Small Area Estimation under the Growth Curve model Innocent Ngaruye, Link¨ oping University, Sweden Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Outline Introduction The model formulation Estimation of model parameters Prediction of random effects Simulation study example Further research Some references Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Introduction The term Growth Curve Modeling has been used in different contexts to refer to a wide array of statistical models for repeated measures data. It has long played a significant role in empirical research within the developmental sciences, particulary in studying between-individual differences and within-individual patterns of change over time. Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Introduction (cont’d) We propose to apply this model in SAE settings to get a model which borrows strength across both small areas and over time by incorporating simultaneously the effects of areas and time interaction. This model accounts for repeated surveys, group individuals and random effects variation. The estimation is discussed with a likelihood based approach and a simulation study is conducted. Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
The model formulation (cont’d) We consider repeated measurements on variable of interest y for p time points, t 1 , ..., t p from the finite population U of size N partitioned into m disjoint subpopulations or domains U 1 , ..., U m called small areas of sizes N i , i = 1 , ..., m such that � m i =1 N i = N . We also assume that in every area, there are k different groups of units of size N ig for goup g such that � m � k g =1 N ig = N . i We draw a sample of size n in all small areas such that the sample of size n i is observed in area i and � m � k g =1 n ig = n and we i suppose that we have auxiliary data x ij of r variables (covariates) available for each population unit j in all m small areas. Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
The model formulation (cont’d) The model at Small Area level is given by Y i = AB i C i + 1 γ ′ X i + 1u ′ i + E i , (1) u i ∼ N N i ( 0 , σ 2 u I ) , E i ∼ N p , N i ( 0 , σ 2 e I , I N i ) , where A and C i are resectively within-individual and between-individual design matrices for fixed effects given by t q − 1 · · · 1 · · · · · · · · · 1 1 0 0 0 0 t 1 1 t q − 1 · · · 0 · · · 0 1 · · · 1 0 · · · 0 1 t 2 2 A = , C i = . . . . . . . . . . . . . . . . · · · · · · · · · · · · . . . . . . . . t q − 1 · · · · · · · · · 0 0 0 0 1 1 1 · · · t p p Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
The model formulation (cont’d) The corresponding model at population level for all small areas can be expressed as + 1 γ ′ [ I r : I r : · · · : I r ] u ′ Y = A B C X + 1 + E ���� ���� ���� ���� ���� ���� ���� ���� � �� � p × N p × q q × mk mk × N mr × N p × 1 1 × N p × N p × mr or Y = ABC + 1 γ ′ DX + 1u ′ + E , (2) for D = [ I r : I r : · · · : I r ] Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Estimation of model parameters In order to transform (2) to a model which is easier to estimate, we transform the design matrix A into a new matrix A 1 with two parts A 1 = [ 1 : H ] and the parameter matrix into a new matrix Ξ = [ ξ 1 : Ξ 2 ] comformably such that C ( A ) = C ( 1 ) ⊕ C ( H ) with C ( H ) = C ( 1 ) ⊥ ∩ C ( A ) One way of this transformation is given below t q − 1 t q − 1 · · · t 1 − t · · · − t q − 1 1 1 t 1 1 1 t q − 1 t q − 1 − t q − 1 1 · · · 1 t 2 − t · · · t 2 2 2 A = − → A 1 = . . . . . . . . . . . . . . · · · . . . · · · . t q − 1 t q − 1 · · · t p − t · · · − t q − 1 1 1 t p p p Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Estimation of model parameters (cont’d) We come up with the model 1 C + HΞ 2 C + 1 γ ′ DX + 1u ′ + E Y = 1 ξ ′ and make a one-to-one transformation 1 C + p γ ′ DX + p u ′ + 1 ′ E 1 ′ Y p ξ ′ = , H ′ Y H ′ HΞ 2 C + H ′ E A o ′ Y A o ′ E where A o for a matrix A is such that A o ′ A = 0 and C ( A o ) = C ( A ) ⊥ . Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Estimation of model parameters (cont’d) After calculation, the maximum likelihood estimators are given by � � − H ′ YC ′ � CC ′ � − � � o � CC ′ � o ′ � H ′ H H ′ H T 1 + H ′ HT 2 Ξ 2 = + � � CC ′ � o CX ′ D ′ � γ ′ =1 1 ′ YX ′ D ′ − 1 ′ YC ′ ( CC ′ ) − CX ′ D ′ − p T 3 � p � � − DXX ′ D ′ − DXC ′ ( CC ′ ) − C × � 1 � � CC ′ � o � C ′ ( CC ′ ) − + T p 1 ′ Y − � ξ ′ 1 = γ ′ DX for some matrices T , T 1 , T 2 and T 3 of proper sizes. Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Estimation of model parameters (cont’d) Once � 1 and � ξ ′ Ξ 2 are obtained, we can then find the parameter matrix B by solving the linear system 1 � 1 C + H � Ξ 2 C = A � ξ ′ B C . Since, the matrices A and C are of full rank, then B = ( A ′ A ) − 1 A ′ � � 1 � � 1 C + H � ξ ′ C ′ ( CC ′ ) − 1 . Ξ 2 C Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Estimation of model parameters (cont’d) Given the covariance structure of Y Σ = 1Σ u 1 ′ + Σ e = m σ 2 u 11 ′ + σ 2 e I p , and its inverse � 11 ′ � m σ 2 Σ − 1 = 1 u I p − . σ 2 mp σ 2 u + σ 2 e e We find the maximum likelihood estimator of the variance component axpressed by � � 11 ′ W − Np σ 2 u = tr σ 2 e � , Nmp 2 where W = ( Y − ABC − 1 γ ′ DX )( Y − ABC − 1 γ ′ DX ) ′ . Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Prediction of random effects Under the theory of linear model and normal distribution, the best linear predictor of u that minimizes the mean square error is the conditional mean E [ u | Y ] given by E [ u | Y ] = E [ u ] + Cov ( u ′ , Y ) Cov − 1 ( Y )( Y − E [ Y ]) . Thus, u 1 ′ � − 1 ( Y − A � BC − 1 � σ 2 γ ′ D ′ X ) � u = � Σ σ 2 � 1 ′ ( Y − A � BC − 1 � u γ ′ D ′ X ) = mp � σ 2 u + σ 2 e Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Simulation study Example We consider 6 small areas and draw a sample with the following sample sizes. Table : Sample sizes Area Group 1 Group 2 Total 1 n 11 =52 n 12 =48 n 1 =100 2 n 21 =60 n 22 =60 n 2 =120 3 n 31 =30 n 32 =40 n 3 =70 4 n 41 =46 n 42 =22 n 4 =68 5 n 51 =65 n 52 =65 n 5 =130 6 n 61 =50 n 62 =62 n 6 =112 m=6 g 1 =303 g 2 =297 n=600 We assume p = 4 and r = 3. Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Simulation study Example (cont’d) The design matrices are 1 1 − 1 . 5 1 2 − 0 . 5 − → H = A = , 1 3 0 . 5 1 4 1 . 5 � � 1 � � 0 �� C 1 0 1 ′ : 1 ′ C = · · · for C i = n i 1 ⊗ n i 2 ⊗ , 0 1 0 C 6 i = 1 , · · · 6; Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Simulation study Example (cont’d) The parameter matrices are � � ξ ′ 1 = 20 21 22 23 24 25 26 27 28 29 30 31 , � � Ξ 2 = 1 2 3 4 5 6 7 8 9 10 11 12 , B = A − � � 1 ξ ′ C − 1 C + HΞ 2 C 17 . 5 , = 16 14 . 5 13 11 . 5 10 8 . 5 7 5 . 5 4 2 . 5 1 1 2 3 4 5 6 7 8 9 10 11 12 and 1 , σ 2 σ 2 γ = 2 u = 5 , e = 6 . 3 Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
Simulation study Example (cont’d) Then, the data are generated from Y ∼ N p , n ( ABC + 1 γ ′ DX , Σ , I n ) , where the matrix of covariates X is generated with random elements. The following MLEs are obtained: � ′ = � ξ 1 20 . 2534 21 . 6548 22 . 5961 23 . 6486 24 . 4233 25 . 0374 25 . 9985 � 28 . 5361 29 . 9077 30 . 3292 31 . 1121 � � 1 . 1151 2 . 0824 3 . 0320 3 . 6376 4 . 6384 5 . 7882 7 . 0238 7 . 8771 Ξ 2 = � 9 . 0386 10 . 1256 10 . 8561 11 . 9422 Innocent Ngaruye, Link¨ oping University, Sweden Small Area Estimation under the Growth Curve model
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