Random Regression Coefficient Models

Abstract

This chapter describes a modification of the nested error regression model having random regression coefficients. We can intuitively expect that the slope parameters of some explanatory variable are not constant and therefore they should take different values in different domains. The random regression coefficient model gives a practical solution to this problem by assuming that the beta parameters are random and therefore they give a more flexible way of modeling. The chapter gives fitting algorithms, derives the empirical best predictors of domain means, and approximates the mean squared errors. The last section provides R codes.

13.1 Introduction

Coefficients of auxiliary variables in the standard nested error regression (NER) model are not allowed to vary across sampling units or domains. This assumption is too rigid in many practical situations. In some small area estimation (SAE) problems, we can intuitively expect that the slope parameters of some explanatory variable are not constant and therefore they should take different values in different domains. The random regression coefficient (RRC) model gives a practical solution to this problem by assuming that the beta parameters are random and therefore they give a more flexible way of modeling.

This section describes a modification of the nested error regression (NER) model having random regression coefficients. In the framework of SAE, Prasad and Rao (1990) derived empirical best linear unbiased predictors (EBLUP) of domain linear parameters under a unit-level RRC model. They also derived a second order approximation of the mean squared error (MSE) of the EBLUP and gave an estimator of the MSE approximation. They considered a special case of the RRC model proposed by Dempster et al. (1981), with a single concomitant variable x and a null intercept parameter (regression through origin).

Moura and Hold (1999) used a class of models allowing for variation between areas because of: (1) differences in the distribution of unit-level or area-level variables between areas, and (2) area-specific components of variance which cannot be explained by covariates. Their family of models contains RRC models as particular cases. These authors derived EBLUPs of linear parameters, gave an approximation to the MSE of the EBLUP, and proposed MSE estimators.

Hobza and Morales (2013) applied a flexible class of RRC models to the prediction of domain linear parameters. They gave a Fisher-scoring algorithm to calculate the residual maximum likelihood estimators of the model parameters and they derived EBLUPs and MSEs estimators. They applied the introduced methodology to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.

This chapter extends the results of Hobza and Morales (2013) by considering a model where the set of random effects has a multivariate normal distribution that includes all variances and covariances as unknown parameters. It also studies the more simple model without covariances and gives some R codes for the last model.

13.2 The RRC Model with Covariance Parameters

We start with a description of the more general model.

13.2.1 The Model

Let us consider the model

$$\displaystyle \begin{aligned} y_{dj}=\sum_{k=1}^p\beta_kx_{kdj}+\sum_{k=1}^pu_{kd}x_{kdj}+e_{dj},\quad d=1,\ldots,D,\,\,j=1,\ldots,n_{d}, \end{aligned} $$

(13.1)

where

• $\boldsymbol {u}_d^\ast =(u_{1d},\ldots ,u_{pd})^\prime \overset {iid}{\sim }N_{p}(\boldsymbol {0},\boldsymbol {V}_{\boldsymbol {u}_d^\ast })$ and $e_{dj}\overset {ind}{\sim }N(0,w_{dj}^{-1}\sigma _{e}^2)$ are independent, d = 1, …, D, j = 1…, n _d,

• $\boldsymbol {V}_{\boldsymbol {u}_d^\ast }$ is a covariance matrix with components $\mbox{cov}(u_{k_1d},u_{k_2,d})=\sigma _{k_1k_2}$, k ₁, k ₂ = 1, …, p.

In models with intercept, the first auxiliary variable is equal to one. The model (13.1) has p regression parameters and $1+p+\frac 12p(p-1)= 1+\frac 12p(p+1)$ variance component parameters. They are β _k, $\sigma _e^2$, $\sigma _{k_1k_2}$, with $\sigma _{k_1k_2}=\sigma _{k_2k_1}$, k, k ₁, k ₂ = 1, …, p.

An example of model (13.1) with intercept, D = 2, n ₁ = n ₂ = 2, n = 4, and p = 2 is

In matrix notation model (13.1) is

$$\displaystyle \begin{aligned} \boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\sum_{k=1}^p\boldsymbol{Z}_k\boldsymbol{u}_k+\boldsymbol{e}, \end{aligned} $$

(13.2)

where $n=\sum _{d=1}^Dn_d$, β = β _p×1, , , , , , , , , , , , , , with w _dj > 0 known, d = 1, …, D, j = 1, …, n _d. Covariance matrices are $\boldsymbol {V}_e=\mbox{var}(\boldsymbol {e})=\sigma _e^2\boldsymbol {W}^{-1}$, $\boldsymbol {V}_{k_1k_2}=\mbox{cov}(\boldsymbol {u}_{k_1},\boldsymbol {u}_{k_2})=\sigma _{k_1k_2}\boldsymbol {I}_D$, k ₁, k ₂ = 1, …, p, and

where

$$\displaystyle \begin{aligned}\boldsymbol{V}_d=\sigma_e^2\boldsymbol{W}_{d}^{-1}+\sum_{k_1=1}^p\sum_{k_2=1}^p\sigma_{k_1k_2}\boldsymbol{x}_{k_1,n_d}\boldsymbol{x}_{k_2,n_d}^\prime,\quad d=1,\ldots,D. \end{aligned}$$

Let and . Under this notation, the variance of u is

$$\displaystyle \begin{aligned}\boldsymbol{V}_u=\mbox{var}(\boldsymbol{u})=\left(\boldsymbol{V}_{k_1k_2}\right)_{k_1k_2=1,\ldots,p} \end{aligned}$$

and the model (13.2) can be written in the general form

$$\displaystyle \begin{aligned}\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{Z}\boldsymbol{u}+\boldsymbol{e}. \end{aligned}$$

If the variance components are known, then the BLUE of β = (β ₁, …, β _p)^′ is (cf. (6.12))

$$\displaystyle \begin{aligned}\tilde{\boldsymbol{\beta}}=(\boldsymbol{X}^\prime{\boldsymbol{V}}^{-1}\boldsymbol{X})^{-1}\boldsymbol{X}^\prime{\boldsymbol{V}}^{-1}\boldsymbol{y} = \Big(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{V}_d^{-1}\boldsymbol{X}_d\Big)^{-1}\Big(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{V}_d^{-1}\boldsymbol{y}_d\Big) \end{aligned}$$

and the BLUP of u is $\tilde {{\boldsymbol {u}}}=\boldsymbol {V}_{u}\boldsymbol {Z}^\prime \boldsymbol {V}^{-1} (\boldsymbol {y}-\boldsymbol {X}\tilde {\boldsymbol {\beta }})$.

