hnp
Half-Normal Plots with Simulation Envelopes

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hnp: Half-Normal Plots with Simulation Envelopes

hnp: Half-Normal Plots with Simulation Envelopes
In hnp: Half-Normal Plots with Simulation Envelopes

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Description

Produces a (half-)normal plot from a fitted model object for a range of different models. Extendable to non-implemented model classes.

Usage

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hnp(object, sim = 99, conf = 0.95, resid.type, maxit,
    halfnormal = T, scale = F, plot.sim = T, verb.sim = F,
    warn = F, how.many.out = F, print.on = F, paint.out = F,
    col.paint.out, newclass = F, diagfun, simfun, fitfun, ...)

Arguments

object

fitted model object or numeric vector.

sim

number of simulations used to compute envelope. Default is 99.

conf

confidence level of the simulated envelope. Default is 0.95.

resid.type

type of residuals to be used; must be one of "deviance", "pearson", "response", "working", "simple", "student", or "standard". Not all model type and residual type combinations are allowed. Defaults are "student" for aov and lm objects, "deviance" for glm, glm.nb, lmer, glmer and aodml objects, "simple" for gamlss objects, "response" for glmmadmb and vglm objects, "pearson" for zeroinfl and hurdle objects.

maxit

maximum number of iterations of the estimation algorithm. Defaults are 25 for glm, glm.nb, gamlss and vglm objects, 300 for glmmadmb, lmer and glmer objects, 3000 for aodml objects, 10000 for zeroinfl and hurdle objects.

halfnormal

logical. If TRUE, a half-normal plot is produced. If FALSE, a normal plot is produced. Default is TRUE.

scale

logical. If TRUE and if object is a numeric vector, simulates from a normal distribution with mean and variance estimated from object. If FALSE, uses a standard normal distribution to simulate from. Default is FALSE.

plot.sim

logical. Should the (half-)normal plot be plotted? Default is TRUE.

verb.sim

logical. If TRUE, prints each step of the simulation procedure. Default is FALSE.

warn

logical. If TRUE, shows warning messages in the simulation process. Default is FALSE.

how.many.out

logical. If TRUE, the number of points out of the envelope is printed. Default is FALSE.

print.on

logical. If TRUE, the number of points out of the envelope is printed on the plot. Default is FALSE.

paint.out

logical. If TRUE, points out of the simulation envelope are plotted in a different color. Default is FALSE.

col.paint.out

If paint.out=TRUE, sets the color of points out of the envelope. Default is "red".

newclass

logical. If TRUE, use diagfun, simfun, and fitfun to extract diagnostics (typically residuals), generate simulated data using fitted model parameters, and fit the desired model. Default is FALSE.

diagfun

user-defined function used to obtain the diagnostic measures from the fitted model object (only used when newclass=TRUE). Default is resid.

simfun

user-defined function used to simulate a random sample from the model estimated parameters (only used when newclass=TRUE).

fitfun

user-defined function used to re-fit the model to simulated data (only used when newclass=TRUE).

...

extra graphical arguments passed to plot.hnp.

Details

A relatively easy way to assess goodness-of-fit of a fitted model is to use (half-)normal plots of a model diagnostic, e.g., different types of residuals, Cook's distance, leverage. These plots are obtained by plotting the ordered absolute values of a model diagnostic versus the expected order statistic of a half-normal distribution,

Φ^{-1}(\frac{i+n-1/8}{2*n+1/2})

(for a half-normal plot) or the normal distribution,

Φ^{-1}(\frac{i+3/8}{n+1/4})

(for a normal plot).

Atkinson (1985) proposed the addition of a simulated envelope, which is such that under the correct model the plot for the observed data is likely to fall within the envelope. The objective is not to provide a region of acceptance, but some sort of guidance to what kind of shape to expect.

Obtaining the simulated envelope is simple and consists of (1) fitting a model; (2) extracting model diagnostics and calculating sorted absolute values; (3) simulating 99 (or more) response variables using the same model matrix, error distribution and fitted parameters; (4) fitting the same model to each simulated response variable and obtaining the same model diagnostics, again sorted absolute values; (5) computing the desired percentiles (e.g., 2.5 and 97.5) at each value of the expected order statistic to form the envelope.

This function handles different model classes and more will be implemented as time goes by. So far, the following models are included:

Continuous data:
Normal: functions lm, aov and glm with family=gaussian
Gamma: function glm with family=Gamma
Inverse gaussian: function glm with family=inverse.gaussian
Proportion data:
Binomial: function glm with family=binomial
Quasi-binomial: function glm with family=quasibinomial
Beta-binomial: package VGAM - function vglm, with family=betabinomial;
package aods3 - function aodml, with family="bb";
package gamlss - function gamlss, with family=BB;
package glmmADMB - function glmmadmb, with family="betabinomial"
Zero-inflated binomial: package VGAM - function vglm, with family=zibinomial;
package gamlss - function gamlss, with family=ZIBI;
package glmmADMB - function glmmadmb, with family="binomial"
and zeroInfl=TRUE
Zero-inflated beta-binomial: package gamlss - function gamlss, with family=ZIBB;
package glmmADMB - function glmmadmb, with family="betabinomial"
and zeroInfl=TRUE
Multinomial: package nnet - function multinom
Count data:
Poisson: function glm with family=poisson
Quasi-Poisson: function glm with family=quasipoisson
Negative binomial: package MASS - function glm.nb;
package aods3 - function aodml, with family="nb"
and phi.scale="inverse"
Zero-inflated Poisson: package pscl - function zeroinfl, with dist="poisson"
Zero-inflated negative binomial: package pscl - function zeroinfl, with dist="negbin"
Hurdle Poisson: package pscl - function hurdle, with dist="poisson"
Hurdle negative binomial: package pscl - function hurdle, with dist="negbin"
Mixed models:
Linear mixed models: package lme4, function lmer
Generalized linear mixed models: package lme4, function glmer with family=poisson or binomial

Users can also use a numeric vector as object and hnp will generate the (half-)normal plot with a simulated envelope using the standard normal distribution (scale=F) or N(mu,sigma^2) (scale=T).

