Sample allocation for efficient model-based small area estimation
Section 3. Some model-free area allocations

Table of contents

The aim of this section is to list the five previously presented allocation methods in order to use them later as references. Depending on which kind of auxiliary information each one uses, they are divided into two groups: number-based and parameter-based allocations.

3.1 Number-based allocations

Two basic allocation solutions commonly used go under the names equal allocation and proportional allocation. Neither of these allocations contains any specific criterion on the area or population level. Their implementation requires only information on the number of strata $D$ and the numbers of units $N_{d}$ in each stratum.

In the equal area allocation the sample size $n_{d}$ is simply a quotient

$n_{d}^{Equ} = n / D . (3.1)$

It is recommended to choose the total sample size $n$ so that the quotient is a whole number. This allocation method does not take differences between the areas into account in any way, which results in inaccurate area estimates. A natural lower limit of the sample size is min $n = 2 D .$

Proportional allocation is a frequently used basic method. Area sample sizes are solved from

$n_{d}^{Pro} = n (N_{d} / N) . (3.2)$

If the sizes of the areas vary strongly, it can lead to situations where the allocated sample size $n_{d}^{Pro} < 2$ for one or more areas. This is an obstacle in calculating direct design-based estimates of standard errors. One solution is to apply the combined allocation proposed by Costa, Satorra and Ventura (2004). The idea is a weighted solution between the equal and proportional allocation depending on the situation. The combined area sample size is

$n_{_{d}}^{Com} = k n_{d}^{Pro} + (1 - k) n_{d}^{Equ} (3.3)$

for a specified constant $k (0 \leq k \leq 1) .$ A minor problem is present if for some areas $n / D > N_{d} .$ A modified solution exists for this case.

3.2 Parameter-based allocations

These allocations use area-level information of the study variable $y$ and in some cases of the auxiliary variable $x$ correlated with $y .$ The values of $x$ are available for all population units. In practice the unknown $y$ is replaced with a proper proxy variable $y^{*}$ such as a study variable obtained from an earlier research of the same subject, or the values of $y^{*}$ are generated with a suitable model developed of a small pre-sample. Also $x$ can be substituted for $y .$ Allocation criteria can be set on population level, only on area level or on combined population and area level.

The Neyman allocation aims at reaching an optimal accuracy concerning population parameters $SD {(y)}_{d}$ (Tschuprow 1923). The standard deviation of the study variable $y$ or some proxy variable and the number of units in each area must be known. Allocation favors large areas with strong variation.

The Bankier or power allocation (1988) is based on a criterion set on the area level. Area CV values of $y$ are weighted by area total transformations $X_{d}^{q}$ which contain a tuning constant $q .$ In practice $y^{*}$ or $x$ must be used in place of $y .$ Allocation favors mainly large areas with high CV.

Choudhry, Rao and Hidiroglou (2012) present the NLP allocation method for direct estimation. This method uses non-linear programming to find a solution. Criteria for the allocation are defined by setting upper limits for CV values of the study variable $y$ in each area and in the population. In practice $y^{*}$ or $x$ replaces $y .$ The program searches the minimum sample size $n = \sum_{d} n_{d}$ satisfying these conditions. The SAS (Statistical Analysis System) procedure NLP with Newton-Raphson option was used to find the solution. The allocation favors areas with high CV regardless of the area size $N_{d} .$

A summary of the model-free allocations and the formulas for calculating area sample sizes are presented in Table 3.1.

Table 3.1
Summary of number-based and parameter-based allocations
Table summary
This table displays the results of Summary of number-based and parameter-based allocations. The information is grouped by Allocation (appearing as row headers), Computing area sample size xxxxx and Optimality level (appearing as column headers).
Allocation	Computing area sample size $n_{d}$	Optimality level
Equal	$n_{_{d}}^{Equ} = n / D$	Area
Proportional	$n_{_{d}}^{Pro} = n (N_{d} / N)$	Population
Neyman	$n_{_{d}}^{Ney} = n (N_{d} S_{d} / \sum_{d = 1}^{D} N_{d} S_{d}),$ where $S_{d}$ is the standard deviation of $y$ (in practise $y^{*}$ or $x)$ in area $d .$	Population
Bankier	$n_{d}^{Ban} = n (X_{d}^{q} CV {(y)}_{d} / \sum_{d = 1}^{D} X_{d}^{q} {CV}_{d} (y)),$ where $X_{d}$ is the area total of $x,$ ${CV}_{d} (y) = S_{d} / {\bar{Y}}_{d}$ and $q$ is a tuning constant. In practise $y^{*}$ or $x$ replace $y .$	Area
NLP	$n_{s t}^{NLP} = \min (\sum_{d = 1}^{D} n_{d})$ satisfying tolerances $CV ({\bar{y}}_{d}) \leq {CV}_{0 d}$ and $CV ({\bar{y}}_{s t}) \leq {CV}_{0} .$ In practise $y^{*}$ or $x$ replace $y .$	Jointly population and area

Some other parameter-based allocation methods are mentioned briefly. For example Longford (2006) introduced inferential priorities $P_{d}$ for the strata $d$ and $G$ for the population and used those constraints for allocation. Another solution is presented by Falorsi and Righi (2008). This solution does not contain a direct imposition of quotas, but tries to solve the comprehensive collection of data by using a multi-stage sampling design, so that the area estimation can be implemented effectively.

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Sample allocation for efficient model-based small area estimation
Section 3. Some model-free area allocations

3.1 Number-based allocations

3.2 Parameter-based allocations

Sample allocation for efficient model-based small area estimation Section 3. Some model-free area allocations

3.1 Number-based allocations

3.2 Parameter-based allocations

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Sample allocation for efficient model-based small area estimation
Section 3. Some model-free area allocations