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The Pearson correlation coefficient is scale-invariant. Therefore the correlation $cor(x,y)$ remains the same if $x$ or $y$ is linearly transformed (i.e., $x - 1$ or $2x$).

I am interested in a correlation coefficient that is sensitive to linear transformation. If $cor(x,y) = 1.00$, I want $cor(x-1, y) < 1$. In other words, I want a strict correlation coefficient that requires $x = y$ in order for $r = 1.00$.

One idea I had was to modify the covariance term $\Sigma (x - \bar x)(y - \bar y)$ to shrink when the means of $x$ and $y$ diverge, but I am uneasy about creating a monster statistic with intractable properties. In principle, I am looking for something like $\Sigma (x - \bar x)(y - \bar y) - (|\bar x - \bar y |)$.

Is there an existing method to achieve this type of "strict" correlation coefficient?

KamRa
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1 Answers1

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There are (at least) two correlation-type measures that measure the absolute agreement between two measurements (i.e., they will only be equal to 1 when $x = y$). The first is the intraclass correlation coefficient and the other is the concordance correlation coefficient. They often give very similar results. For example:

x <- c(2,4,3,6,8)
y <- 1 + x
n <- length(x)

s2x <- sum((x - mean(x))^2) / n
s2y <- sum((x - mean(x))^2) / n
sxy <- sum((x - mean(x)) * (y - mean(y))) / n

cor(c(x,y),c(y,x)) # ICC          
[1] 0.8977505

2*sxy / (s2x + s2y + (mean(x) - mean(y))^2) # concordance correlation
[1] 0.9027237
Wolfgang
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  • You may have access to https://www.sciencedirect.com/science/article/pii/S0169555X05003740 which surveyed not just concordance correlation but the general problem of assessing agreement. – Nick Cox Jan 28 '20 at 12:55
  • @Wolfgang Thank you for your helpful answer. ICC fits the bill. I like your clever implementation using cor(). Where can I read more about this implementation? It is slightly different from other implementations of ICC. – KamRa Jan 29 '20 at 00:53
  • @NickCox Thanks. I found this article to be the most helpful: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4913118/ – KamRa Jan 29 '20 at 01:05
  • @Wolfgang If I may add a follow-up question: What type of ICC does your answer implement? Based on the readings I am looking for ICC with two-way random effects, absolute agreement, and single rater, aka ICC(2,1). – KamRa Jan 29 '20 at 05:05
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    The one that is estimated by `cor(c(x,y),c(y,x))` is the ICC(1,1). This estimator goes back to Fisher. You can also use the ANOVA approach to estimate it. The two approaches often give very similar estimates. If you want to estimate the ICC(2,1), then use the ANOVA approach. – Wolfgang Jan 29 '20 at 07:25
  • @Wolfgang Thanks. This is very helpful. All the best. – KamRa Jan 29 '20 at 18:24