Implementing a new model class is done by providing three functions to hnp: diagfun - to obtain model diagnostics, simfun - to simulate random variables and fitfun - to refit the model to simulated variables. The way these functions must be written is shown in the Examples section.

Value

hnp returns an object of class "hnp", which is a list containing the following components:

x

quantiles of the (half-)normal distribution

lower

lower envelope band

median

median envelope band

upper

upper envelope band

residuals

diagnostic measures in absolute value and in order

out.index

vector indicating which points are out of the envelope

col.paint.out

color of points which are outside of the envelope (used if paint.out=TRUE)

how.many.out

logical. Equals TRUE if how.many.out=TRUE in the hnp call

total

length of the diagnostic measure vector

out

number of points out of the envelope

print.on

logical. Equals TRUE if print.on=TRUE in the hnp call

paint.out

logical. Equals TRUE if paint.out=TRUE in the hnp call

all.sim

matrix with all diagnostics obtained in the simulations. Each column represents one simulation

Note

See documentation on example data sets for simple analyses and goodness-of-fit checking using hnp.

Author(s)

Rafael A. Moral <rafael_moral@yahoo.com.br>, John Hinde and Clarice G. B. Demétrio

References

Moral, R. A., Hinde, J. and Demétrio, C. G. B. (2017) Half-normal plots and overdispersed models in R: the hnp package. Journal of Statistical Software 81(10):1-23.

Atkinson, A. C. (1985) Plots, transformations and regression, Clarendon Press, Oxford.

Demétrio, C. G. B. and Hinde, J. (1997) Half-normal plots and overdispersion. GLIM Newsletter 27:19-26.

Hinde, J. and Demétrio, C. G. B. (1998) Overdispersion: models and estimation. Computational Statistics and Data Analysis 27:151-170.

Demétrio, C. G. B., Hinde, J. and Moral, R. A. (2014) Models for overdispersed data in entomology. In Godoy, W. A. C. and Ferreira, C. P. (Eds.) Ecological modelling applied to entomology. Springer.

See Also

plot.hnp, cbb, chryso, corn, fungi, oil, progeny, wolbachia

Examples

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## Simple Poisson regression
set.seed(100)
counts <- c(rpois(5, 2), rpois(5, 4), rpois(5, 6), rpois(5, 8))
treatment <- gl(4, 5)
fit <- glm(counts ~ treatment, family=poisson)
anova(fit, test="Chisq")

## half-normal plot
hnp(fit)

## or save it in an object and then use the plot method
my.hnp <- hnp(fit, print.on=TRUE, plot=FALSE)
plot(my.hnp)

## changing graphical parameters
plot(my.hnp, lty=2, pch=4, cex=1.2)
plot(my.hnp, lty=c(2,3,2), pch=4, cex=1.2, col=c(2,2,2,1))
plot(my.hnp, main="Half-normal plot", xlab="Half-normal scores",
     ylab="Deviance residuals", legpos="bottomright")

## Using a numeric vector
my.vec <- rnorm(20, 4, 4)
hnp(my.vec) # using N(0,1)
hnp(my.vec, scale=TRUE) # using N(mu, sigma^2)

## Implementing new classes
## Users provide three functions - diagfun, simfun and fitfun,
## in the following way:
##
## diagfun <- function(obj) {
##   userfunction(obj, other_argumens)
##     # e.g., resid(obj, type="pearson")
##   }
##
## simfun <- function(n, obj) {
##   userfunction(n, other_arguments) # e.g., rpois(n, fitted(obj))
##   }
##
## fitfun <- function(y.) {
##  userfunction(y. ~ linear_predictor, other_arguments, data=data)
##    # e.g., glm(y. ~ block + factor1 * factor2, family=poisson,
##    #           data=mydata)
##  }
##
## when response is binary:
## fitfun <- function(y.) {
##  userfunction(cbind(y., m-y.) ~ linear_predictor,
##               other_arguments, data=data)
##    #e.g., glm(cbind(y., m-y.) ~ treatment - 1,
##    #          family=binomial, data=data)
##  }

## Not run: 
## Example no. 1: Using Cook's distance as a diagnostic measure
y <- rpois(30, lambda=rep(c(.5, 1.5, 5), each=10))
tr <- gl(3, 10)
fit1 <- glm(y ~ tr, family=poisson)

