# Grid search over tol for NPML estimation of (generalized) random effect models

### Description

Performs a grid search to select the parameter `tol`

,
which is a tuning parameter for starting point selection of the EM algorithm
for NPML estimation (see e.g. Aitkin, Hinde & Francis, 2009, p. 437)

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
tolfind(formula,
random = ~1,
family = gaussian(),
data,
k = 4,
random.distribution="np",
offset,
weights,
na.action,
EMmaxit = 500,
EMdev.change = 0.001,
lambda = 0,
damp = TRUE,
damp.power = 1,
spike.protect = 1,
sdev,
shape,
plot.opt = 1,
steps = 15,
find.in.range = c(0.05, 0.8),
verbose = FALSE,
noformat = FALSE,
...)
``` |

### Arguments

`formula` |
a formula defining the response and the fixed effects (e.g. |

`random` |
a formula defining the random model. Set |

`family` |
conditional distribution of responses: "gaussian", "poisson", "binomial", "Gamma", or "inverse.gaussian" can be set. |

`data` |
the data frame (mandatory, even if it is attached to the workspace!). |

`k` |
the number of mass points/integration points (supported are up to 600 mass points). |

`random.distribution` |
the mixing distribution, Gaussian Quadrature
( |

`offset` |
an optional offset to be included in the model. |

`weights` |
optional prior weights for the data. |

`na.action` |
a function indicating what should happen when |

`EMmaxit` |
maximum number of EM iterations. |

`EMdev.change` |
stops EM algorithm when deviance change falls below this value. |

`lambda` |
see the help file for |

`damp` |
switches EM damping on or off. |

`damp.power` |
steers degree of damping. |

`spike.protect` |
see the help file for |

`sdev` |
optional fixed standard deviation for normal mixture. |

`shape` |
optional fixed shape parameter for Gamma and IG mixtures. |

`plot.opt` |
For |

`steps` |
number of grid points for the search of |

`find.in.range` |
range for the search of |

`verbose` |
If set to |

`noformat` |
If |

`...` |
further arguments which will be ignored. |

### Details

The EM algorithm for NPML estimation (Aitkin, 1996) uses the Gauss-Hermite masses
and mass points as starting points. The position of the starting points can be
concentrated or extended by setting `tol`

smaller or larger than 1,
respectively. The tuning parameter `tol`

is, as in GLIM4, responsible for this scaling.
A careful selection of `tol`

may be necessary for some data sets.
The reason is that NPML has a tendency to get stuck in local maxima,
as the log-likelihhod function is not concave for fixed `k`

(Boehning, 1999).

For Gaussian, Gamma, and IG mixtures this R implementation uses by default a damping
procedure in the first cycles of the EM algorithm (Einbeck & Hinde, 2006),
which stabilizes the algorithm and makes it less sensitive to the optimal choice
of `tol`

. Application of `tolfind`

to check that the optimal solution has
not been overlooked may nevertheless be advisable.

`tolfind`

works for `alldist`

and `allvc`

. The `tolfind`

function
is mainly designed for NPML (`random.distribution="np"`

). It
can also be applied to Gaussian Quadrature (`random.distribution="gq"`

),
though `tol`

is of little importance for this and primarily influences
the speed of convergence.

### Value

A list of 5 items:

`MinDisparity` |
the minimal disparity achieved (for which EM converged). |

`Mintol` |
the |

`AllDisparities` |
a vector containing all disparities calculated on the grid. |

`Alltol` |
all corresponding |

`AllEMconverged` |
a vector of Booleans indicating
if EM converged for the particular |

### Author(s)

Jochen Einbeck & John Hinde (2006).

### References

Aitkin, M. (1996). A general maximum likelihood analysis of overdispersion in generalized linear models. Statistics and Computing 6 , 251-262.

Aitkin, M., Francis, B. and Hinde, J. (2009). Statistical Modelling in R. Oxford Statistical Science Series, Oxford, UK.

Boehning, D. (1999). Computer-Assisted Analysis of Mixtures and Applications. Meta-Analysis, Disease Mapping and others. Chapman & Hall / CRC, Boca Raton, FL, USA.

Einbeck, J. & Hinde, J. (2006). A note on NPML estimation for exponential family regression models with unspecified dispersion parameter. Austrian Journal of Statistics 35, 233-243.

### See Also

`alldist`

, `allvc`

### Examples

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