13.2.2 REML Estimators

In order to derive formulas for calculating the REML estimates of the unknown variance parameters we consider the alternative parametrization $\sigma ^2=\sigma _e^2$, $\varphi _{k_1k_2}=\sigma _{k_1k_2}/\sigma _e^2$, k ₁, k ₂ = 1…, p, and we define $\boldsymbol {\sigma }=(\sigma ^2,\varphi _{k_1k_2},\,k_1,k_2=1\ldots ,p)$ and where Σ _d = σ ⁻² V _d.

The REML log-likelihood is (cf. (6.32))

$$\displaystyle \begin{aligned}{l}_{REML}(\boldsymbol{\sigma})=-\frac{1}{2}(n-p)\log 2\pi-\frac{1}{2}(n-p)\log\sigma^2-\frac{1}{2}\log|\boldsymbol{K}^\prime\boldsymbol{\Sigma}\boldsymbol{K}| -\frac{1}{2\sigma^2}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y}, \end{aligned}$$

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{P}& =&\displaystyle \boldsymbol{K}(\boldsymbol{K}^\prime\boldsymbol{\Sigma}\boldsymbol{K})^{-1}\boldsymbol{K}^\prime=\boldsymbol{\Sigma}^{-1}-\boldsymbol{\Sigma}^{-1}\boldsymbol{X}(\boldsymbol{X}^\prime\boldsymbol{\Sigma}^{-1}\boldsymbol{X})^{-1}\boldsymbol{X}^\prime\boldsymbol{\Sigma}^{-1}, \\ \boldsymbol{K}& =&\displaystyle \boldsymbol{W}-\boldsymbol{W}\boldsymbol{X}(\boldsymbol{X}^\prime\boldsymbol{W}\boldsymbol{X})^{-1}\boldsymbol{X}^\prime\boldsymbol{W} \end{array} \end{aligned} $$

are such that PX = 0 and P Σ P = P. The matrix Σ can be written in the form

$$\displaystyle \begin{aligned}\boldsymbol{\Sigma}=\boldsymbol{W}^{-1}+\sum_{k_1=1}^{p}\sum_{k_2=1}^{p}\varphi_{k_1k_2}\boldsymbol{A}_{k_1k_2}, \end{aligned}$$

where , k ₁, k ₂ = 1, …, p. In the same way as in (6.43) we obtain $\frac {\partial \boldsymbol {P}}{\partial \varphi _{k_1k_2}}=-\boldsymbol {P}\boldsymbol {A}_{k_1k_2}\boldsymbol {P}$. Thus, by taking partial derivatives of the REML log-likelihood with respect to σ ² and $\varphi _{k_1k_2}$, k ₁, k ₂ = 1, …, p, one gets the scores

$$\displaystyle \begin{aligned}S_{\sigma^2}=-\frac{n-p}{2\sigma^2}+\frac{1}{2\sigma^4}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y},\quad S_{\varphi_{k_1k_2}}=-\frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\}+\frac{1}{2\sigma^2}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{y}, \end{aligned}$$

and the second partial derivatives

$$\displaystyle \begin{aligned} \begin{array}{rcl} H_{\sigma^2\sigma^2}& =&\displaystyle \frac{n-p}{2\sigma^4}-\frac{1}{\sigma^6}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y},\quad H_{\sigma^2\varphi_{k_1k_2}}=-\frac{1}{2\sigma^4}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{y},\\ H_{\varphi_{k_1k_2}\varphi_{i_1i_2}}& =&\displaystyle \frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{A}_{i_1i_2}\} -\frac{1}{2\sigma^2}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{A}_{i_1i_2}\boldsymbol{P}\boldsymbol{y} \\ & -&\displaystyle \frac{1}{2\sigma^2}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_{i_1i_2}\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{y}, \end{array} \end{aligned} $$

where k ₁, k ₂, i ₁, i ₂ = 1, …, p. By taking expectations and multiplying by − 1, we obtain the components of the Fisher information matrix. For k ₁, k ₂, i ₁, i ₂ = 1, …, p, we have

$$\displaystyle \begin{aligned}F_{\sigma^2\sigma^2}=\frac{n-p}{2\sigma^4},\,\, F_{\sigma^2\varphi_{k_1k_2}}=\frac{1}{2\sigma^2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\},\,\, F_{\varphi_{k_1k_2}\varphi_{i_1i_2}}=\frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_{k_1k_2}\boldsymbol{P}\boldsymbol{A}_{i_1i_2}\}. \end{aligned}$$

To calculate the REML estimates, the Fisher-scoring updating formula, at iteration i, is

$$\displaystyle \begin{aligned}\boldsymbol{\sigma}^{(i+1)}=\boldsymbol{\sigma}^{(i)}+\boldsymbol{F}^{-1}(\boldsymbol{\sigma}^{(i)})\boldsymbol{S}(\boldsymbol{\sigma}^{(i)}), \end{aligned}$$

where S(σ) and F(σ) are the vector of scores and the Fisher information matrix evaluated at σ. The following seeds can be used as starting values in the Fisher-scoring algorithm:

$$\displaystyle \begin{aligned}\sigma^{2(0)}=S^2/(p+2),\quad \varphi_{k_1k_2}=\delta_{k_1k_2},\quad k_1,k_2=1,\ldots,p, \end{aligned}$$

where $\delta _{k_1,k_2}=1$ if k ₁ = k ₂, $\delta _{k_1,k_2}=0$ if k ₁ ≠ k ₂, $S^2=\frac {1}{n-p}(\boldsymbol {y}-\boldsymbol {X}\hat {\boldsymbol {\beta }}_0)^\prime \boldsymbol {W}(\boldsymbol {y}-\boldsymbol {X}\hat {\boldsymbol {\beta }}_0)$ and $\hat {\boldsymbol {\beta }}_0=(\boldsymbol {X}^\prime \boldsymbol {W}\boldsymbol {X})^{-1}\boldsymbol {X}^\prime \boldsymbol {W}\boldsymbol {y}$.

13.2.3 EBLUP of the Domain Mean

Let us now consider a finite population U partitioned into D domains U _d, i.e. $U=\cup _{d=1}^DU_d$. Let N and N _d be the sizes of U and U _d, so that $n=\sum _{d=1}^Dn_d$. We assume that the population target vector y = y _N×1 follows the RRC model (13.2) with the obvious size changes, i.e. with N and N _d in the place of n and n _d, respectively.

