### an Application of Small Area Estimation

```Increasing Survey Statistics Precision Using Split Questionnaire
Design: An Application of Small Area Estimation
1
Content
Introduction
Split Questionnaire Design
Population Characteristics Estimation
• Introduction
• Small Area Models
• Nested Error Regression Model
A Simulation Study
• Steps of Procedure
• Measures of Comparisons
• Results
2
Introduction
• Issue:
– Effects of a lengthy survey questionnaire on:
• Increasing response burden
• declining response rate and precision of survey statistics.
• A solution:
Splitting the questionnaire into sub-questionnaires and assigning each one to a group of
sample units.
Procedure of sub-sample selection is at random, therefore, the resulting nonresponse is
completely at random.
The resulting nonresponse would be imputed by the common imputation methods.
• Our method:
– Designing and analyzing the split questionnaire, using Small Area Estimation technique.
– The method is applicable where the efficient survey estimates are required.
– Complete data set is not provided in our method.
3
Content
Introduction
Split Questionnaire Design
Population Characteristics Estimation
• Introduction
• Small Area Models
• Nested Error Regression Model
A Simulation Study
• Steps of Procedure
• Measures of Comparisons
• Results
4
Split Questionnaire Design
(To apply small area estimation)
Design steps:
i.
The original questionnaire is divided to (m) sub-questionnaires. Some common
items as covariates are assigned to the all sub-questionnaires. Therefore, all
sample units respond to them.
ii.
All sample units are classified with respect to a known auxiliary variable.
Consequently, we make homogeneity within classes. Each class is considered as
an area.
iii.
Sample units belong to each area randomly divided into (m) sub-samples. In
each class, each sub-questionnaire is administrated to a sub-sample.
iv.
Step iii is repeated for all classes.
Note: In each class, the number of sub-questionnaires and number of sub samples
should be equal.
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Split Questionnaire Design
(To apply small area estimation) (continued)
Pattern of administering subquestionnaires to sub-samples in small area
estimation approach:
6
Content
Introduction
Split Questionnaire Design
Population Characteristics Estimation
• Introduction
• Small Area Models
• Nested Error Regression Model
A Simulation Study
• Steps of Procedure
• Measures of Comparisons
• Results
7
Population Characteristics Estimation:
Introduction
•
There is not large enough sample to support direct estimates of appropriate
precision based on the proposed design.
•
Small area estimation method as a solution of insufficient sample size in split
questionnaire method would be useful, in order to improve the efficiency of
survey statistics.
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Population characteristics Estimation:
Small Area Models
• Area level models
– Fay-Herriot Model
– Model with Correlated Sampling Errors
– …
• Unit level models
– Nested Error Regression Model
✓
– Random Error Variance Linear Model
– …
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Population characteristics Estimation
Nested Error Regression model
(Rao 2003)
One of the common models which has been used in small area estimation
isNested Error Regression model.
This modelisa special case of unit level linear mixed model with a block diagonal
covariance structure :
: a vector of auxiliary variable
: the response variable
′ : the sample size in the itharea
: the vector of regression coefficients
: an area-specific random effect
: error term
~ (0,  2 )
~ (0,  2 )
and  are assumed to be independent
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Population characteristics Estimation:
Nested Error Regression model (Continued)
(Rao 2003)
The empirical best linear unbiased predictor (EBLUP):
: the auxiliary population mean vector of the ith area
: the auxiliary sample mean vector of the ith area
: the sample-base mean of the itharea
and
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Population characteristics Estimation:
Nested Error Regression model (Continued)
(Rao 2003)
The MSE of EBLUP estimator:
where
Note: and  are the asymptotic variances of the estimators  2 and  2 ,
and  is the asymptotic covariance of  2 and  2 .
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Population characteristics Estimation
Nested Error Regression Model (Continued)
Population mean estimation ( ) using Nested Error Regression Model
i.
The nested error regression model is used for each sub-questionnaire to
compute the EBLUP of population totals for each area.
ii.
Due to obvious independency of each area from the others, we can use
stratified sampling formula for population mean  . Therefore, the estimate
of  for the population of size  ′ takes the form:
The MSE of  :
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Content
Introduction
Split Questionnaire Design
Population Characteristics Estimation
• Introduction
• Small Area Models
• Nested Error Regression Model
A Simulation Study
• Steps of Procedure
• Measures of Comparisons
• Results
14
A Simulation Study:
Steps of Procedure
• Split questionnaire design:
i.
ii.
Creating a questionnaire with 17 questions.
Splitting the questionnaire into five different components (based on
split questionnaire design (Raghunathan and Grizzle 1995)).
iii.
iv.
v.
vi.
Considering five items (which are highly correlated with other
twelve items) as a core part.
Administering the core part to all sample units.
Assigning three items to each component in such a way that the
within component correlation is small whereas, items in different
components are highly correlated.
Creating 6 subquestionnaires consist of each double combination of
four components plus the core part.
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A Simulation Study
Steps of Procedure (Continued)
• Data generator
i.
Generating a multivariate normal random vector (50,000 times), under
the described correlation pattern.
ii. Producing a multinomial variable as a stratification variable which is
strongly correlated with the other variables.
iii. Classifying the population units based on the stratification variable.
iv. Selecting a simple random sample (without replacement) of a fixed size
n=2000 from the population.
v. Assigning sample units in each stratum to the all 6 subquestionnaires.
vi. Estimating the population mean of each item by applying multiple
imputation approach using the predictive mean matching method (Rubin
1987) and the small area estimation technique.
vii. Generating 1000 simulated bootstrap samples to compare two
approaches.
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A simulation Study:
Measures of Comparisons
• Estimated bias
where
with  is the ithbootstrap estimate.
• Estimated absolute relative bias (EARB)
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A simulation Study:
Measures of Comparisons (Continued)
• Estimated mean square error (EMSE)
where
• Estimated relative efficiency ( )
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Results of the Study
• The estimation of absolute relative bias, MSE and relative efficiency for
1000 bootstrap samples using sample auxiliary information
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Results of the Study (Continued)
• Absolute relative bias, MSE and relative efficiency for 1000 bootstrap
samples using population auxiliary information
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Results of the Study (Continued)
• Small area estimators mostly have lower ARB respect to multiple
imputation based estimators.
• There were no cases in which the multiple imputation approach gave a
smaller MSE than the small area method across all items.
• Small area estimates are more efficient than multiple imputation
approach estimates.
• Small Area technique requires less computation compare to multiple
imputation method.
• Small Area method does not require to produce data, Hence it would be
more applicable, where the goals is estimation of population auxiliary and
not the improvement of data quality.
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References
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References
(Continued)
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