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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

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Small Area Estimation under the Growth Curve model

Innocent Ngaruye, Link¨

  • ping University, Sweden

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Outline

Introduction The model formulation Estimation of model parameters Prediction of random effects Simulation study example Further research Some references

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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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¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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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¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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The model formulation (cont’d)

We consider repeated measurements on variable of interest y for p time points, t1, ..., tp from the finite population U of size N partitioned into m disjoint subpopulations or domains U1, ..., Um called small areas of sizes Ni, i = 1, ..., m such that m

i=1 Ni = N.

We also assume that in every area, there are k different groups of units of size Nig for goup g such that m

i

k

g=1 Nig = N.

We draw a sample of size n in all small areas such that the sample

  • f size ni is observed in area i and m

i

k

g=1 nig = n and we

suppose that we have auxiliary data xij of r variables (covariates) available for each population unit j in all m small areas.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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The model formulation (cont’d)

The model at Small Area level is given by Yi =ABiCi + 1γ′Xi + 1u′

i + Ei,

(1) ui ∼ NNi(0, σ2

uI),

Ei ∼ Np,Ni(0, σ2

eI, INi),

where A and Ci are resectively within-individual and between-individual design matrices for fixed effects given by

A =      1 t1 · · · tq−1

1

1 t2 · · · tq−1

2

. . . . . . · · · . . . 1 tp · · · tq−1

p

     , Ci =      1 · · · 1 · · · · · · · · · 1 · · · 1 · · · . . . · · · . . . . . . · · · . . . . . . · · · · · · · · · 1 · · · 1     

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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The model formulation (cont’d)

The corresponding model at population level for all small areas can be expressed as Y

  • p×N

= A

  • p×q

B

  • q×mk

C

  • mk×N

+ 1γ′[Ir : Ir : · · · : Ir]

  • p×mr

X

  • mr×N

+ 1

  • p×1

u′

  • 1×N

+ E

  • p×N
  • r

Y = ABC + 1γ′DX + 1u′+E, (2) for D = [Ir : Ir : · · · : Ir]

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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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 A1 with two parts A1 = [1 : H] and the parameter matrix into a new matrix Ξ = [ξ1 : Ξ2] comformably such that C(A) = C(1) ⊕ C(H) with C(H) = C(1)⊥ ∩ C(A) One way of this transformation is given below

A =      1 t1 · · · tq−1

1

1 t2 · · · tq−1

2

. . . . . . · · · . . . 1 tp · · · tq−1

p

     − → A1 =      1 t1 − t · · · tq−1

1

− tq−1 1 t2 − t · · · tq−1

2

− tq−1 . . . . . . · · · . . . 1 tp − t · · · tq−1

p

− tq−1     

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Estimation of model parameters (cont’d)

We come up with the model Y = 1ξ′

1C + HΞ2C + 1γ′DX + 1u′ + E

and make a one-to-one transformation   1′Y H′Y Ao′Y   =   pξ′

1C + pγ′DX + pu′ + 1′E

H′HΞ2C + H′E Ao′E   , where Ao for a matrix A is such that Ao′A = 0 and C(Ao) = C(A)⊥.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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SLIDE 10

Estimation of model parameters (cont’d)

After calculation, the maximum likelihood estimators are given by

  • Ξ2 =
  • H′H

− H′YC′ CC′− +

  • H′H
  • T1 + H′HT2
  • CC′o′
  • γ′ =1

p

  • 1′YX′D′ − 1′YC′(CC′)−CX′D′ − pT3
  • CC′o

CX′D′ ×

  • DXX′D′ − DXC′(CC′)−C

−

  • ξ′

1 =

1 p1′Y − γ′DX

  • C′(CC′)− + T
  • CC′o

for some matrices T, T1, T2 and T3 of proper sizes.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Estimation of model parameters (cont’d)

Once ξ′

1 and

Ξ2 are obtained, we can then find the parameter matrix B by solving the linear system 1 ξ′

1C + H

Ξ2C = A BC. Since, the matrices A and C are of full rank, then

  • B = (A′A)−1A′

1 ξ′

1C + H

Ξ2C

  • C′(CC′)−1.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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SLIDE 12

Estimation of model parameters (cont’d)

Given the covariance structure of Y Σ =1Σu1′ + Σe = mσ2

u11′ + σ2 eIp,

and its inverse Σ−1 = 1 σ2

e

  • Ip −

mσ2

u

mpσ2

u + σ2 e

11′ . We find the maximum likelihood estimator of the variance component axpressed by

  • σ2

u = tr

  • 11′W
  • − Npσ2

e

Nmp2 , where W = (Y − ABC − 1γ′DX)(Y − ABC − 1γ′DX)′.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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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 =

σ2

u1′

Σ

−1(Y − A

BC − 1 γ′D′X) =

  • σ2

u

mp σ2

u + σ2 e

1′(Y − A BC − 1 γ′D′X)