Let s ⊂ U be a sample of n ≤ N units and let r = U − s be the set of non-sampled units. The domain and subdomain subsets of s and r are denoted by s _d and r _d, respectively. The subindexes s and r in vectors or matrices are used to denote their sampled and the non-sampled parts. Without loss of generality, we renumber the population units and we write

$$\displaystyle \begin{aligned}\boldsymbol{y}=\left(\begin{array}{c} \boldsymbol{y}_{s}\\ \boldsymbol{y}_{r}\end{array}\right),\quad \boldsymbol{X}=\left(\begin{array}{c} \boldsymbol{X}_{s}\\ \boldsymbol{X}_{r}\end{array}\right),\quad \boldsymbol{e}=\left(\begin{array}{c} \boldsymbol{e}_{s}\\ \boldsymbol{e}_{r}\end{array}\right), \quad \boldsymbol{Z}_k=\left(\begin{array}{c} \boldsymbol{Z}_{sk}\\ \boldsymbol{Z}_{rk}\end{array}\right),\ k=1,\ldots, p, \end{aligned}$$

and

$$\displaystyle \begin{aligned}\boldsymbol{V}=\mbox{var}(\boldsymbol{y})=\left(\begin{array}{cc} \boldsymbol{V}_{s}&\boldsymbol{V}_{sr}\\ \boldsymbol{V}_{rs}&\boldsymbol{V}_{r}\end{array}\right). \end{aligned}$$

Using Theorem 4.1, the EBLUP of the linear parameter $\eta =\boldsymbol {a}^\prime \boldsymbol {y}=\boldsymbol {a}_s^\prime \boldsymbol {y}_s+\boldsymbol {a}_r^\prime \boldsymbol {y}_r$ is

$$\displaystyle \begin{aligned}\hat{\eta}=\boldsymbol{a}_s^\prime\boldsymbol{y}_s+\boldsymbol{a}_r^\prime\left[\boldsymbol{X}_r\hat{\boldsymbol{\beta}}+\hat{\boldsymbol{V}}_{rs}\hat{\boldsymbol{V}}_{s}^{-1} (\boldsymbol{y}_s-\boldsymbol{X}_s\hat{\boldsymbol{\beta}})\right]\, , \end{aligned}$$

where

$$\displaystyle \begin{aligned}\hat{\boldsymbol{\beta}} = (\boldsymbol{X}_s^\prime \hat{\boldsymbol{V}}_s^{-1}\boldsymbol{X}_s)^{-1} \boldsymbol{X}_s^\prime\hat{\boldsymbol{V}}_s^{-1} \boldsymbol{y}_s.\end{aligned}$$

As V _ers = 0, $\boldsymbol {V}_{rs}=\boldsymbol {Z}_r\boldsymbol {V}_u\boldsymbol {Z}_s^\prime +\boldsymbol {V}_{ers}=\boldsymbol {Z}_r\boldsymbol {V}_u\boldsymbol {Z}_s^\prime$ and $\hat {\boldsymbol {u}}=\hat {\boldsymbol {V}}_u\boldsymbol {Z}_s^\prime \hat {\boldsymbol {V}}_{s}^{-1}(\boldsymbol {y}_s-\boldsymbol {X}_s\hat {\boldsymbol {\beta }})$, then

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{\eta}& =&\displaystyle \boldsymbol{a}_s^\prime\boldsymbol{y}_s+\boldsymbol{a}_r^\prime\left[\boldsymbol{X}_r\hat{\boldsymbol{\beta}}+\boldsymbol{Z}_r\hat{\boldsymbol{V}}_u\boldsymbol{Z}_s^\prime\hat{\boldsymbol{V}}_{s}^{-1} (\boldsymbol{y}_s-\boldsymbol{X}_s\hat{\boldsymbol{\beta}})\right] = \boldsymbol{a}_s^\prime\boldsymbol{y}_s+\boldsymbol{a}_r^\prime\left[\boldsymbol{X}_r\hat{\boldsymbol{\beta}}+\boldsymbol{Z}_r\hat{\boldsymbol{u}}\right]\\ & =&\displaystyle \boldsymbol{a}^\prime\left[\boldsymbol{X}\hat{\boldsymbol{\beta}}+\sum_{k=1}^p\boldsymbol{Z}_k\hat{\boldsymbol{u}}_k\right]+ \boldsymbol{a}_s^\prime\left[\boldsymbol{y}_s-\boldsymbol{X}_s\hat{\boldsymbol{\beta}}-\sum_{k=1}^p\boldsymbol{Z}_{sk}\hat{\boldsymbol{u}}_k\right]. \end{array} \end{aligned} $$

The domain mean is $\overline {Y}_{d}=\frac {1}{N_{d}}\sum _{j=1}^{N_{d}}y_{dj}=\mu _d+\overline {e}_{d}$, where $\overline {e}_{d}=\frac {1}{N_{d}}\sum _{j=1}^{N_{d}}e_{dj}$ and

$$\displaystyle \begin{aligned}\mu_{d}=\sum_{k=1}^p\overline{X}_{kd}\beta_k+\sum_{k=1}^p\overline{X}_{kd}u_{kd}, \qquad \overline{X}_{kd}=\frac{1}{N_{d}}\sum_{j=1}^{N_d}x_{kdj} . \end{aligned}$$

The domain mean $\overline {Y}_{d}$ can be written in the form η = a ^′ y, where

δ _ab = 1 if a = b and δ _ab = 0 if a ≠ b. It holds that $\boldsymbol {a}^\prime \boldsymbol {X}=\overline {\boldsymbol {X}}_{d}=(\overline X_{1d},\ldots ,\overline X_{pd})$,

If n _d > 0, then the EBLUP of $\overline {Y}_{d}$ is

$$\displaystyle \begin{aligned}\widehat{\overline{Y}}^{eblup}_{d}=\sum_{k=1}^p\overline{X}_{kd}\hat{\beta}_k+\sum_{k=1}^p\overline{X}_{kd}\hat{u}_{kd} + f_{d}\left[\overline{y}_{s,d}-\sum_{k=1}^p\overline{X}_{s,kd}\hat{\beta}_k- \sum_{k=1}^p\overline{X}_{s,kd}\hat{u}_{kd}\right], \end{aligned}$$

where $\overline {y}_{s,d}=\frac {1}{n_{d}}\sum _{j=1}^{n_{d}}y_{dj}$, $\overline {X}_{s,kd}=\frac {1}{n_{d}}\sum _{j=1}^{n_{d}}x_{kdj}$ and $f_{d}=\frac {n_{d}}{N_{d}}$. If f _d ≈ 0, then the EBLUP of $\overline {Y}_{d}$ is approximately equal to

$$\displaystyle \begin{aligned}\hat{\mu}^{eblup}_{d}=\sum_{k=1}^p\overline{X}_{kd}\hat{\beta}_k+ \sum_{k=1}^p\overline{X}_{kd}\hat{u}_{kd}. \end{aligned}$$

The MSE of the EBLUP can be estimated by adapting the steps 1–6 of the parametric bootstrap procedure described in Sect. 8.5.

