CROSS-SECTIONAL INFERENCE BASED ON 

LONGITUDINAL SURVEYS: SOME EXPERIENCES 

WITH STATISTICS CANADA SURVEYS 

Georgia Roberts, Milorad Kovacevic, Harold Mantel, Owen Phillips

Statistics Canada 

1 

Abstract 

This paper focuses on cross-sectional inference based on data from a longitudinal survey which carries some additional 

components to achieve cross-sectional representativity. When inferring about the differences in the cross-sectional 

populations at two different points in time, problems arise with variance estimation for the difference of the respective 

estimates, when the estimates are derived from such a survey. There are several factors contributing to these problems. 

Of these, the most important is the sample overlap at the two time points due to the underlying longitudinal survey 

design; this introduces a strong covariance component which must be included in the estimate of the variance of the 

difference. Also associated with the underlying longitudinal sample is the complexity introduced by longitudinally 

sampled individuals moving from one geographical part of the country to another, and thus being used to represent a 

different part of the cross-sectional population than that for which they were selected. The degree of complication that 

such factors introduce to the variance estimation problem is determined by the manner in which the longitudinal sample 

has been supplemented and adjusted in order to attain cross-sectional samples and by the available design information 

that may be used for cross-sectional inference. 

The variance estimation problem is addressed for Canada’s Survey of Labour and Income Dynamics (SLID) within a 

Taylor linearization approach as well as within the resampling framework with emphasis on the bootstrap method. For 

cross-sectional purposes, SLID combines two independent panels of longitudinal individuals sampled three years apart 

and also includes all members of the families and households with whom the originally selected longitudinal individuals 

live at a certain point in time. A numerical illustration based on SLID is included. 

Key words: bootstrap, combining panels, Taylor linearization, variance estimation 

1. Introduction 

The objective of most cross-sectional surveys is to produce unbiased (or nearly unbiased) estimates 

of levels such as totals or means at a given time point, and, in the case of repeated surveys, to 

produce estimates of the net change that occurred in the population between two time points. These 

estimates are often accompanied by estimated measures of precision. The primary objective of 

longitudinal surveys is the production of longitudinal data series that are appropriate for studying the 

gross change in a population between collection dates, and for research on causal relationships 

among variables. 

In order to improve the cost-effectiveness of surveys, statistical agencies very often derive crosssectional estimates from longitudinal survey data assuming that the survey design takes this 

possibility into account, and that estimation procedures are developed to satisfy cross-sectional as 

well as longitudinal requirements. A good example of such ‘double’ utilization of a longitudinal 

survey is the Canadian Survey of Labour and Income Dynamics (SLID). It was originally designed 

1 Georgia Roberts, Milorad Kovacevic, Harold Mantel, Owen Phillips, Data Analysis Resource Center, Social 

Survey Methods Division, Statistics Canada, Ottawa, Ontario, Canada, K1A 0T6; e-mail: 

georgia.roberts@statcan.ca, milorad.kovacevic@statcan.ca, harold.mantel@statcan.ca, owen.phillips@statcan.ca. 
 

 

 

 

 

to provide longitudinal estimates and analyses. However, recognizing the cross-sectional capabilities 

of SLID, Statistics Canada made it a principal survey for providing annual income data and used 

it to replace a classic cross-sectional Survey of Consumer Finances (SCF) as of 1998. 

In order to achieve cross-sectional representativity, different approaches have been taken in different 

longitudinal surveys. SLID employs overlapping panels, each of six years duration and selected three 

years apart. The cross-sectional sample for a particular year also includes cohabitants of the 

longitudinal individuals from the two panels, i.e., all individuals that are living with the originally 

selected longitudinal individuals at a certain point in time. In this way, only households composed 

entirely of immigrants who have arrived since the last panel selection (at most three years out of 

date) are not represented in the sample. The elaborate cross-sectional weighting scheme that 

includes a non-response adjustment, an optimal combination of the two panels, adjustments for 

interprovincial migration and influential values, and post-stratification to a number of post-stratum 

totals completes the adjustments towards cross-sectional representativity of the population at a given 

time (Levesque and Franklin, 2000). 

