Editing a non-peer-reviewed paper I came across a Excel scatter-plot with an "exponential trend-line" drawn, AND the value of the correlation coefficient presented alongside (calculated for the untransformed data).

My understanding is that the Pearson correlation coefficient is (also) a measure of linear fit and I think it is therefore inappropriate to include it in a chart that displays an exponential curve purportedly as a line of best fit.

I mean the correlation coefficient is a valid description of the relationship between the two variables, but I argued that if they draw an exponential curve, then they are saying a change in the predictor variable leads to a proportional change in the natural log of the response variable, and not the untransformed response variable, So the correlation should therefore also be measured between x and ln(y), as an indication of the drawn model's fit.

The exponential model is (of course) a better fit for the data - see below for a stylised example. Can I insist that the chart portray either the "black" or "red" model and summary statistic but not the "black line" and "red statistic"? Or are the authors OK in keeping the correlation coefficient as is? (they suggest adding a footnote that reads: "The correlation coefficient shown is based on linear regression".)

xz <- c(7,8,12, 14, 19, 20)
yz <- c(500, 400, 280, 250, 200, 200)

plot(xz, yz, pch=20)
exp.model <-lm(log(yz)~ xz)
xvals <- seq(6,20,0.1)
yvals <- exp(predict(exp.model, list(xz = xvals)))
lines(xvals, yvals, lwd=2)
text(16,450,paste("corr.coeff. = ",-summary(exp.model)$r.squared^0.5))
lin.model <- lm(yz~ xz)
yvals <- predict(lin.model, list(xz = xvals))
lines(xvals, yvals, lwd=2, col="red")
text(16,400,paste("corr.coeff. = ",-summary(lin.model)$r.squared^0.5), col="red")

stylised example


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