13.3 The RRC Model Without Covariance Parameters

For the ease of exposition, we consider now a slightly simpler model under which we derive more detailed formulas for the REML estimators and the formulas for the analytic approximation of the MSE of EBLUPs.

13.3.1 The Model

Let us consider the RRC model

$$\displaystyle \begin{aligned} y_{dj}=\sum_{k=1}^p\beta_kx_{kdj}+\sum_{k=1}^pu_{kd}x_{kdj}+e_{dj},\quad d=1,\ldots,D,\,\,j=1,\ldots,n_{d}, \end{aligned} $$

(13.3)

where $u_{kd}\overset {iid}{\sim }N(0,\sigma _{k}^2)$ and $e_{dj}\overset {iid}{\sim }N(0,w_{dj}^{-1}\sigma _{e}^2)$ are independent, d = 1, …, D, j = 1…, n _d, k = 1, …, p. The model variance and covariance parameters are $\sigma _e^2$, $\sigma _k^2$, k = 1, …, p, (p + 1 parameters). In matrix notation model (13.3) is

$$\displaystyle \begin{aligned} \boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\sum_{k=1}^p\boldsymbol{Z}_k\boldsymbol{u}_k+\boldsymbol{e}, \end{aligned} $$

(13.4)

where the meaning of the used symbols is exactly the same as the one given below formula (13.2). The difference with respect to the model (13.2) is only in the variance matrices which are now simpler and have the form $\boldsymbol {V}_e=\mbox{var}(\boldsymbol {e})=\sigma _e^2\boldsymbol {W}^{-1}$, $\boldsymbol {V}_{\boldsymbol {u}_k}=\mbox{var}(\boldsymbol {u}_k)=\sigma _k^2\boldsymbol {I}_D$, k = 1, …, p, and

where

$$\displaystyle \begin{aligned}\boldsymbol{V}_d=\sigma_e^2\boldsymbol{W}_{d}^{-1}+\sum_{k=1}^p\sigma_k^2\boldsymbol{x}_{k,n_d}\boldsymbol{x}_{k,n_d}^\prime,\,\,\, d=1,\ldots,D. \end{aligned}$$

We consider the alternative parameters $\sigma ^2=\sigma _e^2$, $\varphi _k=\sigma _k^2/\sigma _e^2$, k = 1, …, p, in such a way that V  = σ ² Σ and V _d = σ ² Σ _d, where

$$\displaystyle \begin{aligned}\boldsymbol{\Sigma}_d=\boldsymbol{W}_{d}^{-1}+\sum_{k=1}^p\varphi_k\boldsymbol{x}_{k,n_d}\boldsymbol{x}_{k,n_d}^\prime,\,\,\, d=1,\ldots,D. \end{aligned}$$

Let θ = (σ ², φ ₁, …, φ _p)^′ be the vector of variance components, with σ ² > 0, φ ₁ > 0, …, φ _p > 0. Let and . The variance of u is

Using this notation, the model (13.4) can be written in the general form

$$\displaystyle \begin{aligned}\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{Z}\boldsymbol{u}+\boldsymbol{e}. \end{aligned}$$

If θ is known, then the BLUE of β = (β ₁, …, β _p)^′ is (cf. (6.12))

$$\displaystyle \begin{aligned}\tilde{\boldsymbol{\beta}}=(\boldsymbol{X}^\prime{\boldsymbol{V}}^{-1}\boldsymbol{X})^{-1}\boldsymbol{X}^\prime{\boldsymbol{V}}^{-1}\boldsymbol{y} = \left(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\right)^{-1}\left(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d\right) \end{aligned}$$

and the BLUP of u is $\tilde {{\boldsymbol {u}}}=\boldsymbol {V}_{u}\boldsymbol {Z}^\prime \boldsymbol {V}^{-1} (\boldsymbol {y}-\boldsymbol {X}\tilde {\boldsymbol {\beta }})$, i.e.

13.3.2 REML Estimators

In this section we follow the same steps as in the Sect. 13.2.2 and we derive the REML estimators under the model (13.4) with more details concerning the matrix calculations. The REML log-likelihood is

$$\displaystyle \begin{aligned}{l}_{REML}(\boldsymbol{\theta})=-\frac{1}{2}(n-p)\log 2\pi-\frac{1}{2}(n-p)\log\sigma^2-\frac{1}{2}\log|\boldsymbol{K}^\prime\boldsymbol{\Sigma}\boldsymbol{K}| -\frac{1}{2\sigma^2}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y}, \end{aligned}$$

where P = K(K ^′ Σ K)⁻¹ K ^′ = Σ ⁻¹ − Σ ⁻¹ X(X ^′ Σ ⁻¹ X)⁻¹ X ^′ Σ ⁻¹ and K = W −WX(X ^′ WX)⁻¹ X ^′ W are such that PX = 0 and P Σ P = P. The matrix Σ can be written in the form

$$\displaystyle \begin{aligned}\boldsymbol{\Sigma}=\boldsymbol{W}^{-1}+\sum_{k= 1}^{p}\varphi_k\boldsymbol{A}_k, \end{aligned}$$

where , k = 1, …, p. As $\frac {\partial \boldsymbol {P}}{\partial \varphi _k}=-\boldsymbol {P}\boldsymbol {A}_k\boldsymbol {P}$, by taking partial derivatives with respect to σ ² and φ _k, k = 1, …, p, one gets

$$\displaystyle \begin{aligned}S_{\sigma^2}=-\frac{n-p}{2\sigma^2}+\frac{1}{2\sigma^4}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y},\quad S_{\varphi_k}=-\frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_k\}+\frac{1}{2\sigma^2}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_k\boldsymbol{P}\boldsymbol{y},\quad k=1,\ldots,p. \end{aligned}$$

The second partial derivatives are

$$\displaystyle \begin{aligned} \begin{array}{rcl} H_{\sigma^2\sigma^2}& =&\displaystyle \frac{n-p}{2\sigma^4}-\frac{1}{\sigma^6}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{y},\quad H_{\sigma^2\varphi_k}=-\frac{1}{2\sigma^4}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_k\boldsymbol{P}\boldsymbol{y},\\ H_{\varphi_k\varphi_i}& =&\displaystyle \frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_k\boldsymbol{P}\boldsymbol{A}_i\}-\frac{1}{\sigma^2}\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{A}_k\boldsymbol{P}\boldsymbol{A}_i\boldsymbol{P}\boldsymbol{y},\quad k,i=1,\ldots,p. \end{array} \end{aligned} $$

