Best way to check if two variables are correlated (a) I have two variables that I am looking at: Visceral fat levels using MRI, and the visceral fat rating as shown by bioelectrical impedance (BIA) scales. The units for these two variables are different, and the sample size for each is close to 100. For each individual, there will be a single measurement using the MRI and another measurement with the BIA scale.
The MRI is the gold standard, so what I am trying to find is whether the visceral fat ratings given by BIA scales fluctuate in a similar manner to the corresponding MRI measurement. What I want to know is essentially whether the visceral fat ratings given by the BIA scales are accurate enough to give someone a good understanding of their visceral fat levels. 
(b) I want to do the same thing, only across 3 different time points. So this time, I want to check whether the change between time points 1 and 2 for example, are comparable between the two methods.  
 A: This is a kind of calibration problem, with an (expensive, presumably) goldstandard measurement and another cheaper measurement. What would I do? I will name the variables Gold and Cheap.
Plot Gold versus Cheap.  Plot the error Gold - Cheap against Cheap. Is this plot linear? Is the local mean (and local variance) of the error approximately constant? Or is there curvature in the plot?
Then calculate a regression of Gold on Cheap (if there is curvature in the plot above, you could represent Cheap via splines). If the constant term (Intercept) is approximately zero, there is no systematical bias. If the slope is close to zero, the bias is approximately independent of Cheap.
Just the correlation between Cheap and Gold is not a good measure, since it can be close to 1 with a large, but constant, bias.
EDIT---answer to comments
My language above was maybe imprecise, since bias has multiple meanings. What I tried to say is that correlation between Cheap and Gold can be large, even 1, in cases where Cheap is a very bad proxy for Gold. A simple example is Cheap=Gold+100. Here the correlation is 1, but using Cheap as a proxy for Gold gives a constant prediction error (bias) of 100. That is, correlation is a bad measure of agreement!
