Connection between the D-vine based model and linear mixed models

Probably the most popular models for longitudinal data are linear mixed models. In this section we will give a short introduction to this model class and show how they are connected to our approach from Section 5.2.

5.3.1. Linear mixed models for repeated measurements

Linear mixed models have been discussed in detail by many authors, e.g. in Diggle (2002), Verbeke and Molenberghs (2009) and Fahrmeir et al. (2013). Describing the outcome of repeated measurements j, j = 1, . . . , d_i, for individuals i, i = 1, . . . , n, as responses Y_jⁱ, they extend linear models by including random effects γ_i ∈ R^q to the fixed (i.e. non-random) effects β ∈ R^p, p, q ∈ N. These random effects, unlike the fixed effects, are different for each individual. The covariate vectors x_i,j ∈R^p and z_i,j ∈R^q are associated to the fixed and random effects, respectively.

Fori = 1, . . . , n and j = 1, . . . , d_i, the jth measurement for individual i is assumed to decompose to

Y_jⁱ =x^>_i,jβ+z^>_i,jγ_i+ε_i,j, (5.7) where the vector of random effects γ_i ∼ Nq(0, D) is normally distributed with zero expectation and covariance matrixD∈R^q×q and the error vectorε_i = (ε_i,1, . . . , ε_i,di)^>∼ Ndi(0,Σ_i) also follows a centered normal distribution with covariance matrix Σ_i ∈R^di×di. Further, γ₁, . . . ,γ_n, ε₁, . . . ,ε_n are assumed to be independent. Hence,

Y_jⁱ ∼ N(x^>_i,jβ, φ²_i,j) (5.8) with standard deviation φ_i,j := z^>_i,jDz_i,j+σ_i,j² 1/2

, where σ_i,j² := Var(ε_i,j). Using the notation

X_i :=







 x^>_i,1

... x^>_i,di







∈R^di×p, Z_i :=







 z^>_i,1

... z^>_i,di







∈R^di×q, Yⁱ :=







 Y₁ⁱ

... Y_dⁱ_i







∈R^di

we can represent the vector of all measurements belonging to individual i as follows:

Yⁱ =X_iβ+Z_iγ_i+ε_i. (5.9)

We see that due to the independence assumptions of γ_i and ε_i, i = 1, . . . , n, there

ex-ists a correlation between measurements of one individual but measurements of different individuals are independent. Further, the joint distribution of Yⁱ can be determined to be

Yⁱ ∼ N^di(X_iβ, Z_iDZ_i^>+ Σ_i) (5.10) and Y¹, . . . ,Yⁿ are independent. The fixed effects β and random effects γ_i as well as the parameters of the covariance matricesD and Σ_i, i= 1, . . . , n, can be estimated using (restricted) maximum-likelihood estimation as described for example in Diggle (2002) and Fahrmeir et al. (2013).

Linear mixed models are very popular in practice since they are easy to handle and interpret. Further, observations with missing data can also be used for ML estimation as long as the values are missing at random (see e.g. McCulloch et al., 2011; Ibrahim and Molenberghs, 2009).

5.3.2. Aligning linear mixed models and the D-vine based approach

Equation 5.10 implies that all univariate marginal distributions are normal distributions.

Further, the dependence structure is Gaussian and can vary from individual to individual since the correlation matrix R_i of Yⁱ is given by

R_i := Cor(Yⁱ) = diag(φ⁻_i,1¹, . . . , φ⁻_i,d¹

i) Z_iDZ_i^>+ Σ_i

diag(φ⁻_i,1¹, . . . , φ⁻_i,d¹

i),

whereφ_i,j is the standard deviation ofY_jⁱ,j = 1, . . . , d_i,i= 1, . . . , n. In practice, however, this would make estimation infeasible since the number of parameters would be too large;

in many cases one would even have more parameters than observations. Therefore, struc-tural assumptions are made, especially for Σ_i ∈R^di×di, in order to reduce the number of parameters to be estimated.