# diagfun
d.fun <- function(obj) cooks.distance(obj)

# simfun
s.fun <- function(n, obj) {
  lam <- fitted(obj)
  rpois(n, lambda=lam)
}

# fitfun
my.data <- data.frame(y, tr)
f.fun <- function(y.) glm(y. ~ tr, family=poisson, data=my.data)

# hnp call
hnp(fit1, newclass=TRUE, diagfun=d.fun, simfun=s.fun, fitfun=f.fun)

## Example no. 2: Implementing gamma model using package gamlss
# load package
require(gamlss)

# model fitting
y <- rGA(30, mu=rep(c(.5, 1.5, 5), each=10), sigma=.5)
tr <- gl(3, 10)
fit2 <- gamlss(y ~ tr, family=GA)

# diagfun
d.fun <- function(obj) resid(obj) # this is the default if no
                                  # diagfun is provided

# simfun
s.fun <- function(n, obj) {
  mu <- obj$mu.fv
  sig <- obj$sigma.fv
  rGA(n, mu=mu, sigma=sig)
}

# fitfun
my.data <- data.frame(y, tr)
f.fun <- function(y.) gamlss(y. ~ tr, family=GA, data=my.data)

# hnp call
hnp(fit2, newclass=TRUE, diagfun=d.fun, simfun=s.fun,
    fitfun=f.fun, data=data.frame(y, tr))

## Example no. 3: Implementing binomial model in gamlss
# model fitting
y <- rBI(30, bd=50, mu=rep(c(.2, .5, .9), each=10))
m <- 50
tr <- gl(3, 10)
fit3 <- gamlss(cbind(y, m-y) ~ tr, family=BI)

# diagfun
d.fun <- function(obj) resid(obj)

# simfun
s.fun <- function(n, obj) {
  mu <- obj$mu.fv
  bd <- obj$bd
  rBI(n, bd=bd, mu=mu)
}

# fitfun
my.data <- data.frame(y, tr, m)
f.fun <- function(y.) gamlss(cbind(y., m-y.) ~ tr,
                               family=BI, data=my.data)

# hnp call
hnp(fit3, newclass=TRUE, diagfun=d.fun, simfun=s.fun, fitfun=f.fun)

## End(Not run)

Example output

Loading required package: MASS
Analysis of Deviance Table

Model: poisson, link: log

Response: counts

Terms added sequentially (first to last)


          Df Deviance Resid. Df Resid. Dev  Pr(>Chi)    
NULL                         19     39.752              
treatment  3    30.43        16      9.323 1.121e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Poisson model 
Poisson model 
Half-normal plot with simulated envelope generated assuming the residuals are 
        normally distributed under the null hypothesis. 
Half-normal plot with simulated envelope generated assuming the residuals are 
        normally distributed under the null hypothesis. 
Estimated mean: 3.360678 
Estimated variance: 12.67925 
Loading required package: gamlss
Loading required package: splines
Loading required package: gamlss.data
Loading required package: gamlss.dist
Loading required package: nlme
Loading required package: parallel
 **********   GAMLSS Version 5.1-2  ********** 
For more on GAMLSS look at http://www.gamlss.org/
Type gamlssNews() to see new features/changes/bug fixes.

