# Re: [Nlme-help] parameter constraints

Jose Pinheiro (jcp@rice.research.bell-labs.com)
Tue, 22 Jun 1999 09:16:50 -0400

> I'm trying to model longitudinal measurements of children's gross
> motor ability using NLME, for which the expectation function is a
> negative exponential rise y~(a*(1-exp(-b*x)), and the 'a' and 'b'
> parameters have associated random effects. The response
> measurements are only scientifically sensible in the range of 0 to
> 100, and my estimate of the fixed effect of the limiting parameter
> 'a' is in this range. However, with no constraints on the random
> effects, some of the kid's random limit parameters (u_a), when added
> to the fixed effect of 'a' produce expected asymptotes for
> individual children of greater than 100. Can anyone recommend a
> method of imposing constraints on the random effects such that, say,
> a+u_a <= 100?

The optimization algorithm used in nlme does not allow constraints on
the coefficients in the model. You can, however, reparameterize your
model in such a way that the constraints are satisfied within an
unrestricted optimization framework. For example, if you re-express

a = 100/(1+exp(-A))

then A is unrestricted, while 0 < a < 100. In this case, a is a
monotonically increasing function of A and you can derive confidence
intervals for a from the confidence intervals obtained for A (and the
confidence intervals for a are guaranteed to be contained in (0,100)).
You can also assign a random effect to A and still guarantee that the
resulting a will be in the desired range. So, your model would look
something like

y ~ 100 * (1 - exp(-b * x))/(1 + exp(-A)))

after the reparameterization.

Hope this helps,

--Jose'

-----------------------------------------------------------------------------
Jose' Pinheiro
Bell Laboratories jcp@research.bell-labs.com
600 Mountain Avenue, Room 2C-258 office: (908) 582-2390
Murray Hill, NJ 07974 fax: (908) 582-3340
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