# Discrimination between measurements made at different points in time

I would like to ascertain what variables discriminate best between experimental conditions in a repeated-measures experimental design.

I have performed Repeated Measures MANOVA to determine whether the groups of measurements differ significantly at all and found they indeed do. I can get a rough idea of what variables have the highest prediction power, based on their Roy's greatest root value (one could have used another statistic, of course). I would now like to check this via another statistical analysis which would follow the MANOVA.

Speaking loosely, I am looking for a predictive discriminant analysis analogous to repeated measures MANOVA.

From what I have read on Repeated Measures (Longitudinal) Discriminant Analysis, it seems that it is used to discriminate between groups of subjects all of which have been measured on several occasions (Sajobi et al., 2011; Lix and Sajobi, 2010; Komarek et al., 2009; Kohlmann, 2010). On the other hand, what I am looking for would discriminate between these repeated measures themselves, all of which included all of the participants (a single group).

Brief two paragraphs in McLachlan (section 3.7.5, pp. 83-84) were too compressed to give any insight.

# References

Kohlmann, M. (2010). Discriminant Analysis for Longitudinal Data with Application in Medical Diagnostics (Doctoral dissertation, lmu).

Komárek, A., Hansen, B. E., Kuiper, E. M. M., van Buuren, H. R., & Lesaffre, E. (2010). Discriminant analysis using a multivariate linear mixed model with a normal mixture in the random effects distribution. Statistics in Medicine, 29(30), 3267–3283. doi:10.1002/sim.3849

Lix, L. M., & Sajobi, T. T. (2010). Discriminant Analysis for Repeated Measures Data: A Review. Frontiers in Psychology, 1, 1-9. doi:10.3389/fpsyg.2010.00146

McLachlan, G. (2004). Discriminant analysis and statistical pattern recognition (Vol. 544). John Wiley & Sons.

Sajobi, T. T., Lix, L. M., Li, L., & Laverty, W. (2011). Discriminant Analysis for Repeated Measures Data: Effects of Mean and Covariance Misspecification on Bias and Error in Discriminant Function Coefficients. Journal of Modern Applied Statistical Methods, 10(2), 571-582. Available at: digitalcommons.wayne.edu/jmasm/vol10/iss2/15

• I'm trying to grab the gist. You have measurements of several variables on several occassions (t1, t2, t3, ...) on the same group of people. You would like to predict somehow the progress of people, more specifically the progress from t1->t2, t2->t3, t3->.. Is that correct? – spdrnl May 12 '15 at 16:00
• Thanks @spdrnl. It's not a progress, because the (time) sequence is not important. Hence "repeated measures" rather than "longitudinal". The repeated conditions were in random order for different subjects. My task is actually even simpler: I only want to find variables that differentiate (i.e. discriminate) between these occasions most efficiently. – Fato39 May 13 '15 at 6:12

## 1 Answer

I ended up using the multiple factor analysis (Pagès, 2004 /pdf/), which can indeed be calculated for repeated measures (Abdi, Williams, & Valentin, 2013).

This is actually an extension of principle component analysis on multiple tables. It can calculate principle components for each of the repeated measures, but also for grouped variables.

So, I calculated principle components that explained the most variance between different repeated measures. In this way, I got a linear combination of variables that best explained the differences between different experimental conditions (i.e. different repeated measures). If I was lucky, the first dimension was straight-forward to interpret, possibly highly correlated with only one of my variables.

The method is implemented in the R package called FactoMineR (examples and detailed instructions are here).

## References

Pages, J. (2004). Multiple factor analysis: Main features and application to sensory data. Revista Colombiana de Estadística, 27(1), 1-26.

Abdi, H., Williams, L. J. and Valentin, D. (2013), Multiple factor analysis: principal component analysis for multitable and multiblock data sets. WIREs Comp Stat, 5, 149–179. doi:10.1002/wics.1246