likely positively correlated). Is my understanding of the meaning of two observations being correlated in the context of clustered data correct?

We're not talking about imaginary correlations, we are talking about the correlation between 2 observations chosen at random, one from each group, the point being that observations in one cluster are likely to be similar to other observations in the same cluster, rather than to observations from other clusters. Take a simple 2 level model with one grouping variable and no random slopes; so there will be only 1 ICC, and this is the correlation between observations within the same cluster. For nested models with more than one level and for models with crossed random effects there will be several ICCs.

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  • $\begingroup$ Thank you, Robert! How come only two randomly picked observations (say 3.23, and 4.11) can be correlated? I think my understanding of what we mean by correlation among two observations from the same cluster is accurate based on this answer by Ben Bolker as well as the comments below that answer. $\endgroup$
    – rnorouzian
    Commented Jul 19, 2021 at 19:23
  • $\begingroup$ I think it's just a turn of phrase. I don't think there's a misunderstanding. It's not "only" two observations, it's "any" two observations. $\endgroup$
    – Robert Long
    Commented Jul 19, 2021 at 19:46
  • $\begingroup$ Can you possible elaborate a bit more? "any" two observations are still two and cor(3.23, 4.11) is not possible. To view this from another perspective, if the data points were summary statistics (estimates of means, effect sizes etc.), then it would make perfect sense to speak of correlation among any two observations (i.e., in that case, it would have meant that the raw data that were used to compute those summary statistics, effect sizes etc. were correlated). But for raw data, using the phrase "any two observations from the same cluster are literally correlated" seems very confusing. $\endgroup$
    – rnorouzian
    Commented Jul 19, 2021 at 19:57
  • $\begingroup$ I'm still not sure what you mean. First you objected to "only" two, then you objected to "any" two, and now you seem to object to just "two". Well, the usual way to define correlation is a standardisation of covariance, where we choose pairs of observation - one from each variable, and we compute the expectation $\mathbb{E}(X_i - \bar{X})(Y_i - \bar{Y})$. So that's why there are two, and it doesn't matter which two. The result simply tells us about the linear association between the variables. $\endgroup$
    – Robert Long
    Commented Jul 20, 2021 at 9:00
  • $\begingroup$ Robert, maybe OP is saying that when you recourse to "expectation", then, you're referring to what s/he initially suggested which is "repeating" the experiment many times and then inspecting the correlation between "any 2 sets of observations [Xij, Xkj]" (not just two observations) from the same cluster from any two iterations, and discovering that those two sets of observations are positively correlated. So, if you disagree, you may want to clarify the meaning of E(Xi - Xbar)(Yi-Ybar) as it relates to the clustered data in OP's question. $\endgroup$
    – Reza
    Commented Jul 20, 2021 at 11:18

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