# Normal dispersion models.

### Description

This function estimates normal dispersion regression models.

### Usage

1 2 3 |

### Arguments

`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 ‘environment(formula)’, typically the environment from which the function is called. |

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

`epsilon` |
positive convergence tolerance epsilon; the procedure converge when |dev - devold| < epsilon. |

`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 ‘NA’s. The default is set by the ‘na.action’ setting of ‘options’, and is ‘na.fail’ if that is unset. The default is ‘na.omit’. |

`contrasts` |
an optional list. See the ‘contrasts.arg’ of ‘model.matrix.default’. |

`offset` |
this can be used to specify an a priori known component to be included in the linear predictor during fitting. An ‘offset’ term can be included in the formula instead or as well, and if both are specified their sum is used. |

### Details

Normal dispersion models allow to model variance heterogeneity in normal 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 expressed as

*Var(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.
This model can be re-expressed also as

*E(y|x) = β'x*

*Var(y|x) = \exp(λ'z)*

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

### Value

‘lm.dispmod’ returns an object of ‘class’ ‘"dispmod"’.

The function ‘summary’ is used to obtain and print a summary of the results.

An object of class ‘"lm.dispmod"’ is a list containing the following components:

`call` |
the matched call. |

`mean` |
an object of class ‘"glm"’ giving the fitted model for the mean function. |

`var` |
an object of class ‘"glm"’ giving the fitted model for the variance function. |

`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. |

### Note

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

### Author(s)

Luca Scrucca, luca@stat.unipg.it

### References

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

### See Also

`lm`

, `glm`

, `glm.binomial.disp`

, `glm.poisson.disp`

, `ncv.test`

(in the `car`

library).

### Examples

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