On Estimating the Number of Vital Events in Demographic Surveys

On Estimating the Number of Vital Events in Demographic SurveysAuthor(s): Des RajSource: Journal of the American Statistical Association, Vol. 72, No. 358 (Jun., 1977), pp. 377-381Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286802 .

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On Estimating the Number of Vital Events in Demographic Surveys

DES RAJ*

An examination is made of the effectiveness of the Chandrasekar- Deming technique for estimating the number of vital events using both the registration (continuous recording) of events and a periodic retrospective survey. It is shown that, under a general model for response errors, the technique may produce estimates that are considerably biased downwards. A comparison is made with a number of other estimators. The possibility of improving results through double sampling is explored.

KEY WORDS: Dual system estimation; Population growth estimation; Dual record system; Chandrasekar-Deming adjustment.

1. INTRODUCTION The increasing worldwide interest in population con-

trol has generated concern on the part of demographic statisticians that traditional methods of data collection and analysis are inadequate for providing on a timely basis the range and type of population statistics now considered essential for social and economic planning. A census can only be taken at infrequent intervals. In many countries the information obtained from the vital registration system is very defective (Seltzer 1969). In a one-time retrospective survey households are known to understate reports (Horvitz 1966; Raj 1972), and some- times there is no one surviving to declare a death. In view of these difficulties, a number of countries are experiment- ing with simultaneous registration (continuous recording) and retrospective inquiries to gather basic data on births and deaths (Abernathy 1972; Chanlett 1971; Lingner 1973; Wells 1974). The advantage of this dual system is it is possible to check on the coverage of either system. The method of Chandrasekar and Deming (1949; known as the CD adjustment) is used to estimate the number of events missed by both systems. The main purpose of this article is to examine the effectiveness of the CD adjustment under a general model for response errors.

2. THE CD TECHNIQUE

We shall consider the situation in which a random sample of t clusters of households is selected from a population comprising T clusters with each cluster containing M households. The vital events in each sample cluster are recorded as they occur. Simultaneously and independently, a team of interviewers compiles lists of

* Des Raj is Expert in Statistical Methods and Sampling, United Nations Regional Statistical Institute, Box 2048, Baghdad, Republic of Iraq. He wishes to thank Professor H.B. Wells and William Seltzer for providing a number of publications on the subject.

vital events in the sample clusters through periodic retrospective surveying. For simplicity it will be assumed that the variate involved is of the zero-one type,' i.e., the household reports no birth or one birth during the reference period (which is usually a year or less). We shall assume that the population is conceptually divided up into five classes. Class 1 contains households that have no event to report (expected proportion X). Table la gives the other four classes with the associated expected proportions obtaining under the essential conditions of the enquiry. Thus the expected proportion of households that will both register the vital event and report it to the survey interviewer is a and so on.

1. Classification of Households with Some Event

Do noat Total Category Register register

a. Expected proportions

Report a ,8 a+,f Do not report 'y 8 Y +

Total a+y /3+8

b. Observed sample numbers

Report a b Do not report c (u)

It is clear that X + a +0 + -y + 8 = 1, and the expected number of vital events occurring in the population is Y = TM (a + 3 + y + 8) = TMP, where P is the expected proportion of households in which the vital event took place. The purpose of the sample is to estimate Y. When the dual system of registration and retrospective survey is used, the observed sample numbers will be as given in Table lb. Note that the last class of Table la is not observable as it gets mixed up with Class 1 during the investigations. On the assumption that the two systems of continuous recording and retrospective inquiry are independent of each other, u is estimated as

1 This assumption may not be entirely justified in societies where the extended family system is prevalent. There may be more than one birth or death in a household during the reporting period. In such a case the symbol M should apply to the number of persons, and the assumption holds for each person, multiple births being excluded.

? Journal of the American Statistical Association June 1977, Volume 72, Number 358

Theory and Methods Section

377



378 Journal of the American Statistical Association, June 1977

bc/a, which gives an estimate of the total number of vital events occurring in the sample clusters as a + b + c + bc/a = (a + b) (a + c)/a. It follows that the CD estimator of the total number of vital events occurring in the entire population is

T (a + b)(a + c) N (a + b)(a + c) Pi = _ __ __ - (2.1)

t a n a

where N = TM is the total number of households in the population, n = tM is the number of households in the sample, and M is the number of households in a cluster.

