C na (1) a=l. c = CO + Clm + CZ TWO-STAGE SAMPLE DESIGN WITH SMALL CLUSTERS. 1. Introduction
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1 TWO-STGE SMPLE DESIGN WITH SMLL CLUSTERS Robert G. Clark and David G. Steel School of Matheatics and pplied Statistics, University of Wollongong, NSW 5 ustralia. (robert.clark@abs.gov.au) Key Words: saple design, household surveys, telephone surveys. bstract : In two-stage surveys, the saple size of clusters and units within clusters can be chosen to iniise the variance of an estiator, for fixed cost. This paper considers saple designs where the nuber of units to be selected fro each cluster is a function of the cluster size. If there are only a sall nuber of units in each cluster, as in household surveys, then the optiisation should be over integers. n integer prograing ethod is developed which gives significantly lower variances than traditional ethods. non-integer within-cluster saple size can be ipleented by using a ixture of several integers; this can further reduce the variance.. Introduction In two-stage surveys, a saple of clusters is selected, followed by a saple of units fro each selected cluster. There are several possible reasons for this approach. There ay be a list of the clusters in the population, but not of the units: for exaple, there is rarely a list of the people in the general population but there ay be a list of households either for the whole population or within particular areas by a field listing exercise. Two stage surveys are also useful so that the saple can be ade ore geographically clustered, which often reduces the enueration cost. This article assues that clusters are selected by a siple rando saple without replaceent (SR- SWOR) of size. Each cluster g contains N, units. SRSWOR of ng units is selected fro each selected cluster. The population and saple of clusters will be denoted U and s respectively. The values of ng are assued not to depend on sl. The saple design proble is to choose and n,. The allocation of saple to the first and second stages is a well-known proble (e.g. Hansen et al., 953, ch.6,ch.7). The allocation of cluster and unit This research was jointly supported by the ustralian Research Council and the ustralian Bureau of Statistics. saple sizes is a balance between costs and variances. Cost typically consists of a per-cluster coponent (for exaple travel tie if clusters are households or geographic areas) and a per-unit coponent (for exaple interview and processing tie). If the per-cluster costs are uch higher than the per-unit costs, then it is appropriate to select a highly clustered saple; that is the nuber of units selected in each selected cluster should be high. However, a highly clustered saple has higher variance if there are positive correlations between the values of units in the sae cluster. For soe designs, the optial saple sizes n, are proportional to NgS, where Si is the adjusted population variance for cluster g (Hansen et al., 953, ch.6~h.7). In practice there is not usually infora- tion to estiate Sz for every separate cluster g, and the values of Si ay not vary uch between clusters. s a result, choosing n, proportional to Ng will often give a reasonably efficient design. This paper considers saple designs where Ng (and hence n,) are sall integers, so that the realvalued optia derived by Hansen et al. (953) and others ay not be the best designs possible. It is assued that ng are a function of N,, say ng = fia for Ng = a. The proble is to choose and fi, for a =,..., where is the largest cluster size. coon exaple of this is household sapling, where clusters are households and units are people. In practice, either all people or one randoly selected person are usually surveyed; this article will suggest soe ore efficient alternatives. The expected cost of ipleenting the survey is assued to be c = CO + Cl + CZ C na () where: na = %Fia is the expected saple size of units in clusters of size a; Ma is the nuber of clusters of size a in the population; CO is fixed costs; CI is for costs associated with the nuber of clusters in the saple (for exaple travel costs); and C, is for costs associated with the nuber of units in the saple (for exaple interview tie). Suppose that the ean, variance and intraclass 573
2 correlation of the variable of interest do not depend on the cluster size. These assuptions are ade for siplicity; (Clark, 00) gives ore general results. The variance of an inverse selection probability estiator is then proportional to where: M is the nuber of clusters in the population; Na is the nuber of units in the population in clusters of size a; and R is the finite population intracluster correlation coefficient. In Section, the optial values of and fia, which iniize the variance for fixed cost, are discussed. The standard optial allocation, which ignores the fact the and fi, are integers, is stated. n algorith which finds the best integer values of fia is derived. In Section 3, clusters of size a are randoly assigned integer saple sizes, so that the expected saple size within clusters of size a can be a non-integer. Section 4 is a nuerical evaluation of several saple designs using both fixed and rando 79.. Optial Designs with Integer fi, The cost and variance odels, () and (), are algebraically of the sae for as the cost and variance assued in standard optial