Permanent and Collocated Random Number Sampling and

the Coverage of Births and Deaths

Lawrence R. Ernst, Bureau of Labor Statistics

Richard Valliant, Westat

Robert J. Casady, Consultant

U.S. Bureau of Labor Statistics

Room 3160

2 Massachusetts Avenue NE

Washington DC 20212

December 1998
2

Abstract

Permanent random number (PRN) and collocated random number (CRN) sampling are

practical methods of controlling overlap between different samples. The techniques can be

used for overlap control between samples for the same survey selected at different time

periods or between different surveys at the same time period. Although the methods are in

wide use, their properties, when a population is changing due to births and deaths, have not

been studied extensively. Ideally, each technique should produce a sample proportionally

allocated to births and persistent units when equal probability sampling is used. We study

particular PRN and CRN schemes that produce fixed size samples and involve complete,

rather than partial, rotation of units within strata. We present theoretical and empirical results

showing the circumstances where proportional allocation is approximately obtained with these

particular schemes. We also discuss important cases where PRN and CRN sampling are

substantially different in their coverage of birth and persistent units.

Key words: Persistent units; post stratification; sample allocation; sequential simple random

sampling.
3

1. Introduction

The statistical agencies of national governments routinely publish economic statistics

based on surveys of business establishments. Often, different surveys use the same frame of

establishments for sampling, leading to a need to somehow coordinate sampling for the

surveys. Limiting the burden placed on an establishment may be critical to obtaining and

maintaining cooperation when a unit is eligible for several surveys. Controlling the length of

time that a sample unit is in a particular survey and the number of different surveys that a unit

is in are both desirable. Maintaining a frame over time by updating for births and deaths and

properly reflecting these changes in each sample are also important issues. Much of Part B,

“Sample Design and Selection” in Cox, et. al. (1995), for example, is devoted to these topics.

A number of government agencies either currently use or have in the past used

permanent random number (PRN) or collocated random number (CRN) sampling as a way of

facilitating sample coordination among surveys and rotation of units within a survey. The

general methods are described in Section 2. Statistics Sweden (Ohlsson 1992, 1995), the

Institut National de la Statistique et des Etudes Economiques of France (Cotton and Hesse

1992), the Australian Bureau of Statistics (Hinde and Young 1984), and Statistics New

Zealand (Templeton 1990) each have used variations of PRN or CRN sampling. Ohlsson

(1995) summarizes the methods of the different countries.

Though the methods are in common use, there appears to be a limited literature on

their properties, particularly regarding the treatment of population changes due to births and

deaths. There has been some recognition, for instance, that certain implementations may have

a “birth bias,” i.e., births are selected in a sample at more than their proportional rate in the

population (see, e.g., Ohlsson 1995, p.166). How serious the bias is and the parameters that
4

effect it are studied in this paper. The calculations are fairly complex, but, since the PRN and

CRN methods have seen such wide use, we feel that a better understanding of their properties

is worthwhile.

There are a variety of implementations of the methods. Some alternatives lead to

random sizes of sample while others produce fixed sample sizes. Different methods also may

handle births and sample rotation differently. The theory and empirical results we discuss

refer to particular PRN and CRN schemes that (1) yield fixed sample sizes and (2) facilitate

rotation of entire samples within strata. This method of complete rotation is useful in some

types of surveys but, unlike some other methods, does introduce the possibility of a “birth

Section 2 briefly describes the methods and the reasons why collocated sampling was

developed. The third section presents theoretical properties of particular implementations of

the methods when births and deaths can occur in the population. Section 3 also describes the

particular method of complete rotation we consider and reasons for its use. Section 4 gives

some numerical results to illustrate the effects of different population sizes, sample sizes, birth

and death rates, and the method of sampling on the relative misallocation of birth units. The

empirical results also illustrate the theoretical finding that, for the versions studied here,

collocated sampling exercises much tighter control over the achieved sample allocation to

persistent and birth units than does PRN sampling. Section 5 is a conclusion where we briefly

mention some estimation issues.
5

2. Description of the Methods

Denote by F0 the initial (time period 0) frame of N0 units. In the subsequent

sections, we will consider the possibility of births and deaths that occur at later time periods.

The methods described in this section are normally applied within strata but, for simplicity, we

omit most references to stratification. Denote a random variable that is uniformly distributed

on the interval [0,1] by U[0,1].

First, consider equal probability, PRN sampling. A simple random sample of fixed size

n can be selected from the population of size N0 by sorting the population in a random order

and then picking the first n units after some starting point. This can be accomplished as

follows:

(P1) independently assign a realization ui of a U[0,1] random number to each unit in

the population,

(P2) sort the units in ascending order based on ui, and

(P3) beginning at any point a [ ] 0 ∈ 0,1 , include the first n units with u a i > 0. If n units

are not obtained in the interval ( ,1] a0

, then wrap around to 0 and continue.

