# Repeated measures with correlated measures (not time)

Most repeated measures ANOVAs have time as the repeated measure; I was wondering about using a repeated measure that is not time.

Say we fed two groups of animals different diets. At the end of the experiment, we sample the tissues, and measure ~30 different compounds (e.g. different fatty acids [FA]). Animals are sampled but once. Each FA is not independent of each other, as some of these compounds are converted from one form to another. As such, they are frequently moderately correlated with each other.

Would it be fair to treat each compound as a repeated measure, with diet as a between subjects factor? Thus, the interaction between Diet X FA would tell me if the FA content differed among diets?

I note that in many papers, researchers would perform 25-30 separate ANOVAs; one on each compound. Yet, these compounds are not independent of each other, as they are each measured simultaneously on the same animals.

Thanks for any pointers.

Have a look in a Multivariate text for MANOVA--multivariate ANOVA. Here is a website...

http://faculty.chass.ncsu.edu/garson/PA765/manova.htm

Though, that's a lot of dependent variables and it could be hard to interpret. It might be simpler to do some sort of data reduction first, like PCA among your set of fatty acids. I suppose it depends on how much overlap there is and how many PCs you'd need to extract to account for the 30 vars.

• Yes, PCA or Factor Analysis is frequently used with this type of data. But those answer different questions, do they not? I had thought that someone would suggest MANOVA, though the idea of having 30 variables is daunting... I was hoping that repeated measures would be valid, as it is easier to interpret, allows one to compared individual compounds between treatments (post hoc), and it can (?) handle nonindependent variables. I will looking into using MANOVA – P auritus Mar 24 '11 at 20:39
• I was suggesting using PCA as a possible data reduction strategy, then running a MANOVA on the PC scores, if the PCA is helpful. If you could reduce the # of dependents in MANOVA, you could improve your power. Whether or not PCA is helpful, would really depends on how much overlap there is in the Fatty Acid measurements. There are some problems with PCA too, though. For example a PC that explains little variance in the X's may have a strong relationship with Y and by standard conventions, you might throw it away. – Brett Mar 24 '11 at 20:46
• Thanks! As a further aside, I had thought that was one of the advantages of Partial Least Squares (compared with doing regression following PCA) – the first component automatically has the highest correlation with Y, the 2nd component the second highest correlation, etc... – P auritus Mar 24 '11 at 22:05
• Have a look at Haldi and Ling (1998) "Some cautionary notes on the use of principal components regression." in The American Statistician. Bottom line, relationships with variables outside of the PCA can be hidden in even the last PC. – Brett Mar 24 '11 at 22:12
• Yes, I remember that paper from sci.stat.math, posted by the author himself... – P auritus Mar 24 '11 at 22:19

It's interesting... how do you compare the compounds with each other? Repeated measures don't have to be necessarily over time, but they need to be measures of the same thing or things that can be treated as if they are the same. You could certainly measure the same compound in different areas of the body and that would be a repeated measure.

I guess I'm asking the first question out of ignorance of your field. If you can compare the compounds one to another then they are repeated measures grouped by individual animal.

I suppose one could even consider multi-level modelling beyond simple repeated measures because you could take the comparable body part as one level and the animal itself as another.

• Hi, I don't normally see folks use repeated measures in such a fashion. My question is to see if one can. All the compounds are in the same units (eg ng/g). One would not routinely compare compound X with compound Y (though one occasionally does). Following the Diet X FA interaction, a posthoc test could compare compound X between diets, compound Y between diets, etc. My main impetus in asking this question is that none of the compounds are independent of each other. Thus, if compound X varies among diets, then chances are other compounds do as well. What is a better way than 30 ANOVAs? – P auritus Mar 24 '11 at 20:30
• @P auritus: You ask "What is a better way than 30 ANOVAs?" A single MANOVA with 30 dependent variables or a single MANOVA with the first K principal components. Either approach takes into account the interrelationships among your FAs. In fact, just as an aside, a MANOVA on the PCA scores would give you the same result as the MANOVA with the 30 original variables if you use all 30 principal components. – Brett Mar 24 '11 at 21:26
• Appreciate the help. I have already been using multivariate stats on the data – discriminant function analysis. But out of curiosity I had decided to run a repeated measures. Shall I assume that it is the merely the least preferable option compared to your better alternatives, or is it an invalid option entirely? – P auritus Mar 24 '11 at 22:17