allocation theory. Therefore, the values of and na which iniise V for fixed C = Cf are usually large, so rounding to the nearest whole nuber should work well. However,, is a different story, as it is a sall integer, between and a. Rounding of fia ay have a large effect. One ipact of rounding is that the cost of the rounded design ay be significantly different fro the cost constraint Cf. Even if is adjusted so that the cost constraint is et exactly, the resulting design is still not the best possible integer design. For exaple, suppose that all of the, in (4) are equal to an integer plus 0.4. Then all of the, would be rounded down, resulting in a uch lower average saple size per cluster. It is possible that rounding soe of the jia up, and soe down, will give a better solution. To find the best integer-valued,, notice that, for a given set of fia, is deterined by the cost constraint: Substituting into equation () gives = R-'+ ( - R)C (Cochran, 977, pp.96-99). The within-cluster saple sizes are therefore If the intraclass correlations are low or the travel costs (C) are high, then the optial fi, is high, so the saple is highly clustered. The nuber of clusters in saple depends on the cost constraint, Cf: if there are ore funds available then a larger saple will be used. However, the within-cluster saple sizes, fia, do not depend on Cf, so this aspect of the saple design can be chosen without knowing the total budget available for the survey. In practice, saple sizes ust be whole nubers, but allocation (3) will generally give non-integer values of, na and fia. The nuber of clusters,, is [ R + ( - R) c MM;lfi;l) / \ (7) \ The integer optial can be calculated by iniising (7) with respect to integer-valued fia. For each a, there are a possible values for,. So there are! possible cobinations of values of a where is the axiu household size. In any cases,! is sufficiently sall that (7) can be evaluated for every possible cobination. For exaple, in the nuerical study in Section 4,! = 6! = 70. Notice that (7) does not depend on the cost constraint, so that the optial integer saple size is independent of Cf, just like the optial non-integer saple size. 3. Designs with Non-Integer a, It is obviously not possible to select a non-integer saple size fro a particular cluster. It is possible to allocate a range of integer saple sizes, ng, 574
3 to clusters g of the sae size. Then, = n,/, would be the average of these n9, so that R, can be a non-integer. It is proposed that n9 be randoly generated fro an integer-valued distribution with E[ng] = 6,. design of this type ay give lower variances than the integer optial design discussed in Section, because Ria can be set to be equal to, or closer to; the non-integer optial values (4). However, the extra variation in n9 ay result in greater variation in the estiation weights, which ay increase the variance relative to the integer optial design. It is unclear what the net result of these two factors will be: the nuerical study in Section 4 copares the two approaches. If ng are rando variables, there are at least two design unbiased ethods of weighting the saple: 0 The usual two-stage estiation weights for a M a cluster g of size a are --. n9 The inverse of the probability of selection for a unit i in cluster g of size a is 7r:l = P[i E s] = {P [g E sl] P [i E slg E sl]}-l fixed integer 8, with probability, or as a ixture of two neighboring integers. To illustrate, V*, suppose that all clusters are of the sae size, a, and that inverse probability weighting is used. Then V* = V+q%-aR where the relative variance of ng is $, for each g. See the ppendix for proof. 4. Nuerical Study sipler way of calculating 6, would be to approxiate V* by substituting 8, for, in expression 7 for V: / \ Plot illustrates the behaviour of V* copared to v. The plot is based on a population consisting of an equal nuber of clusters of size and size. One unit is selected fro each selected cluster of size. The plot shows the behaviour of v and V* as the expected saple size fro clusters of size, 8, is varied. Plot : Variance for Fixed Cost with HHs of Size and The first weight uses the actual saple sizes, n9, and the second weight uses the expected saple sizes 8,. The first set of weights gives conditionally design unbiased estiators, the second set of weights gives a conditional bias, conditional on n9. Both weights give design unbiased estiators, unconditionally over ng. Clark (00) derives weights which iniise the unconditional design variance; these turn out to be an interpolation between the two weights given above. It ay see ore reasonable to use the first set of weights as they are conditionally unbiased, however this results in greater variation aongst the weights, which inflates the variance of estiators. s a result, the second set of weights often perfor better. The design variance for saple designs with rando n9 will be denoted by V*. The for of V* is coplicated and depends on whether inverse proba- M a bility weights, weighting by --, or an interpola- n9 tion, is used. Clark (00, ch.6) derives V* and the optial design-based weighting schee. It is also shown that it is optial to generate n9 as either a o.5.0 nbarl The plot was calculated assuing: the optial weighting ethod is used (as derived in Clark (00)); C/C = 0.; R = 0.4; and the population variance for units in size clusters is 0.5 ties the population variance for units in size clusters. The last assuption is not realistic and was ade to exaggerate the difference between and V", for presentation. The optial choice for fil is about 0.3; if the approxiation was used, fil would be set at about 0.. Notice that a value of 8, less than eans that clusters of size a are subsapled. 575