Point estimation of parameters of the cross-sectional population based on data from longitudinal 

surveys in general, and from SLID in particular, has been studied and documented (Lavallee 1995, 

Merkouris 1999, Levesque and Franklin 2000). However, variance estimation for these estimates 

hasn’t received as much attention. In particular, the problem of formal comparison of the estimates 

from two years, which requires variance estimation for the difference of the estimates, is seldom 

addressed. This paper focuses on that problem. It is an extension of previous work by Roberts and 

Kovacevic (1999) on the comparison of cross-sectional prevalence rates estimated from the Canadian 

National Population Health Survey. 

The paper is organized into five sections. Section 2 contains a description of the problem and details 

some of its causes. Two approaches to variance estimation as a practical solution to the problem are 

given in Sections 3 and 4. Section 5 contains a numerical illustration and some concluding remarks. 

2. Problem Description 

Statistics Canada conducted the Survey of Consumer Finances (SCF) annually beginning in 1971 

to provide income data for families and individuals. Its output consisted of estimates of a variety of 

income distribution parameters at the national and provincial levels for a number of different 

subpopulations. Due to the near independence of the samples in consecutive years, inference about 

net change from year to year was straightforward and computable from the reported annual estimates 

of levels and their standard errors. Since the survey contents of the SCF and SLID are almost 

identical, Statistics Canada decided to replace the SCF by SLID starting in 1998. The main reason 

was a gain in efficiency. Also the extensive demographic, socio-economic and labour content of 

SLID would allow different perspectives on income distributions through a better fitting of a variety 

of models. 

The longitudinal underpinnings of SLID introduce complexities that cause difficulties when it comes 

to estimation of the variance of the difference of estimates in any two years (that are not more than 

6 years apart). Some of these complexities are the following: 

i) The cross-sectional SLID sample in any year contains all longitudinal individuals and their 

cohabitants who are in-scope for cross-sectional purposes. Thus, the cross-sectional samples are 

not independent at the two time points and have a large degree of overlap. Longitudinal 
 

 

 

individuals are in-scope cross-sectionally for a given year if they are still members of Canada’s 

ten provinces as of December 31 of the reference year, excluding those who live in institutions, 

military barracks or on Indian reserves . 

ii) The cohabitants of the originally selected longitudinal individuals generally stay with these 

individuals for more than a year and thus increase the overlap of the samples. 

iii) At each of the time points after 1995, the cross-sectional samples contain two longitudinal panels 

that were selected three years apart. Each panel represents the entire survey population at the 

time of its selection. These overlapping panels are optimally combined to represent the crosssectional population in a reference year. The optimality criterion was based on minimizing the 

variance of an estimated total and resulted in ‘panel allocation factors’ calculated at the level of 

province for each reference year. These factors were then applied to individual weights. The 

overlapping panels can be thought of as a special case of a dual frame survey (Merkouris, 1999). 

iv) For cross-sectional purposes the longitudinal individuals who changed province of residence 

after being selected into the longitudinal sample are considered to be part of the sample for the

province in which they reside at the time of the cross-section. However, for variance estimation 

these individuals must be considered to be part of their original province, stratum and cluster. 

v) The cross-sectional samples are affected by longitudinal non-response because of the way in 

which longitudinal individuals are included in the cross-sectional samples. 

Many of these complexities are accounted for through SLID’s elaborate weighting scheme so that 

point estimation of cross-sectional parameters and their net change over time is consistent. Bootstrap 

weights specifically created for each cross-sectional sample also account for most of these 

complexities. However, estimation of the variance for the difference of two estimates obtained in 

different years is not straightforward due to the sample overlap. 

For purposes of illustration, suppose we are interested in estimating the variance of the difference 

of estimates for reference years, 1997 and 1998. Here we illustrate some difficulties in determination 

of SLID individuals who are in the cross-sectional samples in these two years through graphical 

presentation of the composition of the cross-sectional and longitudinal SLID samples for 1997 and 

1998. There are 78,532 individuals with positive 1997 cross-sectional weights, and 79,611 with 

positive 1998 cross-sectional weights. An individual with a positive cross-sectional weight in a 

particular year is cross-sectionally in scope and belongs to a responding household in that year. The 

total number of cross-sectional individuals common to both years is 75,351, of which 66,847 are 

longitudinal individuals. The remaining 8,504 common individuals are the cohabitants who were 

with longitudinal individuals in both years. The common individuals represent 96% of the crosssectional sample in 1997 and 95% in 1998. There were 101 (23+78) longitudinal individuals for 

1997 that were not in scope cross-sectionally in 1997; 78 of them remained longitudinal in 1998 and 

also became cross-sectionally valid in 1998, while the remaining 23 individuals are probably 

longitudinal non-respondents in 1998. Another 225 individuals, who were in the longitudinal 

samples in both years and in the 1997 cross-sectional sample, were lost for cross-sectional estimation 

in 1998, most likely by moving out of scope (due to moving into institutions or out of the ten 

provinces, or dying). It is also interesting to observe that 530 (142+388) individuals had positive 

longitudinal weights in 1998 but had zero longitudinal and cross-sectional weights in 1997, most 

likely due to wave nonresponse in 1997. Only 388 of these were cross-sectionally in scope in 1998. 

