hglm2: Fitting Hierarchical Generalized Linear Models

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

Description

hglm2 is used to fit hierarchical generalized linear models. hglm2 is used to fit hierarchical generalized linear models. It extends the hglm function by allowing for several random effects, where the model is specified in lme4 convension, and also by implementing sparse matrix techniques using the Matrix library.

Usage

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hglm2(meanmodel, data = NULL, family = gaussian(link = identity),
      rand.family = gaussian(link = identity), method = "EQL", 
      conv = 1e-6, maxit = 50, startval = NULL,
      X.disp = NULL, disp = NULL, link.disp = "log", 
      weights = NULL, fix.disp = NULL, offset = NULL, 
      sparse = TRUE, vcovmat = FALSE, calc.like = FALSE, 
      RandC = NULL, bigRR = FALSE, verbose = FALSE, ...)

Arguments

meanmodel

formula. A two sided formula specifying the fixed and random terms in lme4 convention, e.g. y ~ x1 + (1|id) indicates y as response, x1 as the fixed effect and (1|id) represent a random intercept for each level of id.

data

data.frame. An optional data frame from where the variables in the meanmodel (and possibly disp) are to be obtained. It is expected that the data frame does not contain any missing value.

family

family. The description of the error distribution and link function to be used in the mean part of the model. (See family for details of family functions.)

rand.family

family. The description of the distribution and link function to be used for the random effect.

method

character. Estimation method where EQL is the method of interconnected GLMs presented in Lee et al (2006). Apart from the default option EQL there is also an EQL1 option, which improves estimation for GLMMs (especially for Poisson models with a large number of levels in the random effects).

conv

numeric. The convergence criteria (change in linear predictor between iterations).

maxit

numeric. Maximum number of iterations in the hglm algorithm.

startval

numeric. A vector of starting values in the following order: fixed effects, random effect, variance of random effects, variance of residuals.

X.disp

matrix. The design matrix for the fixed effects in the dispersion part of the model.

disp

formula. A one-sided formula specifying the fixed effects in the dispersion part of the model.

link.disp

character. The link function for the dispersion part of the model.

weights

numeric. Prior weights to be specified in weighted regression.

fix.disp

numeric. A numeric value if the dispersion parameter of the mean model is known, e.g., 1 for binomial and Poisson model.

offset

An offset for the linear predictor of the mean model.

sparse

logical. If TRUE, the computation is to be carried out by using sparse matrix technique.

vcovmat

logical. If TRUE, the variance-covariance matrix is exported.

calc.like

logical. If TRUE, likelihoods will be computed at convergence and will be shown via the print or summary methods on the output object.

RandC

numeric. Necessary in old versions but can be neglected now. Integers (possibly a vector) specifying the number of column of Z to be used for each of the random-effect terms.

bigRR

logical. If TRUE, and only for the Gaussian model with one random effect term, a specific algorithm will be used for fast fitting high-dimensional (p >> n) problems. See Shen et al. (2013) for more details of the method.

verbose

logical. If TRUE, more information is printed during model fitting process.

...

not used.

Details

Models for hglm are either specified symbolically using formula or by specifying the design matrices ( X, Z and X.disp). Currently, only the extended quasi likelihood (EQL) method is available for the estimation of the model parameters. Only for the Gaussian-Gaussina linear mixed models, it is REML. It should be noted that the EQL estimator can be biased and inconsistent in some special cases e.g. binary pair matched response. A higher order correction might be useful to correct the bias of EQL (Lee et al. 2006). But, those currections are not implemented in the current version. By default, the dispersion parameter is always estimated via EQL. If the dispersion parameter of the mean model is to be held constant, for example if it is desired to be 1 for binomial and Poisson family, then fix.disp=value where, value=1 for the above example, should be used.

Value

It returns an object of class hglm consiting of the following values.

fixef

fixed effect estimates.

ranef

random effect estimates.

RandC

integers (possibly a vector) specified the number of column of Z to be used for each of the random-effect terms.

varFix

dispersion parameter of the mean model (residual variance for LMM).

varRanef

dispersion parameter of the random effects (variance of random effects for GLMM).

iter

number of iterations used.

Converge

specifies if the algorithm converged.

SeFe

standard errors of fixed effects.

SeRe

standard errors of random effects.

dfReFe

deviance degrees of freedom for the mean part of the model.

SummVC1

estimates and standard errors of the linear predictor in the dispersion model.

