# Detecting 'causality' in Likert-time series data

### [Note] I've decided to re-write my question for the sake of brevity. The original question can be found below.

Suppose a number of individuals fill in a questionnaire at a multiple number of time points. In other words, for each person $i \in \{1, \ldots, m\}$ and for each time point $t \in \{1,\ldots,T\}$, we have responses $\mathbf{y}_i^{(t)} = (y_{i1}^{(t)},\ldots,y_{ip}^{(t)})^{T}$ where $p$ is the number of items on the questionnaire.

I am interested in finding out which item responses cause/predict other item responses, preferably using network models. I need a model that can do this under the following conditions:

• all responses $y_{ik}$ are in some finite range (they're Likert responses)
• the data may not be stationary
• I would like to account for feedback loops. That is, at time 1 a response to item A might influence an response to item B at time 2, which might influence item A at time 3.
• An estimation of direction is required, i.e. I'm not only interested in correlation, but also direction.

### Original formulation of my question:

I'm trying to find a suitable model for a problem I've been asked to investigate.

Suppose a number of individuals fill in a questionnaire at a multiple number of time points. In other words, for each person $i \in \{1, \ldots, m\}$ and for each time point $t \in \{1,\ldots,T\}$, we have responses $\mathbf{y}_i^{(t)} = (y_{i1}^{(t)},\ldots,y_{ip}^{(t)})^{T}$ where $p$ is the number of items on the questionnaire.

I am interested in providing these persons with feedback regarding their responses, e.g. the course of outcomes to specific items, and relations between item-responses. In particular, I'm interested in estimating and visualizing a network model which is fitted to this data. The hope is that this network model can indicate which item-responses are predictive for other item-responses.

Granger-causality came to mind but it does not seem the best method here. For one, the data is categorical (Likert-scale) and not necessarily stationary. Also, normality usually cannot be assumed.

I am not interested in predicting anything, the most important feature is that one get's a sense of "what item-responses predict' other item-responses?" What would be a proper model for this setting? Thanks in advance.

Exploratory factor analysis can provide a tool appropriate for your task. It can indicate a) how variables group (co-vary) together, and 2) how individuals compare given their answers.
If the time dimension is not of substantive importance, you can just get the average for each individual on each item from the multiple survey waves. If the time dimension is actually important, you can redefine your variables as the difference in the individual scores on each item between T and t1`. That would give you which variables tend to change together over time. But in any case do not just merge all data before running the model since the observations on each individual over time are obviously non-independent. An alternative which is considered more appropriate for binary variables is item-response models.