By taking expectations and multiplying by − 1, we obtain the components of the Fisher information matrix

$$\displaystyle \begin{aligned}F_{\sigma^2\sigma^2}=\frac{n-p}{2\sigma^4},\, F_{\sigma^2\varphi_k}=\frac{1}{2\sigma^2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_k\},\, F_{\varphi_k\varphi_i}=\frac{1}{2}\,\mbox{tr}\{\boldsymbol{P}\boldsymbol{A}_k\boldsymbol{P}\boldsymbol{A}_i\},\, k,i=1,\ldots,p. \end{aligned}$$

To calculate the REML estimates, the Fisher-scoring updating formula, at iteration i, is

$$\displaystyle \begin{aligned}\boldsymbol{\theta}^{(i+1)}=\boldsymbol{\theta}^{(i)}+\boldsymbol{F}^{-1}(\boldsymbol{\theta}^{(i)})\boldsymbol{S}(\boldsymbol{\theta}^{(i)}). \end{aligned}$$

The following seeds can be used as starting values in the Fisher-scoring algorithm:

$$\displaystyle \begin{aligned}\sigma^{2(0)}=\varphi_1^{(0)}=\ldots=\varphi_p^{(0)}=S^2/(p+2), \end{aligned}$$

where $S^2=\frac {1}{n-p}(\boldsymbol {y}-\boldsymbol {X}\hat {\boldsymbol {\beta }}_0)^\prime \boldsymbol {W}(\boldsymbol {y}-\boldsymbol {X}\hat {\boldsymbol {\beta }}_0)$ and $\hat {\boldsymbol {\beta }}_0=(\boldsymbol {X}^\prime \boldsymbol {W}\boldsymbol {X})^{-1}\boldsymbol {X}^\prime \boldsymbol {W}\boldsymbol {y}$.

13.3.2.1 Matrix Calculations for the RRC Model

In this section we show how to do the matrix calculations in the Fisher-scoring algorithm. We define

such that

For k = 1, …, p, the components of the vector of scores are

$$\displaystyle \begin{aligned} \begin{array}{rcl} S_{\sigma^2}& =&\displaystyle -\frac{n-p}{2\sigma^2}+\frac{1}{2\sigma^{4}}\sum_{d=1}^{D}\boldsymbol{y}_{d}^{t}\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d -\frac{1}{2\sigma^{4}}\left(\sum_{d=1}^{D}\boldsymbol{y}_{d}^{t}\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\right)\boldsymbol{R} \left(\sum_{d=1}^{D}\boldsymbol{X}_{d}^{t}\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d\right), \end{array} \end{aligned} $$

$$\displaystyle \begin{aligned} \begin{array}{rcl} S_{\varphi_k}& =&\displaystyle -\frac{1}{2}\,\mbox{tr}\{\boldsymbol{Z}_k^\prime\boldsymbol{P}\boldsymbol{Z}_k\}+\frac{1}{2\sigma^2}\,\boldsymbol{y}^\prime\boldsymbol{P}\boldsymbol{Z}_k\boldsymbol{Z}_k^\prime\boldsymbol{P}\boldsymbol{y} \\ & =&\displaystyle -\frac{1}{2}\,\sum_{d=1}^D\boldsymbol{x}_{k,n_d}^\prime[\boldsymbol{\Sigma}_d^{-1}-\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\boldsymbol{R}\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}]\boldsymbol{x}_{k,n_d}\\ & +&\displaystyle \frac{1}{2\sigma^2}\,\sum_{d=1}^D\boldsymbol{y}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k,n_d}\boldsymbol{x}_{k,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d \\ & -&\displaystyle \frac{1}{\sigma^2}\bigg(\sum_{d=1}^D\boldsymbol{y}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k,n_d}\boldsymbol{x}_{k,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\bigg)\boldsymbol{R} \bigg(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d\bigg)\\ & +&\displaystyle \frac{1}{2\sigma^2}\,\bigg(\sum_{d=1}^D\boldsymbol{y}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\bigg)\boldsymbol{R} \bigg(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k,n_d}\boldsymbol{x}_{k,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\bigg)\boldsymbol{R} \bigg(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{y}_d\bigg). \end{array} \end{aligned} $$

For k, k ₁, k ₂ = 1, …, p, the components of the REML Fisher information matrix are

$$\displaystyle \begin{aligned} \begin{array}{rcl} F_{\sigma^2\sigma^2}& =&\displaystyle \frac{n-p}{2\sigma^{4}}, \\ F_{\sigma^2\varphi_k}& =&\displaystyle \frac{1}{2\sigma^2}\mbox{tr}\{\boldsymbol{Z}_k^\prime\boldsymbol{P}\boldsymbol{Z}_k\} = \frac{1}{2\sigma^2}\sum_{d=1}^D\boldsymbol{x}_{k,n_d}^\prime\left[\boldsymbol{\Sigma}_d^{-1}-\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d \boldsymbol{R}\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\right]\boldsymbol{x}_{k,n_d}, \\ F_{\varphi_{k_1}\varphi_{k_2}}& =&\displaystyle \frac{1}{2}\,\mbox{tr}\{\boldsymbol{Z}_{k_2}^\prime\boldsymbol{P}\boldsymbol{Z}_{k_1}\boldsymbol{Z}_{k_1}^\prime\boldsymbol{P}\boldsymbol{Z}_{k_2}\} =\frac{1}{2}\!\sum_{d=1}^D(\boldsymbol{x}_{k_2,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k_1,n_d})^2 \\ & -&\displaystyle \sum_{d=1}^D\boldsymbol{x}_{k_2,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k_1,n_d}\boldsymbol{x}_{k_1,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\boldsymbol{R}\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k_2,n_d} \\ & +&\displaystyle \frac{1}{2}\,\sum_{d=1}^D\boldsymbol{x}_{k_2,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\boldsymbol{R} \bigg(\sum_{d=1}^D\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k_1,n_d}\boldsymbol{x}_{k_1,n_d}^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{X}_d\bigg)\boldsymbol{R}\boldsymbol{X}_d^\prime\boldsymbol{\Sigma}_d^{-1}\boldsymbol{x}_{k_2,n_d}. \end{array} \end{aligned} $$

The inverse of matrix Σ _d can be calculated by applying iteratively the formula

$$\displaystyle \begin{aligned}(A+CBD)^{-1}=A^{-1}-A^{-1}C(B^{-1}+DA^{-1}C)^{-1}DA^{-1}. \end{aligned}$$