SummVC2

estimates and standard errors of the linear predictor for the dispersion parameter of the random effects.

dev

individual deviances for the mean part of the model.

hv

hatvalues for the mean part of the model.

resid

studentized residuals for the mean part of the model.

fv

fitted values for the mean part of the model.

disp.fv

fitted values for the dispersion part of the model.

disp.resid

standardized deviance residuals for the dispersion part of the model.

link.disp

link function for the dispersion part of the model.

vcov

the variance-covariance matrix.

likelihood

a list of log-likelihood values for model selection purposes, where $hlik is -2 times the log-h-likelihood, $pvh -2 times the adjusted profile log-likelihood profiled over random effects, $pbvh -2 times the adjusted profile log-likelihood profiled over fixed and random effects, and $cAIC the conditional AIC.

bad

the index of the influential observation.

Author(s)

Moudud Alam, Xia Shen, Lars Ronnegard

References

Lars Ronnegard, Xia Shen and Moudud Alam (2010). hglm: A Package for Fitting Hierarchical Generalized Linear Models. The R Journal, 2(2), 20-28.

Youngjo Lee, John A Nelder and Yudi Pawitan (2006) Generalized Linear Models with Random Effect: a unified analysis via h-likelihood. Chapman and Hall/CRC.

Xia Shen, Moudud Alam, Freddy Fikse and Lars Ronnegard (2013). A novel generalized ridge regression method for quantitative genetics. Genetics.

Moudud Alam, Lars Ronnegard, Xia Shen (2014). Fitting conditional and simultaneous autoregressive spatial models in hglm. Submitted.

See Also

hglm

Examples

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# Find more examples and instructions in the package vignette:
# vignette('hglm')

require(hglm)

# --------------------- #
# semiconductor example #
# --------------------- #

data(semiconductor)

m11 <- hglm(fixed = y ~ x1 + x3 + x5 + x6,
            random = ~ 1|Device,
            family = Gamma(link = log),
            disp = ~ x2 + x3, data = semiconductor)
summary(m11)
plot(m11, cex = .6, pch = 1,
     cex.axis = 1/.6, cex.lab = 1/.6,
     cex.main = 1/.6, mar = c(3, 4.5, 0, 1.5))

# ------------------- #
# redo it using hglm2 #
# ------------------- #

m12 <- hglm2(y ~ x1 + x3 + x5 + x6 + (1|Device),
             family = Gamma(link = log),
             disp = ~ x2 + x3, data = semiconductor)
summary(m12)
     
# -------------------------- #
# redo it using matrix input #
# -------------------------- #

attach(semiconductor)
m13 <- hglm(y = y, X = model.matrix(~ x1 + x3 + x5 + x6),
            Z = kronecker(diag(16), rep(1, 4)),
            X.disp = model.matrix(~ x2 + x3), 
            family = Gamma(link = log))
summary(m13)
     
# --------------------- #
# verbose & likelihoods #
# --------------------- #
     
m14 <- hglm(fixed = y ~ x1 + x3 + x5 + x6,
            random = ~ 1|Device,
            family = Gamma(link = log),
            disp = ~ x2 + x3, data = semiconductor,
            verbose = TRUE, calc.like = TRUE)
summary(m14)

# --------------------------------------------- #  
# simulated example with 2 random effects terms #
# --------------------------------------------- #  
## Not run: 
set.seed(911)
x1 <- rnorm(100)
x2 <- rnorm(100)
x3 <- rnorm(100)
z1 <- factor(rep(LETTERS[1:10], rep(10, 10)))
z2 <- factor(rep(letters[1:5], rep(20, 5)))
Z1 <- model.matrix(~ 0 + z1)
Z2 <- model.matrix(~ 0 + z2)
u1 <- rnorm(10, 0, sqrt(2))
u2 <- rnorm(5, 0, sqrt(3))
y <- 1 + 2*x1 + 3*x2 + Z1%*%u1 + Z2%*%u2 + rnorm(100, 0, sqrt(exp(x3)))
dd <- data.frame(x1 = x1, x2 = x2, x3 = x3, z1 = z1, z2 = z2, y = y)

(m21 <- hglm(X = cbind(rep(1, 100), x1, x2), y = y, Z = cbind(Z1, Z2), 
              RandC = c(10, 5)))
summary(m21)
plot(m21)

(m22 <- hglm2(y ~ x1 + x2 + (1|z1) + (1|z2), data = dd, vcovmat = TRUE))
image(m22$vcov, main = 'Variance-covariance Matrix')
summary(m22)
plot(m22)

m31 <- hglm2(y ~ x1 + x2 + (1|z1) + (1|z2), disp = ~ x3, data = dd)
print (m31)
summary(m31)
plot(m31)

## End(Not run)

Example output

Loading required package: Matrix
Loading required package: MASS
Loading required package: hglm.data

hglm: Hierarchical Generalized Linear Models
Version 2.1-1 (2015-08-28) installed
Authors: Moudud Alam, Lars Ronnegard, Xia Shen
Maintainer: Xia Shen <xia.shen@ki.se>

Use citation("hglm") to know how to cite our work.