This method is known as sequential simple random sampling without replacement (srswor)

and will also be denoted simply as PRN sampling here.

We will consider only fixed sample size plans. These are of interest in survey designs

where the budget is fixed and sample size is closely related to cost. An alternative is to use

PRNs but sample all units with values of ui in an interval [a,b]. This leads to a fixed

sampling fraction but not a fixed sample size, and, thus, makes costs less predictable.
6

The main objection to using unmodified U[0,1] permanent random numbers in

sequential srswor is that the PRNs within detailed strata may not be well distributed. If the

goal is to coordinate two or more surveys by minimizing the overlap among them, the poor

distribution may lead to problems. If the ui’s are, by chance, clumped in one part of the [0,1]

interval, the samples for the surveys may overlap unnecessarily. The problem can be

especially severe in strata where the population size is small. As an illustration, suppose there

are three surveys and that the frame and sample sizes are

N=10, n n 1 2 = = 2 , n3 = 4

Suppose further that the starting points for the three are

a1 = 0, a2 = 0.20, a3 = 0.40

and that, by bad luck, the ui’s for all 10 units are in [0, 0.20).

Using PRN sampling, units 1 and 2 in the sorted frame will be in all three surveys

because survey 1 takes the first two units starting at a1 = 0 , while surveys 2 and 3 wrap

around to 0 since there are no ≥ 0.20 iu . As a result of clumping of the ui’s, only four

distinct units are selected even though, with better placement of the ui’s, the samples could

be completely non-overlapping.. This example is extreme since the probability of all 10 ui’s

being in [0, 0.20) is negligible, but illustrates the general idea that undesirable overlap may

occur unless special measures are taken.

The use of collocated random numbers (Brewer, Early, and Joyce 1972; Brewer,

Early, and Hanif 1984) is one solution to this problem This technique was originally

developed as a way of reducing the randomness of sample size that accompanies Poisson

sampling. The assignment of CRNs is accomplished as follows. A U[0,1] random number is
7

assigned to each unit in the frame. These numbers are sorted in ascending order and the rank

Ri noted for each. A single U[0,1] random number e is then generated and ( ) ui Ri N0 = −e

is calculated for each unit on the frame.

Collocation spaces the random numbers assigned to the population units an equal

distance apart and eliminates the clumping that can occur with PRNs.

Note that in the above example there will be no overlap of the three initial samples if

CRNs are used. However, if for each succeeding time period each of these samples is

completely rotated by choosing as a starting point the CRN of the final unit selected for the

survey for the previous time period, and if there are no births or deaths, then for the fourth

time period, time period 3, every sample unit for survey 3 will also be in sample for either

survey 1 or survey 2. There will never be any overlap of the samples for surveys 1 and 2

because 1 2

n = n .

3. The Effect of Births and Deaths

Let B denote the frame of births at time period 1 and suppose that it contains NB

units. Additionally, let F01 be the set of units in F0 that are “nondeaths” or “persistents,” and

suppose that F01 contains N01 units. The updated frame at time 1 is F1 = F01 ∪ B and

contains N1 = N01 + NB units. The number of deaths is, thus, N00 = N0 − N01 . The true

proportion at time 1 of units that are births is then PT = NB N1

. The sample selected from

the time 0 frame is 0S and the time 1 sample is 1S . In this section we give implementations

of PRN and CRN sampling for handling births and deaths and examine whether the sample

proportion of births, PS, is near PT. If PS differs from PT in expectation, this can be called a
8

“selection bias,” but we emphasize that this is different from the bias of an estimator—a topic

briefly mentioned in Section 5. To avoid the negative connotations of the word “bias,” we

will refer to the quantity ( ) E PS − PT as a measure of “misallocation” rather than bias.

Misallocation is just a measure of how far the sample departs from being proportionally

allocated to births and persistents.

3.1 Permanent Random Numbers

When a frame is periodically updated for population changes, an operationally simple

method is desirable for handling births. One option is to create separate strata for births. If

the same strata definitions are used for the birth strata and the persistents strata, and many of

the strata have few births in the population, then even an allocation of one unit to these birth

strata may result in an overall sampling rate for birth units, in comparison with persistent

units, that is undesirably high. If, however, broader strata are used for the birth units than the

persistents to avoid this problem, this may lead to other undesirable outcomes, such as units

with large differences in size having the same selection probability.

