Random Intercepts Estimation

In this section, the estimates of the random intercepts and their correlations are investigated.

A sample of 250 out of the 1.000 estimated random intercepts is displayed with their 95%

credibility intervals in Figure 5.3. The random intercepts in the model for parameter ν are color-coded by the number of times a zero has been observed for a mother. Figure 5.3 shows

Figure 5.3: Estimated random intercepts with their 95% credibility intervals in the models for the three parameters µ(top panel), σ (middle panel), and ν (bottom panel) that the random intercepts in the model for the parameter µ (top panel) are close to zero

and are estimated with high accuracy as the corresponding credibility intervals are relatively narrow. In the models forµandσ, the random intercepts seem to be approximately normally distributed around zero. This perception is reinforced by the estimated density of the random intercepts in the diagonal of Figure 5.4 . The density of the random intercepts in the model for ν is not normal but rather looks like a mixture of two or more components where one of these components resembles the normal prior. Indeed there is a large proportion of mothers with a positive income over all panel periods for whom no information on the risk of zero earnings is provided from the observed data, and, hence, the random intercept is sampled from the prior distribution. Generally, as described in Section 4.4.2.2, less information is available to estimate the random intercepts in the model for ν, which can be seen by the wide credibility intervals around the random intercepts.

To investigate the main question of interest whether correlation exists between the three random intercepts, their pairwise correlation coefficients are displayed in the upper diagonal in Figure 5.4. Scatter plots of pairs of random intercepts are shown in the lower diagonal.

The correlation coefficients indicate a negative correlation between the estimated random

Figure 5.4: Plot matrix with scatterplots between the estimated random intercepts in the models for the three different parameters including a line for the fit of a linear re-gression (lower off-diagonal), the kernel densities (diagonal) and the pairwise correlation coefficients (upper off-diagonal)

.

intercepts in the models for µ and ν and between the estimated random intercepts in the models for σ and ν. The results for the absolute value of the correlation are similar to the simulation scenario (i) in Figure 4.5 where the true pairwise correlations were 0.5. This outcome suggests that correlation is also present in this application but shrunk to zero due to the assumption of independence between the random intercepts in the model.

Chapter 6 Discussion

As currently only models with independent random intercepts are considered in the Bayesian distributional regression framework, the goal of this Master thesis was to investigate the consequences of a violation of the independence assumption and to outline the necessary adaption in the MCMC inference in a model allowing for correlation between the random intercepts in the models for different distributional parameters.

Firstly, a Bayesian distributional regression model with correlated random intercepts was specified. Next, the MCMC steps for posterior inference in this model were outlined exem-plary for Bayesian distributional regression of normal observations where a random intercept is included in both the models for the mean as well as the variance. As correlated random effects are not yet implemented for mixed models in the Bayesian distributional regression framework, arguments for the implementation of correlation between random intercepts were provided in the second part of this Master thesis. The results of the simulation study suggested that a violation of the independence assumption between the random intercepts in the models for the different distributional parameters impacts the model fitting. It was shown that the correlation between random intercepts in the data resulted in estimates which are correlated but with a correlation shrunk to zero. These estimation results imply to implement correlated random effects for mainly two reasons: Firstly, it allows to model independent as well as correlated random effects, and secondly, the covariance or, respec-tively, the correlation matrix is estimated explicitly which introduces greater flexibility in the modeling approach. A drawback of the simulation study was that the response distribution has not been varied among the simulations and was fixed to the ZAGA distribution. More-over, due to the capability of BayesX, the sample size was restricted to 2.000 observations per simulation. Finally, evidence for the presence of correlation between random intercepts in the models for the different parameters in real-world data was provided by fitting a ZAGA mixed model on the yearly income of mothers over time. It was observed that the size of the absolute correlation coefficients between the estimated random intercepts were similar to the result of one model in the simulation study where the underlying correlation was 0.5.

This shows that correlation between random intercepts is indeed present in real-world data, and consequently, emphasizes the necessity of including and estimating the correlation of random effects in mixed models for the Bayesian distributional regression framework.

Appendix

A.1 Notation

The following list summarizes the notation used in the thesis.

