[R] Finding points with equal probability between normal distributions

[Chris Evans]

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