Another obvious approach, that we will study, is to repeat for the birth units the

procedure used earlier for the old units. For PRN sampling, a U[0,1] random number is

assigned independently to each birth unit. Birth units and persistents are then sorted together

based on PRN. Let a0

∗

 be the PRN of the last unit in the time 0 sample and suppose that the

time 1 sample consists of the first n units with u a i >

∗

0

. (As can be seen from the proofs of

Propositions 1 and 2 below, it is this assumption about the time 1 sample that results in the
9

misallocation of birth units.) This approach appears to be quite similar to one used by

Statistics Netherlands (Van Huis, Koeijers, and de Ree 1994).

This type of sampling is appropriate when the entire sample in a stratum is being

rotated. The Bureau of Labor Statistics (BLS), for example, is currently using this method

for its Occupational Employment Statistics survey. Data are collected annually for this survey

and BLS promises respondents that they will not be in sample more than once every three

years, necessitating full sample rotation annually.

An alternative to full sample rotation that is used many surveys is to rotate a part of

each stratum—a topic not considered in detail here. The problem with extending the fixed

sample size plan to partial rotation is that if the first n′ of the time 0 sample units are replaced

at time 1 by the first n′ units with u a i >

∗

0

 and the remaining units at time 0 are retained at

time 1, then while the expected proportion of births among the n′ new units at time 1 is the

same as that given in the propositions below, there is of course no births among the units

retained at time 1. In addition, although there would be deaths among the retained units they

would not be compensated for in sample size by taking additional units. The PRN shift

method, described by Ohlsson (1995), in which a moving fixed-length sampling window is

used, avoids this problem with births, but leads to a random sample size.

The full-stratum rotation method, that we do analyze, has a slight selection bias

toward births as shown in Propositions 1 and 2.

To make the exposition clearer, we have separated the case of no deaths (Proposition

1) from one having both births and deaths (Proposition 2). This separation will be especially

useful when considering CRN sampling in Section 3.2.
10

Proposition 1. Assume that n < N1 and that there are no deaths, i.e., N0 = N01 . Using the

PRN method of sampling described above, the expected proportion of the time 1 sample that

is in B is

1 1

PRN

−

=

N

N

P

B

. (1)

Proof: The final unit selected for the S0 sample is a unit on the F0 frame. This unit is not

among the first N1 −1 units that can be selected for the S1 sample. Consequently, among

these 1 N1 − units, exactly NB are in B and, by the nature of PRNs, each of these 1 N1 − units

has a probability ( 1) NB N1 − of being in B. This establishes (1). n

The relative misallocation in the proportion of birth units in the S1 sample for PRN

sampling is

( 1)

1

T 1

PRN T

−

=

−

P N

P P

. (2)

Thus, the relative misallocation does not depend on n and is small when the population size

N1 is large.

Proposition 2. Assume that n < N1

 and that there may be deaths, that is N01 ≤ N0. The

expected proportion of the time 1 sample that is in B is

















−

= +

( 1)

1

0 1

01

1

PRN N N

N

N

N

P

B

. (3)

Proof. As in the first proof, if the final unit selected for the S0

 sample is in F01, then each of

the first 1 N1 − units that can be selected for the S1

 sample has a probability ( 1) NB N1 − of
11

being in B. If, however, this final unit is a death, and hence not in F1, then each unit in the

frame F1

 has a probability 1 NB/ N of being in B. Since the probability that this final unit is in

F01 is N01 N0

/ , we have

0 1

01

0 1

01

PRN 1

( 1) N

N

N

N

N N

N N

P

B B

















+ −

−

=

from which (3) follows after simplification. n

The relative misallocation in the proportion of birth units in the S1

 sample for PRN in

the general case is

1

1

( 1) 0 1 1

01

T

PRN T

−

≤

−

=

−

N N N

N

P

P P

. (4)

As in the case when there are no deaths, the relative misallocation does not depend on n and is

small for large N1. The relative misallocation also decreases as the death rate, 01 11− N / N ,

increases.

3.2 Collocated Random Numbers

Assigning collocated random numbers has the advantage of spreading the numbers

evenly across the unit interval, but the analysis becomes quite complicated. The CRN method

can also lead to some unexpected results for small populations as we show in this section.

Assume that the births are handled as the original units were. A U[0,1] random number is

assigned to each birth. These numbers are sorted in ascending order and the rank RBi noted

for each unit. A single U[0,1] random number B

e is then generated and
12

( ) uBi RBi B NB = −e is calculated for every birth unit on the frame. The original CRNs and

the new birth CRNs are then sorted together.

The results for collocated random number sampling are considerably more

complicated to derive, and we have placed proofs in the Appendix. Assume that NB ≤N 0.

