# How to quantify intraindividual variability between paired measurements?

I've collected data measuring several different parameters in several subjects at two time-points (t1 and t2).

X and Y are meausurements of two anatomical structures; X is an area measured in mm2, Y is a volume measured in mm3. For each one I can describe at least 4 variables: measuments at t1 and t2 (eg. X1, X2), absolute difference between the two measurements (X2-X1), percentual difference between the two.

I want to compare the "reliability" of X and Y in term of intraindividual variability. My intuitive choice would be to compare standard deviation or coefficient of variation between the two (or their variation). But:

• In the single time point measure (say, X1 vs Y1), a low SD/CV tells me only that different individuals tend to have roughly the same values, while high SD/CV tells me that different individuals tend to have very different values.
• Considering absolute difference between the two measurements does not seems to have much sense.
• Considering percentual difference between the two, low SD/CV in e.g. X tells me only that, in a population, X tend to change of roughly the same % in all the individuals, while an high SD/CV tells me that there is more variability in the entity of the change between individuals.

So, I'm not sure which statistic will tell me what I want to know. I was thinking of Pearson's r (or Spearman's rho) to quantify correlation between X1-X2 vs correlation between Y1-Y2 (if there is a stronger correlation between X1 and X2 than with Y1-Y2, it means X1 and X2 tend to increase/decrease in the same direction more thant Y1 and Y2 do, so they is less intraindividual variability).

• Am I correct in this assumption?
• If I were to calculate this in R, is it just a matter of cor.test(X1,X2) cor.test(Y1,Y2)

and then compare results?

## 1 Answer

Let $\rho_1$ and $\rho_2$ be Pearson's correlation between X1 and X2, and Y1 and Y2, respectively. By comparing the difference between two correlations, you test the $H_0: \rho_1 = \rho_2$. You are correct in that you reject the null if one correlation has a bigger value given both have the same sign (or if the two correlations have different signs). To do this, you need to perform the so-called Steiger Test because the two correlations in your study have no shared variable. In R you can use r.test in the psych package.

But I am not sure if this is what you are actually looking for. Pearson correlation tells you how consistent subjects are across T1 and T2 (e.g., the highest score in X1 at T1 remains the highest at T2). I suspect, however, that you may be looking for a measure of agreement (e.g., values in X1 remain the same across T1 and T2). If this is the case, you may want to look into intraclass correlation (i.e., a repeatability measure). For detailed information, see Nakagawa and Schielzeth (2010).

• Thank you. Pearson correlation is what I was looking for; however, I found your mention of intraclass correlation extremely interesting, and I will surely look into it. Aug 5 '14 at 17:00