np.boxcoxmix {boxcoxmix} R Documentation

## Response Transformations for Random Effect and Variance Component Models

### Description

The function `np.boxcoxmix()` fits an overdispersed generalized linear model and variance component models using nonparametric profile maximum likelihood.

### Usage

```np.boxcoxmix(
formula,
groups = 1,
data,
K = 3,
tol = 0.5,
lambda = 1,
steps = 500,
EMdev.change = 1e-04,
plot.opt = 1,
verbose = TRUE,
start = "gq",
...
)
```

### Arguments

 `formula` a formula describing the transformed response and the fixed effect model (e.g. y ~ x). `groups` the random effects. To fit overdispersion models , set `groups` = 1. `data` a data frame containing variables used in the fixed and random effect models. `K` the number of mass points. `tol` a positive scalar (usually, 0< `tol` <= 2) `lambda` a transformation parameter, setting `lambda`=1 means 'no transformation'. `steps` maximum number of iterations for the EM algorithm. `EMdev.change` a small scalar, with default 0.0001, used to determine when to stop EM algorithm. `plot.opt` Set `plot.opt=1`, to plot the disparity against iteration number. Use `plot.opt=2` for `tolfind.boxcox()` and `plot.opt=3` for `optim.boxcox()`. `verbose` If set to FALSE, no printed output on progress. `start` a description of the initial values to be used in the fitted model, Quantile-based version "quantile" or Gaussian Quadrature "gq" can be set. `...` extra arguments will be ignored.

### Details

The Box-Cox transformation (Box & Cox, 1964) is applied to the overdispersed generalized linear models and variance component models with an unspecified mixing distribution. The NPML estimate of the mixing distribution is known to be a discrete distribution involving a finite number of mass-points and corresponding masses (Aitkin et al., 2009). An Expectation-Maximization (EM) algorithm is used for fitting the finite mixture distribution, one needs to specify the number of components `K` of the finite mixture in advance. To stop the EM-algorithm when it reached its convergence point, we need to defined the convergence criteria that is the absolute change in the successive log-likelihood function values being less than an arbitrary parameter such as `EMdev.change` = 0.0001 (Einbeck et at., 2014). This algorithm can be implemented using the function `np.boxcoxmix()`, which is designed to account for overdispersed generalized linear models and variance component models using the non-parametric profile maximum likelihood (NPPML) estimation.

The ability of the EM algorithm to locate the global maximum in fewer iterations can be affected by the choice of initial values, the function `np.boxcoxmix()` allows us to choose from two different methods to set the initial value of the mass points. When option "gq" is set, then Gauss-Hermite masses and mass points are used as starting points in the EM algorithm, while setting start= "quantile" uses the Quantile-based version to select the starting points.

### Value

 `mass.point` the fitted mass points. `p` the masses corresponding to the mixing proportions. `beta` the vector of coefficients. `sigma` the standard deviation of the mixing distribution (the square root of the variance). `se` the standard error of the estimate. `w` a matrix of posterior probabilities that element i comes from cluster k. `loglik` the log-likelihood of the fitted regression model. `complete.loglik` the complete log-likelihood of the fitted regression model. `disparity` the disparity of the fitted regression model. `EMiteration` provides the number of iterations of the EM algorithm. `EMconverged` TRUE means the EM algorithm converged. `call` the matched call. `formula` the formula provided. `data` the data argument. `aic` the Akaike information criterion of the fitted regression model. `bic` the Bayesian information criterion of the fitted regression model. `fitted` the fitted values for the individual observations. `fitted.transformed` the fitted values for the individual transformed observations. `residuals` the difference between the observed values and the fitted values. `residuals.transformed` the difference between the transformed observed values and the transformed fitted values. `predicted.re` a vector of predicted residuals.

The other outcomes are not relevant to users and they are intended for internal use only.

### Author(s)

Amani Almohaimeed and Jochen Einbeck

### References

Box G. and Cox D. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), pages 211-252.

Aitkin, M. A., Francis, B., Hinde, J., and Darnell, R. (2009). Statistical modelling in R. Oxford University Press Oxford.

Jochen Einbeck, Ross Darnell and John Hinde (2014). npmlreg: Nonparametric maximum likelihood estimation for random effect models. R package version 0.46-1.

`optim.boxcox`, `tolfind.boxcox`.

### Examples

```#Pennsylvanian Hospital Stay Data
data(hosp, package = "npmlreg")
test1 <- np.boxcoxmix(duration ~ age + wbc1, data = hosp, K = 2, tol = 1,
start = "quantile", lambda = 1)
round(summary(test1)\$w, digits = 3)
# [1,] 1.000 0.000

# Refinery yield of gasoline Data
data(Gasoline, package = "nlme")
test2.vc <- np.boxcoxmix(yield ~ endpoint + vapor, groups = Gasoline\$Sample,
data = Gasoline, K = 3, tol = 1.7, start = "quantile", lambda = 0)
test2.vc\$disparity
# [1] 176.9827

```

[Package boxcoxmix version 0.28 Index]