Eleni Rapsomaniki sent the following at 07/08/2006 11:35: > Dear mailing list, > > For two normal distributions, e.g: > > r1 =rnorm(20,5.2,2.1) > r2 =rnorm(20,4.2,1.1) > plot(density(r2), col="blue") > lines(density(r1), col="red") > > Is there a way in R to compute/estimate the point(s) x where the density of > the > two distributions cross (ie where x has equal probability of belonging to > either of the two distributions)?
I worry about showing my statistical incompetence or incomprehension but isn't what you need Jacobson et al.'s criterion C for clinical change? I.e. the point at which the misclassification rates in two Normal distributions, one with a higher mean than the other, match. It's at (sd1*mean2 + sd2*mean1)/(sd1 + sd2) So for Eleni's example I think that comes out at 4.544 and if I use: > r1b <- rnorm(200,5.2,2.1) > r2b <- rnorm(200,4.2,1.1) > plot(density(r2b), col="blue") > plot(density(r1b), col="red") > plot(density(r2b), col="blue") > lines(density(r1b), col="red") > cscc <- 4.544 > abline(v=cscc) It happened to work out beautifully: > sum(r1b > cscc) [1] 126 > sum(r2b < cscc) [1] 126 of course, set a different seed (I broke the posting rules and didn't set one, yes, I know) you won't get such a nice result every time and with n=20 in each group you'll get much more wobble. Or am I missing something. The original paper, which got reliable change wrong, was: Jacobson, N. S., Follette, W. C. & Revenstorf, D. (1984) Psychotherapy outcome research: methods for reporting variability and evaluating clinical significance. Behavior Therapy, 15, 336-352. There's a summary most people cite at: Jacobson, N. S. & Truax, P. (1991) Clinical significance: a statistical approach to defining meaningful change in psychotherapy research. Journal of Consulting and Clinical Psychology, 59, 12-19. and shameless self-promotion here, I tried to summarise it: Evans, C., Margison, F. & Barkham, M. (1998) The contribution of reliable and clinically significant change methods to evidence-based mental health. Evidence Based Mental Health, 1, 70-72. I hadn't twigged that what the criterion gives is balanced missclassification when I wrote that. I've played with some simulations and it's not as vulnerable to non-Gaussian distributions as I'd expected but someone can probably point to published work, simulation or analytic, on that. Cheers all, Chris ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.