GAMLSS-RS iteration 1: Global Deviance = 70.338 
GAMLSS-RS iteration 2: Global Deviance = 70.338 
GAMLSS-RS iteration 1: Global Deviance = 49.6386 
GAMLSS-RS iteration 2: Global Deviance = 49.6386 
GAMLSS-RS iteration 1: Global Deviance = 62.7893 
GAMLSS-RS iteration 2: Global Deviance = 62.7893 
GAMLSS-RS iteration 1: Global Deviance = 75.0953 
GAMLSS-RS iteration 2: Global Deviance = 75.0953 
GAMLSS-RS iteration 1: Global Deviance = 54.6353 
GAMLSS-RS iteration 2: Global Deviance = 54.6353 
GAMLSS-RS iteration 1: Global Deviance = 64.3118 
GAMLSS-RS iteration 2: Global Deviance = 64.3118 
GAMLSS-RS iteration 1: Global Deviance = 62.2811 
GAMLSS-RS iteration 2: Global Deviance = 62.2811 
GAMLSS-RS iteration 1: Global Deviance = 44.5703 
GAMLSS-RS iteration 2: Global Deviance = 44.5703 
GAMLSS-RS iteration 1: Global Deviance = 51.3957 
GAMLSS-RS iteration 2: Global Deviance = 51.3957 
GAMLSS-RS iteration 1: Global Deviance = 63.1949 
GAMLSS-RS iteration 2: Global Deviance = 63.1949 
GAMLSS-RS iteration 1: Global Deviance = 73.5997 
GAMLSS-RS iteration 2: Global Deviance = 73.5997 
GAMLSS-RS iteration 1: Global Deviance = 73.5247 
GAMLSS-RS iteration 2: Global Deviance = 73.5247 
GAMLSS-RS iteration 1: Global Deviance = 69.6304 
GAMLSS-RS iteration 2: Global Deviance = 69.6304 
GAMLSS-RS iteration 1: Global Deviance = 78.5506 
GAMLSS-RS iteration 2: Global Deviance = 78.5506 
GAMLSS-RS iteration 1: Global Deviance = 65.1351 
GAMLSS-RS iteration 2: Global Deviance = 65.1351 
GAMLSS-RS iteration 1: Global Deviance = 60.8973 
GAMLSS-RS iteration 2: Global Deviance = 60.8973 
GAMLSS-RS iteration 1: Global Deviance = 58.2045 
GAMLSS-RS iteration 2: Global Deviance = 58.2045 
GAMLSS-RS iteration 1: Global Deviance = 55.71 
GAMLSS-RS iteration 2: Global Deviance = 55.71 
GAMLSS-RS iteration 1: Global Deviance = 37.9664 
GAMLSS-RS iteration 2: Global Deviance = 37.9664 
GAMLSS-RS iteration 1: Global Deviance = 71.3821 
GAMLSS-RS iteration 2: Global Deviance = 71.3821 
GAMLSS-RS iteration 1: Global Deviance = 77.8963 
GAMLSS-RS iteration 2: Global Deviance = 77.8963 
GAMLSS-RS iteration 1: Global Deviance = 60.9955 
GAMLSS-RS iteration 2: Global Deviance = 60.9955 
GAMLSS-RS iteration 1: Global Deviance = 45.579 
GAMLSS-RS iteration 2: Global Deviance = 45.579 
GAMLSS-RS iteration 1: Global Deviance = 59.431 
GAMLSS-RS iteration 2: Global Deviance = 59.431 
GAMLSS-RS iteration 1: Global Deviance = 75.4688 
GAMLSS-RS iteration 2: Global Deviance = 75.4688 
GAMLSS-RS iteration 1: Global Deviance = 46.5143 
GAMLSS-RS iteration 2: Global Deviance = 46.5143 
GAMLSS-RS iteration 1: Global Deviance = 75.0105 
GAMLSS-RS iteration 2: Global Deviance = 75.0105 
GAMLSS-RS iteration 1: Global Deviance = 58.4968 
GAMLSS-RS iteration 2: Global Deviance = 58.4968 
GAMLSS-RS iteration 1: Global Deviance = 81.5422 
GAMLSS-RS iteration 2: Global Deviance = 81.5422 
GAMLSS-RS iteration 1: Global Deviance = 61.8046 
GAMLSS-RS iteration 2: Global Deviance = 61.8046 
GAMLSS-RS iteration 1: Global Deviance = 65.7101 
GAMLSS-RS iteration 2: Global Deviance = 65.7101 
GAMLSS-RS iteration 1: Global Deviance = 55.8205 
GAMLSS-RS iteration 2: Global Deviance = 55.8205 
GAMLSS-RS iteration 1: Global Deviance = 51.2587 
GAMLSS-RS iteration 2: Global Deviance = 51.2587 
GAMLSS-RS iteration 1: Global Deviance = 67.9337 
GAMLSS-RS iteration 2: Global Deviance = 67.9337 
GAMLSS-RS iteration 1: Global Deviance = 70.374 
GAMLSS-RS iteration 2: Global Deviance = 70.374 
GAMLSS-RS iteration 1: Global Deviance = 71.8968 
GAMLSS-RS iteration 2: Global Deviance = 71.8968 
GAMLSS-RS iteration 1: Global Deviance = 74.3043 
GAMLSS-RS iteration 2: Global Deviance = 74.3043 
GAMLSS-RS iteration 1: Global Deviance = 48.7273 
GAMLSS-RS iteration 2: Global Deviance = 48.7273 
GAMLSS-RS iteration 1: Global Deviance = 78.2913 
GAMLSS-RS iteration 2: Global Deviance = 78.2913 
GAMLSS-RS iteration 1: Global Deviance = 62.408 
GAMLSS-RS iteration 2: Global Deviance = 62.408 
GAMLSS-RS iteration 1: Global Deviance = 64.443 
