Under-coverage is one of the most common problems of sampling frames. To reduce the impact of coverage error on survey estimates several frames can be combined in order to achieve a complete (or nearly complete) coverage of the target population. Multiple frame estimators have been developed to be used in the context of multiple frame surveys. Sampling frames may overlap which is the case when a single unit of the sampling frame is related with more than one element of the target population. Indirect sampling (Lavallée, 1995) is an alternative approach to classical sampling theory in dealing with the overlapping problem of sampling frames on survey estimates. Not infrequently a survey may need more than one sampling frame in order to improve coverage and simultaneously the sampling frame overlap. In this paper a new class of estimators is presented which is the result from merging multiple frames estimators (only the particular case of dual frames will be presented) with indirect sampling estimators in order to bring together in a single estimator the effect of several frames on survey estimates.Indirect Sampling, Generalized Weight Share Method, Dual Frame Surveys

Manuela Maia

Paula Vicente

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
DOCUMENTOS DE TRABALHO 
 
WORKING PAPERS 
GESTÃO 
 
MANAGEMENT 
 
 
 
 
 
 
 
 
 
 
Nº 08/2009 
 
 
 
INDIRECT SAMPLING IN THE CONTEXT OF DUAL FRAME 
SURVEYS 
 
Manuela Maia 
Universidade Católica Portuguesa (Porto) 
 
Paula Vicente  
ISCTE 
 
 
 Manuela Maia  
Faculdade de Economia e Gestão - Universidade Católica Portuguesa | Porto 
 UNIDE- Unidade de investigação em Desenvolvimento Empresarial 
mmaia@porto.ucp.pt 
 
Paula Vicente 
Departamento de Métodos Quantitativos – ISCTE Business School 
UNIDE- Unidade de investigação em Desenvolvimento Empresarial 
paula.vicente@iscte.pt 
 
 
Indirect sampling in the context of dual frame surveys 
Abstract:  
Under-coverage is one of the most common problems of sampling frames. To reduce the 
impact of coverage error on survey estimates several frames can be combined in order to 
achieve a complete (or nearly complete) coverage of the target population. Multiple frame 
estimators have been developed to be used in the context of multiple frame surveys. Sampling 
frames may overlap which is the case when a single unit of the sampling frame is related with 
more than one element of the target population. Indirect sampling (Lavallée, 1995) is an 
alternative approach to classical sampling theory in dealing with the overlapping problem of 
sampling frames on survey estimates. 
Not infrequently a survey may need more than one sampling frame in order to improve 
coverage and simultaneously the sampling frame overlap. In this paper a new class of 
estimators is presented which is the result from merging multiple frames estimators (only the 
particular case of dual frames will be presented) with indirect sampling estimators in order to 
bring together in a single estimator the effect of several frames on survey estimates. 
 
Keywords: Indirect Sampling, Generalized Weight Share Method, Dual Frame Surveys  
AMS Subject Classification: 62; 62D05  
  2 
 
 
1. Introduction 
  In any survey the random selection of the sample requires that a sampling frame is 
available. The sampling frame is used to identify the elements of the target population. The 
frames may be maps of areas in which elements can be found, among others. At their simplest, 
sampling frames consist of a list of population elements (Groves et al, 2007). There are 
populations for which lists are readily available, such as members of a professional 
organization, hospitals or schools. There are many populations, though, for which lists of 
individuals elements are not readily available. For example the adults  living in a country, or 
the students attending school on a specific district.  
  When available, one central statistical concern for the survey researcher is how well 
the sampling frame actually covers the target population. A sampling frame is perfect when 
there is a one-to-one mapping of frame elements to target population elements. In practice, 
perfect frames seldom exist; there are always problems that disrupt the desired one-to-one 
mapping, namely: a) under coverage, b) duplication and c) clustering. Under coverage happens 
when some elements of the target population do not appear in the sampling frame; therefore 
such elements cannot appear in any sample drawn for the survey. Duplication happens when 
several frame units are mapped onto the single elements in the target population but the 
mapping is not unique, not one-to-one. Clustering is the term used when multiple elements of 
the target population are linked to the same single frame unit, i.e., a one-to-many mapping. 
There are also cases that combine the duplication and the clustering problems in which 
multiple frame units map to multiple target population elements, i.e., many-to-many 
mappings.  
  3 
   
Selecting a sample from a sampling frame that suffers from under coverage can cause 
coverage error on survey statistics. One of the strategies to reduce coverage error is to use  
multiple frames. A principal frame that provide nearly complete coverage of the target 
population may be supplemented by a frame that provides better or unique coverage for the 
population elements absent or poorly covered in the principal frame. In most cases 
supplemental frames overlap with the principal frame which requires estimation procedures to 
be adapted in order to correct probabilities of selection and possibly to yield improved 
precision for survey estimates. 
  Selecting a sample from a sampling frame that suffers from either clustering or 
duplication poses several difficulties to estimation, namely in what concerns sample weights 
computation.  
 
