Models with Mixed Response Types • galamm

library(galamm)
library(lme4)

This vignette describes how galamm can be used to estimate models with mixed response types.

Mixed Normal and Binomial Response

We start with the mresp dataset, which comes with the package. The variable “itemgroup” defines the response type; it equals “a” for normally distributed responses and “b” for binomially distributed responses. They are link through a common random intercept.

head(mresp)
#>   id         x          y itemgroup
#> 1  1 0.8638214  0.2866329         a
#> 2  1 0.7676133  2.5647490         a
#> 3  1 0.8812059  1.0000000         b
#> 4  1 0.2239725  1.0000000         b
#> 5  2 0.7215696 -0.4721698         a
#> 6  2 0.6924851  1.1750286         a

In terms of the GALAMM defined in the introductory vignette, and for simplicity assuming we use canonical link functions, we have the response model

$f\left(y_{ij} | \nu_{ij}, \phi\right) = \exp \left( \frac{y_{ij}\nu_{ij} - b\left(\nu_{ij}\right)}{\phi} + c\left(y_{ij}, \phi\right) \right)$

for the $i$ th observation of the $j$ subject. Although we don’t show this with a subscript, when the variable itemgroup = "a" we have a Gaussian response, so $b(\nu) = \nu^{2}/2$ and the support of the distribution is the entire real line $\mathbb{R}$ . The mean in this case is given by $\mu_{ij} = \nu_{ij}$ . When itemgroup = "b" we have a binomial response, so $b(\nu) = \log(1 + \exp(\nu))$ and the support is $\{0, 1\}$ . In this binomial case we also have $\phi=1$ . The mean in this case is given by $\mu_{ij} = \exp(\nu_{ij}) / (1 + \exp(\nu_{ij}))$ . The function $c(y_{ij}, \phi)$ also differs between these cases, but is not of the same interest, since it does not depend on the linear predictor. It hence matters for the value of the log-likelihood, but not for its derivative with respect to the parameters of interest.

Next, the nonlinear predictor is given by

$\nu_{ij} = \beta_{0} + x_{ij}\beta_{1} + \mathbf{z}_{ij}^{T}\boldsymbol{\lambda} \eta$

where $x_{ij}$ is an explanatory and $\mathbf{z}_{ij}$ is a dummy vector of length 2 with exactly one element equal to one and one element equal to zero. When itemgroup = "a", $\mathbf{z} = (1, 0)^{T}$ and when itemgroup = "b", $\mathbf{z} = (0, 1)^{T}$ . The parameter $\boldsymbol{\lambda} = (1, \lambda)^{T}$ is a vector of factor loadings, whose first element equals zero for identifiability. $\eta$ is a latent variable, in this case representing an underlying trait “causing” the observed responses.

The structural model is simply

$\eta = \zeta \sim N(0, \psi),$

where $N(0, \psi)$ denotes a normal distribution with mean 0 and variance $\psi$ .

We define the loading matrix as follows, where the value 1 indicates that the first element $\boldsymbol{\lambda}$ is fixed, and the value NA indicates that its second element is unknown, and to be estimated.

(loading_matrix <- matrix(c(1, NA), ncol = 1))
#>      [,1]
#> [1,]    1
#> [2,]   NA

We set load.var = "itemgroup" because all rows of the data with the same value of itemgroup will receive the same factor loading. In formula, we state (0 + level | id) to specify that each subject, identified by id, has a given level. level is not an element of the mresp data, but is instead the factor onto which the loading matrix loads. We identify this with the argument factor = "level". We could have chosen any other name for "level", except for names that are already columns in mresp.