Most of cross-sectional individuals in SLID stayed in the province where they were originally 

selected: in 1997 only 3.2% lived in a different province and in 1998 only 3.9%. 
1977 

CS 97 

979 

225 

66,847 

8,504 

78 

388 

3794 

23 

142 

L 97 

0 

In order to address the problem of variance 

estimation for the difference of cross-sectional 

estimates obtained in 1997 and 1998, we now 

introduce some notation. Let s

t be the 

individuals on the cross-sectional SLID sample 

at time t, where t=1 for 1997 and t=2 for 1998. 

Suppose that we are interested in the mean 

after-tax income within a domain of the 

population at each time point. The mean 

income within the domain at time t may be 

estimated by θˆ ˆ 

t /Xˆ t Y t , with Yˆ t Ds w

Xˆ ti yti and 

t s D wti xti , where w

t

onal 

weight ot

ti is the cross-secti

f the ith individual in s

t (who will be 

called called the itth individual); yti income if 

the itth individual is in the domain, and yti 0 

otherwise; and xti 1 if the itth individual is in 

the domain, and x 0 otherwise. Then

∆ˆ θˆ θˆ

ti 

1 2 estimates the net change in the mean 

income between the two time periods. The main problem addressed in this paper is the estimation 

of the variance of ∆ˆ . In the next two sections we present two possible methods, Taylor linearization 

and a pseudo-coordinated bootstrap method. 

3. Variance Estimation: Taylor Linearization Approach 

3.1 Linearization of ∆ˆ

One possible approach to obtaining a design-based variance estimate of ∆ˆ is Taylor linearization. 

In developing this approach, for ease of presentation, adjustments to the final weights will be 

ignored. Since ∆ˆ is a non-linear function of the data from both samples s

t , t=1,2, the first step is 

to linearize ∆ˆ by expansion into a Taylor series around the true net change in means. Assuming that 

the remainder term is negligible for a sufficiently large sample, the following approximation holds: 

ˆ

1

∆ K

1

∆  (Yˆ ) ˆ ˆ ˆ

1 Y1

θ1(X1 X1) (Y2 Y2) θ2(X2 X2) , 

X1 X2 

(1)

where θt Yt /Xt , t=1,2. This implies that 

Var (∆ˆ) K

1 1 Var ˆ w1i (y1i θ1 x1i

) ˆ w2i (y )

X

. 2i θ2 x2i

1 s

1 X2 s2 

(2)

10 

Sample s

t can be expressed as st

 { s

tk where stk represents those observations in st forming the 

k1 

cross-sectional sample for province k at time t. It then follows that 

Var (∆ˆ) 10 K Var ˆk1 Zˆ

10

ˆ

1

(s

1k

) ˆk1 Z2

(s

2k

) , (3) 
 

where Zˆ 

t (stk )  ˆ wti Zti , and Zti  X

1

t ( yti θt xti ), t=1,2. 

s

tk 

If we ignore, for the moment, longitudinal individuals who are residing in a different province than 

the one for which they were selected, the provincial samples s

tk are independent due to design of 

SLID, where there was independent sample selection in different provinces . This then implies that 

10 Var (∆ˆ) K ˆk1 Var Zˆ

1

(s

1k

)Zˆ

2

(s

2k

) . (4) 

The kth provincial component of this variance, Var Zˆ

1

(s

1k

)Zˆ

2

(s

2k

) , can be expanded further as 

Var Zˆ

1

(s

1k

)Zˆ

2

(s

2k

) Var Zˆ

1

(s

1k

)  Var Zˆ

2

(s

2k

) 2Cov Zˆ

1

(s

1k

),Zˆ

2

(s

2k . (5) 

The problem of estimating the variance of ∆ˆ then reduces to estimating the terms on the right hand 

side of (5). 