Step 1: Define $\boldsymbol {L}_{1d}=\boldsymbol {W}_d^{-1}+\varphi _1\boldsymbol {x}_{1,n_{d}}\boldsymbol {x}_{1,n_{d}}^\prime$. Take $A=\boldsymbol {W}_d^{-1}$, $C=\varphi _1\boldsymbol {x}_{1,n_{d}}$, B = I ₁, and $D=\boldsymbol {x}_{1,n_{d}}^\prime$, so that

$$\displaystyle \begin{aligned}\boldsymbol{L}_{1d}^{-1}=\boldsymbol{W}_d-\boldsymbol{W}_d\varphi_1\boldsymbol{x}_{1,n_{d}}(1+\boldsymbol{x}_{1,n_{d}}^\prime\boldsymbol{W}_d\varphi_1\boldsymbol{x}_{1,n_{d}})^{-1}\boldsymbol{x}_{1,n_{d}}^\prime\boldsymbol{W}_d. \end{aligned}$$

Step 2: Define $\boldsymbol {L}_{2d}=\boldsymbol {L}_{1d}+\varphi _2\boldsymbol {x}_{2,n_d}\boldsymbol {x}_{2,n_d}^\prime$. Take A = L _1d, $C=\varphi _2\boldsymbol {x}_{2,n_d}$, B = I ₁ and $D=\boldsymbol {x}_{2,n_d}^\prime$, so that

$$\displaystyle \begin{aligned}\boldsymbol{L}_{2d}^{-1}=\boldsymbol{L}_{1d}^{-1}-\boldsymbol{L}_{1d}^{-1}\varphi_2\boldsymbol{x}_{2,n_d}(1+\boldsymbol{x}_{2,n_d}^\prime\boldsymbol{L}_{1d}^{-1}\varphi_2\boldsymbol{x}_{2,n_d})^{-1} \boldsymbol{x}_{2,n_d}^\prime\boldsymbol{L}_{1d}^{-1}. \end{aligned}$$

…

Step p: Finally, $\boldsymbol {L}_{pd}=\boldsymbol {L}_{p-1d}+\varphi _p\,\boldsymbol {x}_{p,n_d}\boldsymbol {x}_{p,n_d}^\prime$. Take A = L _p−1d, $C=\varphi _p\,\boldsymbol {x}_{p,n_d}$, B = I ₁, and $D=\boldsymbol {x}_{p,n_d}^\prime$, so that

$$\displaystyle \begin{aligned}\boldsymbol{L}_{pd}^{-1}=\boldsymbol{L}_{p-1d}^{-1}-\boldsymbol{L}_{p-1d}^{-1}\varphi_p\,\boldsymbol{x}_{p,n_d}(1+\boldsymbol{x}_{p,n_d}^\prime\boldsymbol{L}_{p-1d}^{-1}\varphi_p\boldsymbol{x}_{p,n_d})^{-1} \boldsymbol{x}_{p,n_d}^\prime\boldsymbol{L}_{p-1d}^{-1}. \end{aligned}$$

13.3.3 EBLUP of a Domain Mean

The formulas for the EBLUP of the linear parameter $\overline {Y}_d$ under model (13.4) have exactly the same form as the ones derived under model (13.2) in Sect. 13.2.3. Namely, if n _d > 0, then the EBLUP of $\overline {Y}_{d}$ is

$$\displaystyle \begin{aligned}\widehat{\overline{Y}}^{eblup}_{d}=\sum_{k=1}^p\overline{X}_{kd}\hat{\beta}_k+\sum_{k=1}^p\overline{X}_{kd}\hat{u}_{kd} + f_{d}\left[\overline{y}_{s,d}-\sum_{k=1}^p\overline{X}_{s,kd}\hat{\beta}_k- \sum_{k=1}^p\overline{X}_{s,kd}\hat{u}_{kd}\right], \end{aligned}$$

where $\overline {X}_{kd}=\frac {1}{N_{d}}\sum _{j=1}^{N_d}x_{kdj}$, $\overline {y}_{s,d}=\frac {1}{n_{d}}\sum _{j=1}^{n_{d}}y_{dj}$, $\overline {X}_{s,kd}=\frac {1}{n_{d}}\sum _{j=1}^{n_{d}}x_{kdj}$ and $f_{d}=\frac {n_{d}}{N_{d}}$. If n _d = 0, then the EBLUP of $\overline {Y}_{d}$ is the synthetic part

$$\displaystyle \begin{aligned}\hat{\mu}^{eblup}_{d}=\sum_{k=1}^p\overline{X}_{kd}\hat{\beta}_k+ \sum_{k=1}^p\overline{X}_{kd}\hat{u}_{kd}. \end{aligned}$$

13.3.4 MSE of the EBLUP

Let θ = (σ ², φ ₁, …, φ _p)^′ be the vector of variance components and $\hat {\boldsymbol {\theta }}$ be the corresponding REML estimate. The MSEs of the EBLUPs of $\overline {Y}_{d}$ and μ _d are (cf. pp. 12 and 9, respectively)

$$\displaystyle \begin{aligned} MSE(\widehat{\overline{Y}}_{d}^{eblup})= g_1(\boldsymbol{\theta})+g_2(\boldsymbol{\theta})+g_3(\boldsymbol{\theta})+g_4(\boldsymbol{\theta}),\quad MSE(\hat{\mu}_{d}^{eblup})= g_1(\boldsymbol{\theta})+g_2(\boldsymbol{\theta})+g_3(\boldsymbol{\theta}), \end{aligned}$$

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} g_1(\boldsymbol{\theta})& =&\displaystyle \boldsymbol{a}_r^\prime\boldsymbol{Z}_r\boldsymbol{T}_s\boldsymbol{Z}_r^\prime\boldsymbol{a}_r,\\ g_2(\boldsymbol{\theta})& =&\displaystyle [\boldsymbol{a}_r^\prime\boldsymbol{X}_r-\boldsymbol{a}_r^\prime\boldsymbol{Z}_r\boldsymbol{T}_s\boldsymbol{Z}_s^\prime\boldsymbol{V}_{es}^{-1}\boldsymbol{X}_s]\boldsymbol{Q}_s [\boldsymbol{X}_r^\prime\boldsymbol{a}_r-\boldsymbol{X}_s^\prime\boldsymbol{V}_{es}^{-1}\boldsymbol{Z}_s\boldsymbol{T}_s\boldsymbol{Z}_r^\prime\boldsymbol{a}_r],\\ g_3(\boldsymbol{\theta})& \approx&\displaystyle \mbox{tr}\left\{(\nabla\boldsymbol{b}^\prime)\boldsymbol{V}_s(\nabla\boldsymbol{b}^\prime)^\prime E\left[(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})^\prime\right]\right\},\\ g_4(\boldsymbol{\theta})& =&\displaystyle \boldsymbol{a}_r^\prime\boldsymbol{V}_{er}\boldsymbol{a}_r \end{array} \end{aligned} $$