We also need to define the response families, with the vector

families <- c(gaussian, binomial)

We also need a way of telling galamm which row belongs to which family, which we do with a family mapping. The values in family_mapping refer to the elements of families.

family_mapping <- ifelse(mresp$itemgroup == "a", 1, 2)

We are now ready to estimate the mixed response model.

mixed_resp <- galamm(
  formula = y ~ x + (0 + level | id),
  data = mresp,
  family = families,
  family_mapping = family_mapping,
  load.var = "itemgroup",
  lambda = loading_matrix,
  factor = "level"
)

We can also look at its summary output.

summary(mixed_resp)
#> GALAMM fit by maximum marginal likelihood.
#> Formula: y ~ x + (0 + level | id)
#>    Data: mresp
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>   9248.7   9280.2  -4619.3   3633.1     3995 
#> 
#> Lambda:
#>         level      SE
#> lambda1 1.000       .
#> lambda2 1.095 0.09982
#> 
#> Random effects:
#>  Groups Name  Variance Std.Dev.
#>  id     level 1.05     1.025   
#> Number of obs: 4000, groups:  id, 1000
#> 
#> Fixed effects:
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept)    0.041    0.05803  0.7065 4.799e-01
#> x              0.971    0.08594 11.2994 1.321e-29

Mixed Response and Heteroscedastic Residuals

Mixed response models can be combined with heteroscedastic residuals, which are described in the vignette on linear mixed models with heteroscedastic residuals. However, some care is needed in this case. We illustrate with the dataset mresp_hsced, whose first few lines are shown below:

head(mresp_hsced)
#>   id         x           y itemgroup grp isgauss
#> 1  1 0.8638214   0.2866329         a   b       1
#> 2  1 0.7676133   2.5647490         a   b       1
#> 3  1 0.8812059   1.0000000         b   b       0
#> 4  1 0.2239725   1.0000000         b   b       0
#> 5  2 0.7215696 -11.0020148         a   a       1
#> 6  2 0.6924851   1.1750286         a   b       1

This dataset has the same structure as mresp, except that the residual standard deviation of the normally distributed responses vary according to the variable grp, which has two levels. Two fit a model on this dataset with heteroscedastic residuals for the normally distributed variable, we must use the dummy variable isgauss when setting up the weights. This is illustrated below. By writing ~ (0 + isgauss | grp) we specify that the heteroscedasticity is assumed between levels of grp, but that this only applies when isgauss is nonzero. Hence, the binomially distributed responses don’t have any heteroscedastic residuals estimated, since for these observations the dispersion parameter $\phi$ is fixed to 1. We also ignore the factor loadings for simplicity.

family_mapping <- ifelse(mresp$itemgroup == "a", 1L, 2L)
mod <- galamm(
  formula = y ~ x + (1 | id),
  weights = ~ (0 + isgauss | grp),
  family = c(gaussian, binomial),
  family_mapping = family_mapping,
  data = mresp_hsced
)

The summary output now shows that we also have estimated a variance function.

summary(mod)
#> GALAMM fit by maximum marginal likelihood.
#> Formula: y ~ x + (1 | id)
#>    Data: mresp_hsced
#> Weights: ~(0 + isgauss | grp)
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>  12356.9  12388.3  -6173.4  31426.4     3995 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  id     (Intercept) 0        0       
#> Number of obs: 4000, groups:  id, 1000
#> 
#> Variance function:
#>       a       b 
#> 1.00000 0.07756 
#> 
#> Fixed effects:
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept)  0.01523    0.06303  0.2416 8.091e-01
#> x            0.99563    0.11153  8.9267 4.388e-19

Covariate Measurement Error Model

This example is taken from Chapter 14.2 in Skrondal and Rabe-Hesketh (2004), and follows the analyses in (Rabe-Hesketh, Pickles, and Skrondal 2003; Rabe-Hesketh, Skrondal, and Pickles 2003). No originality is claimed with respect to the analyses; but for the sake of understanding the galamm code, we explain it in quite some detail. The original dataset comes from Morris, Marr, and Clayton (1977).

The scientific question concerns the impact of fiber intake on risk of coronary heat disease. 337 middle aged men were followed, all of whom either worked as bank staff or as London Transport staff (indicated by dummy variable bus). All men had a first measurement of fiber intake, and some had an additional measurement six months later. In addition, all had a binary variable indicating whether or not they had developed coronary heart disease.