We first consider the case N01 = N0, i.e., there are no deaths. Example 1 illustrates a

disconcerting phenomenon that occurs when the birth rate is extremely high, and the sample

size is small.

Example 1. Suppose that ( N0, NB

, n) = (4,4,1). Let the rounded CRNs for the N0 =4 old

units be (0.20, 0.45, 0.70, 0.95) and the sample at time 0 be the first unit—the one with

CRN=0.20. The CRN assigned to the first birth unit will be in [0, 0.25). If it is less than

0.20, then the next birth unit will receive a CRN somewhere in the interval [0.20, 0.45). If the

CRN for the first birth is greater than 0.20, it will have to be in (0.20, 0.25). In either case,

the sample unit at time period 1 must be a birth. In fact, this forced selection of a birth holds

regardless of the particular CRNs used.

The general result for the expected proportion of births is given in Proposition 3,

which shows that the problem disappears when the sample size is large.

Proposition 3. Assume that N01 = N0

 and n < N1. For CRN sampling, the expected

proportion, denoted PCRN , of the sample that is birth units is





































=

1 0 1

0

CRN min ,

1

N

nN

N

N

N

nN

n

P

B B

(5)
13

where x is the smallest integer ≥ x .

Note that (5) implies that

0 PCRN − PT ≥ (6)

and that

N n

N

N

N

N n

N

N

nN

N

nN

N

N

N

nN

N

nN

P n

P P

B B

B B 2

min ,

1

min ,

1 1

0

1 1

0 1 1

1

1

0

1

0

T

CRN T ≤













≤

























−

























−













=

−

. (7)

It follows from (6) and (7) that the CRN misallocation is nonnegative and that the relative

misallocation is bounded above by 2 n . As n varies, the expected number of excess birth

units in sample fluctuates within these bounds, but the general trend in the misallocation is

downward as n increases and is small for large n.

The proof of Proposition 3 in the Appendix shows (see expressions A.2 - A.4) that

nB − nNB N1 < 1 and nB n − NB N1 <1 n . In other words, the realized number of births

selected with CRN will be within 1 unit of the expected number. Consequently, for large n,

being off from the expectation by 1 unit is nothing to worry about. On the other hand, when

n is small, being off by 1 may be a large percentage misallocation. For Example 1 we have,

PCRN = min{4 / 8, 4 / 8}=1, reflecting the fact that, in this extreme case, we have no

choice but to select a birth at time 1. Note that if PRN sampling was used, then Proposition 1

implies that 4 7 PPRN = compared to the proportion of births in the population which is 1 2 .

Thus, the degree of misallocation is less for PRNs.

An advantage of CRN sampling is that it offers tighter control over the sample

allocation than PRN because of the way the CRNS are spaced in the interval. That is, while
14

the realized number of births for CRN sampling is always within 1 of the expected number

when there are no deaths, the only restrictions on Bn for PRN sampling are that

max{ 1, 0} min{ , } 0n − N + ≤ nB ≤ NB n .

The following two examples illustrate the ideas behind the proof of Proposition 3. In

particular, they illustrate the key results (A.4) and (A.5) with the first expression following

“min” in (A.5) applicable in Example 2 and the second expression applicable in Example 3.

Example 2. Suppose that ( N0, NB

, n) = (5,4,3) and the CRN of the last sample unit in 0S is

0 = .27 ∗

a . The smallest interval of the form (.27, x] for which there must be CRNs of at least

n −1 units in the interval is (.27, .52]. The CRN of 1 unit in F0

 and 1 unit in B is in this

interval. The third unit to be selected for 1

S is in B if and only if there is a unit in B with

CRN in the interval (.52, .67) since the CRN of the first unit in F0 with CRN >.52 is .67. The

probability that there is such a unit in B is .6. Hence =1 Bn or 2 and ( = 2) = .6 B P n

Example 3. The only change from Example 2 is that n = 4. Then with x defined as in

Example 2, we have x = .67, since there are in (.27, .67] the CRNS of 2 units in F0 and the

CRNs of either 1 or 2 units in B. Furthermore, there will be 2 units in B in 1

S if and only if

there is at least 1 unit in B with CRN in the interval (.52, 87), which is always the case.

Hence ( = 2) =1 B P n .

We next consider the general case for CRNs, that is N01 ≤ N0

. We proceed to

derive an expression for PCRN , which is much more complex than for the case N01 = N0

.
15

For each positive integer m let S1m denote the first m units in F0 ∪ B (that is including

deaths) following the last unit in 0S . CRN sampling begins at the first unit after the last one

in the time 0 sample and marches through the updated frame until the desired sample of size n

is obtained, skipping over a death whenever one is encountered. In symbols, we seek the

smallest m such that S1m ∩ F1 has exactly n elements, and hence S1 = S1m ∩ F1. The number

of deaths between times 0 and 1 is N00 = N0 − N01. The range of m is given by the set

M = {m : n ≤ m ≤ n + N00 } since, with 0 deaths, we have to traverse only n units to obtain the

sample, but with deaths, we may need to skip over all N00 of them before getting a sample of

n.