GAMLSS-RS iteration 2: Global Deviance = 64.443 
GAMLSS-RS iteration 1: Global Deviance = 70.6256 
GAMLSS-RS iteration 2: Global Deviance = 70.6256 
GAMLSS-RS iteration 1: Global Deviance = 56.6855 
GAMLSS-RS iteration 2: Global Deviance = 56.6855 
GAMLSS-RS iteration 1: Global Deviance = 63.3335 
GAMLSS-RS iteration 2: Global Deviance = 63.3335 
GAMLSS-RS iteration 1: Global Deviance = 79.7926 
GAMLSS-RS iteration 2: Global Deviance = 79.7926 
GAMLSS-RS iteration 1: Global Deviance = 51.2723 
GAMLSS-RS iteration 2: Global Deviance = 51.2723 
GAMLSS-RS iteration 1: Global Deviance = 64.2482 
GAMLSS-RS iteration 2: Global Deviance = 64.2482 
GAMLSS-RS iteration 1: Global Deviance = 69.4916 
GAMLSS-RS iteration 2: Global Deviance = 69.4916 
GAMLSS-RS iteration 1: Global Deviance = 49.7801 
GAMLSS-RS iteration 2: Global Deviance = 49.7801 
GAMLSS-RS iteration 1: Global Deviance = 83.6285 
GAMLSS-RS iteration 2: Global Deviance = 83.6285 
GAMLSS-RS iteration 1: Global Deviance = 72.1972 
GAMLSS-RS iteration 2: Global Deviance = 72.1972 
GAMLSS-RS iteration 1: Global Deviance = 47.3143 
GAMLSS-RS iteration 2: Global Deviance = 47.3143 
GAMLSS-RS iteration 1: Global Deviance = 62.6432 
GAMLSS-RS iteration 2: Global Deviance = 62.6432 
GAMLSS-RS iteration 1: Global Deviance = 88.1662 
GAMLSS-RS iteration 2: Global Deviance = 88.1662 
GAMLSS-RS iteration 1: Global Deviance = 69.7228 
GAMLSS-RS iteration 2: Global Deviance = 69.7228 
GAMLSS-RS iteration 1: Global Deviance = 76.7393 
GAMLSS-RS iteration 2: Global Deviance = 76.7393 
GAMLSS-RS iteration 1: Global Deviance = 66.008 
GAMLSS-RS iteration 2: Global Deviance = 66.008 
GAMLSS-RS iteration 1: Global Deviance = 66.8624 
GAMLSS-RS iteration 2: Global Deviance = 66.8624 
GAMLSS-RS iteration 1: Global Deviance = 83.9569 
GAMLSS-RS iteration 2: Global Deviance = 83.9569 
GAMLSS-RS iteration 1: Global Deviance = 67.7432 
GAMLSS-RS iteration 2: Global Deviance = 67.7432 
GAMLSS-RS iteration 1: Global Deviance = 60.4655 
GAMLSS-RS iteration 2: Global Deviance = 60.4655 
GAMLSS-RS iteration 1: Global Deviance = 64.9151 
GAMLSS-RS iteration 2: Global Deviance = 64.9151 
GAMLSS-RS iteration 1: Global Deviance = 60.1031 
GAMLSS-RS iteration 2: Global Deviance = 60.1031 
GAMLSS-RS iteration 1: Global Deviance = 70.4794 
GAMLSS-RS iteration 2: Global Deviance = 70.4794 
GAMLSS-RS iteration 1: Global Deviance = 72.162 
GAMLSS-RS iteration 2: Global Deviance = 72.162 
GAMLSS-RS iteration 1: Global Deviance = 70.0347 
GAMLSS-RS iteration 2: Global Deviance = 70.0347 
GAMLSS-RS iteration 1: Global Deviance = 71.8012 
GAMLSS-RS iteration 2: Global Deviance = 71.8012 
GAMLSS-RS iteration 1: Global Deviance = 65.4826 
GAMLSS-RS iteration 2: Global Deviance = 65.4826 
GAMLSS-RS iteration 1: Global Deviance = 58.9921 
GAMLSS-RS iteration 2: Global Deviance = 58.9921 
GAMLSS-RS iteration 1: Global Deviance = 47.6986 
GAMLSS-RS iteration 2: Global Deviance = 47.6986 
GAMLSS-RS iteration 1: Global Deviance = 69.3959 
GAMLSS-RS iteration 2: Global Deviance = 69.3959 
GAMLSS-RS iteration 1: Global Deviance = 62.0146 
GAMLSS-RS iteration 2: Global Deviance = 62.0146 
GAMLSS-RS iteration 1: Global Deviance = 69.0488 
GAMLSS-RS iteration 2: Global Deviance = 69.0488 
GAMLSS-RS iteration 1: Global Deviance = 69.3584 
GAMLSS-RS iteration 2: Global Deviance = 69.3584 
GAMLSS-RS iteration 1: Global Deviance = 49.6307 
GAMLSS-RS iteration 2: Global Deviance = 49.6307 
GAMLSS-RS iteration 1: Global Deviance = 79.9691 
GAMLSS-RS iteration 2: Global Deviance = 79.9691 
GAMLSS-RS iteration 1: Global Deviance = 66.2107 
GAMLSS-RS iteration 2: Global Deviance = 66.2107 
GAMLSS-RS iteration 1: Global Deviance = 74.1087 
GAMLSS-RS iteration 2: Global Deviance = 74.1087 
GAMLSS-RS iteration 1: Global Deviance = 54.582 
GAMLSS-RS iteration 2: Global Deviance = 54.582 
GAMLSS-RS iteration 1: Global Deviance = 52.5192 
GAMLSS-RS iteration 2: Global Deviance = 52.5192 
GAMLSS-RS iteration 1: Global Deviance = 73.577 
GAMLSS-RS iteration 2: Global Deviance = 73.577 
GAMLSS-RS iteration 1: Global Deviance = 67.4348 
