A step by step example
In this section I am going to present a step by step example whose
aim is to show how the R package ptmixed can
be used to estimate the Poisson-Tweedie GLMM, as well as a few simpler
models.
Data preparation
All functions in the package assume that data are in the so-called
“long
format”. Let’s generate an example dataset (already in long format)
using the function simulate_ptglmm:
example.df = simulate_ptglmm(n = 14, t = 4, seed = 1234,
beta = c(2.3, -0.9, -0.2, 0.5),
D = 1.5, a = -1,
sigma2 = 0.7)## Loading required namespace: tweeDEseqdata.long = example.df$data
head(data.long)## y id group time
## 1 6 1 0 0
## 2 0 1 0 1
## 3 2 1 0 2
## 4 1 1 0 3
## 5 10 2 0 0
## 6 10 2 0 1In this example I have generated a dataset of 14 subjects with 4
repeated measurements each. y is the response variable,
id denotes the subject identicator, group is a
dummy variable and time is the time at which a measurement
was taken.
Data visualization
Before fitting a model, it is often useful to make a few plots to get a feeling of the data that you would like to model. Below I use two simple plots to visualize the distribution of the response variable and its relationship with the available covariates.
We can view the marginal distribution of the response variable
y using the function pmf, and visualize the
individual trajectories of subjects over time using the function
make.spaghetti:
pmf(data.long$y, xlab = 'y', title = 'Distribution of y')
make.spaghetti(x = time, y = y, id = id,
group = group, data = data.long,
title = 'Trajectory ("spaghetti") plot',
legend.title = 'GROUP')
The Poisson-Tweedie generalized linear mixed model
The most important function of the package, ptmixed, is
a function that makes it possible to carry out maximum
likelihood (ML) estimation of the Poisson-Tweedie GLMM. This
function employs the adaptive Gauss-Hermite quadrature (AGHQ) method to
evaluate the marginal likelihood of the GLMM, and then maximizes this
likelihood using the Nelder-Mead and BFGS methods. Finally, if
hessian = T (default value) a numerical evaluation of the
hessian matrix (needed to compute the standard errors associated to the
parameter estimates) in correspondance of the ML estimate is
performed.
Model Estimation
Estimation of the Poisson-Tweedie GLMM can be carried out using
ptmixed:
pt_glmm = ptmixed(fixef.formula = y ~ group*time, id = id,
data = data.long, npoints = 3,
hessian = T, trace = F)## Loading required namespace: GLMMadaptive## Loading required namespace: moments## Loading required namespace: lme4## Loading required namespace: mvtnorm##
##
## Total number of NM iterations = 570
## Convergence reached. Computing hessian...## Loading required namespace: numDerivThe code above requires to estimate a GLMM
- with
yas response,group,timeand their interaction as fixed effects (as specified infixef.formula), and a subject-specific random intercept (id = id) - using an AGHQ with 3 quadrature points
(
npoints = 3) - and to further evaluate the Hessian matrix at the MLE
(
hessian = T)
Note that the function comprises several other arguments, detailed in the function’s help page. In particular, there are four remarks that I’d like to make here:
- an offset term, if relevant, can be included through the
offsetargument - a higher number of quadrature points can be specified by changing
the value of
npoints(my recommendation is to usenpoints = 5) - detailed information about the optimization can be printed on screen
by specifying
trace = T(here I have settrace = Fto prevent a long tracing output to be printed in the middle of the vignette) - ML estimation for this model is not trivial, and optimizations may
sometimes fail to converge. If this happens, you may try to use a
different number of quadrature points (
npoints), to change the default values of the argumentsfreq.updates,reltol,maxitandmin.var.init, or to supply an alternative (sensibly chosen) starting value (theta.start)
Viewing parameter estimates, standard errors and p-values
The results of the fitted model can be viewed using
summary(pt_glmm)## Loading required namespace: matrixcalc## Loglikelihood: -140.623
## Parameter estimates:
## Estimate Std. error z p.value
## (Intercept) 2.0059 0.2865 7.0010 0.0000
## group -1.4523 0.4641 -3.1292 0.0018
## time -0.1359 0.0762 -1.7826 0.0746
## group:time 0.5105 0.1458 3.5019 0.0005
##
## Dispersion = 1.63
## Power = -0.23
## Variance = 0.41that reports the ML estimates of the regression coefficients (column
Estimate), the associated standard errors (column
Std. error) and univariate Wald tests (columns
z and p.value), as well as the ML estimates of
the dispersion and power parameters of the Poisson-Tweedie distribution,
and the ML estimate of the variance of the random effects.
Testing more complex hypotheses
More complex hypotheses can be tested using the multivariate Wald test or, when possible, the likelihood ratio test.
For example, one may want to test the null hypothesis that there are
no differences between the two groups, that is to say that the
regression coefficients of group and
group:time are both = 0.