4 Table shows the variance of several alternative designs, for regression estiation of eployent, with auxiliary variables agegroup by sex. Clusters are households and units are people. This table was calculated using data fro the 99 ustralian Census of Population and Housing. Households contained between and 6 adults. It is assued that Cl/C is about 0.5; this is probably soewhat lower than the ratio for face to face interviewing but ay be appropriate for telephone interviewing. The intra-class correlation of eployent, adjusted for age and sex, is about 0.. ll/household sapling is usually used in practice for labour force data. If the real-valued optial Oa, given by (4) in Section, are rounded to the nearest integer, the resulting variance is 0.96 (all variances are relative to the all/household design). The best integer optial design, calculated using the ethod in Section, turned out to be half of the people in each household, rounding down; this gave a variance of The best design allowing rando ng was found by nuerically iniising V* with respect to (0, : a =,...,) in Splus, using the NLMIKB procedure (e.g. Venables & Ripley, 994). This design took approxiately half of the people in the household in expectation. Its variance was Table : Various Saple Designs, Regression Estiation of Eployent Design Var. 9a ll HH OnLlHH Rnded Real Best Integer Rando ng Y Siilar tables were calculated for different variables and different cost ratios. The best gains fro the new ethods are when Cl/C is sall, or when R is sall. 5. Conclusions and Further Work It is possible to ake useful reductions in the variance in household surveys, by explicitly allowing for integer saple sizes. For the eployent variable and a particular cost odel, the variance was reduced by about 0% by using the best integer design rather than the usual all/household design. further reduction of about 3% was ade by allowing the expected within household saple sizes, Oa, to be non-integers. In general, the new designs work best if the variable of interest is not highly correlated within households, and the costs associated with households are sall copared to interview costs. The new saple designs require interviewers to decide on the within-household saple size after identifying the size of the household. randoly generated saple size ay be required for soe households. This coplicates the interviewer's task but is probably feasible if coputer assisted personal interviewing or telephone interviewing is used. Whether the gains justify the extra coplication requires further study. The ain benefit of randoizing ng sees to be to allow soe subsapling of sall households. It is counter-intuitive that an interviewer should knock on the door, find out the household contains only one or two people, and then terinate the interview with soe probability. However, if the cost of this initial contact is sall, it is sensible to devote resources to interviewing people fro larger households, rather than have an excessive nuber of sole person households in saple. This research could be extended to ore sophisticated ethods of sapling within households, for exaple stratification. Stratification within households has not been uch used in practice, possibly because any strata would contain 0 or units. The ethods described in this paper could be used to design an effective within-household stratification schee by allocating non-integer expected withinhousehold saple sizes. ppendix: Derivation of V* for a Siple Case The estiator is T=--CCy" Ma e gesl ies, where: y" is the variable for interest for unit i; s is the saple of clusters g; and sg is the saple of units i in cluster g. Let n be the vector containing all ng for g E sl. Let Si and pg be the variance and ean, respectively, of yi over units i in cluster g. The design variance of T is r
5 = Ep [. [$$ (ng - ) gesl r r r \ ges $IS]] References +-as(i + (a - )R) -asr+ --S( u - - R) e +c$~-u~us~r = V + q5 -u~us~r. Clark, R. G. (00). Saple design and estiation for household surveys [PhD Thesis]. University of Wollongong. Cochran, W. G. (977). Sapling techniques (3 ed.). New York: Wiley. Hansen, M., Hurwitz, W., & Madow, W. (953). Saple survey ethods and theory u0.l and. New York: Wiley. Venables, W., & Ripley, B. (994). Modern applied statistics with splus. New York: Springer- Ver lag. (6) where 5 ; is the population variance of Yg over all clusters g. It is assued that the ean of Y over all units, 7, is zero, and that all clusters are of size a. In this case, the following identities hold: c sg M g e l c Y; = g e l MS(-R) c (Fg - F) x Ma-lS( + (U - )R) g sl s; = a-ls(+ (u - )R) where S is the overall population variance of J$ and R is the population intraclass correlation of Y. The approxiations are of order /M. See for exaple Hansen et al. (953). Substituting into (6) gives - M u +--84Ma-S( 8 + (a- )R) +-as( + (u- )R) u -- s( - R) - -as( e - R) as( - R) + 4-uuS( + (a - )R) 577
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