GAMLSS-RS iteration 2: Global Deviance = 67.4348 
GAMLSS-RS iteration 1: Global Deviance = 58.4658 
GAMLSS-RS iteration 2: Global Deviance = 58.4658 
GAMLSS-RS iteration 1: Global Deviance = 65.5043 
GAMLSS-RS iteration 2: Global Deviance = 65.5043 
GAMLSS-RS iteration 1: Global Deviance = 67.6032 
GAMLSS-RS iteration 2: Global Deviance = 67.6032 
GAMLSS-RS iteration 1: Global Deviance = 71.118 
GAMLSS-RS iteration 2: Global Deviance = 71.118 
GAMLSS-RS iteration 1: Global Deviance = 58.9473 
GAMLSS-RS iteration 2: Global Deviance = 58.9473 
GAMLSS-RS iteration 1: Global Deviance = 61.9704 
GAMLSS-RS iteration 2: Global Deviance = 61.9704 
GAMLSS-RS iteration 1: Global Deviance = 58.1423 
GAMLSS-RS iteration 2: Global Deviance = 58.1423 
GAMLSS-RS iteration 1: Global Deviance = 75.5422 
GAMLSS-RS iteration 2: Global Deviance = 75.5422 
GAMLSS-RS iteration 1: Global Deviance = 74.8513 
GAMLSS-RS iteration 2: Global Deviance = 74.8513 
GAMLSS-RS iteration 1: Global Deviance = 53.5539 
GAMLSS-RS iteration 2: Global Deviance = 53.5539 
GAMLSS-RS iteration 1: Global Deviance = 51.5939 
GAMLSS-RS iteration 2: Global Deviance = 51.5939 
GAMLSS-RS iteration 1: Global Deviance = 76.6603 
GAMLSS-RS iteration 2: Global Deviance = 76.6603 
GAMLSS-RS iteration 1: Global Deviance = 72.0876 
GAMLSS-RS iteration 2: Global Deviance = 72.0876 
GAMLSS-RS iteration 1: Global Deviance = 67.1662 
GAMLSS-RS iteration 2: Global Deviance = 67.1662 
GAMLSS-RS iteration 1: Global Deviance = 68.1357 
GAMLSS-RS iteration 2: Global Deviance = 68.1357 
GAMLSS-RS iteration 1: Global Deviance = 62.0032 
GAMLSS-RS iteration 2: Global Deviance = 62.0032 
GAMLSS-RS iteration 1: Global Deviance = 51.9242 
GAMLSS-RS iteration 2: Global Deviance = 51.9242 
GAMLSS-RS iteration 1: Global Deviance = 67.6385 
GAMLSS-RS iteration 2: Global Deviance = 67.6385 
GAMLSS-RS iteration 1: Global Deviance = 142.445 
GAMLSS-RS iteration 2: Global Deviance = 142.445 
GAMLSS-RS iteration 1: Global Deviance = 146.0183 
GAMLSS-RS iteration 2: Global Deviance = 146.0183 
GAMLSS-RS iteration 1: Global Deviance = 139.978 
GAMLSS-RS iteration 2: Global Deviance = 139.978 
GAMLSS-RS iteration 1: Global Deviance = 145.5484 
GAMLSS-RS iteration 2: Global Deviance = 145.5484 
GAMLSS-RS iteration 1: Global Deviance = 144.4861 
GAMLSS-RS iteration 2: Global Deviance = 144.4861 
GAMLSS-RS iteration 1: Global Deviance = 144.7136 
GAMLSS-RS iteration 2: Global Deviance = 144.7136 
GAMLSS-RS iteration 1: Global Deviance = 136.2759 
GAMLSS-RS iteration 2: Global Deviance = 136.2759 
GAMLSS-RS iteration 1: Global Deviance = 153.2467 
GAMLSS-RS iteration 2: Global Deviance = 153.2467 
GAMLSS-RS iteration 1: Global Deviance = 139.2379 
GAMLSS-RS iteration 2: Global Deviance = 139.2379 
GAMLSS-RS iteration 1: Global Deviance = 134.0985 
GAMLSS-RS iteration 2: Global Deviance = 134.0985 
GAMLSS-RS iteration 1: Global Deviance = 142.0257 
GAMLSS-RS iteration 2: Global Deviance = 142.0257 
GAMLSS-RS iteration 1: Global Deviance = 128.216 
GAMLSS-RS iteration 2: Global Deviance = 128.216 
GAMLSS-RS iteration 1: Global Deviance = 134.6136 
GAMLSS-RS iteration 2: Global Deviance = 134.6136 
GAMLSS-RS iteration 1: Global Deviance = 132.2147 
GAMLSS-RS iteration 2: Global Deviance = 132.2147 
GAMLSS-RS iteration 1: Global Deviance = 143.6439 
GAMLSS-RS iteration 2: Global Deviance = 143.6439 
GAMLSS-RS iteration 1: Global Deviance = 149.8976 
GAMLSS-RS iteration 2: Global Deviance = 149.8976 
GAMLSS-RS iteration 1: Global Deviance = 132.4189 
GAMLSS-RS iteration 2: Global Deviance = 132.4189 
GAMLSS-RS iteration 1: Global Deviance = 137.5186 
GAMLSS-RS iteration 2: Global Deviance = 137.5186 
GAMLSS-RS iteration 1: Global Deviance = 139.043 
GAMLSS-RS iteration 2: Global Deviance = 139.043 
GAMLSS-RS iteration 1: Global Deviance = 138.5819 
GAMLSS-RS iteration 2: Global Deviance = 138.5819 
GAMLSS-RS iteration 1: Global Deviance = 130.3707 
GAMLSS-RS iteration 2: Global Deviance = 130.3707 
GAMLSS-RS iteration 1: Global Deviance = 158.7694 
