Linked Michaelis-Menten type models for
methanogenesis data
“A stable isotope titration method for measurement of
the contribution of acetate and carbon dioxide reduction to methane production”
By N.D. Gray, J.N.S. Matthews and I.M. Head
MMLINK:
a program to fit the linked Michaelis-Menten type equations in this report
Numerical
details
As described in the document on statistical methods, the parameters of the
various models are estimates by choosing those values which maximise the
likelihood of the data, given the model.
The three models have four (Model I), five (Model II) or six (Model III)
parameters and the likelihoods, or more conveniently, the log-likelihoods,
cannot be maximised analytically. Consequently
we need to use numerical methods. These types
of methods require the user to supply some initial guesses about the values of
the parameters (starting values) and then the methods successively choose new
values, each time attempting to make the value of the log-likelihood
larger. The methods also incorporate
methods for deciding when a maximum has been reached.
Although the different models have between
four and six parameters, some analytical progress can be made and in all cases
the log-likelihoods can be maximised analytically with respect to all but two
parameters, B and K (in fact, analytical maximization with
respect to B is probably also
possible but has not been pursued). The profile log-likelihoods thus obtained can
be maximised using the Simplex Method (Nelder and Mead, 1965). The user therefore needs to specify starting
values for B and K: how this can be done sensibly is described in How to get starting values.
The simplex algorithm is implemented through the
routine MINIM written by D.E. Shaw and amended by Wedderburn and Miller, which can
be found as the second routine in the file lib.stat.cmu.edu/apstat/47. This is located in the software archive Statlib. The MINIM algorithm minimizes
functions, so it is, in fact, the negative of the profile log-likelihood that
is presented to the routine. The routine
used has been adapted by one of the authors so that the size of the lower
triangle of the covariance matrix is passed as an additional argument and the
whole of the lower triangle of the covariance matrix is returned, rather than
just the diagonal elements. In this
application this feature is not used.
The values of the various control parameters used by this routine can be
found by inspecting the lines near the beginning of the FORTRAN
code.
The p-values for the likelihood ratio tests have been
computed using the routine NPROB (
References
Adams, A.G. (1969) Areas under the
Nelder, J.A. and Mead, R. (1965) A simplex method for
function minimization. Computer Journal, 4, 308-313.