Symbol/Notation Description

i= 1, . . . , n Index fori^thindividual

j= 1, . . . , ni Index forj^thobservation of individuali

k= 1, . . . , K Index fork^thparameter

l= 0, . . . , Lk Index forl^thunspecified function ofk^thparameter

r= 1, . . . , Rk,l Index forr^th basis function of unspecified function lin the model for parameterk

d Index for the order of the random walk dependence

t= 1, . . . , T Number of MCMC iterations

y= (y1, . . . ,yi, . . . ,yn)⁰ Vector of response observations

yi= (yi,1, . . . , yi,n_i)⁰ Vector of response observations of individuali(in case of e.g.

repeated measurements)

X= (x1, . . . ,xi, . . . ,xn)⁰ Covariate matrix

xi= (xi,1, . . . , xi,j. . . , xi,n_i)⁰ Covariate vector of individuali(in case of e.g. repeated mea-surements)

Z= (Z1, . . . ,Zk, . . . ,ZK)⁰ Design matrix

Zk= (Zk,1, . . . ,Zk,i, . . . ,Zk,n)⁰ Vector of design matrices of all individuals in the model for parameterk

Zk,i= (zk,i,1, . . . ,zk,i,j, . . . ,zk,i,n_i)⁰ Vector of design matrices of individualiin the model for pa-rameterk

Zk,l Design matrix ofl^thpredictor in the model for parameterk

vk,i= (vk,i,1, . . . , vk,i,j, . . . , vk,i,n_i) Vector of design matrices of the random intercepts for individual iin the model for parameterk

β= (β₁, . . . ,β_k, . . . ,β_K)⁰ Vector of regression coefficients in the models for all parameters β_k= (β_k,1, . . . ,β_k,l, . . . ,β_k,L

k)⁰ Vector of regression coefficients in the model for parameterk β_k,−l= (β_k,1, . . .β_k,l−1,β_k,l+1, . . . ,β_k,Lk): Regression coefficients in the model for parameterkwithout

l^thcoefficient

γ= (γ₁, . . . ,γ_k, . . . ,γ_K)⁰ Vector of random intercepts in the models for all parameters γ_k= (γk,1, . . . , γk,i, . . . , γk,n)⁰ Vector of random intercepts in the model for parameterk

γ_i= (γ1,i, . . . , γk,i, . . . , γK,i)⁰ Vector of random intercepts for individuali

γ_−k,i= (γ1,i, . . . , γk−1,i, γk+1,i. . . , γK,i): Vector of random intercepts for individualiexcept ofγk,i

τ²= (τ²₁, . . . ,τ²_k, . . . ,τ²_K)⁰ Vector of optional prior smoothing variances of regression coef-ficients

τ_k²= (τ_k,1² , . . . , τ_k,l² , . . . , τ_k,L²

K)⁰ Vector of optional prior smoothing variances of regression coef-ficientsβ_k

τ²_γ= (τ_1,γ² , . . . , τ_k,γ² , . . . , τ_K,γ² )⁰ Vector of prior variances of random intercepts

ρk,k⁰ Correlation between the random intercepts in the model for

parameterkand parameterk⁰

η= (η₁, . . . ,η_k, . . .η_K)⁰ Vector of linear predictors of all parameters η_k= (η_k,1, . . . ,η_k,i, . . .η_k,n)⁰ Vector of linear predictors of parameterk

η_k,−l=η_k−Z⁰_k,lβ_k,l Vector of linear predictors of parameterkwithout thel^thterm η_k,i= (ηk,i,1, . . . , ηk,i,j, . . . , ηk,i,n_i)⁰ Vector of linear predictors of individualiof parameterk η_k,−i = (ηk,−i,1, . . . , ηk,−i,j, . . . , ηk,−i,n_i)⁰ = η_k,i−

vk,iγk,i

Vector with the linear predictors of individualifor parameterk without thei^thrandom intercept

hk(.) Monotonic, twice differentiable link function of parameterk

a= (a1,1, . . . , a1,L₁, . . . , aK,L_K)⁰ (1) Vector of optional hyperparameters of hyperprior ofτ² b= (b1,1, . . . , b1,L₁, . . . , bK,L_K)⁰ (2) Vector of optional hyperparameters of hyperprior ofτ² aγ= (a1,γ, . . . , ak,γ, . . . , aK,γ)⁰ (1) Vector of hyperparameter of hyperprior ofτ²_γ bγ= (b1,γ, . . . , b_k,γ, . . . , bK,γ)⁰ (2) Vector of hyperparameter of hyperprior ofτ²_γ

V Covariance matrix for random intercepts

ν (1) Hyperparameter of hyperior ofV

S⁻¹ (2) Hyperparameter (matrix) of hyperior ofV

θ= (β,γ,τ²,V) Vector of all regression parameters

Σ=diag

σ²(x1,1,β₂,γ_2,1), . . . , σ²(x1,n1,β₂,γ_2,1), . . . , σ²(xn,nn,β₂,γ_2,n)

Diagonal matrix of variances of each observation

fk,l(.) l^thunspecified function for the linear predictor in the model for

parameterk

Bk,l(.) = (Bk,l,1(.), . . . ,Bk,l,r(.), . . . ,Bk,l,R(.))⁰ l^th(B-Spline) basis function in the model for parameterk