The first few rows of the dataset are printed below. All of these had only a single measurement of fiber intake.

head(diet)
#>   id   age bus   item         y chd fiber fiber2
#> 1  1 -0.38   1 fiber1 17.814280   0     1      0
#> 2  1 -0.38   1    chd  0.000000   1     0      0
#> 3  2  0.54   1 fiber1  9.487736   0     1      0
#> 4  2  0.54   1    chd  0.000000   1     0      0
#> 5  3  8.78   1 fiber1 15.958630   0     1      0
#> 6  3  8.78   1    chd  0.000000   1     0      0

It is instructive to also look at some men who had two measurements of fiber intake. Here are three of them.

diet[diet$id %in% c(219, 220, 221), ]
#>      id   age bus   item        y chd fiber fiber2
#> 429 219 -8.13   0 fiber1 15.64263   0     1      0
#> 430 219 -8.13   0 fiber2 14.87973   0     1      1
#> 431 219 -8.13   0    chd  0.00000   1     0      0
#> 432 220 -1.13   0 fiber1 13.46374   0     1      0
#> 433 220 -1.13   0 fiber2 14.73168   0     1      1
#> 434 220 -1.13   0    chd  0.00000   1     0      0
#> 435 221  3.67   0 fiber1 15.95863   0     1      0
#> 436 221  3.67   0 fiber2 17.28778   0     1      1
#> 437 221  3.67   0    chd  0.00000   1     0      0

Structural Model

Our first goal is to model the effect of fiber intake on risk of heart disease. Since this variable is very likely to be subject to measurement error, it is well known that simply averaging the two measurements, where available, and using the single measurement otherwise, will lead to bias and lack of power (Carroll et al. 2006). Instead, we let $\eta_{j}$ denote the true (latent) fiber intake of person $j$ , and define the structural model

$\eta_{j} = \mathbf{x}_{ij}'\boldsymbol{\gamma} + \zeta_{j}.$

where $\mathbf{x}_{ij}$ is a vector of covariates age, bus, and their interaction, as well as a constant term for defining the intercept. The R formula for setting up the model matrix would be model.matrix(~ age * bus, data = diet). The term $\zeta_{j}$ is a normally distributed disturbance, $\zeta_{j} \sim N(0, \psi)$ .

Measurement Model

It is assumed that the fiber measurements are normally distributed around the true value $\eta_{j}$ , and allow for a drift term $d_{ij}\beta_{0}$ , where $d_{2ij}$ is a dummy variable whose value equals one if the $i$ th row of the $j$ th person is a replicate fiber measurement, and zero otherwise. This allows for a drift in the measurements, and the model becomes

$y_{ij} = d_{2ij} \beta_{0} + \eta_{j} + \epsilon_{ij}, \qquad \epsilon_{ij} = N(0, \theta).$

Disease Model

Next, for the rows corresponding to coronary heart disease measurement, we assume a binomial model with a logit link function, i.e., logistic regression. This model is given by

$\text{logit}[P(y_{ij}=1 | \eta_{j})] = \mathbf{x}_{ij}'\boldsymbol{\beta} + \lambda \eta_{j},$

where $\mathbf{x}_{ij}$ is the same as before, but where $\boldsymbol{\beta}$ now represents the direct effect of age and bus on the probability of coronary heart disease. The factor loading $\lambda$ is the regression coefficient for latent fiber intake on the risk of coronary heart disease.

Nonlinear Predictor

Stacking the three responses, which are fiber intake at times 1 and 2, and coronary heart disease, we can define the joint model as a GLLAMM with nonlinear predictor

$\nu_{ij} = d_{2ij} \beta_{0} + d_{3ij} \mathbf{x}_{j}'\boldsymbol{\beta} + \mathbf{x}_{j}'\boldsymbol{\gamma}\left[(d_{1i} + d_{2i}) + \lambda d_{3i}\right] + \zeta_{j} \left[(d_{1i} + d_{2i}) + \lambda d_{3i}\right],$

where $d_{1i}$ is a dummy variable for fiber measurements at timepoint 1 and $d_{3i}$ is an indicator for coronary heart disease.

Estimating the Model with galamm

In the measurement model for fiber above, note that we implicitly have factor loadings equal to one. Letting the item variable define which rows should receive which loading, we note that the factor levels are as follows:

levels(diet$item)
#> [1] "fiber1" "fiber2" "chd"

We could of course have changed this, but given these levels, we define the following loading matrix:

(loading_matrix <- matrix(c(1, 1, NA), ncol = 1))
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]   NA

That is, “fiber1” and “fiber2” receive a loading fixed to one, whereas the loading for “chd” remains to be estimated.