In Proposition 4 below h(x,t,a,b) denotes the hypergeometric probability of x

successes in x+t trials when there are a successes and b failures in the population, i.e.,

( )

















+

+

































=

x t

a b

t

b

x

a

h x,t, a,b .

Proposition 4. Let nBm denote the number of units in S1m ∩ B , N ′ = N0 + NB

, and

nBm mNB N 

′ = ′ . Next, let n0m , n01m denote the number of elements in S1m ∩ F0

,

S1m ∩ F01 , respectively, and s1 fm denote the final sample unit in S1m . For each m there are

at most three different ways that m can be the smallest integer for which S1m ∩ F1 has exactly

n elements, namely:

nBm nBm n m n nBm = ′ = − ′

01 , , and s1 fm ∈F01 (8)

1, 1 nBm = n′Bm + n01m = n − n′Bm − , and s1 fm ∈ B (9)
16

1, 1 nBm = n′Bm + n01m = n − n′Bm − , and s1 fm ∈F01. (10)

Then, the expected proportion of a sample of size n that is birth units is the sum of the number

of births in the events (8), (9), and (10) times their respective probabilities of occurrence

divided by the total sample size. Symbolically, this is

∑[ ( ) ( ) ]

∈

= ′ + ′ + + ′ +

m M

nBm PLF m nBm PUBm nBm PUF m

n

P

01 01

1 1

1

CRN (11)

where PLF01m , PUBm , and PUF01m are the probabilities associated with (8), (9), and (10)

respectively and are shown to be

( ) ( ) 01 00

0

01 1, , 1,

01

h n n m n N N

N

N

P P n n LF m = Bm = Bm′ − Bm′ − − − (12)

( ) ( ) [ ( )] ( ) 1 01 00 1 1 1, , ,

1

PUB m = P nBm = nBm′ + − P nB m− = nBm′ + h n − n′Bm − m − n N N (13)

( ) ( ) ( ) 01 00

0

01

1

1 2, , 1,

01

h n n m n N N

N

N

P P n n UF m = B m− = ′Bm + − Bm′ − − − (14)

where ( ), ( 1), ( 1) P nBm = nBm′ P nBm = nBm′ + P nB(m−1) = n′Bm + are computed from (A.8),

(A.9) and (A.14).

Proposition 4 can be interpreted as follows. At time 1 we update the frame with

births but, for the moment, we just note which units are deaths without removing them. To

select the time 1 sample, we start with the first unit beyond where the time 0 sample left off.

If we go some arbitrary number m of units further on the list (including deaths) and the

number of births, nBm , in this sample plus the number of persistents, n01m , equals the desired

sample size n (after throwing away deaths), then this sample is a possibility for being the one

with the smallest m. Because of the random ordering of the collocated units, a probability is

associated with each possible value of m. Depending on the last unit in the S1m sample, the
17

probability of obtaining a particular number of persistents and passing over a particular

number of deaths is hypergeometric. For instance, associated with (8) and (12) is

( )

















− ′ −

+ −

















−

















− ′ −

−

− ′ − − − =

1

1

1

1

1, , 1,

01 00

01 00

01 00

Bm

Bm

Bm

m n

N N

m n

N

n n

N

h n n m n N N

which is the probability of (a) selecting n − nBm′ −1 persistents from the N01 − 1 population

persistents (given that the last unit in S1m is a persistent) and (b) having to pass over m − n

deaths from the N00 population deaths. The remaining two terms in (12) are obtained as

follows. It is shown in the proof in the Appendix that if nBm nBm = ′ then 1 F0s fm ∈ and hence

1 01 01 0 P(s fm ∈ F nBm = nBm′ ) = N / N . Finally ( ) P nBm nBm = ′ , which is given in (A.8) and

(A.9), is obtained from (A.4) and (A.5) in the proof of Proposition 3 and the fact that the

distribution of nBm is independent of the set of deaths among units in F0. The remainder of

the proof of Proposition 4 uses similar ideas.

