Re: [R] Finding points with equal probability between normal distributions

From: Chris Evans <>
Date: Mon 07 Aug 2006 - 22:00:07 EST

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 mailing list PLEASE do read the posting guide and provide commented, minimal, self-contained, reproducible code. Received on Mon Aug 07 22:04:49 2006

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