A Generalization of the Lavallée and Hidiroglou Algorithm for Stratification in Business Surveys

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1 Survey Metodology, December Vol. 8, No., pp Statistics Canada A Generalization of te Lavallée and Hidiroglou Algoritm for Stratification in Business Surveys LOUIS-PAUL RIVEST 1 ABSTRACT Tis paper suggests stratification algoritms tat account for a discrepancy between te stratification variable and te study variable wen planning a stratified survey design. Two models are proposed for te cange between tese two variables. One is a log-linear regression model; te oter postulates tat te study variable and te stratification variable coincide for most units, and tat large discrepancies occur for some units. Ten, te Lavallée and Hidiroglou (1988 stratification algoritm is modified to incorporate tese models in te determination of te optimal sample sizes and of te optimal stratum boundaries for a stratified sampling design. An example illustrates te performance of te new stratification algoritm. A discussion of te numerical implementation of tis algoritm is also presented. KEY WORDS: Neyman allocation; Power allocation; Stratified random sampling. 1. INTRODUCTION te Lavallée and Hidiroglou algoritm may fail to meet te precision criterion. Stratification in situations were te survey variable and te stratification variable differ is considered in Dalenius and Gurney (1951, see also Cocran (1977, capter 5A. Many autors ave studied approximate formulae for determining stratum boundaries, and for evaluating te gain in precision resulting from stratification on an auxiliary variable. Some relevant contributions are Serfling (1968, Sing and Sukatme (1969, Sing (1971, Sing and Parkas (1975, Anderson, Kis and Cornell (1976, Oslo (1976, Wang and Aggarwal (1984 and Yavada and Sing (1984. Hidiroglou and Srinat (1993 and Hidiroglou (1994 suggest tecniques to update stratum boundaries using a new stratification variable. However tese papers do not explicitly provide stratification algoritms accounting for te discrepancy between te stratification variable and te survey variable. Tis paper fills tis gap by constructing generalizations of te Lavallée and Hidiroglou (1988 algoritm tat express te difference between tese two variables in terms of a statistical model. A brief review of stratified sampling and of sample allocation metods is first given. Models for te difference between stratification and survey variables are ten proposed. Te implementation of Seti s algoritm, wen te stratification and te survey variable differ, is ten presented. Numerical illustrations are provided. Te construction of stratified sampling designs as a long istory in te statistical sciences. Capters 5 and 5A in Cocran (1977 review several tecniques for splitting a population into strata. Te construction of strata is a topic of current interest in te statistical literature. Recent contributions include Hedlin (000 wo revisits Ekman (1959 rule for stratification, and Dorfman and Valiant (000 wo compare model-based stratification wit balanced sampling. Model based stratification, is discussed in Godfrey, Roswalb, and Wrigt (1984 and in capter 1 of Särndal, Swensson, and Wretman (199. In business surveys, populations ave skewed distributions; a small number of units accounts for a large sare of te total of te study variable. It is terefore appropriate to include all large units in te sample (Dalenius 195; Glasser 196. A good sampling design as one take-all stratum for big firms, were te units are all sampled, togeter wit take-some strata for businesses of medium and small sizes. Typically te sampling fraction goes down wit te size of te unit; small businesses get large sampling weigts. Te Lavallée and Hidiroglou (1988 stratification algoritm is often used to determine te stratum boundaries and te stratum sample sizes in tis context (see for instance Slanta and Krenzke 1994, Tis algoritm uses a stratification variable, known for all te units of te population. It gives te stratum boundaries and te stratum sample sizes tat minimize te total sample size required to acieve a target level of precision. It uses an iterative procedure, due to Seti (1963, to determine te optimal stratum boundaries. Te Lavallée and Hidiroglou algoritm does not account for a difference between te stratification and te survey variables. As time goes by, tis difference increases and te sampling design provided by. A REVIEW O STRATIIED RANDOM SAMPLING Some of te standard notation of stratified random sampling tat will be used in tis paper is L te number of strata; 1 Louis-Paul Rivest, Département de matématiques et de statistique, Université Laval, Ste-oy, Québec, Canada, G1K 7P4.