It should be pointed out that the sample numbers a, b, and c can only be obtained by a case-by-case matching of the lists of vital events provided by the two systems and there may be errors of matching. Also, there may be events recorded or reported that occurred outside the sample area or the reference period employed. Such errors may cause considerable bias (Marks, Seltzer, and Krotki 1974; Seltzer and Adlakha 1974). However, these errors are disregarded here in order to focus attention on the role played by the strength of association between the two systems. There is a general belief that it is very difficult to keep the two systems independent over a long period and that the events omitted in one system are also likely to be omitted in the other.

3. BIAS AND VARIANCE OF THE CD ESTIMATOR

Suppose the population contains T clusters (each containing M households) from which a random sample of t clusters is selected with replacement and with equal probabilities for both continuous recording and periodic retrospective survey. The recording or reporting of vital events is done according to the probabilities given in Table la. Let a = E ai, b = E bi, c = E ci, where ai, bi, ci are the numbers observed in the sample from the ith cluster, i = 1, -2, ..., t. We shall define ei, as 1, if the vital event occurred in the jth household of the ith cluster, and 0 if otherwise. With each eij are associated the random variables Hij, Hij,'Hij", Hij"'. The random variable Hij takes the value of 1 with probability a/P when the vital event (that has actually occurred) has been both registered and reported in the survey; Hij = 0 otherwise. The same definition applies to the other situa- tional variables (i.e., reported but not registered, registered but not reported, and neither registered nor reported) with the associated probabilities 3/P, y/P, and i/P, respectively. It is clear that

a= i eiHij, bi = eiH j i

and so on. We shall define p, the intracluster correlation of vital events, as

T M

p V = S E E Ev (ejP(t P)/ t=1 j-i

*[TM(M -l)P(1 -P)]-

Using the symbols E, V", and C for the expected value,

variance, and covariance, respectively, we have

E(ai) = (a/PT) , ei = Ma

E(ai2) = E e ei12Hjj2 + E eijHij E eij,Hiji j i H

= (a/PT) E E e,j2 + (a2/TP2) eijei, j i i i Hi

= Ma + (a2/P2)M(M -1)P[P + p(l - P)]

V(ai) = Ma + Ma 2(K -1),

where K = (M - l)p( -P)/P. Similarly, it is easy to see that

C(ai, bi) = Mao (K - 1) Hence

E(a) = na

V (a) = na + na2 (K -1)

C (a, b) = naf3(K - 1) Similarly,

E(a + b) = n(a? +)

V(a + b) = n(a + 3) + n(a + 13)2(K - 1)

In order to find the bias and the variance of the CD

estimator FI, we shall use the following result. Let there be random variables x, y, and z, with finite first and second order moments. Then, large sample approximations to the mean and variance of xy/z are

E xy\ E(x)E(y) + C(x, Y) < z, E(z) L E(x)E(y)

_C(x, z) _C(y, z) V(z) 1 (31) _ _ ~ ~+ 1 31 E(x)E(z) E(y)E(z) E2(z) 1

(xy E2(x)E2(y) - V(X) V(y) V(z) V + +

z Z, E2 (Z) _E2 (x) E2 (y) E2 (Z)

2C(x, y) 2C(x, z) 2C(y, z) I + __.(3. 2)

E(x)E(y) E(x)E(z) E(y)E(z)J

Setting x = a + b, y = a + c, and z = a in formulas (3.1) and (3.2), we have

E( 1)--(a + #)(a + y+ 2 (3.3) a n a

V(1 2 N(a + )(a + y)2

na2

rl a + K- -- : +--7 ] . (3.4) La (a + #)(a + -y)

Hence the bias of the estimator for estimating the total number of vital events is

B ( 1) - (N/a) (a3 - /y) + (N/n)O/y/a2 . (3.5)

Defining 0, the measure of association in Table 1, as

I= (a - I3y)/[(at + 3) ( + 8) (a + I) \ + \)]i,

it is easily seen that the CD estimlator iS subject to a constant bias that depends on the strength of association 4f



Estimation in Demographic Surveys 379

of the registration system and the retrospective survey. If 4 is zero, the bias of 2l, is negligibly small in large samples. In practice 4 is expected to be positive for the reason that the proportion of registered events among those reported is supposed to be higher than the corresponding proportion among those not reported. In this case the estimator could be seriously biased downwards when q is not low. These findings are in line with the elegant results of Jabine and Bershad (1 970)who considered the situation, 3 = 'y, in which the coverage rates of both procedures are the same. Experience from studies in a number of countries shows that the two procedures can produce coverage rates that differ considerably (Lingner and Wells 1973; Myers 1976).