GAMLSS-RS iteration 2: Global Deviance = 158.7694 
GAMLSS-RS iteration 1: Global Deviance = 144.417 
GAMLSS-RS iteration 2: Global Deviance = 144.417 
GAMLSS-RS iteration 1: Global Deviance = 144.8647 
GAMLSS-RS iteration 2: Global Deviance = 144.8647 
GAMLSS-RS iteration 1: Global Deviance = 133.1535 
GAMLSS-RS iteration 2: Global Deviance = 133.1535 
GAMLSS-RS iteration 1: Global Deviance = 158.762 
GAMLSS-RS iteration 2: Global Deviance = 158.762 
GAMLSS-RS iteration 1: Global Deviance = 139.8287 
GAMLSS-RS iteration 2: Global Deviance = 139.8287 
GAMLSS-RS iteration 1: Global Deviance = 146.1165 
GAMLSS-RS iteration 2: Global Deviance = 146.1165 
GAMLSS-RS iteration 1: Global Deviance = 146.151 
GAMLSS-RS iteration 2: Global Deviance = 146.151 
GAMLSS-RS iteration 1: Global Deviance = 132.3584 
GAMLSS-RS iteration 2: Global Deviance = 132.3584 
GAMLSS-RS iteration 1: Global Deviance = 142.3326 
GAMLSS-RS iteration 2: Global Deviance = 142.3326 
GAMLSS-RS iteration 1: Global Deviance = 135.1846 
GAMLSS-RS iteration 2: Global Deviance = 135.1846 
GAMLSS-RS iteration 1: Global Deviance = 158.6723 
GAMLSS-RS iteration 2: Global Deviance = 158.6723 
GAMLSS-RS iteration 1: Global Deviance = 135.4198 
GAMLSS-RS iteration 2: Global Deviance = 135.4198 
GAMLSS-RS iteration 1: Global Deviance = 132.9159 
GAMLSS-RS iteration 2: Global Deviance = 132.9159 
GAMLSS-RS iteration 1: Global Deviance = 143.8727 
GAMLSS-RS iteration 2: Global Deviance = 143.8727 
GAMLSS-RS iteration 1: Global Deviance = 135.3703 
GAMLSS-RS iteration 2: Global Deviance = 135.3703 
GAMLSS-RS iteration 1: Global Deviance = 130.2431 
GAMLSS-RS iteration 2: Global Deviance = 130.2431 
GAMLSS-RS iteration 1: Global Deviance = 154.0567 
GAMLSS-RS iteration 2: Global Deviance = 154.0567 
GAMLSS-RS iteration 1: Global Deviance = 151.7925 
GAMLSS-RS iteration 2: Global Deviance = 151.7925 
GAMLSS-RS iteration 1: Global Deviance = 153.1577 
GAMLSS-RS iteration 2: Global Deviance = 153.1577 
GAMLSS-RS iteration 1: Global Deviance = 142.2882 
GAMLSS-RS iteration 2: Global Deviance = 142.2882 
GAMLSS-RS iteration 1: Global Deviance = 145.2743 
GAMLSS-RS iteration 2: Global Deviance = 145.2743 
GAMLSS-RS iteration 1: Global Deviance = 156.6035 
GAMLSS-RS iteration 2: Global Deviance = 156.6035 
GAMLSS-RS iteration 1: Global Deviance = 153.0664 
GAMLSS-RS iteration 2: Global Deviance = 153.0664 
GAMLSS-RS iteration 1: Global Deviance = 137.2799 
GAMLSS-RS iteration 2: Global Deviance = 137.2799 
GAMLSS-RS iteration 1: Global Deviance = 144.2358 
GAMLSS-RS iteration 2: Global Deviance = 144.2358 
GAMLSS-RS iteration 1: Global Deviance = 135.6286 
GAMLSS-RS iteration 2: Global Deviance = 135.6286 
GAMLSS-RS iteration 1: Global Deviance = 156.0447 
GAMLSS-RS iteration 2: Global Deviance = 156.0447 
GAMLSS-RS iteration 1: Global Deviance = 155.485 
GAMLSS-RS iteration 2: Global Deviance = 155.485 
GAMLSS-RS iteration 1: Global Deviance = 138.9901 
GAMLSS-RS iteration 2: Global Deviance = 138.9901 
GAMLSS-RS iteration 1: Global Deviance = 144.0765 
GAMLSS-RS iteration 2: Global Deviance = 144.0765 
GAMLSS-RS iteration 1: Global Deviance = 140.1216 
GAMLSS-RS iteration 2: Global Deviance = 140.1216 
GAMLSS-RS iteration 1: Global Deviance = 154.8097 
GAMLSS-RS iteration 2: Global Deviance = 154.8097 
GAMLSS-RS iteration 1: Global Deviance = 150.4456 
GAMLSS-RS iteration 2: Global Deviance = 150.4456 
GAMLSS-RS iteration 1: Global Deviance = 137.0094 
GAMLSS-RS iteration 2: Global Deviance = 137.0094 
GAMLSS-RS iteration 1: Global Deviance = 142.4653 
GAMLSS-RS iteration 2: Global Deviance = 142.4653 
GAMLSS-RS iteration 1: Global Deviance = 161.6731 
GAMLSS-RS iteration 2: Global Deviance = 161.6731 
GAMLSS-RS iteration 1: Global Deviance = 141.087 
GAMLSS-RS iteration 2: Global Deviance = 141.087 
GAMLSS-RS iteration 1: Global Deviance = 133.4353 
GAMLSS-RS iteration 2: Global Deviance = 133.4353 
GAMLSS-RS iteration 1: Global Deviance = 144.1005 
GAMLSS-RS iteration 2: Global Deviance = 144.1005 
GAMLSS-RS iteration 1: Global Deviance = 148.4537 