We also need to define the families, making sure it is binomial for “chd” and Gaussian for “fiber1” and “fiber2”.

families <- c(gaussian, binomial)
family_mapping <- ifelse(diet$item == "chd", 2, 1)

The model formula also needs some care in this case. We define it as follows, and note that the whole part 0 + chd + fiber + fiber2 could have been replaced by item. We use the former for ease of explanation.

formula <- y ~ 0 + chd + (age * bus):chd + fiber +
  (age * bus):fiber + fiber2 + (0 + loading | id)

The initial zero specifies that we don’t want R to insert an intercept term. Next, chd + (age * bus) : chd applies to “chd” rows only, and corresponds to $\mathbf{x}_{ij}^{T} \boldsymbol{\beta}$ in the disease model. The part fiber + (age * bus) : fiber corresponds to the term $\mathbf{x}_{ij}^{T}\boldsymbol{\gamma}$ in the disease model, and fiber2 corresponds to the drift term $d_{2ij}\beta_{0}$ .

We can inspect the model implied by the formula by using the nobars function from lme4 (Bates et al. 2015) to remove the random effects, and then model.matrix. It looks correctly set up.

head(model.matrix(nobars(formula), data = diet))
#>   chd fiber fiber2 chd:age chd:bus age:fiber bus:fiber chd:age:bus age:bus:fiber
#> 1   0     1      0    0.00       0     -0.38         1        0.00         -0.38
#> 2   1     0      0   -0.38       1      0.00         0       -0.38          0.00
#> 3   0     1      0    0.00       0      0.54         1        0.00          0.54
#> 4   1     0      0    0.54       1      0.00         0        0.54          0.00
#> 5   0     1      0    0.00       0      8.78         1        0.00          8.78
#> 6   1     0      0    8.78       1      0.00         0        8.78          0.00

Finally, the random effects part (0 + loading | id) specifies that for each id there should be a single random intercept, which corresponds to latent fiber intake $\eta_{j}$ . Writing 0 + loading means that $\eta_{j}$ should be multiplied by the factor loadings defined in the loading matrix.

We are now ready to fit the model.

mod <- galamm(
  formula = formula,
  data = diet,
  family = families,
  family_mapping = family_mapping,
  factor = "loading",
  load.var = "item",
  lambda = loading_matrix
)

Looking at the summary output, however, we see a warning that the Hessian matrix is rank deficient.

summary(mod)
#> Warning in vcov.galamm(object, parm = "lambda"): Rank deficient Hessian matrix.Could not compute covariance matrix.
#> Warning in vcov.galamm(object, "beta"): Rank deficient Hessian matrix.Could not compute covariance matrix.
#> GALAMM fit by maximum marginal likelihood.
#> Formula: formula
#>    Data: diet
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>   2837.6   2892.9  -1406.8  12529.3      730 
#> 
#> Lambda:
#>         loading SE
#> lambda1   1.000  .
#> lambda2   1.000  .
#> lambda3  -2.026  .
#> 
#> Random effects:
#>  Groups Name    Variance Std.Dev.
#>  id     loading 0        0       
#> Number of obs: 742, groups:  id, 333
#> 
#> Fixed effects:
#>               Estimate Std. Error z value Pr(>|z|)
#> chd           -1.78692         NA      NA       NA
#> fiber         17.96184         NA      NA       NA
#> fiber2        -0.64927         NA      NA       NA
#> chd:age        0.06682         NA      NA       NA
#> chd:bus       -0.06882         NA      NA       NA
#> fiber:age     -0.20480         NA      NA       NA
#> fiber:bus     -1.69601         NA      NA       NA
#> chd:age:bus   -0.04934         NA      NA       NA
#> fiber:age:bus  0.16097         NA      NA       NA

We suspect that this is a convergence issue of the algorithm, and start by simply turning a lever to make the algorithm more verbose. The argument trace = 3 is passed onto optim.