In Section 5.2.2 we assumed that the dependence structure is basically the same for all individuals and only differs due to the number of measurements d_i that individual i has had so far. In order to obtain the same for linear mixed models, we simply have to require the following homogeneity condition:

Homogeneity condition: We call correlation matrices R_i homogeneous if they are the same for all individuals i= 1, . . . , nexcept for the dimension, i.e.R_i = (r_{k,`</sub>)<sup>d</sup><sub>k,`=1}ⁱ ∈R^di×di is a (d_i×d_i)-submatrix of a correlation matrix R =R_d= (r_{k,`</sub>)<sup>d</sup><sub>k,`=1} ∈R^d×d.

This condition is in particular fulfilled if the covariance matrices of the errors Σ_i ∈R^di×di and the design matrices of the random effects Z_i ∈ R^di×q are constant in i except for the dimension. Despite being a restriction, linear mixed models meeting this requirement still comprise a wide range of models used in practice. The assumption on the covariance

matrices Σ_i is for example fulfilled if errors

• are assumed to be i.i.d., i.e. the (k, `)th entry of Σ<sub>i</sub> is given by σ<sup>2</sup>1<sub>{k=`}</sub>, where 1_{·}

denotes the indicator function;

• exhibit a compound symmetry structure, i.e. the (k, `)th entry of Σ<sub>i</sub> is σ<sup>2</sup>ρ<sup>1{k6=`}</sup> for some ρ∈(−1,1);

• follow an autoregressive structure of order 1 (AR(1)), i.e. the (k, `)th entry of Σ<sub>i</sub> is given by σ<sup>2</sup>ρ<sup>|k−`|</sup> for someρ∈(−1,1);

• have an exponential decay structure, i.e. the (k, `)th entry of Σ<sub>i</sub> can be written as σ<sup>2</sup>exp{− |k−`|/r}, where r >0 is the constant “range” parameter.

These are typical simplifications that are made anyway for modeling longitudinal data in most applications if the number of individuals is large with respect to the number of measurements. The assumption on the design matrices Z_i is also often satisfied, e.g. for the popular class of so-called random intercept models, where Z_i = (1, . . . ,1)^> ∈ R^di×1 for j = 1, . . . , d_i and i= 1, . . . , n. Further, the assumption includes any model where the covariates associated with the random effect only depend on the (common) measurement times t_j, j = 1, . . . , d, i.e. for example Z_i = (t₁, . . . , t_di)^> ∈ R^di×1 or more generally Z_i = (h(t₁), . . . , h(t_di))^> ∈ R^di×1 for some function h: R → R. Thus, assuming that Z_i only depends on the number of measurements d_i for individual i is also not uncommon such that there is in fact a wide class of linear mixed models sharing the property that the correlation matrix R_i of Yⁱ only depends on the number of measurements.

If R_i is homogeneous in i, we have that all individuals i share the same Gaussian dependence structure, i.e. correlation matrix. This scenario is a special case of the D-vine based model since we can represent any Gaussian correlation matrix using a D-vine with Gaussian pair-copulas and the corresponding (partial) correlations as parameters (see for example St¨ober et al., 2013, Theorem 4.1). The univariate margins F_jⁱ can be chosen arbitrarily for the copula approach such that we can simply use N(x^>_i,jβ, φ²_i,j)-margins (cf. Equation 5.8) to end up with a model describing the same joint distribution of Yⁱ as the corresponding linear mixed model (Equation 5.10). Since we can use arbitrary distributions for the margins and/or any D-vine copula for the dependence structure, our approach can be seen as an extension of linear mixed models with common correlation structure for all individuals. Figure 5.2 illustrates the link between our D-vine based model and linear mixed models.

For the application in Section 5.6 we will compare how well both model classes perform fitting real life data.

Linear mixed model

LMM with common correlation structure for all individuals

Gaussian copula with Gaussian regression margins

Gaussian copula with arbitrary margins

D-vine copula with Gaussian regression margins

D-vine copula with arbitrary margins

Figure 5.2.: Flow chart illustrating how the D-vine based model is linked to linear mixed models.

Im Dokument Model distances, block maxima and repeated measurements in the context of vine copulas  (Seite 98-101)