2. Multiple frame estimators 
  The estimation under multiple frame designs was originally proposed by Hartley (1962) 
and others. They suggested that the union of the frames be used in estimation to obtain a 
more efficient estimator. They proposed that a dual frame design be examined as a set of no 
overlapping domains and results from each domain combined to obtain a target population 
estimate. By taking Q sampling frames - A1, A2, ..., Aq - (that may overlap) to cover the target 
population 2
Q-1 domains mutually exclusive can be defined. In the particular case of Q=2 three 
mutually exclusive domains can be defined: D1, contains elements exclusively from frame 1, 
that is  2 1 1 A A D Ç = , D2, contains the elements that belong simultaneously to both frames, 
that is,  2 1 2 A A D Ç = , and D3, contains elements exclusively from frame 2, that is  
  
  4 
 
2 1 3 A A D Ç = . In this context, the dual frame estimator of the total population proposed by 
Hartley (1974) is based on the weighted average of the total estimates from the domains: 
2
3
2
2
1
2
1
1
ˆ ˆ ) 1 ( ˆ ˆ ) ( ˆ A
D
A
D
A
D
A
D Y Y Y Y Y + - + + = q q q         (1) 
where q is a parameter chosen to maximize precision (0≤q≤1), 
1
1
ˆ A
D Y  is the total estimate from 
D1, 
2
3
ˆ A
D Y  is the total estimate from D3, 
2
2
ˆ A
D Y , the total estimate from D2 using a sample from A2 
and 
1
2
ˆ A
D Y  is the total estimate of from D2 using a sample from A1. 
  Alternatively, the population total Y may be represented by the following expression 
(Hartley 1962, 1974): 
i i y ) ( y
q q q q
 
A     A     ∑ ∑ ∑
Î Î
= =
K i
i
i
K Y
U U
d           (2) 
where di(K) is an indicator of domain variable: 


 Î
= d
otherwise , 0
D i , 1
) K (
k
i  
  The sample selected from each frame is then used to produce an estimate for the total 
in each domain, which in turn, is combined to produce a single estimate for the population 
total. The estimator is given by 
i
K
) ( y ) ( ˆ
q q
 
       A     ∑ ∑ ∑
Î Î
=
K q i
i
q
i K w Y
U
d              (3) 
and it requires the weights 
) (q
i w  to be computed.  
   
  
  5 
 
In the literature there are two approaches to estimate these weights: the Domain Membership 
approach and the Multiplicity Unit approach. According to Domain Membership approach a 
partition of domains is defined in the frames, in such a way that it is always possible to 
correctly identify to which domain belongs each element of the sample. There are three types 
of estimators in this class of estimators: 
  (a) The Optimal Estimator 
) (
,
q
opt i w - presents good theoretical properties - it has minimal 
variance (Hartley 1962, 1974; Lund 1968; Fuller and Burmeister 1972) - but, in operational 
terms, is very complex.  
  (b) The Single Based Estimator 
) (
,
q
SF i w  - uses fixed weights guaranteeing unbiased 
estimates (Bankier 1986; Kalton and Anderson 1986; Skinner 1991; Skinner, Holmes and Holt 
1994), however, are less efficient than the optimal estimator (Lohr and Rao 2000).  
  (c) The Pseudo Maximum Likelihood Estimator
) (
,
q
PML i w  - extends the applicability of the 
optimal estimator increasing its efficiency when compared with the single based estimator 
(Skinner and Rao 1996; Lohr and Rao 2000).  
  The Unit Multiplicity estimators are based on the concept of unit multiplicity which 
reflects the number of frames to which a sample element belongs (Mecatti 2007). This concept 
was first used by Casady and Sirken (1980). Under this approach, the population total can be 
written as: 
Y m   y y   m
y   m   y
Q
1 q A     i
1
i i i
A     i
i
i
A     i
i
q A     i
i
q q q
q q q
= =
=
∑∑ ∑
∑ ∑∑
= Î
-
Î
Î Î
U
U
          (4)  
  6 
and involves solely the frames to which the sample element belongs. The expression of the 
population total estimator is given by: 
1 -
i i
1
) ( y ˆ
q
m w Y
Q
q i
q
i M    
s     ∑∑
= Î
=           (5) 
where mi is the number of frames in which each unit is include among the frames involved in 
the survey.  
  Mecatti (2007) provides argument to apply Unit Multiplicity estimators in surveys with 
more than two sampling frames based on overlapping.  
 
