Small Area Estimation for Multivariate Repeated Measures Data - PowerPoint PPT Presentation

Small Area Estimation for Multivariate Repeated Measures Data Innocent Ngaruye Department of Mathematics, University of Rwanda Department of Mathematics, Link¨ oping University 2nd Science day-Stockholm 13th December 2016 1 / 11

My Supervisors Martin Singull Dietrich von Rosen Main Supervisor Co-supervisor Link¨ oping University Swedish University of Agricultural Sciences 2 / 11

Introduction Small Area Estimation (SAE) theory is concerned with solving the following problems 1 How to produce reliable estimates of characteristics of interest, (total, means, quantiles, etc...) for small areas or domains, based on small samples or even no samples taken from these areas. 2 How to assess the estimation or prediction error 3 / 11

Introduction (cont’d) Censuses and surveys have limited scope and often provide very little information for subpopulations (domains) such that direct survey estimates on a target small area are not reliable due to a small sample size connected to this area We propose a multivariate linear regression model for repeated measurements in SAE settings to get a model which borrows strength across both small areas and over time. We consider repeated measurements on variable of interest y for p time points from the finite population of size N partitioned into m disjoint subpopulations or domains called small areas of sizes N i , i = 1 , ..., m . 4 / 11

The working model The corresponding model at small area level is given by Y i = ABC i + 1 p γ ′ X i + u i z ′ i + E i , u i ∼ N p ( 0 , Σ u ) , E i ∼ N p , N i ( 0 , Σ e , I N i ) . The working model combining all small areas is expressed by Y = ABHC + 1 p γ ′ X + UZ + E , (1) E ∼ N p , N ( 0 , Σ e , I N ) , U ∼ N p , m ( 0 , Σ u , I m ) , � Ngaruye et al. (2016). Small Area Estimation under a Multivariate Linear Model for Repeated measures Data. Communications in Statistics - Theory and Methods 5 / 11

Estimation of model parameters Theorem ( Ngaruye et al. (2016)) Consider the corresponding model for sample data for model defined in (1). Then, the likelihood based estimators of γ , B and Σ u can be expressed as 1 p ( PX ′ ) − PY ′ 1 p + ( PX ′ ) o � � = γ t 2 , � ( A ′ A ) − 1 A ′ Y R 2 K ′ 2 ( K 2 K ′ 2 ) − B = − 1 − XR 2 K ′ p ( A ′ A ) − 1 A ′ 1 p 1 ′ p Y P ′ ( XP ′ ) 2 ( K 2 K ′ 2 ) − 2 ) − + � ′ 2 ( PX ′ ) o ′ XR 2 K ′ 2 ) o ′ , − ( A ′ A ) − 1 A ′ 1 p � 2 ( K 2 K ′ T 1 ( K 2 K ′ t 1 C 2 ) ′ − Σ e , � m ( V 3 − A � T 1 � ′ 2 � C 2 )( V 3 − A � T 1 � ′ 2 � C 1 − 1 p � C 1 − 1 p � Σ u = t t 6 / 11

Theorem (cont.) where K 1 = H ( CC ′ ) 1 / 2 Γ 1 , K 2 = H ( CC ′ ) 1 / 2 Γ 2 , R 1 = C ′ ( CC ′ ) − 1 / 2 Γ 1 , R 2 = C ′ ( CC ′ ) − 1 / 2 Γ 2 , ′ + XR 2 R ′ P = XC ′ o ( C ′ o ) 2 − XR 2 K ′ 2 ( K 2 K ′ 2 ) − K 2 R ′ 2 , o ′ ′ 1 ) − + A ′ T 11 � � ′ 2 � C 2 ) � 1 ( � C 1 � T 1 = ( A ′ S − 1 2 A ) − 1 A ′ S − 1 2 ( V 3 − 1 p � t C C C 1 , 2 ) − � ′ 1 ) o ′ t ′ 1 1 p ) − 1 ( � � 1 1 p + ( � p S − 1 3 S − 1 � t 2 = ( 1 ′ 1 V ′ C 2 Q � C C 2 Q � C 2 Q � 21 1 p , ′ ′ ′ C C C 1 2 ) V ′ 2 V ′ 3 Q ′ S 1 = V 3 Q ( � S 2 = S 1 + Q 1 p , S − 1 1 V 3 P Q � 3 , 1 . 1 : � ′ ′ � ′ 1 p , S − 1 C C C C ′ 1 � 2 ) o ′ K 1 , C 1 = ( K 2 K ′ C 2 = ( PX ′ ) o ′ X ( I − R 2 K ′ � 2 ( K 2 K ′ 2 ) − ) R 1 . 7 / 11

Prediction of random effects and small area means Theorem ( Ngaruye et al. (2016)) Consider the model defined by (1). Then, the prediction of random effects and target small area means at each time point for each group and across all time points are given by � � − 1 − 1 � Σ e � ( Y − A � γ ′ X ) Z ′ U = Σ + I p BHC − 1 p � u � � � � µ ig = 1 Y ( s ) γ ′ X ( r ) u i z ( r ) ′ A � β g 1 ′ � ig 1 n ig + N ig − n ig + 1 p � ig + � 1 N ig − n ig , ig N ig µ ig =1 p 1 ′ � p � g = 1 , . . . , k , t = 1 , . . . , p . µ ig , 8 / 11

Crop yield estimation at district level for SAS 2014 in Rwanda The variable of interest is crop yield. We are interested in estimating average yield for beans (bush beans and climbing beans varieties) at district level during two agricultural seasons 2014, A and B. Note that in Rwanda, there are 30 districts. Figure: Grain beans, Bush beans, Climbing beans � Ngaruye, I., von Rosen, D., and Singull, M. (2016). Crop yield estimation at district level for agricultural seasons 2014 in Rwanda. African Journal of Applied Statistics , 3(1):69-90 . 9 / 11

Figure: Distribution of average beans yield estimates at district level during agricultural seasons A and B, 2104 10 / 11

Thank you for your attention! 11 / 11