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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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 n11=52 n12=48 n1=100 2 n21=60 n22=60 n2=120 3 n31=30 n32=40 n3=70 4 n41=46 n42=22 n4=68 5 n51=65 n52=65 n5=130 6 n61=50 n62=62 n6=112 m=6 g1=303 g2=297 n=600 We assume p = 4 and r = 3.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Simulation study Example (cont’d)

The design matrices are A =     1 1 1 2 1 3 1 4     − → H =     −1.5 −0.5 0.5 1.5     , C =   C1 · · · C6   for Ci =

  • 1′

ni1 ⊗

1

  • : 1′

ni2 ⊗

1

  • ,

i = 1, · · · 6;

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Simulation study Example (cont’d)

The parameter matrices are ξ′

1 =

  • 20

21 22 23 24 25 26 27 28 29 30 31

  • ,

Ξ2 =

  • 1

2 3 4 5 6 7 8 9 10 11 12

  • ,

B =A− 1ξ′

1C + HΞ2C

  • 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 γ =   1 2 3   , σ2

u = 5,

σ2

e = 6.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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SLIDE 17

Simulation study Example (cont’d)

Then, the data are generated from Y ∼ Np,n(ABC + 1γ′DX, Σ, In), where the matrix of covariates X is generated with random elements. The following MLEs are obtained:

  • ξ1

′ =

  • 20.2534

21.6548 22.5961 23.6486 24.4233 25.0374 25.9985 28.5361 29.9077 30.3292 31.1121

  • Ξ2 =
  • 1.1151

2.0824 3.0320 3.6376 4.6384 5.7882 7.0238 7.8771 9.0386 10.1256 10.8561 11.9422

  • Innocent Ngaruye, Link¨
  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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SLIDE 18

Simulation study Example (cont’d)

  • B =

17.4657 17.3902 14.0748 14.5546 12.8274 10.5669 8.4390 1.1151 2.0824 3.0320 3.6376 4.6384 5.7882 7.0238 5.9397 4.5936 3.1890 1.2566 9.0386 10.1256 10.8561 11.9422

  • σ2

u = 5.0061,

  • γ =

  1.0093 1.9501 3.0469   , ABC =     18.5 · · · 18 · · · 13 19.5 · · · 20 · · · 25 20.5 · · · 22 · · · 37 21.5 · · · 24 · · · 49    

  • ABC =

    18.5808 · · · 18.4726 · · · 13.1988 19.6959 · · · 20.5550 · · · 25.1410 20.8110 · · · 22.6373 · · · 37.0833 21.9261 · · · 24.7197 · · · 49.0255    

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Further research

After obtaining all unkown parameters, then we can find directly the target small area characteristics of interest such as the small area totals and samall area means In further research, we want to test the efficiency, the distribution and all properties of the estimators We wish also to study the possible time correlation

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Some references

Bai Peng, Exact distribution of MLE of covariance matrix in GMANOVA-MANOVA model, J. Science in China, 2005 Battese, G.E, R.M. and W.A. Fuller, An error-components model for prediction of county crop areas using survey and satellite data, American Statistical Association, 1988. Danny Pfeffermann Small Area Estimation-New Develooments and

  • Directions. J. International Statistical Review, 2002.
  • G. Datta, P. Lahiri, T. Maiti, K. Lu, Hierarchical Bayes estimation of

unemployment rates for the states of the US, Journal of the American Statistical Association 94 (1999) G.K. Robinson, That BLUP Is a Good Thing: The estimation of Random Effects, Statistical Science, Vol. 6, 1991 J.N.K. Rao, Small Area Estimation. Willey, 2003.

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Some references

  • T. Kollo and D. von Rosen, Advanced Multivariate Statistics with
  • matrices. Springer, 2005.

Kari Nissinen, Small Area Estimation with Linear Mixed Models from Unit-level panel and Rotating panel data. PhD Thesis, Jyv¨ askyl¨ a University, 2009..

  • M. Ghosh and J.N.K. Rao, Small Area Estimation: An Appraisal, J.

Statistical Science, 1994.

  • R. Chambers and R. G. Clark, An introduction to Model-Based

Survey Sampling with Applications. Oxford, 2012. Robb J. Muirhead, Aspects of Multivariate Statistical theory , Wiley 2005. Tatsuya Kubokawa and Muni S. Srivastava, Prediction in Multivariate Mixed Linear Models, J. Japan Statist. Soc, 2003

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model

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Note: This work is being jointly conducted in the framework of my PhD research together with my supervisors: Dietrich von Rosen and Martin Singull to whom I owe my aknowledgements.

THANKS !!!!!!!!

Innocent Ngaruye, Link¨

  • ping University, Sweden

Small Area Estimation under the Growth Curve model