3. Indirect Sampling and the Generalized Weight Share Method 
  In Classical Sampling Theory the weight for each sampled element is related to the 
inverse of its selection probability. The Horvitz-Thompson estimator for the population total - 
HT Y ˆ  - resumes this principle: 
∑
Î
=
N
S k k
k HT y
Y
p
ˆ             (6) 
where  ( ) S k P k Î = p  is the probability of the k element be selected in the samples. This theory 
assumes that the sampling frame is a perfect representation of the target population, i.e., a 
one-to-one mapping and is difficultly applied outside this condition. 
  Indirect Sampling was first proposed by Lavallée (1995) to deal with the problem of 
Cross-sectional weighting for longitudinal household surveys. 
Indirect sampling assumes that a sampling frame U
A with M
A units is available to represent the 
target population U
B. U
B contains M
B elements, divided into N clusters, each one with 
B
i M    
  7 
 
elements. A sample sA with mA units is then selected from the frame U
A in order to estimate 
some parameter of the target population U
B. The Generalized Weight Share Method (GWSM), 
developed by Lavellée (1995) in the context of indirect sampling, uses the links between the 
units j Î U
A and the elements k of the i
th cluster of U
B to compute the weight for each element 
in the sample. 
Under the GWSM the estimator for the population total is given by  
            ∑∑
= =
=
n
1 i
M
1 k
ik ik
B
B
i
y w Y ˆ         (7) 
where  ik w  is the weight attached to the element k of cluster i, defined by 
B
i
M
k
ik i L w w
B
i
∑
=
=
1
'
           (8) 
where 
'
ik w  corresponds to the inverse of the selection probability of units j of s
A that have non-
zero link with unit k of cluster i of 
B Y ˆ .
 
The process to compute  ik w  can be resumed in four steps: 
1) Compute the number of links 
B
ik L between the units’ j Î U
A and the element k of the i
th 
cluster of U
B by 
∑
=
=
A M
j
ik j
B
ik l L
1
,
 
where  
 


 Î Î
=
otherwise 0
exists U ik    and U j between link    a   if 1
l
B A
ik , j  
2) Obtain the total number of links in cluster i:  ∑
=
=
B
i M
k
B
ik
B
i L L
1
.  
  8 
3) Compute an initial weight  
A
j
M
j
j ik j
ik
A
t l
w
p
∑
= =
1
,
'
 
where 



Ï
Î
=
A
A
j s   j if 0
s   j if 1
t  and 
A
j p  is the selection probability of unit j Î A s . 
4) Compute the final weight wi to each sampled element k Î Ui
B. 
  The GWSM estimator can be re-written as (Lavallée 1995): 
j
M
j
A
j
j
i
n
i
ik B Z
t
y w Y
A
∑ ∑
= =
= =
1 1
ˆ
p
 
where  ik
N
i
M
k
ik j j z l Z
B
i
∑∑
= =
=
1 1
,  and  B
i
i
ik L
Y
z =  
  The application of the GWSM requires the matching between sampling frame and 
target population and needs to satisfy the follow constraint: 
There exists, at least, one link between the unit j Î U
A and the elements k of i
th 
cluster of U
B i.e.  1   l = L
N
i
M
k ik j
A
j
B
i ³ ∑ ∑ = = 1 1 ,  for every U j
A Î ; 
This constraint is essential to ensure de unbiasedness of the GWSM 
Lavallé (1995) proved that the GWSM estimator is unbiased and its variance is directly given 
by: 
  Z Z
) - (
= ) Y Var( j j A
j
A
j
A
j
A
j
A
j j
M
1 = j
M
1 = j
B
A A
¢
¢
¢ ¢
¢ ∑ ∑
p p
p p p ˆ   
  (9) 
where p
A
j j ¢  is the joint probability of selecting units  j  and j¢ . 
  
  9 
 
4. Combining Multiple Frame Estimators with Indirect Sampling 
  Both multiple frame designs and indirect sampling seek to improve estimation in 
surveys where a “perfect” sampling frame does not exist.  Suppose for example a RDD survey 
is used to reach the general adult population of a country. RDD will, in principle, cover all 
adults living in households with fixed line telephone access but it fails to cover adults living in 
households without a fixed line telephone. A remedy to under coverage may be a 
supplementary frame of mobile phone numbers. Under such a dual frame design the two 
frames together will likely provide a complete (or nearly complete) coverage of the adult 
population, however an important statistical problem will raise researcher’ concern: some 
adults of the target population may be reachable both by mobile phone and fixed line phone, 
which means there is a many -to- one mapping. Under these circumstances the estimation 
approach should merge the solutions coming from dual frame estimation and indirect 
sampling. 
  Our proposal is to put dual frame estimators – both the Domain Membership 
estimators and Unit Multiplicity estimators – to the context of indirect sampling and thus 
provide an estimation approach adequate for surveys where the sampling frame suffers from 
under coverage (and several frames are combined to reduce the coverage error). 
4.1 The Domain Membership estimator 
  In the context of Indirect Sampling the Domain Membership estimator for the 
population total can be expressed by: 
j
A j
A
j
j
j
A j
A
j
j
DM y
) ( x
y
) ( z
Y ˆ
1
2
1
1 ∑ ∑
Î Î
+ =
p
q
p
q
          (10) 
  