Discussion: https://r-forge.r-project.org/forum/?group_id=558
BugReports: https://r-forge.r-project.org/tracker/?group_id=558
VideoTutorials: http://www.youtube.com/playlist?list=PLn1OmZECD-n15vnYzvJDy5GxjNpVV5Jr8
Call: 
hglm.formula(family = Gamma(link = log), fixed = y ~ x1 + x3 + 
    x5 + x6, random = ~1 | Device, disp = ~x2 + x3, data = semiconductor)

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept) -4.71168    0.06696 -70.368  < 2e-16 ***
x1           0.20979    0.06638   3.160  0.00263 ** 
x3           0.32893    0.06696   4.913 9.34e-06 ***
x5          -0.17314    0.06638  -2.608  0.01185 *  
x6          -0.35690    0.06633  -5.380 1.80e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 52 degrees of freedom

Summary of the random effects estimates:

                   Estimate Std. Error
as.factor(Device)1   0.2724     0.1787
as.factor(Device)2   0.0097     0.1787
as.factor(Device)3  -0.2697     0.1584
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Model estimates for the dispersion term:

Link = log 

Effects:
            Estimate Std. Error
(Intercept)  -2.5887     0.1972
x2           -0.6861     0.1971
x3           -0.5024     0.1971

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 0.0486

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
   -3.0242     0.5172 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 4 iterations.
dev.new(): using pdf(file="Rplots1.pdf")
dev.new(): using pdf(file="Rplots2.pdf")
Call: 
hglm2.formula(meanmodel = y ~ x1 + x3 + x5 + x6 + (1 | Device), 
    data = semiconductor, family = Gamma(link = log), disp = ~x2 + 
        x3)

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept) -4.71168    0.06696 -70.368  < 2e-16 ***
x1           0.20979    0.06638   3.160  0.00263 ** 
x3           0.32893    0.06696   4.913 9.34e-06 ***
x5          -0.17314    0.06638  -2.608  0.01185 *  
x6          -0.35690    0.06633  -5.380 1.80e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 52 degrees of freedom

Summary of the random effects estimates:

                      Estimate Std. Error
(Intercept)| Device:1   0.2724     0.1787
(Intercept)| Device:2   0.0097     0.1787
(Intercept)| Device:3  -0.2697     0.1584
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Model estimates for the dispersion term:

Link = log 

Effects:
            Estimate Std. Error
(Intercept)  -2.5887     0.1972
x2           -0.6861     0.1971
x3           -0.5024     0.1971

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 0.0486

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
   -3.0242     0.5172 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 4 iterations.
Call: 
hglm.default(X = model.matrix(~x1 + x3 + x5 + x6), y = y, Z = kronecker(diag(16), 
    rep(1, 4)), family = Gamma(link = log), X.disp = model.matrix(~x2 + 
    x3))

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept) -4.71168    0.06696 -70.368  < 2e-16 ***
x1           0.20979    0.06638   3.160  0.00263 ** 
x3           0.32893    0.06696   4.913 9.34e-06 ***
x5          -0.17314    0.06638  -2.608  0.01185 *  
x6          -0.35690    0.06633  -5.380 1.80e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 52 degrees of freedom

Summary of the random effects estimates:

    Estimate Std. Error
Z.1   0.2724     0.1787
Z.2   0.0097     0.1787
Z.3  -0.2697     0.1584
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Model estimates for the dispersion term:

Link = log 

Effects:
            Estimate Std. Error
(Intercept)  -2.5887     0.1972
x2           -0.6861     0.1971
x3           -0.5024     0.1971

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 0.0486

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
   -3.0242     0.5172 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 4 iterations.