2 19 Rivest: A Generalization of te Lavallée and Hidiroglou Algoritm for Stratification in Business Surveys W N /N is for 1,..., L te relative weigt of 3. SOME MODELS OR THE DISCREPANCY stratum, N is te size of stratum, andn 3N is BETWEEN THE STRATIICATION AND te total population size; THE SURVEY VARIABLE n is for 1,..., L te sample size in stratum and f n /N is te sampling fraction; In tis section {x i, i 1, ÿ, N } denotes te known YG stratification variable for te N units in te population. and yg are te population and sample means of Y Many stratification algoritms, including Lavallée and witin stratum ; Hidiroglou, suppose tat {x i, i 1, ÿ, N } also represents S y is te population standard deviation of Y witin te values of te study variable. Tis section suggests stratum. statistical models to account for a difference between tese two variables. In tis paper te strata are constructed using X, a stratification variable. Stratum consists of all units wit an continuous random variables and to let f (x, x0r denote or te sequel, it is convenient to look at X and Y as X-value in te interval (b 1, b ], were te density of X. Te data {x i, i 1, ÿ, N } can be viewed 4 b 0 <b 1 <ÿ<b <b L 4 are Y G te stratum boundaries. as N independent realizations of te random variable X. Te survey estimator for can be expressed as Since stratum consists of te population units wit an yg st 3W G y ; its variance is given by: X-value in te interval (b 1, b ], te stratification process uses te values of E(Y*b $X > b 1 and L Var( G y st j W 1 1 S y n N Var(Y*b $X > b 1, te conditional mean and variance of (.1 Y given tat te unit falls in stratum, for 1, ÿ,. Tree models for te difference between X and Y are next given along wit teir conditional means and variances for Y. In business surveys, all te big firms are sampled; we coose stratum L as te take-all stratum so tat n L N L. or < L, n, tesamplesizeintake-somestratum, canbe written as (n N L a were n is te total sample size and a depends on te allocation rule. Te two allocation rules tat are considered in tis paper are Te power allocation rule a (W Y G p j (W k YG k p k1 (. 3.1 A Log-linear Model Te first model considers tat log(y α β log log(xε, were ε is a normal random variable wit mean 0 and variance σ log, wic is independent from X, and α and β log are parameters to be determined. Wen α 0, β log 1 and σ log 0, one as X Y; te survey and te stratification variables are te same. In general, Y e α X β log e ε. Te conditional moments of Y can be evaluated using te basic properties of te lognormal distribution (see Jonson and Kotz 1970, tat is were p is a positive number in (0, 1]; Te Neyman allocation rule Solving (.1 for n leads to n NW L a W S y. j W k S yk k1 j W S y /a. Var( G y st j W S y /N (.3 (.4 Te optimal stratum boundaries are te values of b 1, ÿ, b tat minimize n subject yg st Var ( G y st Y G to a requirement on te precision of suc as c were c is te target coefficient of variation (CV. Te range c 1 to 10 is often used for business surveys. E (e ε e σ log / and Var(e ε e σ log (e σ log 1. One as E(Y*b exp(ασ log / E(X β log *b wile Var(Y*b $X > b 1 is equal to Var(E(Y*X*b E(Var(Y*X*b exp(ασ log {Var(X β log *b $ X>b 1 (e σ log 1E(X β log *b } exp(ασ log {e σ log E(X β log *b E(X β log *b $ X>b 1 }.

3 Survey Metodology, December β log Y β lin X ε, σ log Te parameter values and can sometimes be calculated from istorical data. Simple ad oc values are β log 1 and σ log (1ρ Var(log(X. Here ρ is te assumed correlation between log(x and log(y. It can be set equal to predetermined values suc as 0.95 or A Linear Model In te survey sampling literature, te discrepancy between Y and X is often modeled wit a eteroscedastic linear model, (3.5 E(Y*b (1ε E(X*b $ X>b 1 εe(x, wile its conditional variance is equal to Var(Y*b (1εE(X *b εe(x {(1εE(X*b εe(x}. were te conditional distribution of ε, given X, as mean 0 and variance σ lin X γ, for some non negative parameter γ. Straigtforward calculations lead to E(Y*b β lin 4. AN EXAMPLE E(X*b wile Var(Y*b β lin Before addressing te tecnical details underlying te {Var(X*b (σ lin /β lin E(X γ *b }. construction of te algoritms, it is convenient to look at an or an arbitrary γ $ 0, te conditional variance of Y example. Consider te MU84 population of Särndal, depends on tree conditional moments of X. Te generalization Swensson and Wretman (199, presenting data on 84 of Seti s