Consider a factorial experiment with three independent variables (A, B, Time) and one continuous dependent variable (length), and consider that the experiment is repeated multiple times (e.g. 2 or 3). In order to carry out a correct data analysis, should I consider the 'experiment repetition' as a new independent variable? Or could I pool data from multiple experiments in order to obtain a single database composed of one virtual experiment with more replications?
It depends on the charactheristics of the repetition itself. If you repeat the experiments on the same sample unit, you can combine the experiments as technical replication. If, on the other hand, each experiment repetition is performed on different sample units, then you can combine all experiments as one bigger experiment.
All this is considering that you have all experiments performed with the same experimental protocol, of course!
I think it is interesting to consider repetition as an IV, then, to prove the success of replication, you would expect all the interaction involving repetition insignificant. An alternative way is to analyze each experiment individually, then conduct a meta-analysis to evaluate the whole effect.
I believe that if you want to analyze the development of A & B throught Time, you should consider a Longitudinal Data Analysis instead of a typical design of experiments analysis. In addition, to consider a longitudinal analysis you need to repeat the experiments on the same sample unit.
Also, you can consider your experiment as a Repeated Measures Design (RMD) and analyze your results (wich is simple). The main difference between RMD and Longitudinal Analysis is the correlation structure of the repeated measures. Thus, if you want to analyze the interaction between your treatments and Time, you should consider a longitudinal data analysis.
LDA seems interesting. I should read something about that. Can you explain briefly what does it means? What difference from RMD in terms of data presentation, apart from correlation structure?
Another question has arisen in my mind. Could the variable 'Experiment' be treated as a random effect? If yes, would this be different from the variable 'Replication' (subject) in terms of contribution to variance?
You can get more information about LDA by reading this classic book Diggle, et al "Analysis of Longitudinal Data"
There is no difference between RMD and LDA in terms of data representation.
In RMD you use the interchangeable correlation structure, that means that every correlation between two time points is the same, rho.
On the other hand, in LDA you can use different types of correlation structures such as:
- Order 1 Autocorrelacion: You use it when the correlation depends on the space between time points.
- General. You use it when you assume that every correlation between two time points is different.
- Interchangeable. Explained above.
- Among others.
You can view RMD as a special case of LDA.
So, if you think that the correlation structure is not an Interchangeable one, you should use LDA to model your data.
What are the correspondences between the correlation structure types that you mentioned and those in the attached table (SAS)?
I think that there are no correspondence between the correlation structure types and those in the attached tables. Maybe because those in the table are covariance structures to model heteroscedasticity. You have to look for correlation structures, instead.
I dont use SAS, I use R and the correlation that I mentioned are contained in the 'nlme''. In addition, I recommend you this book Pinheiro & Bates "Mixed Models in S and S-PLUS"
Yes those asre covariance structure. Ooops! I assumed that correlation structure and covariance structures meant the same. Now, that table comes from proc MIXED, but the attached one comes from proc GENMOD
Ok, this new table is much better.
Here is the correspondence:
AR | AR(1) - Order 1 Autorregresive
EXCH - Exchangeable
IND - Independent (no correlation among time points)
UNSTR - General
MDEP - You use this structure when you assume that only some pairs of time points have the same correlation (rho) and the rest is zero.
USER - You state all the values of the correlation structure and then you fit the model. On the other hand, with the rest of structures you only state the structure and then estimate the parameters of that structure.
Many thanks Alan. This is intriguing, because it is very useful to do a correct analysis of repeated mesures data. Since I'm not a statistician/matematician, I can't understand the difference between correlation structure and covariance structure, but I realized that they are important for a correct model. Also, I can't understand the matrixes used to explain covariances types, and books are too much deep for me :(
Ok, well, I will recommend you this book that is for people who are not deeply related to LDA.
Twisk, "Applied Longitudinal Data Analysis for Epidemiology"
I'm pretty sure you'll like it..
Repetition is not an actual variable, but rather constitutes a replicate or pseudo-replicate. You should keep the data separate to show variation across replicates imo :)
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