-------------------
Iteration 1 
-------------------
Sum of squared linear predictor: 1437.809 
Convergence: 0.001198849 

-------------------
Iteration 2 
-------------------
Sum of squared linear predictor: 1441.496 
Convergence: 9.225213e-05 

-------------------
Iteration 3 
-------------------
Sum of squared linear predictor: 1441.963 
Convergence: 1.078477e-06 

-------------------
Iteration 4 
-------------------
Sum of squared linear predictor: 1442.064 
Convergence: 8.388541e-08 
Call: 
hglm.formula(family = Gamma(link = log), fixed = y ~ x1 + x3 + 
    x5 + x6, random = ~1 | Device, disp = ~x2 + x3, data = semiconductor, 
    calc.like = TRUE, verbose = TRUE)

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept) -4.71168    0.06696 -70.368  < 2e-16 ***
x1           0.20979    0.06638   3.160  0.00263 ** 
x3           0.32893    0.06696   4.913 9.34e-06 ***
x5          -0.17314    0.06638  -2.608  0.01185 *  
x6          -0.35690    0.06633  -5.380 1.80e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 52 degrees of freedom

Summary of the random effects estimates:

                   Estimate Std. Error
as.factor(Device)1   0.2724     0.1787
as.factor(Device)2   0.0097     0.1787
as.factor(Device)3  -0.2697     0.1584
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Model estimates for the dispersion term:

Link = log 

Effects:
            Estimate Std. Error
(Intercept)  -2.5887     0.1972
x2           -0.6861     0.1971
x3           -0.5024     0.1971

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 0.0486

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
   -3.0242     0.5172 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

---------------
LOG-LIKELIHOODS
---------------

h-likelihood: 307.1976 
Adjusted profile likelihood 
   Profiled over random effects: 286.9197 
   Profiled over fixed and random effects: 277.9467 
Conditional AIC: -577.9375 

EQL estimation converged in 4 iterations.
Call: 
hglm.default(X = cbind(rep(1, 100), x1, x2), y = y, Z = cbind(Z1, 
    Z2), RandC = c(10, 5))

---------------------------
Estimates of the mean model
---------------------------

Fixed effects:
               x1       x2 
1.593223 2.163230 2.875127 

Random effects:
       z1A        z1B        z1C 
 0.5876582  0.7925926 -1.3893579 
...
      z1I       z1J 
-1.071900  1.615409 
NOTE: to show all the random effects estimates, use print(hglm.object, print.ranef = TRUE).

Random effects:
       z2a        z2b        z2c        z2d        z2e 
 1.0972865 -1.2008286  0.5139767 -0.8425195  0.4320849 

Dispersion parameter for the mean model: 1.574819 

Dispersion parameter for the random effects: 2.079812 1.597671 

Estimation converged in 4 iterations
Call: 
hglm.default(X = cbind(rep(1, 100), x1, x2), y = y, Z = cbind(Z1, 
    Z2), RandC = c(10, 5))

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

   Estimate Std. Error t-value Pr(>|t|)    
     1.5932     0.7410    2.15   0.0343 *  
x1   2.1632     0.1321   16.38   <2e-16 ***
x2   2.8751     0.1210   23.77   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 88 degrees of freedom

Summary of the random effects estimates:

    Estimate Std. Error
z1A   0.5877     0.8920
z1B   0.7926     0.8911
z1C  -1.3894     0.8918
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

Summary of the random effects estimates:

    Estimate Std. Error
z2a   1.0973     0.9263
z2b  -1.2008     0.9254
z2c   0.5140     0.9254
z2d  -0.8425     0.9255
z2e   0.4321     0.9252

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Dispersion parameter for the mean model:
[1] 1.574819

Model estimates for the dispersion term:

Link = log 

Effects:
  Estimate Std. Error 
    0.4541     0.1503 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 2.080 1.598

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
    0.7323     0.5707 

.|Random2 
  Estimate Std. Error 
    0.4685     0.9162 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 4 iterations.
dev.new(): using pdf(file="Rplots3.pdf")
dev.new(): using pdf(file="Rplots4.pdf")
Call: 
hglm2.formula(meanmodel = y ~ x1 + x2 + (1 | z1) + (1 | z2), 
    data = dd, vcovmat = TRUE)

---------------------------
Estimates of the mean model
---------------------------

Fixed effects:
(Intercept)          x1          x2 
   1.593223    2.163230    2.875127 

Random effects:
(Intercept)| z1:A (Intercept)| z1:B (Intercept)| z1:C 
        0.5876582         0.7925926        -1.3893579 
...
(Intercept)| z1:I (Intercept)| z1:J 
        -1.071900          1.615409 
NOTE: to show all the random effects estimates, use print(hglm.object, print.ranef = TRUE).