4. Numerical Comparisons

Because the effects of different parameters on the expected proportions of births are

difficult to discern in some of the earlier formulas, we present some numerical results in this

section. First, we calculated the relative misallocation for PRN sampling in (4) using various

population sizes ranging from 5 to 100. Equal birth and death rates, from 0.2 to 0.8, were

used so that the population was stable ( N0 = N1). The relative misallocation

( ) PPRN − PT PT

 is plotted in Figure 1 versus the N1 population size. The four panels show
18

the different birth rates. The relative misallocation, which is independent of sample size, can

be as large as 0.20 for 5 N1 = but decreases rapidly as the population size increases.

Figure 2 shows the relative misallocations for CRN sampling plotted versus the

sample size for the same four birth rates. Equal birth and death rates were again used and

relative misallocations were computed as ( ) PCRN − PT PT with PCRN computed from (11).

Population sizes of N0 =5, 10, 50, 100, and 200 were used. Expression (11) was evaluated

for samples of n=1, 3, 5, 20, 35, and 50 in cases where N0n < . The results for the different

population sizes are shown in Figure 2 with different shades of gray. The points are jittered

slightly to minimize overplotting. For a given sample size, the shading goes from dark gray

for the smallest value of N0 to light gray for the largest. For example, for n = 5, there are

four population sizes having n < N0: N0 = 10, 50, 100, 200. The darkest gray dot is for

N0 =10, the lightest gray dot is for N0 =200, while N0 =50 and 100 are intermediate shades.

As the figure shows, the main determinant of misallocation is the sample size with population

size much less important. For samples of size 1 the relative misallocation can be as much as

50%, but decreases rapidly as n increases.

Since the earlier analytic work was confined to two time periods, we conducted a

simulation study to examine the performance of PRN and CRN sampling over three periods

(t=0, 1, and 2). An initial population of N0 =200 was used and equal birth and death rates of

0.2 were assumed to generate the populations at t=1, 2. Persistents at t=1 were identified by

generating a Bernoulli random variable for each of the N0 =200 time 0 units. If the random

variable was greater then 0.2, then the unit was a persistent; otherwise, it was a death. To

generate the number of births at t=1, a realization from a Poisson distribution was generated
19

with parameter 0.2 N0. For the t=2 population the procedure was repeated: each t=1

persistent was given a 0.2 chance of death and a Poisson number of births generated with

parameter 0.2 N1. PRNs and CRNs were assigned to original units and births as described in

Sections 2 and 3. At both times, samples of n=1, 3, 5, 20, 35, and 50 were selected in cases

where n < N0 (and N1). At time t +1 (t = 0,1) the sample consisted of the first n units with

PRNs (or CRNs) greater than the ui associated with the last sample unit at time t. This

procedure of population generation and sample selection was repeated 10,000 times for every

sample size.

Relative misallocations like those above were then computed. Let NB1 be the

number of births in the t=1 population, N012 the number of units that persist through t=0, 1,

and 2, NB12 be the number of time 1 birth units that persist at t=2, and let NB2 equal the

number of births at time 2. Further, let B1n , n012 , B12 n , and B2n be the corresponding

numbers of units in a sample of size n. The relative misallocations in the simulations were

computed as ( ) PS − PT PT

, where P S = ∑PSi 10,000 and PSi is sample proportion of units

of a specified type (births or persistents) in sample i, and the summation is across the 10,000

simulations. The average population proportion was calculated as PT = ∑PTi 10,000 where

PTi is the true population proportion in simulation i. Due to the way that the number of births

and deaths were randomly generated in the simulations, these population proportions can vary

among the runs.

Figure 3 is a dotchart of the relative misallocations for B1n , B12 n , and B2n . A panel

for 012 n is omitted since n = 012 n + B12 n + B2n . Note that B1n corresponds to t =1, and
20

B12 n and B2

n to t = 2 . For sample sizes of n=1, 3, and 5, the misallocation is much less for

PRN sampling than for CRN. The CRN technique tends to over-allocate the new births ( B1n

and B2n ) at both t =1 and 2 for the small sample sizes. For sample sizes of 20 and larger the

discrepancy between PRN and CRN sampling disappears since the relative misallocations

approach 0 for both techniques. Note that for n=3 and 5 in the panel for nB1, the relative

misallocations for PRN sampling are slightly negative although the theoretical expected value

in (4) is positive. However, both are well within simulation error of (4).