To test this hypothesis with the multivariate Wald test, we first need to specify it in the form \(L \beta = 0\), where \(L\) is specified as follows:
L.group = matrix(0, nrow = 2, ncol = 4)
L.group[1, 2] = L.group[2, 4] = 1
L.group## [,1] [,2] [,3] [,4]
## [1,] 0 1 0 0
## [2,] 0 0 0 1Then, we can proceed with computing the multivariate Wald test:
wald.test(pt_glmm, L = L.group, k = c(0, 0))## chi2 df P
## 1 14.22634 2 0.0008143105Alternatively, the same hypothesis can be tested using the
likelihood ratio test (LRT). To do so, you first need
to estimate the model under the null hypothesis (note that for the
purpose of this computation, evaluating the hessian matrix is not
necessary, so we can avoid to compute it by setting
hessian = F):
null_model = ptmixed(fixef.formula = y ~ time, id = id,
data = data.long, npoints = 3,
hessian = F, trace = F)##
##
## Total number of NM iterations = 398Then, we can proceed to compare the null and alternative model by computing the LRT test statistic, whose asymptotic distribution is in this case a chi-squared with two degrees of freedom, and the corresponding p-value:
lrt.stat = 2*(pt_glmm$logl - null_model$logl)
lrt.stat## [1] 14.17753p.lrt = pchisq(lrt.stat, df = 2, lower.tail = F)
p.lrt## [1] 0.0008344269Computing the predicted random effects
To computate the predicted random effects, simply use
ranef(pt_glmm)## 1 2 3 4 5 6
## -0.74838728 0.31472249 1.13391844 -1.17000951 0.49289844 0.40439346
## 7 8 9 10 11 12
## -0.22676683 0.31639414 -0.35419327 -0.02760355 0.17689645 -0.27431201
## 13 14
## -0.18584234 0.65265844Simpler models
The Poisson-Tweedie GLMM is an extension of three simpler models:
- negative binomial GLMM (this is obtained by fixing the power parameter a = 0)
- Poisson-Tweedie GLM (this is obtained by removing the random effects from the model)
- negative binomial GLM (this is obtained by fixing the power parameter a = 0 and removing the random effects from the model)
For this reason, the package also offers the possibility to estimate these simpler models, as illustrated below.
Negative binomial generalized linear mixed model
The syntax to estimate the negative binomial GLMM is
the same as that used for the Poisson-Tweedie GLMM. Just make sure to
replace the function ptmixed with nbmixed:
nb_glmm = nbmixed(fixef.formula = y ~ group*time, id = id,
data = data.long, npoints = 3,
hessian = T, trace = F)##
## Total number of iterations = 362
## Convergence reached. Computing hessian...To view the model summary and compute the predicted random effects, once again you can use
summary(nb_glmm)## Loglikelihood: -140.623
## Parameter estimates:
## Estimate Std. error z p.value
## (Intercept) 1.9855 0.2890 6.8699 0.0000
## group -1.4244 0.4645 -3.0666 0.0022
## time -0.1338 0.0761 -1.7579 0.0788
## group:time 0.5057 0.1442 3.5067 0.0005
##
## Dispersion = 1.63
## Power = 0
## Variance = 0.41ranef(nb_glmm)## 1 2 3 4 5 6
## -0.73856990 0.33249113 1.15104776 -1.16599968 0.51121344 0.42102715
## 7 8 9 10 11 12
## -0.21148564 0.31620528 -0.35946848 -0.02816143 0.17458088 -0.27611762
## 13 14
## -0.18756259 0.65244576Poisson-Tweedie generalized linear model
Estimation of the Poisson-Tweedie GLM can be done
using the ptglm function:
pt_glm = ptglm(formula = y ~ group*time, data = data.long, trace = F)
summary(pt_glm)## Loglikelihood: -152.931
## Parameter estimates:
## Estimate Std. error z p.value
## (Intercept) 2.2396 0.2128 10.5265 0.0000
## group -1.3792 0.4083 -3.3778 0.0007
## time -0.1661 0.1244 -1.3353 0.1818
## group:time 0.5111 0.2046 2.4984 0.0125
##
## Dispersion = 4.24
## Power = -0.16Negative binomial generalized linear model
Finally, estimation of the negative binomial GLM can
be done using the nbglm function:
nb_glm = nbglm(formula = y ~ group*time, data = data.long, trace = F)
summary(nb_glm)## Loglikelihood: -152.955
## Parameter estimates:
## Estimate Std. error z p.value
## (Intercept) 2.2401 0.2146 10.4362 0.0000
## group -1.3751 0.4070 -3.3788 0.0007
## time -0.1701 0.1241 -1.3701 0.1707
## group:time 0.5170 0.2030 2.5473 0.0109
##
## Dispersion = 4.37
## Power = 0