mod <- galamm(
  formula = formula,
  data = diet,
  family = families,
  family_mapping = family_mapping,
  factor = "loading",
  load.var = "item",
  lambda = loading_matrix,
  control = galamm_control(optim_control = list(trace = 3))
)
#> N = 11, M = 20 machine precision = 2.22045e-16
#> At X0, 0 variables are exactly at the bounds
#> At iterate     0  f=         2148  |proj g|=       122.68
#> At iterate    10  f =       1770.3  |proj g|=        30.656
#> At iterate    20  f =       1467.2  |proj g|=        11.286
#> At iterate    30  f =       1417.6  |proj g|=        73.904
#> At iterate    40  f =       1406.8  |proj g|=       0.21144
#> 
#> iterations 43
#> function evaluations 51
#> segments explored during Cauchy searches 44
#> BFGS updates skipped 0
#> active bounds at final generalized Cauchy point 1
#> norm of the final projected gradient 0.0123301
#> final function value 1406.8
#> 
#> F = 1406.8
#> final  value 1406.801105 
#> converged

Now we got some information, but the final estimate is the same, since we otherwise used the same parameters. We now instead try to increase the initial estimate of the random effect parameter $\theta$ . In general $\theta$ is a vector containing elements of the Cholesky factor of the covariance matrix (see, Bates et al. (2015)), but in this case it is just the standard deviation of the latent variable. By default, its initial value is 1, but we now try to increase it to 10, with the start argument.

mod <- galamm(
  formula = formula,
  data = diet,
  family = families,
  family_mapping = family_mapping,
  factor = "loading",
  load.var = "item",
  lambda = loading_matrix,
  start = list(theta = 10),
  control = galamm_control(optim_control = list(trace = 3))
)
#> N = 11, M = 20 machine precision = 2.22045e-16
#> At X0, 0 variables are exactly at the bounds
#> At iterate     0  f=       2827.6  |proj g|=       123.31
#> At iterate    10  f =       1764.5  |proj g|=        103.86
#> At iterate    20  f =       1623.6  |proj g|=        131.81
#> At iterate    30  f =       1442.4  |proj g|=        35.169
#> At iterate    40  f =         1388  |proj g|=        20.654
#> At iterate    50  f =       1372.4  |proj g|=        1.4042
#> 
#> iterations 57
#> function evaluations 69
#> segments explored during Cauchy searches 58
#> BFGS updates skipped 0
#> active bounds at final generalized Cauchy point 0
#> norm of the final projected gradient 0.0119052
#> final function value 1372.16
#> 
#> F = 1372.16
#> final  value 1372.160387 
#> converged

The output printed is the negative log-likelihood, and since 1372.16 is less than 1406.8, increasing the initial value to 10 led us to a local optimum with higher likelihood value. Ideally we should have tried other initial values as well, but since Skrondal and Rabe-Hesketh (2004) obtained a negative log-likelihood value of 1372.35 while using adaptive quadrature estimation, we are pretty happy about finding a value close to theirs.

We can now use the summary method, where we find that our estimates are very close to the corresponding model in Table 14.1 of Skrondal and Rabe-Hesketh (2004).

summary(mod)
#> GALAMM fit by maximum marginal likelihood.
#> Formula: formula
#>    Data: diet
#> Control: galamm_control(optim_control = list(trace = 3))
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>   2768.3   2823.6  -1372.2   2002.9      730 
#> 
#> Lambda:
#>         loading      SE
#> lambda1  1.0000       .
#> lambda2  1.0000       .
#> lambda3 -0.1339 0.05121
#> 
#> Random effects:
#>  Groups Name    Variance Std.Dev.
#>  id     loading 23.64    4.862   
#> Number of obs: 742, groups:  id, 333
#> 
#> Fixed effects:
#>               Estimate Std. Error  z value   Pr(>|z|)
#> chd           -1.91520    0.27229 -7.03361  2.013e-12
#> fiber         17.94834    0.48686 36.86566 1.641e-297
#> fiber2         0.22406    0.41783  0.53624  5.918e-01
#> chd:age        0.06613    0.05931  1.11515  2.648e-01
#> chd:bus       -0.02900    0.34355 -0.08441  9.327e-01
#> fiber:age     -0.21207    0.10090 -2.10172  3.558e-02
#> fiber:bus     -1.68278    0.63721 -2.64084  8.270e-03
#> chd:age:bus   -0.04998    0.06507 -0.76812  4.424e-01
#> fiber:age:bus  0.16821    0.11223  1.49882  1.339e-01

Now we were able to reproduce results very close to those in Table 14.1 on page 420 of Skrondal and Rabe-Hesketh (2004). This is a useful assurance that the algorithm used by galamm is correctly implemented.