CRN sampling offers the possibility of tighter control over the sample allocation

than PRN because of the way the CRNs are spaced on the unit interval. To investigate this,

we calculated, for both the PRN and CRN simulations, a relative misallocation for simulation

run i as ( ) PSi − PTi PTi where the proportions are for the four types of units mentioned above

subscripted by B1, 012, B12, and B2. Figure 4 gives box plots of these quantities for samples

of sizes 20, 35, and 50 for the 10,000 simulation runs. The box plots for the combination

(PRN, n = 20 ), for example, are labeled on the horizontal axis as PRN20. Other

combinations use the same convention. The whiskers in the plot extend to the extreme values

of the data or a distance 1.5 times the interquartile range from the center, whichever is

smaller. The horizontal white line across each box is at the median and outlying points are

shown as dots. The key point to note is that the distributions of relative misallocations are

much tighter for the CRN samples than for PRN. PRN sampling produces noticeably larger

interquartile ranges and generates more extreme misallocations for all four types of units.
21

5. Conclusions

Permanent random number sampling and collocated random number sampling are

appealing methods because they are simple to execute and offer practical ways of controlling

sample overlap between different surveys and between time periods for a single survey. CRN

sampling was developed to eliminate the clumping that can occur with PRNs and to provide

more control over sample allocations. Although intuitively reasonable, the CRN method leads

to much more complicated theoretical analysis than does PRN sampling.

We have studied particular implementations of PRN and CRN sampling that yield

fixed sample sizes and rotate entire stratum samples at once. There are instances where equal

probability PRN or CRN sampling can yield samples that in expected value are far from

proportionally allocated to births and persistent units. The closeness of the PRN allocation to

proportionality, for example, depends on the size of the population. The creation of small

strata combined with the use of a fixed sample size PRN method with complete sample

rotation should be avoided if a proportional allocation is high priority in a survey. For CRN

sampling the large departures from proportionality occur at small sample sizes.

If at time 1 all units are incorrectly assumed to have a selection probability of n / N ,

and hence weighted by N / n , a biased estimator of total will generally result when using the

PRN and CRN implementations considered here. This bias can be avoided by using the

Horvitz-Thompson estimator, which differentially weights the birth and persistent units.

However, calculation of the selection probabilities for the births and persistents requires the

use of either Proposition 1, 2, 3, or 4, which, particularly in the case of Proposition 4, is

cumbersome. An alternative, easily calculable, estimator is the post-stratified estimator with
22

F01 and B the two post-strata. However, if it is possible that either 0 n01 = or = 0 B

n , then

this post-stratified estimator is not unconditionally unbiased without modification.

Finally, we note that different variants of PRN and CRN sampling have been used by

different countries. The most commonly used procedures appear to be ones that allow the

sample size to be random, perhaps because of a realization that statistical properties of these

methods are easier to derive. Each variant may require separate theory to describe its

properties. We hope that the methods presented here will be useful in analyzing other

alternatives that are in use.
23

Appendix

Proof of Proposition 3. Let n nB, 0 denote the random number of sample units in a CRN

sample of size n that are in F0,B , respectively. Note that to establish (5) it is sufficient to

show that





































=

1 0 1

0

( ) min ,

N

nN

N

N

N

nN

E n

B B

B

. (A.1)

Define

n0′ = nN0/ N1, nB = nNB/ N1

′ (A.2)

where x is the greatest integer ≤ x . Since there are no deaths, N1 = N0 + NB

. Note that

if N1nNB is an integer, then so is nN0 N1 and also n′ + nB

′ = n 0

. Otherwise,

1 n0

′ + n′B = n − . We will show that

if N1nNB is an integer then B Bn = n′ ; (A.3)

and if N1

nNB is not an integer then B Bn = n′ or = ′ +1 B Bn n , (A.4)

and













− ′













( = ′ +1) = min , 1

1 0

0

B

B

B B n

N

N

N

nN

P n n . (A.5)

Observe that (A.2) and (A.3) establish (A.1) in the integer case and that (A.2), (A.4) and

(A.5) establish (A.1) in the noninteger case. To establish (A.3), (A.4), and (A.5) let:

l′ = ′ 0 0 0n / N , B B NBl′ = n′ / , l′0

′ = 0′ + 0

(n 1) / N , l′B

′ = B′ + B

(n 1) / N (A.6)

and for any l > 0 let I( ) (a , a ]

* *

l = + l 0 0, where a0

*

 is the CRN for the last sample unit in S0.

Now if nNB N1

 is an integer then l′ = l′ B 0 by (A.6), and hence I B(l′ ) contains n′0

CRNs from F0 and n′B CRNs from B and, since n′ + n′ = n 0 B, (A.3) follows.
24

To establish (A.4), let l′ = max{l′0

, l′B} and observe the following. I(l′) contains at

least n′0 CRNs from F0 and n′B CRNs from B by the definitions of l′ l′ 0, B. Furthermore,

I(l′) contains no more than n′0 CRNs from F0, since I(l′′) 0 is the smallest interval of the

form I(l) containing n0

′ + 1 CRNs from F0

 and l′ < n / N < l′′ 1 0. Similarly, I(l′) contains

no more than nB

′ +1 CRNs from B since l′ < l′′B

 (Note that while I B(l′′ ) contains nB

′ +1

CRNs from B , it is not necessarily the smallest interval of the form I(l) to do so, which is

why it is possible for I(l′) to contain nB

′ +1 CRNs from B .) Thus I(l′) contains no more

than n′ + n′ + = n 0 B1 CRNs from F0 ∪ B , and (A.4) follows.