3.2 Notation and Assumptions Required for Variance Estimation 

The following detailed notation is required for explanation of the variance estimation: 

Htk # of strata in the cross-sectional sample in province k at time t,

ntkh # of sampled clusters in the hth stratum in province k at time t,

ntkhc # of sampled individuals in cth cluster of hth stratum in province k at time t,

wtkhci weight on the ith individual in cth cluster of hth stratum in province k at time t, and

z Xˆ 1

ˆ

tkhci  t (ytkhci θt xtkhci). 

It should be noted that the strata and weights are those in use after the combining of the two panels. 

See Levesque and Franklin (2000) and Merkouris (1999) for details. 

The following standard assumptions for variance estimation for data from a survey with a stratified 

multistage design are considered to hold for each of the cross-sectional SLID samples: 

i) The design of each cross-sectional sample is approximately stratified with selection of psu’s with 

replacement. 

ii) Each psu is selected at most once (because of small sampling fractions). 

n

n ˆ

tkhc 

tkh i1 wtkhciz  tkhci ntkh ztkhc iii) (i.e., ntkh × weighted cluster total) is approximately unbiased as 

an estimator for the stratum total Ztkh for any z variable and for any value of t, k, h, and c.

Under these assumptions, there is a straightforward approach to estimate a stratum total and the 

variance of stratum total at each time point. As well, if the same psu’s are represented in a stratum 

at both time points, there is a straightforward approach to estimating a covariance between stratum 

totals at the two time points. In particular, under these assumptions: 

where

iii) If, at times t=1 and t=2, the same psu’s are observed in a stratum sample, (which implies that 

i) An (approximately) unbiased estimate for Ztkh is Ztkh  ˆc

tkh 

1 z

tkhc . 

ˆ n

ii) An (approximately) unbiased estimate of the variance of Zˆ 

tkh is 

varˆ [Zˆ  ̄

tkh]ntkh /(ntkh 1)ˆ

n

c

tkh 

1

(z

tkhc Ztkh)

2 

,

Z ̄

tkh Zˆ 

tkh /ntkh .
n1kh  n2kh ), an (approximately) unbiased estimate of the covariance of Zˆ

1kh andZˆ2kh is given 

by

n

covˆ [Zˆ

1kh  ̄  ̄

1kh ,Zˆ2kh]n1kh /(n1kh 1)ˆc1

(z

1khc Z1kh)(z2khc Z2kh). 

 

 

 

 

 

 

3.3 Application 

These results can then be readily applied to the cross-sectional SLID samples for 1997 and 1998. By 

the design of SLID, cross-sectional samples for those two years should consist of the same strata and 

psu’s within each province at both time points, even though there are several reasons to expect that 

the individuals within a particular psu would not be exactly the same at the two time points (such 

as nonresponse of a longitudinal individual to the income questions at one of the time points or a 

longitudinal person entering an institution between the two time points). The following variance and 

covariance estimates would follow in a straightforward manner from the results above: 

Htk ntkh 

varˆ Zˆ 

t

(s

tk) ˆh1 ntkh /(ntkh 1)ˆc1

(z

tkhc Ztkh)

2  ̄ , and 

H1kn1kh 

cov ˆ Zˆ  ̄  ̄

1

(s

tk),Zˆ2

(s

2k

) ˆh1 n1kh /(n1kh 1)ˆc1(z

1khc Z1kh)(z2khc Z2kh), 

while z

1khc and z

2khc would consist of weighted sums over different individuals if the khc-th psu

contained different individuals at the two time points. 

3.4 Accounting for Movers Between Provinces 

In the development above, it was assumed that individuals continue to reside in the province for 

which they were selected into the sample. Modifications need to be made to the Taylor linearization 

variance approach when there are movers, that is, people who, for either time point, are crosssectionally representing a different province than the one for which they were drawn into the sample. 

This can be done by first decomposing s { { tk into s { tk s

t1k st2k ... st10k where stjk are those people 

in s

tk who were selected into the sample in province j. Then, Var(∆ˆ) can be expanded in the stjk , and 

terms be grouped according to the province of selection. Making use of the fact that independent 

sampling was done by province, formulae similar to those in 3.3 above may be developed readily 

for calculating the required variances and covariances among the s

tjk domains. While theoretically 

straightforward, implementation could be tedious if many of the s

tjk, j£k are non-empty. 

