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POQ-Q line in log-log space (log="xy"): abline's argument 'untf' does not work as expected
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copied!<p>I try to understand why <code>qqline</code> does not give me what I expect if (and only if) <code>log="xy"</code>:</p> <pre><code>th <- 0.5 n <- 1e3 set.seed(271) x <- rgamma(n, shape=th) qF <- function(p) qgamma(p, shape=th) ## 1) Q-Q plot in normal space plot(qF(ppoints(n)), sort(x)) qqline(y=sort(x), distribution=qF) # fine ## 2) Q-Q plot in log10-x space plot(qF(ppoints(n)), sort(x), log="x") qqline(y=sort(x), distribution=qF, untf=TRUE) # with untf=TRUE, fine ## 3) Q-Q plot in log10-y space plot(qF(ppoints(n)), sort(x), log="y") qqline(y=sort(x), distribution=qF, untf=TRUE) # with untf=TRUE, fine </code></pre> <p>Now the <code>?abline</code> addresses <code>untf</code>. According to that, the following should also work (in my opinion):</p> <pre><code>## 4) Q-Q plot in log10-log10 space plot(qF(ppoints(n)), sort(x), log="xy") qqline(y=sort(x), distribution=qF, untf=TRUE) # ... but this is wrong qqline(y=sort(x), distribution=qF, col="red") # ... (still) wrong qqline(y=sort(log10(x)), distribution=function(p) log10(qF(p)), col="blue") # seems to be correct (but draws a line, not the curve resulting from transforming the Q-Q line in 'non-log-log' space) </code></pre> <p>My questions are: 1) Why does <code>untf=TRUE</code> not give the right answer for <code>log="xy"</code> although it did for <code>log="x"</code> and <code>log="y"</code>? 2) On a more general basis, what's the 'right' Q-Q line in log-log-space (anyways)?</p> <p>Please note that I have posted a similar question to <a href="https://www.stat.math.ethz.ch/pipermail/r-help/2013-June/356127.html" rel="nofollow">R-help</a>, but have not received an answer.</p> <p><strong>UPDATE</strong></p> <p>I did more experiments, reproducing (by foot) the strange curve that <code>qqline()</code> in log-log space produces. I also figure that <code>log10()</code> instead of <code>log()</code> might be the reason for an incorrect line. So the blue line indeed seems to be the correct one <em>if</em> one wants to have a <em>line</em> in log-log space. What is not clear to me is why the points of the standard Q-Q line displayed in log-log scale do not give the right curve, although the points of the Q-Q plot are constructed this way, too. </p> <pre><code>th <- 0.5 n <- 1e2 # note the difference for n=1e2 vs n=1e4! set.seed(271) x <- rgamma(n, shape=th) qF <- function(p) qgamma(p, shape=th) ## Q-Q plot qq.x <- qF(ppoints(n)) qq.y <- sort(x) plot(qq.x, qq.y, log="xy", main=substitute("Q-Q plot with sample size n="*n., list(n.=n)), xlab="Theoretical quantiles", ylab="Sample quantiles") abline(v=quantile(qq.x, probs=0.25), col="gray50") # vertical line through 25%-quantile abline(v=quantile(qq.x, probs=0.75), col="gray50") # vertical line through 75%-quantile qqline(y=qq.y, distribution=qF, untf=TRUE) # ... this is wrong (*) qqline(y=qq.y, distribution=qF, col="red") # ... still wrong ## => doesn't pass first and third quartile, but closer to 'right' line (blue) for large n qqline(y=sort(log10(x)), distribution=function(p) log10(qF(p)), col="blue") ## => seems to be correct (but draws a line, not the curve resulting from transforming ## the Q-Q line in 'non-log-log' space) ## by foot (as will be seen, this reproduces (*)) probs <- c(0.25, 0.75) q.x <- qF(probs) q.y <- quantile(x, probs=probs, names=FALSE) slope <- diff(q.y)/diff(q.x) int <- q.y[1L] - slope * q.x[1L] f <- function(x) slope * x + int # line in *normal* space qq.y. <- f(qq.x) # points of the Q-Q line evaluated in normal space lines(qq.x, qq.y., col="darkgreen") ## => reproduces qqline(.., untf=TRUE), see (*) above, that is, ## standard Q-Q line added to a plot in log-log scale = wrong (although points ## of the Q-Q plot are constructed this way!) ## legend legend("topleft", lty=rep(1,5), col=c("darkgreen", "black", "red", "blue", "gray50"), legend=c("Standard Q-Q line with 'untf=TRUE'", "Standard Q-Q line plotted in log-log scale", "Standard Q-Q line", "Q-Q line of log10(x) vs log10-quantiles", "1st and 3rd quartiles")) </code></pre>

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