To obtain (A.5), we observe that since I B(l′ ) contains n′B CRNs from B and since

I(l′′) 0

 is the smallest interval of the form I(l) containing n0

′ + 1 CRNs from F0

, then

P n n B B( = ′ +1) is the probability that I I B(l′′) ~ (l′ ) 0 contains at least 1 CRN from B . (The

notation I(l a ) ~ I(l b ) means the interval l a excluding l b.) However, since the length of

I I B(l′′) ~ (l′ ) 0 is 

nN

N N

n

N

B

B

0

1 0

 1









 −

′

 and there is a distance of 1 NB

 between CRNs in B,

(A.5) follows by taking the quotient of the last two expressions. n

Proof of Proposition 4. As in the statement of Proposition 4, nBm is the number of units in

S1m ∩ B , N′ = N0 + NB,

nBm mNB N 

′ = / ′ (A.7)

and

( ) Lm Bm Bm P = P n = n′ , = ( = ′ +1) Um Bm Bm P P n n . (A.8)
25

Then it follows from (A.2), (A.3), (A.4), (A.5), (A.7) and (A.8) that













− ′













′

= min , 1

0

0

Bm

B

Um n

N

N

N

mN

P and PLm = 1 − PUm . (A.9)

Recall that n0m , n01m denote the number of elements in S1m ∩ F0

, S1m ∩ F01 , respectively,

and s1 fm denotes the final sample unit in S1m . For each m the three different ways that m can

be the smallest integer for which S1m ∩ F1

 has exactly n elements were given in (8), (9), and

(10). Note that it is not possible to have nBm nBm = ′ and s1 fm ∈ B . This is because if

s1 fm ∈ B and l is the length of an interval with left end point the CRN for the last sample

unit in S0 and right end point the CRN of s1 fm , then n m N nBm NB/ / 0 0 < l < . Consequently,

Bm

B

Bm m

B

B

Bm

Bm B

n

N N

n n

N

N

n

n N > ′

+

+

= >

0

0

.

To compute the probability of (8), PLF01m , first note that

1 01 01 0 P(s fm ∈ F nBm = n′Bm ) = N / N (A.10)

since s1 fm ∈ B from the above discussion. Next observe that,

( , ) ( 1, , 1, )

01 1 01 N01 N00 P n n n n n s F h n n m n m = − ′Bm Bm = ′Bm fm ∈ = − ′Bm − − − . (A.11)

Combining (8), (A.8), (A.10) and (A.11), we obtain that

( / ) ( 1, , 1, ) PLF01m PLm N01 N0 h n nBm − m − n N01 − N00 = − ′ . (A.12)

To obtain the probability of (9), PUBm , we observe that

( 1)

( 1and ) ( 1and )

( 1)

1 1

= − = ′ +

= ′ + ∈ = − = ′ + ∉

Um B m− Bm

Bm Bm fm Um Bm Bm fm

P P n n

P n n s B P P n n s B

(A.13)

and
26

( )





 ′ = ′

= ′ + =

− −

−

0 otherwise

 if 

( 1)

( 1) 1

( 1)

U m B m Bm

B m Bm

P n n

P n n . (A.14)

We then combine (A.13) with

( 1 1, ) ( 1, , , ) 01 1 B h n n m n N01 N00 P n n n n n s m = − ′Bm − Bm = Bm′ + fm ∈ = − Bm′ − − (A.15)

to obtain

[ ( 1)] ( 1, , , ) PUBm PUm P nB(m 1) nBm h n nBm − m − n N01 N00 = − = ′ + − ′ −. (A.16)

Similarly we obtain that

( 1)( / ) ( 2, , 1, ) ( 1) 01 0 01 00 01PUF m = P nB m− = nBm′ + N N h n − nBm′ − m − n N − N . (A.17)

We finally combine all of the above to conclude.

∑[ ( ) ( ) ]

∈

= ′ + ′ + + ′ +

m M

nBmPLF m nBm PUBm nBm PUF m

n

P

01 01

1 1

1

CRN (A.18)

where M = {m : n ≤ m ≤ n + N00 }. n

Authors’ Note

Much of this research was conducted while the second and third authors were

employed by the U.S. Bureau of Labor Statistics (BLS). Any opinions expressed are those of

the authors and do not constitute policy of the BLS.
27

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