4. BOUNDS ON THE BIAS OF THE CD ESTIMATOR Formula (3.5) provides a large sample approximation

to the bias of the CD estimator. In view of the important role the estimator bias plays in investigations of this type, we shall find bounds on the bias of the estimator (relative to Y) for any sample size. Writing YPi as

p = (N/n) (a + b) (a + c)/a = h/a we have

C(h/a, a) = E(h) - E(P?)E(a)

which gives

E(111) - E(h)/(na) = -(na)-1*-( Y1)oa(a)po(h/a, a) or

I E (,1) -E (h) / (na) I < (c (a) /na) c(fi)

where po and a- refer to simple correlation coefficient and standard deviation, respectively. Noting that

E(h) N - (a + #)(a?+ y) __ - Y - 1 +(K -1)

na. n La

-n za

the bounds on the relative bias are found to be

A-D < (E( 1)-Y)/Y < A + D, (4.1) where

A ~ ab - fly a + (K -1)(a + 3)(a+,y) A =- a-- r + _ _ _ _ _ _ _ _

a(a+3+y+ 6) na(a+/3+y+ y ) D = [(1/n) (1/a + K -l)]*(l Y .

Formula (4.1) is useful inasmuch as it enables us to speculate on the bounds of the relative bias for various values of the parameters involved in the model. Table 2 gives some numerical values in this regard when the intracluster correlation coefficient is zero.

5. OTHER RELEVANT ESTIMATORS In forming the CD estimator we begin by calculating

the ratio of the number of vital events reported in the sample (a + b) and the events registered among these (a). This ratio is multiplied by the number of events registered

2. Relative Bias of the CD Estimator

Relative Bias (percent)

Case Lower Upper a /3 y 8 n Eq. limit limit

(3.5) (A-D) (A +D)

(i) .049 .024 .011 .016 5,000 -10.6 -10.7 -10.2 (ii) .080 .005 .005 .010 5,000 - 9.7 - 9.7 - 9.3 (iii) .050 .035 .005 .010 5,000 - 6.5 - 6.6 - 6.1

by all the households in the sample (a + c). If nation- wide civil registration exists, one may like to use as the multiplier, X, the total number of vital events registered in the country. In this case the estimator becomes

2 = X(a + b)/a . (5.1)

Using large sample approximations to the mean and the variance of the ratio estimator (Raj 1968, p. 88), we have

E(fY2) (N/a)(a + 13)(a + y) + (N/n)3 (a + y) /a2 , (5.2)

V ( P2) (N2/n) (A3/a) (a + y)2 2(a + 3) /a2 (5.3)

which gives

B(P2) -(N/a) (a6 - 3y) + (N/n)3 (a + y)/a2 (5.4)

The bias of 12 is about the same as that of Pi, but the variance is smaller if

K > 1 - (ac-13/E(a +0) (a + -y)],

which will usually hold. Another estimator of interest is

13 = (N/n) (a + b + c) , (5.5)

based on the total number of events obtained through the retrospective survey and the registration system without using the CD adjustment. This estimator is being used in the Indian Sample Registration System (Lingner and Wells 1973). Its bias and variance are

B (V3) = -NS, (5.6)

V( 3) = (N2/n)[(a + A + y) + (K-1) (a +3 + y)2]. (5.7)

The absolute value of the bias of P3 iS greater than the corresponding constant component of the bias of f2 or Pi. Finally, if the registration system is not used at all, the single-system estimator based on events reported in the periodic survey is

4- (N/n)(a + b) ' (5.8)

with its bias and variance given by

B (4) = -N(- + X) , (5.9)

V( 4) = (N2/n) [a + 3 + (K -1) (a + /3)2] . (5.10)

The absolute value of the bias of IX is larger than that of IX. Actually, the mean square error of IX will be found to exceed that of f3 in large samples.



380 Journal of the American Statistical Association, June 1977

6. NUMERICAL EXAMPLES

With the model from Section 2 it is possible to speculate on the results likely to be obtained under different situations. We shall assume that 25 clusters each containing 200 households are selected with replacement and with equal probabilities from a universe comprising a large number of clusters of the same size. The purpose is to estimate the total number of deaths that occurred last year. Assume that ten percent of the households have a death to report. The method used is the dual system of registration and retrospective survey in the sample clusters. The following hypothetical situations will be considered.

