VGAM (version 0.8-2)

micmen: Michaelis-Menten Model

Description

Fits a Michaelis-Menten nonlinear regression model.

Usage

micmen(rpar = 0.001, divisor = 10, init1 = NULL, init2 = NULL,
       method.init = 1, oim = TRUE,
       link1 = "identity", link2 = "identity",
       firstDeriv = c("nsimEIM", "rpar"),
       earg1 = list(), earg2 = list(), prob.x = c(0.15, 0.85),
       nsimEIM = 500,
       dispersion = 0, zero = NULL)

Arguments

rpar

Numeric. Initial positive ridge parameter. This is used to create positive-definite weight matrices.

divisor

Numerical. The divisor used to divide the ridge parameter at each iteration until it is very small but still positive. The value of divisor should be greater than one.

init1, init2

Numerical. Optional initial value for the first and second parameters, respectively. The default is to use a self-starting value.

link1, link2

Parameter link function applied to the first and second parameters, respectively. See Links for more choices.

earg1, earg2

List. Extra argument for each of the links. See earg in Links for general information.

dispersion

Numerical. Dispersion parameter.

firstDeriv

Character. Algorithm for computing the first derivatives and working weights. The first is the default.

method.init, prob.x

See CommonVGAMffArguments for more information.

nsimEIM

See CommonVGAMffArguments for more information.

oim

Use the OIM? See CommonVGAMffArguments for more information.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2}. A NULL means none. See CommonVGAMffArgume

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

This function is not (nor could ever be) entirely reliable. Plotting the fitted function and monitoring convergence is recommended.

Details

The Michaelis-Menten model is given by $$E(Y_i) = (\theta_1 u_i) / (\theta_2 + u_i)$$ where $\theta_1$ and $\theta_2$ are the two parameters.

The relationship between iteratively reweighted least squares and the Gauss-Newton algorithm is given in Wedderburn (1974). However, the algorithm used by this family function is different. Details are given at the Author's web site.

References

Seber, G. A. F. and Wild, C. J. (1989) Nonlinear Regression, New York: Wiley.

Wedderburn, R. W. M. (1974) Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439--447.

Bates, D. M. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications, New York: Wiley.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

enzyme.

Examples

fit = vglm(velocity ~ 1, micmen, enzyme, trace = TRUE, crit = "coef",
           form2 = ~ conc - 1)
summary(fit)

plot(velocity ~ conc, enzyme, xlab = "concentration", las = 1,
     col = "blue", main = "Michaelis-Menten equation for the enzyme data",
     ylim = c(0, max(velocity)), xlim = c(0, max(conc)))
points(fitted(fit) ~ conc, enzyme, col = "red", pch = "+", cex = 1.5)

# This predicts the response at a finer grid:
newenzyme = data.frame(conc = seq(0, max(with(enzyme, conc)), len = 200))
fit@extra$Xm2 = newenzyme$conc   # This assignment is needed for prediction
lines(predict(fit, newenzyme, "response") ~ conc, newenzyme, col = "red")