and definitions of the symbols T _s, Q _s, and b ^′ are given in Sect. 9.2 and will be revised on the following pages. The Prasad-Rao (PR) estimator of $MSE(\hat {\overline {Y}}_{d}^{eblup})$ is

$$\displaystyle \begin{aligned}mse_d=mse(\hat{\overline{Y}}_{d}^{eblup})= g_1(\hat{\boldsymbol{\theta}})+g_2(\hat{\boldsymbol{\theta}})+2g_3(\hat{\boldsymbol{\theta}}) + g_4(\hat{\boldsymbol{\theta}})\, . \end{aligned}$$

Now, we present a detailed description of calculation of the terms g _i(θ), i = 1, …, 4, under the present model.

Calculation of g ₁(θ)

To calculate $g_1(\boldsymbol {\theta })=\boldsymbol {a}_r^\prime \boldsymbol {Z}_r\boldsymbol {T}_s\boldsymbol {Z}_r^\prime \boldsymbol {a}_r$, basic elements are

where

and $\delta _{k_1k_2}=0$ if k ₁ ≠ k ₂, $\delta _{k_1k_2}=1$ if k ₁ = k ₂. Therefore

where f _d = n _d∕N _d and $\overline {X}^{*}_{kd}=\frac {1}{N_d-n_d}\sum _{j\in r_d}x_{kdj}=(1-f_d)^{-1}(\overline {X}_{kd}-f_d\overline {x}_{kd})$.

Calculation of g ₂(θ)

We recall that

$$\displaystyle \begin{aligned} \begin{array}{rcl} g_2(\boldsymbol{\theta})& =&\displaystyle [\boldsymbol{a}_r^\prime\boldsymbol{X}_r-\boldsymbol{a}_r^\prime\boldsymbol{Z}_r\boldsymbol{T}_s\boldsymbol{Z}_s^\prime\boldsymbol{V}_{es}^{-1}\boldsymbol{X}_s]\boldsymbol{Q}_s [\boldsymbol{X}_r^\prime\boldsymbol{a}_r-\boldsymbol{X}_s^\prime\boldsymbol{V}_{es}^{-1}\boldsymbol{Z}_s\boldsymbol{T}_s\boldsymbol{Z}_r^\prime\boldsymbol{a}_r] \\ & =&\displaystyle [\boldsymbol{a}_{1}^\prime-\boldsymbol{a}_{2}^\prime]\boldsymbol{Q}_s[\boldsymbol{a}_{1}-\boldsymbol{a}_{2}], \end{array} \end{aligned} $$

where $\boldsymbol {Q}_s=(\boldsymbol {X}_s^\prime \boldsymbol {V}^{-1}\boldsymbol {X}_s)^{-1}=\sigma ^2\left (\sum _{d=1}^D\boldsymbol {X}_{sd}^\prime \boldsymbol {\Sigma }_{sd}^{-1}\boldsymbol {X}_{sd}\right )^{-1}$ and $\boldsymbol {V}_{es}^{-1}=\sigma ^{-2}\boldsymbol {W}_s$. The second vector is

The first vector is

$$\displaystyle \begin{aligned}\boldsymbol{a}_{1}^\prime=\boldsymbol{a}_r^\prime\boldsymbol{X}_r=\frac{1}{N_{d}}\boldsymbol{1}_{N_{d}-n_{d}}^\prime\boldsymbol{X}_{rd}=\frac{1}{N_{d}}\sum_{j\in r_d}\boldsymbol{x}_{dj} =(1-f_{d})\overline{\boldsymbol{X}}_{d}^*,\quad \overline{\boldsymbol{X}}_{d}^*=(\overline{X}^{*}_{1d},\ldots,\overline{X}^{*}_{pd}). \end{aligned} $$

Calculation of g ₃(θ)

We recall that

$$\displaystyle \begin{aligned}g_3(\boldsymbol{\theta})\approx \mbox{tr}\left\{(\nabla\boldsymbol{b}^\prime)\boldsymbol{V}_s(\nabla\boldsymbol{b}^\prime)^\prime E\left[(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})^\prime\right]\right\}, \end{aligned}$$

where

The first derivative is $\frac {\partial \boldsymbol {b}^{\prime }}{\partial \sigma ^2}=0$. As $\frac {\partial \boldsymbol {\Sigma }_{s\ell }}{\partial \varphi _k}=\boldsymbol {x}_{k,n_\ell }\boldsymbol {x}_{k,n_\ell }^\prime$, the remaining derivatives are

k = 1, …, p, where the formula for derivative of an inverse matrix given in Appendix A was used. As , then

Let us define $\boldsymbol {Q}(\boldsymbol {\theta })=(q_{k_1,k_2})_{k_1,k_2=0,1,\ldots ,p}$ , where q _0,k = q _k,0 = 0, k = 0, 1, …, p and

for any k ₁, k ₂ = 1, …, p. After further simplifications we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} q_{k_1,k_2}& =&\displaystyle \sigma^2(1-f_d)^2\,\overline{X}_{k_1d}^*\boldsymbol{x}_{k_1,n_d}^\prime\boldsymbol{\Sigma}_{sd}^{-1}\boldsymbol{x}_{k_2,n_d}\overline{X}_{k_2d}^* \\ & -&\displaystyle \sigma^2(1-f_d)^2\,\overline{X}_{k_1d}^*\boldsymbol{x}_{k_1,n_d}^\prime\boldsymbol{\Sigma}_{sd}^{-1}\boldsymbol{x}_{k_2,n_d}\boldsymbol{x}_{k_2,n_d}^\prime\boldsymbol{\Sigma}_{sd}^{-1} \left(\sum_{i=1}^p\varphi_i\boldsymbol{x}_{i,n_d}\overline{X}_{id}^*\right) \\ & -&\displaystyle \sigma^2(1-f_d)^2\left(\sum_{i=1}^p\varphi_i\overline{X}_{id}^*\boldsymbol{x}_{i,n_d}^\prime\right) \boldsymbol{\Sigma}_{sd}^{-1}\boldsymbol{x}_{k_1,n_d}\boldsymbol{x}_{k_1,n_d}^\prime\boldsymbol{\Sigma}_{sd}^{-1}\boldsymbol{x}_{k_2,n_d}\overline{X}_{k_2d}^* \\ & +&\displaystyle \sigma^2(1-f_d)^2\left(\sum_{i=1}^p\varphi_i\overline{X}_{id}^*\boldsymbol{x}_{i,n_d}^\prime\right)\boldsymbol{\Sigma}_{sd}^{-1}\,\boldsymbol{x}_{k_1,n_d}\boldsymbol{x}_{k_1,n_d}^\prime \boldsymbol{\Sigma}_{sd}^{-1}\boldsymbol{x}_{k_2,n_d}\boldsymbol{x}_{k_2,n_d}^\prime\boldsymbol{\Sigma}_{sd}^{-1} \\ & \cdot&\displaystyle \left(\sum_{i=1}^p\varphi_i\boldsymbol{x}_{i,n_d}\overline{X}_{id}^*\right), \quad k_1,k_2=1,\ldots,p. \end{array} \end{aligned} $$