algoritm presented in section 5 does not Swedis municipalities. work in tis situation. Note owever tat wen γ, te To build a stratified design for estimating te average of conditional mean and variance of Y are proportional to RMT85, te revenues from te 1985 municipal taxation, tose for te log-linear model wit REV84, te real estate value according to 1984 assessment, β log 1andσ log log(1 (σ lin / β is used as a stratification variable. One takes L 5 and set lin ; (3.6 te target CV at 5. Two stratified designs obtained wit te proportionality factors are exp(α σ log //β lin and te Lavallée and Hidiroglou algoritm are given in Table 1, exp(ασ log /β p 0.7 lin for te conditional expectations and te for te power allocation wit and te Neyman conditional variances respectively. Tus te two models for allocation. Bot ave n 19. Wen applied on survey te discrepancy between te stratification and te survey variable RMT85, tese two designs give estimators of total variable, eiter te log-linear model of section 3.1 or te revenue wit coefficients of variation of 8.3 and 7.3 linear model (3.5 wit parameter γ, lead, in section 5, respectively. ailing to account for a cange between te to te same stratified design provided tat (3.6 olds. In te survey and te stratification variables yields estimators tat later sections, te log-linear model is used to represent te aremorevariabletanexpected. cange between X and Y. It sould give good results wen te true relationsip between Y and X is modeled by (3.5 Table 1 wit γ.. Wen model (3.5 is assumed to old wit a Stratified designs obtained wit te Lavallée and smaller value of γ, te algoritm of section 5 can still be Hidiroglou algoritm for te MU84 population using implemented wen γ is set to eiter 0 or 1. Tis is owever REV84 as stratification variable and a target CV of 5 not pursued in tis paper. Power allocation wit p A Random Replacement Model b mean variance N n f n Tis model assumes tat te stratification variable is stratum 1 1, , equal to te survey variable, i.e., X Y, for most units. stratum,35 1, , Tere is owever a small probability ε tat a unit canged stratum 3 4,603 3, , drastically; its Y value ten as f(x as density and is distributed independently of its X value. Tis is te approac stratum 4 10,606 6,44,07, stratum 5 59,878 19,631 75,50, used in Rivest (1999 to model te occurrence of stratum jumpers for wic X is not representative of Y. More Neyman allocation formally, tis can be written as, b mean variance N n f n stratum 1 1, , Y X wit probability 1ε stratum,336 1,701 99, X new wit probability ε, stratum 3 4,619 3, , were X new density f(x stratum 5 59,878 8,418 46,851, distributed independently of X. Te conditional mean for Y represents a random variable wit stratum 4 11,776 6,91 3,74, under tis model is given by

4 194 Rivest: A Generalization of te Lavallée and Hidiroglou Algoritm for Stratification in Business Surveys To model te discrepancy between REV84 and RMT85, same as te CVs obtained wit te designs of Table. Te we use te log-linear model of section 3.1. Tere are main difference between tese designs is te size of te outliers in te linear regression of log(rmt85 on take-all stratum. Te design constructed wit te Lavallée log(rev84; tey make te least squares estimates of β log and Hidiroglou algoritm as a take-all stratum of size and σ log unrepresentative of te relationsip between te N 5 13 as compared to N 5 5 and N 5 6 for te designs two variables. Robust estimates obtained wit te Splus of Table. Allowing te stratification and te survey function lmrobmm are used instead. Tey are given by variables to differ appears to reduce te relative importance ˆβ log 1.1 and ˆσ log Table gives te stratified of te take-all stratum in te sampling design. urter designs obtained wit te generalized Lavallée and investigations are needed to ascertain tis ypotesis. Hidiroglou algoritm for two allocation rules. Tey bot Te stratification algoritm for te random replacement give estimators of te total of RMT85 aving a CV of 5.7. model of section 3.3 (wit Neyman allocation was also Tis CV is still larger tat 5. Since tere are outliers in applied to REV84. Assuming canges in of te units te log-linear regression, te assumption of normal errors ( ε 0.0, te generalized Lavallée and Hidiroglou algomade in section 3.1 is not met. Tis migt explain te ritm yields a stratified design wit n 37 sample units; failure to reac te target CV