# “Mixed” Feature Reduction (with Random Grouping Factors)

I have a dataset that contains 6 continuous variables (V1-V6), reported by participants in two conditions (A and B). For each participant, I have two lines of observations of the 6 features, corresponding to each condition (each participant has two observations of the same variable, one for each condition).

I would like to reduce these 6 features into 1 (or more) variable that best represents my data. However, I have the intuition that it is important that the algorithm "knows" about the Condition grouping factor ("random factor" in the mixed-modelling framework).

Are there any feature reduction techniques that can deal with such hierarchically nested data and take into account a grouping structure?

I've looked for "mixed principal component analysis" but didn't find much :(

• Why cannot you do PCA on all the 12 features (v1a, v1b, v2a, v2b...)? – David Dale Nov 9 '17 at 11:36
• I could indeed... Thanks for the suggestion. However, as I have relatively small sample, doesn't it decrease the robustness of the estimates (as it increases the number of reatures and decreases the number of observations?). Moreover, I would like, further, to do a regression with, let's say, the condition factor as predictor (or any other variable) and the principal component. That seems only possible if I keep the grouping structure instead of "spreading" the grouping factor... right? – Dominique Makowski Nov 9 '17 at 11:52
• Isn't that the question of nonhomogeneity (such as Simpson's paradox od similar)? Results of PCA done on each condition separately might drastically differ from results of PCA done on the combined dataset. – ttnphns Nov 9 '17 at 12:05
• Do I understand correctly that you record your target variable twice for each participant (once for each of 2 conditions)? Or it is recorded only once for each respondent? – David Dale Nov 9 '17 at 12:42
• Dear David, indeed each variable is recorder twice (for example, the emotional arousal in two conditions, negative and neutral)... – Dominique Makowski Nov 9 '17 at 16:53

There are different ways how to deal with this data structure. Which one is best depends on what your aim of the analysis is. From your comments, it would be to run a regression of a response variable on "condition" and further covariables $X$ derived from the original data.

1. Run a PCA on $X$ to reduce the number of dimensions, ignoring the grouping structure implied by "condition". The more similar the distribution of $X$ across different conditions, the better this would work. How could you check this? Try to predict "condition" by $X$, e.g. using a mixed effects logistic regression. It the model is bad, then it means the distributions of $X$ across condition is not too different and the PCA approach could be fine.

2. Run separate PCAs per condition. No. Don't do it...

3. No PCA, but regularized mixed-effects regression, e.g. mixed effects Lasso. While elegant, it will be quite impossible to get valid inference from a model like this.

4. Be creative (A). Maybe dimension reduction is not necessary at all. Just run your mixed effects model with $X$ and "condition" as covariables. If the $X$ are strongly correlated, you can still transform them smartly, like considering ratios etc.

5. Be creative (B). Compress the two lines per object to one: Take the difference in the model response variable (Condition 1 minus Condition 2). Take average and difference in each $X$ variable (across conditions). Run PCA on the derived variables from $X$ and use the first few to explain your new response by OLS or so.

6. Be creative (C)
7. Be creative (D)