Description Usage Arguments Details Value Note References See Also Examples

This function estimates Gaussian dispersion regression models.

1 2 3 |

`formula` |
a symbolic description of the mean function of the model to be fit. For the details of model formula specification see |

`var.formula` |
a symbolic description of the variance function of the model to be fit. This must be a one-sided formula; if omitted the same terms used for the mean function are used. For the details of model formula specification see |

`data` |
an optional data frame containing the variables in the model. By default the variables are taken from |

`maxit` |
integer giving the maximal number of iterations for the model fitting procedure. |

`epsilon` |
tolerance value for checking convergence. See |

`subset` |
an optional vector specifying a subset of observations to be used in the fitting process. |

`na.action` |
a function which indicates what should happen when the data contain |

`contrasts` |
an optional list as described in the |

`offset` |
this can be used to specify an a priori known component to be included in the linear predictor during fitting. An |

Gaussian dispersion models allow to model variance heterogeneity in Gaussian regression analysis using a log-linear model for the variance.

Suppose a response *y* is modelled as a function of a set of *p* predictors *x* through the linear model

*y_i = β'x_i + e_i*

where *e_i ~ N(0, σ^2)* under homogeneity.

Variance heterogeneity is modelled as

*V(e_i) = σ^2 = exp(λ'z_i)*

where *z_i* may contain some or all the variables in *x_i* and other variables not included in *x_i*; *z_i* is however assumed to contain a constant term.

The full model can be expressed as

*E(y|x) = β'x*

*V(y|x) = exp(λ'z)*

and it is fitted by maximum likelihood following the algorithm described in Aitkin (1987).

`lm.dispmod()`

returns an object of class `"dispmod"`

.

The `summary`

method can be used to obtain and print a summary of the results.

An object of class `"dispmod"`

is a list containing the following components:

`call` |
the matched call. |

`mean` |
an object of class |

`var` |
an object of class |

`initial.deviance` |
the value of the deviance at the beginning of the iterative procedure, i.e. assuming constant variance. |

`deviance` |
the value of the deviance at the end of the iterative procedure. |

Based on a similar procedure available in Arc (Cook and Weisberg, http://www.stat.umn.edu/arc)

Aitkin, M. (1987), Modelling variance heterogeneity in normal regression models using GLIM, *Applied Statistics*, 36, 332–339.

`lm`

, `glm`

, `glm.binomial.disp`

, `glm.poisson.disp`

, `ncvTest`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
data(minitab)
minitab <- within(minitab, y <- V^(1/3) )
mod <- lm(y ~ H + D, data = minitab)
summary(mod)
mod.disp1 <- lm.disp(y ~ H + D, data = minitab)
summary(mod.disp1)
mod.disp2 <- lm.disp(y ~ H + D, ~ H, data = minitab)
summary(mod.disp2)
# Likelihood ratio test
deviances <- c(mod.disp1$initial.deviance,
mod.disp2$deviance,
mod.disp1$deviance)
lrt <- c(NA, abs(diff(deviances)))
cbind(deviances, lrt, p.value = 1-pchisq(lrt, 1))
# quadratic dispersion model on D (as discussed by Aitkin)
mod.disp4 <- lm.disp(y ~ H + D, ~ D + I(D^2), data = minitab)
summary(mod.disp4)
r <- mod$residuals
phi.est <- mod.disp4$var$fitted.values
plot(minitab$D, log(r^2))
lines(minitab$D, log(phi.est))
``` |

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