4. Variance Estimation: Bootstrap Methods 

Replication methods for variance estimation are becoming increasingly popular for analysis of data 

from complex surveys. Methods suitable for data from stratified multistage survey designs are now 

available, and their properties have been investigated both theoretically and empirically. One 

attractive feature of these methods is that the relatively difficult task of deriving replicate survey 

weights only needs to be done once by the methodologists most familiar with the survey design and 

weighting. In particular, complexities due to multistage sampling, multiple frame estimation, 

interprovincial migration of longitudinal panel members, adjustments to the weights to account for 

non-response, etc., can be incorporated into these replicate weights. Use of the replicate weights by 

any analyst to derive valid design based variance estimates is then relatively simple, and does not 

require any direct knowledge of the complex survey design or weighting procedures. 
In this section we first briefly describe a bootstrap method, called the coordinated bootstrap, which 

is suitable for overlapping samples on two occasions. We then describe an approximation to the 

coordinated bootstrap, called the pseudo-coordinated bootstrap, which may be used when 

coordinated bootstrap weights are unavailable. 

4.1 Coordinated Bootstrap Method 

In this subsection we describe a coordinated bootstrap method for estimation of the variance of the 

difference of two cross-sectional estimates. The bootstrap resampling method for iid samples has 

been extensively studied (see Efron, 1982). It was extended by Rao and Wu (1988) to stratified 

multistage designs and again by Rao, Wu and Yue (1992) to include nonsmooth statistics. Yung 

(1997) contains a concise description of the procedure. To summarize, for each bootstrap replicate 

a sample of PSUs is drawn with replacement from the set of sampled PSUs in each stratum. 

Sampling weights of each sample unit are then adjusted to reflect this resampling; this is called the 

bootstrap adjustment to the sampling weights. Any further adjustments to the sampling weights, 

such as nonresponse adjustments or calibration of the weights, should also be applied to each 

bootstrap replicate to produce what we will call a set of bootstrap weights. The bootstrap variance 

estimator for a weighted estimator θˆ is then calculated as 

θˆ



θˆ



vB(θˆ) 

1 2 

(b) (])

B ˆb (6)

where θˆ



 is the estimate of θ based on the bth set of bootstrap weights, and θˆ



(b) (.) is the mean of θˆ



(b)

over the B bootstrap replicates. Alternatively, θˆ is often substituted for θˆ



(.) in (6).

The same method may be used for multistage sampling on two occasions with overlapping samples. 

The following procedure is used for each bootstrap replicate. For sample PSUs that are common to 

the two occasions by design, the bootstrap samples for the two occasions must be “coordinated”; i.e., 

the same bootstrap samples of PSUs should be used for each occasion. For the sample PSUs that 

are chosen independently on either occasion, bootstrap samples of PSUs should also be chosen 

independently. Bootstrap adjustments to the sampling weights would be applied as usual, and any 

further adjustments to the weights would be applied independently in each sample. Now, if θˆ  ∆ˆ

is the difference between two cross-sectional estimates, one from each of the samples, then its 

variance can be estimated consistently from (6) using these coordinated sets of bootstrap weights. 

4.2 Pseudo-Coordinated Bootstrap Method 

Although the coordinated bootstrap offers a neat solution to the problem of variance estimation for 

the difference of two cross-sectional estimates, it cannot be applied when the bootstrap samples were 

drawn independently for each of the two samples, as is often the case for cross-sectional files 

produced from longitudinal surveys. Recalling that Var(∆ˆ )  Var(θˆ )Var(θˆ )2Cov(θˆ ,θˆ

1 2 1 2 ), 

we propose here a method to produce approximate coordinated bootstrap weights which may be used 

for estimation of the covariance of the two cross-sectional estimates. Because of the approximations 

and assumptions involved it is recommended that the original bootstrap weights, wti(b), be used for 

estimation of the variances of the cross-sectional estimates. 

In the coordinated bootstrap approach, for individuals in PSUs that are common to the two samples 

the bootstrap adjustment of the basic sampling weights would be the same for both samples. Thus 
 

 

for an individual in the overlap of the two samples, the ratio of the bth coordinated bootstrap weight 

to the final estimation weight should be approximately the same for both samples, with any 

differences in these ratios due only to differences in the other adjustments to the weights. If we also 

assume that individuals not in the sample overlap were sampled independently of the overlap, and 

independently on each occasion, then their contribution to the covariance should be zero. Under 

these conditions the procedure described below should yield reasonable results. For SLID, crosssectional individuals who are not in the overlap are not independent of the overlap; however, the 

number of such individuals is relatively small. 