Random effects:
(Intercept)| z2:a (Intercept)| z2:b (Intercept)| z2:c (Intercept)| z2:d 
        1.0972865        -1.2008286         0.5139767        -0.8425195 
(Intercept)| z2:e 
        0.4320849 

Dispersion parameter for the mean model: 1.574819 

Dispersion parameter for the random effects: 2.079812 1.597671 

Estimation converged in 4 iterations
Call: 
hglm2.formula(meanmodel = y ~ x1 + x2 + (1 | z1) + (1 | z2), 
    data = dd, vcovmat = TRUE)

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept)   1.5932     0.7410    2.15   0.0343 *  
x1            2.1632     0.1321   16.38   <2e-16 ***
x2            2.8751     0.1210   23.77   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 88 degrees of freedom

Summary of the random effects estimates:

                  Estimate Std. Error
(Intercept)| z1:A   0.5877     0.8920
(Intercept)| z1:B   0.7926     0.8911
(Intercept)| z1:C  -1.3894     0.8918
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

Summary of the random effects estimates:

                  Estimate Std. Error
(Intercept)| z2:a   1.0973     0.9263
(Intercept)| z2:b  -1.2008     0.9254
(Intercept)| z2:c   0.5140     0.9254
(Intercept)| z2:d  -0.8425     0.9255
(Intercept)| z2:e   0.4321     0.9252

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Dispersion parameter for the mean model:
[1] 1.574819

Model estimates for the dispersion term:

Link = log 

Effects:
  Estimate Std. Error 
    0.4541     0.1503 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 2.080 1.598

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
    0.7323     0.5707 

.|Random2 
  Estimate Std. Error 
    0.4685     0.9162 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 4 iterations.
dev.new(): using pdf(file="Rplots5.pdf")
dev.new(): using pdf(file="Rplots6.pdf")
Call: 
hglm2.formula(meanmodel = y ~ x1 + x2 + (1 | z1) + (1 | z2), 
    data = dd, disp = ~x3)

---------------------------
Estimates of the mean model
---------------------------

Fixed effects:
(Intercept)          x1          x2 
   1.676184    2.148076    2.989964 

Random effects:
(Intercept)| z1:A (Intercept)| z1:B (Intercept)| z1:C 
        0.1914595         1.3496934        -1.2882400 
...
(Intercept)| z1:I (Intercept)| z1:J 
        -1.327242          1.596106 
NOTE: to show all the random effects estimates, use print(hglm.object, print.ranef = TRUE).

Random effects:
(Intercept)| z2:a (Intercept)| z2:b (Intercept)| z2:c (Intercept)| z2:d 
        0.8394165        -0.9463058         0.5146495        -0.5542017 
(Intercept)| z2:e 
        0.1464415 

------------------------------------------
Estimates of the residual dispersion model
------------------------------------------

Link = log 

Effects:
(Intercept)          x3 
 -0.1065797   0.9969769 

Dispersion parameter for the random effects: 2.041197 1.067874 

Estimation converged in 6 iterations
Call: 
hglm2.formula(meanmodel = y ~ x1 + x2 + (1 | z1) + (1 | z2), 
    data = dd, disp = ~x3)

----------
MEAN MODEL
----------

Summary of the fixed effects estimates:

            Estimate Std. Error t-value Pr(>|t|)    
(Intercept)  1.67618    0.65467    2.56   0.0122 *  
x1           2.14808    0.08218   26.14   <2e-16 ***
x2           2.98996    0.07374   40.55   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note: P-values are based on 88 degrees of freedom

Summary of the random effects estimates:

                  Estimate Std. Error
(Intercept)| z1:A   0.1915     0.8087
(Intercept)| z1:B   1.3497     0.8047
(Intercept)| z1:C  -1.2882     0.8179
...
NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).

Summary of the random effects estimates:

                  Estimate Std. Error
(Intercept)| z2:a   0.8394     0.8051
(Intercept)| z2:b  -0.9463     0.8055
(Intercept)| z2:c   0.5146     0.8051
(Intercept)| z2:d  -0.5542     0.8049
(Intercept)| z2:e   0.1464     0.8070

----------------
DISPERSION MODEL
----------------

NOTE: h-likelihood estimates through EQL can be biased.

Model estimates for the dispersion term:

Link = log 

Effects:
            Estimate Std. Error
(Intercept)  -0.1066     0.1516
x3            0.9970     0.1485

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

Dispersion parameter for the random effects:
[1] 2.041 1.068

Dispersion model for the random effects:

Link = log

Effects:
.|Random1 
  Estimate Std. Error 
    0.7135     0.5449 

.|Random2 
  Estimate Std. Error 
    0.0657     0.9867 

Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).

EQL estimation converged in 6 iterations.
dev.new(): using pdf(file="Rplots7.pdf")
dev.new(): using pdf(file="Rplots8.pdf")
dev.new(): using pdf(file="Rplots9.pdf")

hglm documentation built on May 2, 2019, 6:11 p.m.