Comparison to a Model with Indirect Effect Only

The model above assumed that the age and bus variables had both an indirect effect on the risk of coronary heart disease through its impact on fiber intake, as well as a direct effect. Still following Skrondal and Rabe-Hesketh (2004), we now estimate an alternative model which only has indirect effects. The disease model is now

$\text{logit}[P(y_{ij}=1 | \eta_{j})] = d_{ij3}\beta_{03} + \lambda \eta_{j},$

The formula becomes

formula0 <- y ~ 0 + chd + fiber + (age * bus):fiber + fiber2 +
  (0 + loading | id)

Everything else is the same as before, so we fit the new model with

mod0 <- galamm(
  formula = formula0,
  data = diet,
  family = families,
  family_mapping = family_mapping,
  factor = "loading",
  load.var = "item",
  lambda = loading_matrix,
  start = list(theta = 10),
  control = galamm_control(optim_control = list(trace = 3))
)
#> N = 8, M = 20 machine precision = 2.22045e-16
#> At X0, 0 variables are exactly at the bounds
#> At iterate     0  f=       2827.6  |proj g|=       123.31
#> At iterate    10  f =       1664.7  |proj g|=        25.235
#> At iterate    20  f =       1427.9  |proj g|=        37.186
#> At iterate    30  f =       1373.5  |proj g|=        5.1449
#> 
#> iterations 37
#> function evaluations 40
#> segments explored during Cauchy searches 38
#> BFGS updates skipped 0
#> active bounds at final generalized Cauchy point 0
#> norm of the final projected gradient 0.000304544
#> final function value 1373.01
#> 
#> F = 1373.01
#> final  value 1373.013263 
#> converged

Looking at the summary output, the estimates are now close to the first column in Table 14.1 of Skrondal and Rabe-Hesketh (2004).

summary(mod0)
#> GALAMM fit by maximum marginal likelihood.
#> Formula: formula0
#>    Data: diet
#> Control: galamm_control(optim_control = list(trace = 3))
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>   2764.0   2805.5  -1373.0   2007.9      733 
#> 
#> Lambda:
#>         loading      SE
#> lambda1   1.000       .
#> lambda2   1.000       .
#> lambda3  -0.136 0.05148
#> 
#> Random effects:
#>  Groups Name    Variance Std.Dev.
#>  id     loading 23.64    4.862   
#> Number of obs: 742, groups:  id, 333
#> 
#> Fixed effects:
#>               Estimate Std. Error  z value   Pr(>|z|)
#> chd            -1.9668    0.18248 -10.7783  4.360e-27
#> fiber          17.9741    0.48123  37.3504 2.502e-305
#> fiber2          0.2238    0.41779   0.5358  5.921e-01
#> fiber:age      -0.1896    0.09914  -1.9122  5.585e-02
#> fiber:bus      -1.6991    0.62529  -2.7173  6.581e-03
#> fiber:age:bus   0.1515    0.11012   1.3760  1.688e-01

We can use anova method for galamm objects to compare the models. AIC and BIC favor the simpler model with only indirect effects.

anova(mod, mod0)
#> Data: diet
#> Models:
#> mod0: formula0
#> mod: formula
#>      npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
#> mod0    9 2764.0 2805.5 -1373.0   2002.9                     
#> mod    12 2768.3 2823.6 -1372.2   2002.9 1.7058  3     0.6357

References

Bates, Douglas M, Martin Mächler, Ben Bolker, and Steve Walker. 2015. “Fitting Linear Mixed-Effects Models Using Lme4.” Journal of Statistical Software 67 (1): 1–48. https://doi.org/10.18637/jss.v067.i01.

Carroll, Raymond J., David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu. 2006. Measurement Error in Nonlinear Models: A Modern Perspective. 2nd ed. Monographs on Statistics and Applied Probability. Boca Raton, Florida: John Wiley & Sons.

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