A correlation is not a very good measure of agreement. You will get a perfect agreement between cholesterol measured in US units (mg%) and SI units but, of course, the individual values will be completely different, by a factor of about 37, if I can remember back to the 1980s when SI replaced mg%.
You can use a Bland and Altman approach to calculate agreement, which is, the authors have argued, applicable even when one of the measures is a gold standard.
However, this theory-free measure ignores the consequences of error. If you want to replace a gold standard measure with a simpler, easier method, then you need to be sure that the effects on decision making are not worrying. And to do this, you need to know what the test is used for. I’ve worked on some physiotherapy protocols where we classified errors into
no consequence – clinical management would be the same based on either test
significant disagreement – the difference between the two tests will result in a difference in patient management’ and
major error – test fails to detect something that would be serious of missed
This is just one kind of approach, but it has the advantage of giving the prevalence of errors of several types.
I suggest that you don’t use a correlation, which no-one in real life can interpret, and focus on the effects of method change on the decisional process in which the measurement is used. That’s what people really need to know before they switch.