Then

$$\displaystyle \begin{aligned}g_3(\boldsymbol{\theta})\approx \mbox{tr}\left\{\boldsymbol{Q}(\boldsymbol{\theta})\boldsymbol{F}^{-1}(\boldsymbol{\theta})\right\}, \end{aligned}$$

where F(θ) is the REML Fisher information matrix.

Calculation of g ₄(θ)

We recall that $g_4(\boldsymbol {\theta })=\boldsymbol {a}_r^\prime \boldsymbol {V}_{er}\boldsymbol {a}_r$, where

Therefore

13.4 R Codes for EBLUPs

This section gives R codes for fitting the RRC model to the survey data file LFS20.txt. The target variable y is the variable income. As auxiliary variables we take registered and education. The function dir2 is employed for calculating direct estimators. The domains are defined by the variable area crossed by sex. The parameters of interest are the income means by domains.

We install and/or load some R packages: Matrix, lme4, and sae.

if(!require(Matrix)){   install.packages("Matrix")   library(Matrix) } if(!require(lme4)){   install.packages("lme4")   library(lme4) } if(!require(sae)){   install.packages("sae")   library(sae) }

The following code reads the data files and calculate some variables:

# Read unit-level data dat <- read.table("LFS20.txt", header=TRUE, sep = "\t", dec = ".") # Education level 2 edu2 <- as.numeric(dat$EDUCATION==2) # Education level 3 edu3 <- as.numeric(dat$EDUCATION==3) # Read domain-level data aux <- read.table("Nds20.txt", header=TRUE, sep = "\t", dec = ".") # Prop. of registered people aux$mreg <- aux$reg/aux$N # Proportion of edu2 people aux$medu2 <- aux$edu2/aux$N # Proportion of edu3 people aux$medu3 <- aux$edu3/aux$N

We calculate direct estimators of domain average incomes and the population sizes by domain, by using dir2 function described in Sect. 2.8.4. We also define some new variables.

income.dir <- dir2(data=dat$INCOME, w=dat$WEIGHT, domain=list(sex=dat$SEX,                    area=dat$AREA)) diry <- income.dir$mean         # Direct estimates of domain means hatNd <- income.dir$Nd          # Direct estimates of population sizes nd <- income.dir$nd             # Sample sizes fd <- nd/aux$N                  # Sample fractions

The following code calculates sample means by domains:

dat2 <- data.frame(income=dat$INCOME, edu2, edu3, reg=dat$REGISTERED) smeans <- aggregate(dat2, by=list(sex=dat$SEX,area=dat$AREA), mean) meany <- smeans$income            # Sample means of income meanedu2 <- smeans$edu2           # Sample means of edu2 meanedu3 <- smeans$edu3           # Sample means of edu3 meanreg <- smeans$reg             # Sample means of registered

We fit a random regression coefficient model with income as dependent variable and registered and education as explanatory variables. The fitted model has a random intercept and random slopes on the coefficients of the categories edu2 and edu3 of the variable education. The employed R code is

dat$EDUCATION <- as.factor(dat$EDUCATION) rrc <- lmer(formula=INCOME ~ REGISTERED + EDUCATION + (EDUCATION|AREA:SEX),             data=dat, REML=FALSE) summary(rrc)                              # Summary of the fitting procedure anova(rrc)                                # Analysis of Variance Table beta <- fixef(rrc); beta                  # Regression parameters var <- as.data.frame(VarCorr(rrc))        # Variance parameters ref <- ranef(rrc)[[1]]                    # Modes of the random effects head(fitted(rrc))                         # Predicted values residuals <- resid(rrc)                   # Residuals p.values <- 2∗pnorm(abs(coef(summary(rrc))[,3]), low=F) p.values                                  # p values

Table 13.1 presents the estimated regression parameters and p-values.

Table 13.1 Estimated regression parameters of RRC model

We calculate the EBLUPs of income means by domain.

Xbeta <- beta[1] + beta[2]∗aux$mreg + beta[3]∗aux$medu2 + beta[4]∗aux$medu3 Xubeta <- ref[,1] + aux$medu2∗ref[,2] + aux$medu3∗ref[,3] mu <- Xbeta + Xubeta              # Projective estimates of income means xbeta <- beta[1] + beta[2]∗meanreg + beta[3]∗meanedu2 + beta[4]∗meanedu3 xubeta <- ref[,1] + meanedu2∗ref[,2] + meanedu3∗ref[,3] mu.s <- meany - xbeta - xubeta eb <- mu + fd∗mu.s                # EBLUPs of income means

Summary of results

output <- data.frame(Nd=aux$N[c(T,F)], hatNd=hatNd[c(T,F)] , nd=nd[c(T,F)],            meany=round(meany[c(T,F)],0), dir=round(diry[c(T,F)],0),            hatmu=round(mu[c(T,F)],0), eblup=round(eb[c(T,F)],0)) head(output, 10)

Figure 13.1 (left) plots the RRC model residuals $\hat {e}_d=y_d-\hat {y}_d$. The residuals are situated symmetrically around 0. Figure 13.1 (right) plots the EBLUPs and direct estimates of men income means by areas. The EBLUPs behave more smoothly than the direct estimators.

Fig. 13.1

Plots of residuals (left) and estimated men income means (right)

For the ten first areas, Table 13.2 gives a summary of results for men. The population sizes, the estimated population sizes, and the sample sizes are denoted by N _d, $\hat {N}_d$, and n _d, respectively. The columns meany and dir contain the sample means and the direct estimates of the population mean of the variable income. The projective predictors and the EBLUPs of the population means are labelled by hatmu and eblup, respectively.

Table 13.2 Estimates of domain mean incomes for men