Case (i): a = 0.049, 3 = 0.024, y = 0.011, a = 0.016, 0 = 0.24. In this case the coverage of the retrospective survey is 73 percent and that of the registration system 60 percent. Only 49 percent of events are common to both systems, and the measure of association (of the two systems) is moderate. The results in Table 3a show that the CD technique brings down the bias to 10.6 percent, although the bias remains the major component of the mean square error (MSE). Confidence intervals for Y based on the sampling variance alone may be misleading in this case. This model could possibly apply to the Turkish Demographic Survey (Chanlett 1971; Seltzer 1969) if the finding of Brass (1971) is true that the observed number of events should be increased by 20 percent to take into account events missed by both registration and the retrospective survey.

3. Numerical Results

p=O p=0.005 Relative

parameter V1 V3 k4 31 V, k4

a. Case (i)

Bias -10.6% -16% -27% -10.6% -16% -27% (Bias)2 .0112 .0256 .0729 .0112 .0256 .0729 Variance .0018 .0015 .0014 .0033 .0028 .0023 MSE .0130 .0271 .0743 .0145 .0284 .0752

b. Case (ii)

Bias -9.7% -10% -15% -9.7% -10% -15% (Bias)2 .0094 .0100 .0225 .0094 .0100 .0225 Variance .0016 .0016 .0016 .0031 .0031. .0028 MSE .0110 .0116 .0241 .0125 .0131 .0253

c. Case (iii)

Bias -6.5% -10% -15% -6.5% -10% -15% (Bias)2 .0042 .0100 .0225 .0042 .0100 .0225 Variance .0018 .0016 .0016 .0034 .0031 .0028 MSE .0060 .0:116 .0241 .0076 .0131 .0253

Case (ii): a = 0.080, i = 0.005, y = 0.005, 8 = 0.010, q = 0.61. In this situation both the registration system and the retrospective survey have a high coverage rate of 85 percent, but the strength of association between the two systems is high. The results in Table 3b show that the CD adjustment is not very effective.

Case (iii): a = 0.050, A = 0.035, y = 0.005, 6 = 0.010, = 0.18. This is a situation in which the coverage of the

retrospective survey is the same as in (ii), but the registration system is very defective. The strength of association is low. Because of the low value of X, the CD technique is effective (see Table 3c). However, the bias component continues to dominate the mean square error.

7. THE USE OF DOUBLE SAMPLING

It is clear that the purpose of the retrospective survey is to provide a correction factor for the recording system and vice versa. Thus it appears that it is not essential to carry out the retrospective survey in all areas covered by the recording system. A better method of using available resources may be to limit the survey to a subsample of clusters while the continuous recording is done on the entire sample. When the interviewer sample is small, there is greater opportunity to insure that there is no col- laboration between the interviewers and the recording personnel. Also, interviewers can be given adequate training in handling certain segments of the population that are more likely to fail to register births or deaths. These include the very poor households or those in very remote rural areas. The improvement of survey reports in this part of the population should lead to lower de- pendence between the two systems.

Suppose that a random sample of t' clusters is selected with replacement and with equal probabilities for continuous recording. The retrospective survey is done on a subsample of to clusters. Let n' = t'M, no = toM, where M is the number of households in a cluster. Further, let ao, bo, co be the observed number of events based on the matching of records of the two systems, while a' + c' is the number of events obtained by continuous recording in the t' clusters. An estimator (of Y) to employ in this situation is

6 = (N/n')(a' + c')(ao + bo)/ao . (7.1)

Using (3.1) and (3.2), the large sample bias and the variance of the estimator are

B (25)- (N/a) (ad - y) + (N/a) (a + 3)j *[(1/noa) - 1/n' (a + y)] , (7.2)

V(1%)- (N2/a2)(a + 0)2(a + y)2

[(1/no) (1/a - 1/ (a + A))

+ (1/n') (2a/ (a + /) (a + y) -1 /(CY + y) + K-1)]. (7.3)

Assuming the cost function for the dual system as

C = c'n' + cono , (7.4)

it is easy to see that the best values of n' and no can be obtained from the relation

, [1/ -1/( + A3)]* flu/fl = (c'/cO)* [(a - /)/(a + /3) ( + ay) + K- 1]I

(7.5)



Estimation in Demographic Surveys 381

This shows that the retrospective survey subsample (no) should be small if the unit cost of the survey is large or the variance contribution from the retrospective survey is small.

