On Models for Small Area Compositions Li-Chun Zhang Angela Luna L.Zhang@soton.ac.uk A.Luna-Hernandez@soton.ac.uk Social Statistics & Demography University of Southampton Acknowledgements: This work has been funded by the Office for National Statistics - ONS and the Economic and Social Research Council - ESRC. 1 / 22
Motivation Compositions Area 1 . . . Total J 1 Y 1 · 2 Y 2 · 3 Y 3 · Y aj . . . . . . A Y A · Total . . . Y · 1 Y · J Y ·· 2 / 22
Motivation Compositions Area 1 . . . Total J 1 Y 1 · 2 Y 2 · 3 Y 3 · Y aj . . . . . . A Y A · Total . . . Y · 1 Y · J Y ·· Target: Estimate the within area cell counts Y aj , using proxy information and fixed row/column margins. 2 / 22
Motivation Two different (fixed-effects) approaches to this problem are considered: ⊲ Structure Preserving Models: Long tradition in SAE. Assumptions about the relationship between the interactions of two compositions in the log-linear scale. (Proxy information is required). 3 / 22
Motivation Two different (fixed-effects) approaches to this problem are considered: ⊲ Structure Preserving Models: Long tradition in SAE. Assumptions about the relationship between the interactions of two compositions in the log-linear scale. (Proxy information is required). ⊲ Regression (Generalized Linear) Models: Multinomial-Logistic: Assumptions about the relationship between the log-odds with respect to a reference category and a set of covariates. (Proxy information can be used as covariate). 3 / 22
Motivation In this work, we: 1 Introduce a generalization of the Structure Preserving approach, covering the SPREE and GSPREE models and also the logit-multinomial (using proxy information) as particular cases. 4 / 22
Motivation In this work, we: 1 Introduce a generalization of the Structure Preserving approach, covering the SPREE and GSPREE models and also the logit-multinomial (using proxy information) as particular cases. 2 Use data from 2001 and 2011 Population Censuses in England to compare the different models in terms of their Prediction Error. 4 / 22
Motivation In this work, we: 1 Introduce a generalization of the Structure Preserving approach, covering the SPREE and GSPREE models and also the logit-multinomial (using proxy information) as particular cases. 2 Use data from 2001 and 2011 Population Censuses in England to compare the different models in terms of their Prediction Error. 3 Show some ongoing work on a model using a mapping matrix between the proxy and desired compositions, which allows to incorporate auxiliary information at the aggregate level. 4 / 22
Outline 1 Structure Preserving Models 2 Model using a Mapping matrix 5 / 22
Structure Preserving Models 6 / 22
Structure Preserving Models Denote by θ X aj = X aj / X a · an auxiliary composition of exactly the same dimension as θ Y aj = Y aj / Y a · , its log-linear representation given by: γ X aj = α X 0 + α X a + α X j + α X aj where γ X aj = log θ X α X γ X α X γ X γ X α X γ X γ X ·· and aj , 0 = ¯ ·· , a = ¯ a · − ¯ ·· , j = ¯ · j − ¯ α X aj = γ X γ X γ X γ X aj − ¯ a · − ¯ · j − ¯ ·· . 6 / 22
Structure Preserving Models Denote by θ X aj = X aj / X a · an auxiliary composition of exactly the same dimension as θ Y aj = Y aj / Y a · , its log-linear representation given by: γ X aj = α X 0 + α X a + α X j + α X aj where γ X aj = log θ X α X γ X α X γ X γ X α X γ X γ X ·· and aj , 0 = ¯ ·· , a = ¯ a · − ¯ ·· , j = ¯ · j − ¯ α X aj = γ X γ X γ X γ X aj − ¯ a · − ¯ · j − ¯ ·· . The log-linear representation satisfies the constraints: aj = 0 . Analogous for θ Y aj . a α X j α X a α X j α X � a = 0 , � j = 0 , � aj = � 6 / 22
Structure Preserving Models Denote by θ X aj = X aj / X a · an auxiliary composition of exactly the same dimension as θ Y aj = Y aj / Y a · , its log-linear representation given by: γ X aj = α X 0 + α X a + α X j + α X aj where γ X aj = log θ X α X γ X α X γ X γ X α X γ X γ X ·· and aj , 0 = ¯ ·· , a = ¯ a · − ¯ ·· , j = ¯ · j − ¯ α X aj = γ X γ X γ X γ X aj − ¯ a · − ¯ · j − ¯ ·· . The log-linear representation satisfies the constraints: aj = 0 . Analogous for θ Y aj . a α X j α X a α X j α X � a = 0 , � j = 0 , � aj = � The modelling process is focused on the relationship between α Y aj and α X aj . a ˆ j ˆ Marginal constraints such as � Y aj = Y · j for j = 1 , . . . , J and � Y aj = Y a · for a = 1 , . . . , A can be considered using IPF without modifying the parameter estimates. Proxy information (not just covariates) is required. 6 / 22