  10 
 
where 
( )
 
D j if 1
D j if 1
) ( x         and        
D j if
D j if 1
) ( z
2
3
j
2
1
j



Î -
Î
=



Î
Î
=
q
q
q
q   are indicators of domain 
variables, (0 ≤ q ≤ 1) and q A
j p  represents the selection probability of unit j from the q
th frame. 
 
4.2 The Unit Multiplicity estimator 
  In the context of Indirect Sampling the Unit Multiplicity estimator for the population 
total can be expressed by: 
 
(11) 
where 
B
i L  represents the total number of links between the unit j Î Aq, (q=1,2) and the 
element i from U
B and 
q A
j p  represents the selection probability of unit j from Aq, (q=1,2).  q , ji L  
is given by: 



=
otherwise 0
 U from unit    i    the and , A   from   unit, th  - j beteween  link    a   is    there if 1
L
B
q
q , ji  
 
4.3 The Dual Frame estimator 
  The estimator proposed by Hartley (1974) (eq. ??) can, in the same way, be converted 
to Indirect Sampling context: 
j
D
A
j
A
A
s j
A
j
j
C
A
j
A
A
s j
A
j
H y
N ˆ
N 1
y
N ˆ
N 1
Y ˆ
j
2
2
2
2 A
2
j
1
1
1
1 A
1
43 4 2 1 4 3 4 2 1
j
p
j
p ∑ ∑
Î Î
+ =        (15) 
B
2
ji,q
B
1 1 i   U i
L 1 ˆ      
L
Aq
q
m
M j A
q j j
Y y
p = = Î
=∑∑ ∑ 
  11 
 
where   
 



=
=
Û







=
+
=
=
1 if
~
0 if 1
1 if
0 if
L
1
2 1
2
2
2 1
1
2
1
1
A
j
A
1 j
A
j
A
j A
2 j
A
1 j
A
j
A
j A
j A
j d q
d
d
q q
q
d
j  
and   
 



= -
=
Û







=
- + -
-
=
=
1 if
~
1
0 if 1
1 if
) 1 ( ) 1 (
1
0 if
L
1
1 1
1
1
2 2
2
1
2
2
A
j
A
2 j
A
j
A
j A
2 j
A
1 j
A
j
A
j A
j A
j d q
d
d
q q
q
d
j  
 
1 A
1 j
~
q -  
1 A
2 j
~
1 q - -  
q A
j d  is an indicator of frame variable and 
q A
j p  represents the selection probability of unit j 
from U
Aq with q=1,2. 
  From equation (1) is possible to obtain the classes of estimators above described. 
Considering that  ) ( x D     and     ) ( z C j j j j q q = = the class of Domain Membership estimators 
can be deduced. Replacing Cj and Dj by the proportion of the links from the frames A1 and A2, 
respectively, i.e.,   
L
L
D    nd a     
L
L
C
B B U     i
B
i
ji,2
j
U     i
B
i
ji,1
j ∑ ∑
Î Î
= =
 
we obtain the class of Unit Multiplicity 
estimators. 
 
5. Concluding remarks  
  Indirect sampling is a viable approach to deal with estimation in sample surveys when 
the available sampling frames overlap. The under coverage problem of some sampling frames 
or the increased use of mixed modes to collect data for the same   
  12 
 
 
survey makes inevitable the combination of two or more sampling frames to better cover the 
target population of the survey.  
  In surveys where more than one frame is used, the estimation becomes complex since 
it requires that both the effect of multiple frames and the specific links between frames and 
target population be accounted for in the estimators. The transformation of the classical dual 
frames estimators into indirect sampling estimators is a way of making one solution for a 
double problem. The properties of this new class of estimators and its impact on estimates’ 
precision are a subject that warrants future research.  
  13 
 
 
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to Multiple Frame Surveys, Journal of the American Statistical Association, Vol. 81, pp.1074-
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Skinner C. J (1991), On the Efficiency of Ratio Estimation for Multiple Frame Surveys Journal of 
the American Statistical Association, Vol. 86, nº 415, pp.779-784.  
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Estimators in Multiple Frame,

Estimators of samples selected from two overlapping frames,

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Weighting issues for longitudinal and family estimates, Panel Surveys,

INDIRECT SAMPLING in the CONTEXT of DUAL FRAME SURVEYS

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INDIRECT SAMPLING in the CONTEXT of DUAL FRAME SURVEYS

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