# A statistical model for a sample of independent networks

Based on the lack of responses to my previous network question, perhaps this is not quite the place to ask this question, but I'll give it a try.

I am planning a series of studies that involve small groups of people. In a typical study, the participants will meet in groups of 4 to have a discussion about a particular topic. As a group, the (student) participants will reach a decision about the topic (e.g., whether to support or oppose a proposed change in university policy). The discussion will be videotaped, and we will use transcripts of the discussion to obtain information about who is talking to whom about what topics. In addition, the participants will complete ratings of themselves and each other (e.g., What is your opinion about the proposed change? What do you think is this person's opinion of the proposed change?). I will also have a variety of data about the individual characteristics of all the participants. Thus, each group from each study will give me a 4-node network with directed, valued edges and a variety of attributes about each node. In a typical study, I might have 20 - 30 of these small groups (i.e., small, 4-node networks).

In all the studies, I will be interested in group-level, relationship-level, and individual-level outcomes. I have provided below a sampling of the kinds of questions I would want to ask from these data:

1. Which groups express the most agreement in their discussions? Which groups express the most disagreement? Is group composition related to agreement / disagreement and to the final decision outcome?
2. Which pairs of people like each other during the discussion? Which pairs of people dislike each other? How are the pairwise patterns of liking related to the final decision outcome?
3. Who has the most influence on the course of the discussion? Who has influence over the final decision of each group? Are there any individual difference characteristics that are related to influence over the discussion and / or final decision?

And so on. What I am looking is the proper statistical model to use to investigate samples of networks, rather than individual networks.

I have read a bit about single-network methods (like exponential random graph models, for which there is a nice R package available), but these methods do not seem appropriate, since I am dealing with a set of independent networks, rather than a single network. In addition, a plain linear mixed model does not seem appropriate because I am collecting explicitly relational data. Finally, although a method like the social relations model seems appropriate for this situation at first blush, the social relations model seems to only partition the variance in round-robin ratings into perceiver, target, and relationship sources rather than allowing me to, for example, relate the patterns within a set of networks to overall network outcomes.

Could anyone offer some advice about which statistical model might be appropriate for my situation? Any readings suggestions and / or software recommendations would be greatly appreciated. (FYI, I am a proficient R user, so R package recommendations would be especially appreciated).

Edit:

At the request of Alex Williams, I have posted some example data here in csv format to give everyone a concrete example of the type of data I'm working with. In the sample data, participants, identified by participant_id, are each in 4-person groups, identified by group_id. The participants discuss a proposal and decide as a group whether the support or oppose the proposal. group_decision is the result of the group discussion. The participants also rate their own attitude toward the proposal (pers_att), their perceptions of the attitudes of the other group members (p1_att through p4_att; person_id tracks who p1 through p4 are within each group), and their own enjoyment of the discussion.

In the sample data, I might be interested in the following sorts of questions:

1. Are personal attitudes related to group-level decisions?
2. Are personal attitudes related to personal enjoyment of the group discussion?
3. Were people accurate in their ratings of other peoples' attitudes?
4. Were the ratings of others related to others' enjoyment of the discussion?
5. Does disagreement within a group (quantified, for example, by the difference between the minimum and maximum attitude ratings) relate to group-level decisions?
6. Does disagreement within a group cause people to enjoy the discussion less?
7. Did people who enjoyed the discussion more have more influence over the outcome?

Questions #1 and #3 (and maybe #2) seem like they could be formulated as simple regression problems. Based on your question, it seems like you may have considered this and rejected it already. Here's a stab in the dark anyways...

(Q1) Quantify the amount of agreement/disagreement in each group as the dependent variable ($y$). I assume you have a way of doing this through the questionnaires? Then compute various characteristics about the graph for each group and independent/predictor variables. For example, here are 7 possible independent variables:

• Total number of directed edges ($x_1$)
• Maximum out-degree of nodes ($x_2$)
• Minimum out-degree of nodes ($x_3$)
• Maximum and minimum in-degree of nodes ($x_4,x_5$)
• Maximum and minimum difference in out- and in-degree across nodes ($x_6,x_7$). By this I mean subtract the out-degree of a node from the in-degree.

I'm sure you could think of many better examples that I can, seeing as you know the data better. Then you could use multiple linear regression with LASSO or ridge regularization. Using LASSO would probably be better in this case, because it does variable selection -- you have many possible independent variables to choose from, but you aren't sure which are important or not. LASSO will help you identify the important ones.

(Q3) Measure influence of each node in each group; this is your dependent variable ($y$). In Q1 each group had one dependent variable observation, but now there are four observations per group (one for each node). The important independent variables will be different than in Q1, and will focus more on individual characteristics than group characteristics. The following independent variables may help predict the influence of a node:

• An quantitative estimate of a personality trait for that node ($x_1$)
• The in-degree of that node ($x_2$)
• The out-degree of that node ($x_3$)
• The number of directed edges in the graph that are not to/from that node ($x_4$)

(Q2) I will cover your second question last, as it is potentially more complicated. You could repeat a similar procedure where you quantify the mutual friendliness of all possible pairs of nodes in a group ($y$), and come up with relevant independent variables to predict this. This may be tricky because there will probably be situations where one person likes somebody but is not liked in return.

Hope this helps. Let me know if this is completely off-base.

• Alex, thank you very much for your answer! What I'm concerned about in all these models is the fact that my individual-level measures obtained during the group discussions are dependent on the other group measures, so the independence assumption will be violated if I use most conventional statistical models. Perhaps this isn't a concern for the group-level analysis, but it will be for the individual-level and pairwise analyses. This is why I was considering a linear mixed effects model, some variation of which might be most appropriate in this situation. – Patrick S. Forscher Jan 13 '14 at 18:25
• I see, this potential problem did occur to me while I was writing my answer, but I was unsure whether this was the case with your data. I think you are right that the group-level analysis doesn't suffer from this problem. Maybe you could edit your question (or ask a new one) to provide an explicit individual-level problem you are interested in. Exactly how are the individual-level measures dependent on the other group measures? I'd be happy to have another go at it. – Alex Williams Jan 13 '14 at 19:08
• I just posted an edit to my initial question with a link to some sample data posted on pastebin. Let me know if I can clarify my question any further. – Patrick S. Forscher Jan 14 '14 at 18:19