Structure Preserving Models In the context of SAE, the following Structure Preserving models have been used: 7 / 22
Structure Preserving Models In the context of SAE, the following Structure Preserving models have been used: � � θ X 1. Given , { Y 1 · , . . . , Y A · } : aj 7 / 22
Structure Preserving Models In the context of SAE, the following Structure Preserving models have been used: � � θ X 1. Given , { Y 1 · , . . . , Y A · } : aj Synthetic Estimator: Adapted from Gonzalez & Hoza (1978), ˆ Y aj = θ X aj Y a · The underlining model is α Y j = α X j , α Y aj = α X aj . The estimated composition is a rescaled version of the auxiliary composition. 7 / 22
Structure Preserving Models � � θ X 2. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } : aj 8 / 22
Structure Preserving Models � � θ X 2. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } : aj SPREE: Purcell & Kish (1980) use IPF to fit the two margins, Y (1) ˆ Y (1) Y (2) ˆ ˆ aj = θ X aj Y a · , = Y · j , ... aj aj Y (1) ˆ · j until convergency is achieved. This estimator minimizes the distance between the compositions X and ˆ Y satisfying the marginal constraints. The underlining model is α Y aj = α X aj . 8 / 22
Structure Preserving Models � � � � 3. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } and an estimated : θ X θ Y aj aj 9 / 22
Structure Preserving Models � � � � 3. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } and an estimated : θ X θ Y aj aj Generalized Linear Structural Model (GSPREE): Zhang & Chambers (2004) propose the model α Y aj = βα X aj . 9 / 22
Structure Preserving Models � � � � 3. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } and an estimated : θ X θ Y aj aj Generalized Linear Structural Model (GSPREE): Zhang & Chambers (2004) propose the model α Y aj = βα X aj . β can be estimated using ML under the multinomial distribution, when expressing the model as: µ Y aj = λ j + βµ X aj for µ aj = log θ aj − 1 � log θ aj = α j + α aj . J l 9 / 22
Structure Preserving Models � � � � 3. Given , { Y 1 · , . . . , Y A · } , { Y · 1 , . . . , Y · J } and an estimated : θ X θ Y aj aj Generalized Linear Structural Model (GSPREE): Zhang & Chambers (2004) propose the model α Y aj = βα X aj . β can be estimated using ML under the multinomial distribution, when expressing the model as: µ Y aj = λ j + βµ X aj for µ aj = log θ aj − 1 � log θ aj = α j + α aj . J l Given the sum-zero constraint of the α j , the λ j are nuisance parameters with no practical interest. 9 / 22
Extension of the Structure Preserving approach All the previous models can be seen as particular cases of the more general model: α Y α X a 1 a 1 . . . . = B β B . . α Y α X aJ aJ 10 / 22
Extension of the Structure Preserving approach All the previous models can be seen as particular cases of the more general model: α Y α X a 1 a 1 . . . . = B β B . . α Y α X aJ aJ Where B J × J = I − J − 1 11 ′ and β J × J = { β jk } contains all the parameters. 10 / 22
Extension of the Structure Preserving approach All the previous models can be seen as particular cases of the more general model: α Y α X a 1 a 1 . . . . = B β B . . α Y α X aJ aJ Where B J × J = I − J − 1 11 ′ and β J × J = { β jk } contains all the parameters. The multiplication on left and right by B ensure that the sum zero constraints are satisfied by the predicted α Y aj , as well as the uniqueness of G = B β B . Denoting by { g jk } the components of G we can write, � α Y g jk α X aj = ak . k As in the GSPREE, the estimation of β can be done using ML under the multinomial distribution writing the model as η Y a = λ + B β B η X a . 10 / 22
Extension of the Structure Preserving approach Some particular cases: a) SPREE: β = I aj − 1 � α Y aj = α X α X ak J k 11 / 22
Extension of the Structure Preserving approach Some particular cases: a) SPREE: β = I aj − 1 � α Y aj = α X α X ak J k b) GSPREE: With parameter φ , β = φ I aj − φ 1 α Y aj = φα X � α X ak J k 11 / 22
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