From the bth set of bootstrap weights associated with s

1 we define a set of pseudo-coordinated 

bootstrap (PCB) replicate weights as follows: 

w1,1i(b) 

w1i(b) iMs

1 , iMs2 

w1i iMs

1 , i¬Ms2 

0 i¬Ms

1 

w1,2i(b) 

w2iw1i(b) /w1i 

w2i 

iMs

1 , iMs2 

i¬Ms

1 , iMs2 

0 i¬Ms

2 

(7) 

We can similarly define PCB weights, w2,1i(b) and w2,2i(b) , corresponding to the bth set of bootstrap

weights associated with s

2

. If PSU identifiers were available, then we could replace the PCB 

adjustment factor w1i(b)/w1i in (7) by D D jMPSU(i)w1j(b)/

jMPSU(i)w1j , which would be more stable. If we 

have B replicates in each set of bootstrap weights then it may be reasonable to construct B/2 sets of 

PCB weights based on s

1

bootstraps and B/2 based on s

2

; however, we may have as many as B sets 

based on each sample. If the original bootstrap weights are benchmarked to some population totals, 

then we may wish to similarly benchmark the PCB weights, assuming that the benchmarking 

procedure is known. The covariance of two cross-sectional estimates, θˆ

1 and θˆ

2

, would then be 

estimated by 

θˆ



θˆ



θˆ



θˆ



covB(θˆ

1

,θˆ

2

) 

1 ˆb 1(b) 1(]) 2(b) 2(])

BPC 

(8)

where the summation is over the B sets of PCB weights, and θˆ



PC 1(b) and θˆ



1(b) are calculated using, 

respectively, either w1,1(b) and w1,2(b) from (7), or w2,1(b) and w2,2(b) .

4.2.1 Pseudo-Coordinated Bootstrap for Non-independent Non-overlap 

If the assumption of independence of the sampling of individuals not in the overlap is not reasonable, 

then the above procedure would tend to underestimate the magnitude of the covariance. However, 

the procedure could be modified in various ways. 

The first approach to accounting for dependence of the non-overlapping part of the sample is based 

on identifying PSUs within the samples. For s

2 individuals whose PSU intersects the common 

sample, PCB weights based on s

1 bootstrap weights could be constructed by multiplying w2i by

DjMPSU(i) 1 w j(b)/DjMPSU(i)w1j . For PSUs that do not intersect the common sample at all, it might be 

reasonable to assume that such PSUs from s

1 are sampled independently of those in s2

. 

Alternatively, if such PSUs from s

1 can be linked to corresponding PSUs from s2

, then a similar 

type of adjustment can be used to construct PCB weights. 

For a second, somewhat simpler approach, if θ is a smooth function of population totals, then some 
 

 

 

 

 

of the extra covariance due to the non-overlapping parts of the samples could be captured using a 

linear approximation. Suppose for example that θ  θ(X) where X is the population total of a 

variable x. If we write Xˆ o 

k Xˆ no

 Xˆ 

k  k , where the superscript “o” denotes the overlap part of the 

sample, and the superscript “no” denotes the non-overlap part, then we may write an approximation: 

jθ1 jθ2 Xˆ no Xˆ no Cov(θˆ

1

,θˆ

2

)  Cov(Xˆ

1

o

,Xˆ o) Cov(Xˆ no,Xˆ o) Cov(Xˆ

1

o

, ) Cov(Xˆ

no

, ) . 2 1 2 2 1 2 jX1 jX2 

If we now define PCB weights based on s

1 bootstraps as 

w1,1i(b) 

w1i(b) iMs

1 

0 i¬Ms

1 

w1,2i(b) 

w2iw1i(b)/w1iiMs

1

, iMs

2 

w2ii¬Ms

1

, iMs

2 

0 i¬Ms

2 

then these weights could be used to estimate Cov(Xˆ

o

1

,Xˆ o) and Cov(Xˆ no,Xˆ o

2 1 2 ). Similarly defined 

PCB weights based on s X Xˆ no

2 bootstraps could be used to estimate Cov( ˆ

o

,Xˆ

o

) and Cov(Xˆ

o

,

ˆ ˆ 1 2 1 2 ). 

However, estimation of the component Cov(X

no

1

,X

no

2 ) requires PCB weights that simultaneously 

adjust the weights for both of the non-overlapping parts of the samples. 