Another estimator of interest is the difference estimator. Denote by a + c the number of events recorded in those clusters in which the retrospective survey is not done. The difference estimator of Y that is theoretically more convenient, is

1% = N(a + c)/(n' - no) + (N/no) *[(ao + bo)(ao + co)/ao - (ao + co)]

= N(a + c)/(n' - no) + (N/no) (ao + co)bo/ao . (7.6)

The first term in (7.6) estimates the number of events based on recording in the main sample. The second term provides an adjustment obtained from the retrospective survey. The large sample bias and the variance of the estimator are

B (Y6) - (N/a) (a8 - Oy) + (N/no)fry/a'2, (7.7)

and

V(f'6) (N2/no)[(a + y)232/a2][(l - A

+ y/[a(a + y)] + K] + [N2/(n' - no)] -Ea+ 7+ (a + y)2 (K- .(7.8)

To take a numerical example, consider the following hypothetical situation in which a random sample of t'= 50 clusters (each containing 250 households) is selected with replacement and with equal probabilities for continuous recording, and the retrospective survey is done on a subsample of to = 35 clusters. This gives n'= 12,500 and no = 8,750 households for the two phases of the inquiry. We shall assume the following parameters in the model: a = 0.13, d = 0.05, y = 0.01, and 8 = 0.01, which leads to c = 0.15 as the measure of association. The relative bias and the mean square error of the estimator 15 are given in Table 4 for p = 0 and p = 0.005. Since the difference estimator does not make use of the entire information in the main sample,

4. Comparison of Double Sampling and Single Sampling

p =O p =0.005 Relative

parameter V5 V6 V, V5 V6 V,

Bias -3.1% -3.1% -6.7% -3.1% -3.1% -6.7% (Bias)2 .0009 .0009 .0044 .0009 .0009 .0044 Variance .0004 .0010 .0004 .0008 .0017 .0008 MSE .0013 .0019 .0048 .0017 .0026 .0052

the performance of P6 is not as good as that of 15. With c' = 1, co = 2, the total cost of the inquiry is 30,000 in some units. If double sampling is not used, both con- tinuousrer recrdinga the retrospective survey can be done on 40 clusters for the same cost. However, the coverage of reporting is then expected to decrease and the measure of association to increase. We shall assume the

values of the parameters as ae = 0.12, 3 = 0.04, y = 0.02, and 6 = 0.02, which gives 4 = 0.22. As is clear from Table 4, the mean square error of 1 has more than trebled. This shows that double sampling can be effective provided the smaller size of the sample for the retrospective survey can help in reducing the strength of association 4. It must be pointed out, however, that there is the danger of bias (due to conditioning) creeping into the investigations over time because the recording may improve in areas where the retrospective survey is taking place but not in other areas. In such a situation the subsample for the retrospective survey should be suitably rotated over time.

[Received May 1974. Revised October 1976.]

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Agrawal, Bhanwar L. (1969), "Sample Registration in India," Population Studies, 23, 379-94.

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Chandrasekar, C., and Deming, William E. (1949), "On a Method of Estimating Birth and Death Rates and the Extent of Registra- tion," Journal of the American Statistical Association, 44, 101-15.

Chanlett, Eliska (1971), "Organization and Methods of the Dual- Report System in Turkey," Scientific Report Series No. 2, Chapel Hill, N.C., International Program of Laboratories for Population Statistics.

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Jabine, Thomas B., and Bershad, Max A. (1970), "Some Comments on the Chandrasekar-Deming Technique for the Measurement of Population Change," CENTO Symposium on Demographic Sta- tistics, Turkey.

Linder, Forrest E. (1971), "Vital Event Enumeration System as a New Tool for Measuring Population Change," Bulletin of Interna- tional Statistical Institute, 44, 377-93.

Lingner, Joan W., and Wells, H. Bradley (1973), "Organization and Methods of the Dual Record System in India," Scientific Report Series No. 9, Chapel Hill, N.C., International Program of Labora- tories for Population Statistics.

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Myers, Robert J. (1976), "The Dual Record System: an Overview of Experience in Five Countries," Scientific Report Series No. 26, Chapel Hill, N.C., International Program of Laboratories for Population Statistics.

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Seltzer, William (1969), "Some Results from Asian Population Growth Studies," Population Studies, 23, 395-406. -, and Adlakha, Arjun (1974), "On the Effect of Errors in the Application of the Chandrasekar-Deming Technique," Reprint Series No. 14, Chapel Hill, N.C., International Program of Labora- tories for Population Statistics.

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On Estimating the Number of Vital Events in Demographic Surveys