GAMLSS-RS iteration 2: Global Deviance = 148.4537 
GAMLSS-RS iteration 1: Global Deviance = 139.6583 
GAMLSS-RS iteration 2: Global Deviance = 139.6583 
GAMLSS-RS iteration 1: Global Deviance = 145.9598 
GAMLSS-RS iteration 2: Global Deviance = 145.9598 
GAMLSS-RS iteration 1: Global Deviance = 143.8283 
GAMLSS-RS iteration 2: Global Deviance = 143.8283 
GAMLSS-RS iteration 1: Global Deviance = 141.4765 
GAMLSS-RS iteration 2: Global Deviance = 141.4765 
GAMLSS-RS iteration 1: Global Deviance = 138.3281 
GAMLSS-RS iteration 2: Global Deviance = 138.3281 
GAMLSS-RS iteration 1: Global Deviance = 137.4648 
GAMLSS-RS iteration 2: Global Deviance = 137.4648 
GAMLSS-RS iteration 1: Global Deviance = 141.2079 
GAMLSS-RS iteration 2: Global Deviance = 141.2079 
GAMLSS-RS iteration 1: Global Deviance = 132.0084 
GAMLSS-RS iteration 2: Global Deviance = 132.0084 
GAMLSS-RS iteration 1: Global Deviance = 138.8554 
GAMLSS-RS iteration 2: Global Deviance = 138.8554 
GAMLSS-RS iteration 1: Global Deviance = 136.5733 
GAMLSS-RS iteration 2: Global Deviance = 136.5733 
GAMLSS-RS iteration 1: Global Deviance = 141.6205 
GAMLSS-RS iteration 2: Global Deviance = 141.6205 
GAMLSS-RS iteration 1: Global Deviance = 140.9606 
GAMLSS-RS iteration 2: Global Deviance = 140.9606 
GAMLSS-RS iteration 1: Global Deviance = 132.6981 
GAMLSS-RS iteration 2: Global Deviance = 132.6981 
GAMLSS-RS iteration 1: Global Deviance = 140.103 
GAMLSS-RS iteration 2: Global Deviance = 140.103 
GAMLSS-RS iteration 1: Global Deviance = 131.8618 
GAMLSS-RS iteration 2: Global Deviance = 131.8618 
GAMLSS-RS iteration 1: Global Deviance = 136.1863 
GAMLSS-RS iteration 2: Global Deviance = 136.1863 
GAMLSS-RS iteration 1: Global Deviance = 142.6953 
GAMLSS-RS iteration 2: Global Deviance = 142.6953 
GAMLSS-RS iteration 1: Global Deviance = 143.7413 
GAMLSS-RS iteration 2: Global Deviance = 143.7413 
GAMLSS-RS iteration 1: Global Deviance = 162.5236 
GAMLSS-RS iteration 2: Global Deviance = 162.5236 
GAMLSS-RS iteration 1: Global Deviance = 152.9845 
GAMLSS-RS iteration 2: Global Deviance = 152.9845 
GAMLSS-RS iteration 1: Global Deviance = 146.0983 
GAMLSS-RS iteration 2: Global Deviance = 146.0983 
GAMLSS-RS iteration 1: Global Deviance = 144.9088 
GAMLSS-RS iteration 2: Global Deviance = 144.9088 
GAMLSS-RS iteration 1: Global Deviance = 143.2483 
GAMLSS-RS iteration 2: Global Deviance = 143.2483 
GAMLSS-RS iteration 1: Global Deviance = 154.0586 
GAMLSS-RS iteration 2: Global Deviance = 154.0586 
GAMLSS-RS iteration 1: Global Deviance = 149.0875 
GAMLSS-RS iteration 2: Global Deviance = 149.0875 
GAMLSS-RS iteration 1: Global Deviance = 131.6215 
GAMLSS-RS iteration 2: Global Deviance = 131.6215 
GAMLSS-RS iteration 1: Global Deviance = 143.1584 
GAMLSS-RS iteration 2: Global Deviance = 143.1584 
GAMLSS-RS iteration 1: Global Deviance = 142.5994 
GAMLSS-RS iteration 2: Global Deviance = 142.5994 
GAMLSS-RS iteration 1: Global Deviance = 140.0903 
GAMLSS-RS iteration 2: Global Deviance = 140.0903 
GAMLSS-RS iteration 1: Global Deviance = 152.2458 
GAMLSS-RS iteration 2: Global Deviance = 152.2458 
GAMLSS-RS iteration 1: Global Deviance = 144.924 
GAMLSS-RS iteration 2: Global Deviance = 144.924 
GAMLSS-RS iteration 1: Global Deviance = 149.1011 
GAMLSS-RS iteration 2: Global Deviance = 149.1011 
GAMLSS-RS iteration 1: Global Deviance = 140.4613 
GAMLSS-RS iteration 2: Global Deviance = 140.4613 
GAMLSS-RS iteration 1: Global Deviance = 143.4972 
GAMLSS-RS iteration 2: Global Deviance = 143.4972 
GAMLSS-RS iteration 1: Global Deviance = 145.2543 
GAMLSS-RS iteration 2: Global Deviance = 145.2543 
GAMLSS-RS iteration 1: Global Deviance = 143.4959 
GAMLSS-RS iteration 2: Global Deviance = 143.4959 
GAMLSS-RS iteration 1: Global Deviance = 136.7692 
GAMLSS-RS iteration 2: Global Deviance = 136.7692 
GAMLSS-RS iteration 1: Global Deviance = 138.8718 
GAMLSS-RS iteration 2: Global Deviance = 138.8718

hnp documentation built on May 2, 2019, 12:40 p.m.

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