5. Illustration 

The proposed methods are applied to SLID data where the average after-tax incomes for individuals 

aged 16 and over with income for 1997 and 1998 are compared. There were 60,901 and 62,272 such 

individuals in the 1997 and 1998 cross-sectional samples, respectively. The averages, their difference 

and the corresponding standard errors obtained by the proposed methods are given in the Table 

below. 

Estimated averages of income-after-tax, their

 difference and standard errors 

Estimates 

Standard Errors 

Taylor 

Bootstrap 

Coordinated PseudoCoordinated 

θˆ

97 20285 137 132 

θˆ

98 21125 142 137 

ˆ ˆ θ -

97 θ98 -840 79 82 81 

For application of the Taylor method all 

longitudinal individuals, and so their 

cohabitants, were associated with their 

province of residence at their time of 

selection. Also, it was assumed that the 

weights of all individuals from a stratum 

were multiplied by the same panel 

allocation factor (PAF). In such a case the 

stratum total can be estimated unbiasedly 

and the basic assumptions for variance 

estimation by Taylor linearization method 

as stated in Section 3 are satisfied. 

This, however, may not be exactly true since the weights of individuals that joined the population 

after the selection of the first panel are not modified by the PAF, meaning that within a Panel 2 

stratum some individual weights may be multiplied and some may not. However, the number of such 

individuals represents less then 0.6% of the Panel 2 size. 

The bootstrap calculations are based on 500 replicates. The bootstrap weights that were produced 

for SLID for the 1997 and 1998 cross-sectional samples are already coordinated. The PCB weights 
 

 

for this empircal comparison are defined as in (7), using an individual level PCB adjustment factor, 

with no subsequent benchmarking, and based on the assumption of independence of individuals not 

in the sample overlap. The first 250 sets PCB weights were based on the first 250 sets of bootstrap 

weights for s

1

, while the second 250 were based on the second 250 sets of bootstrap weights for s

2

. 

The estimate of Cov( θˆ ˆ

97 ,θ98 ) based on the Taylor linearization method was 16346, while that based 

on coordinated bootstrap was 14806, and that based on the pseudo-coordinated bootstrap was 14838. 

The preferred method for variance estimation in this set-up is the coordinated bootstrap, as it can 

take explicit account of all of the complexities of the survey design and estimation. The pseudocoordinated bootstrap performed well in our example. Some additional empirical investigation is 

needed to assess its properties. Standard errors estimated by Taylor method are very close to those 

obtained by the bootstrap methods despite the approximations involved, including the ingnoring of 

weight adjustments. 

Acknowledgment The authors would like to acknowledge the computational input by Michael Lo. 

REFERENCES: 

Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia. 

Lavallee, P. (1995). Cross-Sectional Weighting of Longitudinal Surveys of Individuals and 

Households Using the Weight Share Method. Survey Methodology, Vol.21, 1, 25-32. 

Levesque, I. and Franklin, S. (2000). Longitudinal and Cross-Sectional Weighting of the Survey of 

Labour and Income Dynamics 1997 Reference Year. Income Statistics Division Paper 

No.75F0002MIE-00004, Statistics Canada 

Merkouris, P. (1999). Cross-Sectional Estimation in Multiple-Panel Household Surveys. 

Methodology Branch Working Paper HSMD 99-004E, Statistics Canada 

Rao, J.N.K. and Wu, C.F.J. (1988). Resampling Inference with complex Survey Data. Journal of 

American Statistical Association, 83, 231-241. 

Rao, J.N.K. , Wu, C.F.J.. and Yue (1992). Some Recent Work on Resampling Methods for Complex 

Surveys. Survey Methodology, 18, 209-217. 

Roberts, G. and Kovacevic, M. (1999). Comparison of Cross-Sectional Estimates from Two Waves 

of a Longitudinal Survey. Proceedings of the Survey Methods Section, Statistical Society of 

Canada, 153-158 

Stukel, D.M., Mohl, C.A., and Tambay, J.-L. (1997). Weighting for Cycle Two of Statistics 

Canada’s National Population Health Survey. Proceedings of the Survey Methods Section, 

Statistical Society of Canada, 111-116. 

Yung, W. (1997). Variance Estimation for Public Use Microdata Files. Proceedings of Symposium 

97 “New Directions in Surveys and Censuses”, Statistics Canada, 91-95.