R help - Aug 2006 - Finding points with equal probability between normal distributions

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)?

Many Thanks

Eleni Rapsomaniki
PhD student
Birkbeck College, UK

Eleni Rapsomaniki wrote:
> 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")

dr1 <- density(r1, from=-3, to=14)
dr2 <- density(r2, from=-3, to=14)
w <- which(diff(sign(dr2$y -  dr1$y)) != 0)
dr1$x[w]

plot(dr1, ylim = c(0, 0.33))
lines(dr2, col = "red")
abline(v = dr1$x[w], col = "blue")

Uwe Ligges


> 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)?
> 
> Many Thanks
> 
> Eleni Rapsomaniki
> PhD student
> Birkbeck College, UK
> 
> ______________________________________________
> R-help at 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.

if you want to base it on Normality, then you can use:

prob <- function(x){
    dnorm((x - 5.2) / 2.1) / 2.1 - dnorm((x - 4.2) / 1.1) / 1.1
}

uniroot(prob, c(-3, 3))

otherwise if you want to estimate it you can use Uwe's solution.


I hope it helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/(0)16/336899
Fax: +32/(0)16/337015
Web: http://med.kuleuven.be/biostat/
     http://www.student.kuleuven.be/~m0390867/dimitris.htm


----- Original Message ----- 
From: "Eleni Rapsomaniki" <e.rapsomaniki at
mail.cryst.bbk.ac.uk>
To: <r-help at stat.math.ethz.ch>
Sent: Monday, August 07, 2006 12:35 PM
Subject: [R] Finding points with equal probability between 
normaldistributions

> 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)?
>
> Many Thanks
>
> Eleni Rapsomaniki
> PhD student
> Birkbeck College, UK
>
> ______________________________________________
> R-help at 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.
> 

Disclaimer: http://www.kuleuven.be/cwis/email_disclaimer.htm

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