# Function to obtain the maximum likelihood estimates of the parameters for linear mixed effects models with random intercept and a stationary or non-stationary stochastic process component, under multivariate t response distribution

### Description

Obtains the maximum likelihood estimates of the parameters by expectatio-maximisation (E-M) algorithm for linear mixed effects models with random intercept and a stationary or non-stationary stochastic process component, under multivariate normal t distribution

### Usage

1 2 3 4 |

### Arguments

`formula` |
a typical |

`data` |
a data frame from which the variables are to be extracted |

`id` |
a numerical vector for subject identification |

`timeVar` |
a numerical vector for the time variable |

`init.em` |
a vector of inital values for the E-M algorithm |

`maxiter.em` |
a numerical value for the maximum number of iterations for the E-M algorithm |

`tol.em` |
a numerical value for the maximum tolerance to assess the convergence of the E-M algorithm |

`process` |
a character string for the stochastic process: |

`silent` |
a character string, if set to |

`dof.est` |
a vector of three elements, to be passed to |

`tol.cd` |
a numerical value for the tolerance of central-difference approximation |

`tol.lmenssp` |
a numerical value for the maximum tolerance to assess the convergence, to be passed to |

`init.lmenssp` |
a vector of inital values, to be passed to |

`maxiter.lmenssp` |
a numerical value of the number of iterations for the Fisher-Scoring or Nelder-Mead algorithms, to be passed to |

`silent.lmenssp` |
a character string, if set to |

### Details

`lmenssp.heavy`

calls `lmenssp`

inside.

Whilst theoretical standard errors are calculated and reported only for the fixed effects, central-difference based standard errors are calculated and reported for all the parameter estimates.

There are more than one way of specifying `init.em`

, it can be set to:

1) fixed effects, random effects parameters and degrees-of freedom,

2) only the degrees-of-freedom, and

3) NULL; for this specification, `lmenssp.heavy`

finds the intials internally.

For the details of `"process"`

, see `lmenssp`

.

In `dof.est`

, first and second elements are the minimum and maximum values of the search and the
third element is the tolerance. It is passed to `optimize`

.

### Value

Returns a list of results.

### Author(s)

Ozgur Asar, Peter J. Diggle

### References

Asar O, Ritchie J, Kalra P, Diggle PJ (2015) Acute kidney injury amongst chronic kidney disease patients: a case-study in statistical modelling. To be submitted.

Pinheiro JC, Liu C, Wu YN. (2001) Efficient algorithms for robust estimation in linear mixed-effects
models using the multivariate t distribution. *Journal of Computational and Graphical Statistics* **10**, 249-276.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
# loading the data set and subsetting it for the first 20 patients
# for the sake illustration of the usage of the functions
data(data.sim.ibm.heavy)
data.sim.ibm.heavy.short <- data.sim.ibm.heavy[data.sim.ibm.heavy$id <= 20, ]
# estimating the parameters
# tol.em is set to 10^-1 and tol.lmenssp to 10^-2 only for illustration,
# decrease these values in your applications
fit.heavy <- lmenssp.heavy(formula = log.egfr ~ sex + bage + fu + pwl,
data = data.sim.ibm.heavy.short, id = data.sim.ibm.heavy.short$id,
timeVar = data.sim.ibm.heavy.short$fu, init.em = 5, maxiter.em = 1000,
tol.em = 10^-1, process = "ibm", silent = FALSE,
dof.est = c(0.1, 10, 0.0001), tol.cd = 0.001, tol.lmenssp = 10^-2,
silent.lmenssp = FALSE)
fit.heavy
``` |