exactly. Te increase in te resulting estimator of total RMT85 as a CV of 5.5. sample size for n 19 to n 8 is noteworty! or bot An interesting property of tis stratified design is tat te allocation metods te design obtained using te log-linear smallest sampling fraction is min f 9.3; it is muc model as smaller take-all strata tan Lavallée and larger tan min f for te designs of Tables 1 and. Hidiroglou. Despite te presence of outliers, te random replacement model does not describe te canges between REV84 and RMT85 as well as te log-linear model. Tis explains wy a larger sample size, 37 instead of 8, is needed to get an estimator wit a variance comparable to tat obtained wit te stratification based on a log-linear model. Table Stratified designs obtained wit te generalized Lavallée and Hidiroglou algoritm for te MU84 population using REV84 as stratification variable, a log-linear wit β log 1.1 and σ log for te discrepancy between REV84 and RMT85, and a target CV of 5 Log-linear model stratification algoritm wit power allocation wit p 0.7 b mean variance N n f n stratum 1 1,558 1,03 97, stratum 3,031,19 168, stratum 3 5,706 4,0 464, stratum 4 11,107 7,60,659, stratum 5 59,878 5,536 39,131, Log-linear model stratification algoritm wit Neyman allocation b mean variance N n f n stratum 1 1,58 1,03 97, stratum 3,040,19 168, stratum 3 5,608 4,0 464, stratum 4 11,476 7,709,95, stratum 5 59,878 8,418 46,851, An alternative to te generalized Lavallée and Hidiroglou algoritm for te construction of stratified designs is to us teir original algoritm wit a smaller target CV. Tis increases te sample size tereby reducing te variance of te estimator of te total of te survey variable. Wen constructing a design for RMT85 using REV84 as a stratification variable, te standard Lavallée and Hidiroglou algoritm wit power allocation rule ( p 0.7 and a target CV of 3.6, yields a stratified design wit n 8. Tis design as te same sample size as tose presented in Table. Te CV of te estimator of te total RMT85 is 5.7, te 5. A METHOD OR CONSTRUCTING STRATIICATION ALGORITHMS Te aim of a stratification algoritm is to determine te optimal stratum boundaries and sample sizes for sampling Y using te known values {x i ; i 1,..., N} of variable X for all te units in te population. A model, suc as tose given in section 3, caracterizes te relationsip between X and Y. Tis section extends te stratification algoritm of Lavallée and Hidiroglou (1988 to situations were X and Y differ. It uses te log-linear model of section 3.1 to account for te differences between Y and X. Modifications to andle te random replacement model are easily carried out (see Rivest A Generalization of Seti s (1963 Stratification Metod It is convenient to consider an infinite population analogue to equation (.4 for n. Since te random variable X as a density f (x, te first two conditional moments of Y given tat b 1 < X #b can be written in terms of W m b b 1 f(xdx,φ m b b 1 α β f(xdx, and ψ m b b 1 x β f(xdx, were β is te slope of te log-linear model given in section 3.1 (in tis section β and σ represent parameters of te log-linear model of section 3.1, since tere is no risk of

5 Survey Metodology, December confusion te subscript log is not used anymore. or stratification purposes, it is useful to rewrite (.4 in terms of te conditional means and variances for Y, n NW L a, X j Y G c j W Var(Y*b $ X > b 1 /a, X W Var(Y*b /N were denotes te allocation rule written in terms of te known X. or instance, under power allocation, a, X {W E(Y*b $ X>b 1 }p, j {W k E(Y*b k $X>b k1 } p k1 for 1,...,. Given a model for te relationsip between Y and X, Var(Y *b $ X > b 1 and E (Y *b $ X >b 1 can be written in terms of W, φ, ψ. Tus, te partial derivatives of n wit respect tob and can be evaluated, for MW Observe tat <, using te cain rule, MW Mφ Mψ Mφ Mψ MW 1 MW 1 Mφ 1 Mφ 1 Mψ 1 Mψ 1 MW MW 1 f(b Mφ Mφ 1 b β f(b Mψ Mψ 1 b β Mb f(b Tis leads to te following result, for f(b b β MW MW 1 Mφ Mφ 1 Mψ Similarly, Mb f(b N MW <, Mφ b β Mψ 1, (5.7 b β. Mψ b β. Te Seti s (1963 algoritm is used to solve / 0. It considers tat te partial derivatives are proportional to quadratic functions in b β. Te updated value for b β is given by te largest root of te corresponding quadratic function. Wen <, tis gives Mφ Mφ 1 Mφ b β new Mφ 1 Mφ 4 b β new Mψ Mψ Mψ wile for L 1 we ave Mφ φ 1 N Mψ 1 Mψ 1 Mψ 1 4 Mψ Mψ j i:b 1 < x i #b MW x β i. MW 1 MW N Te partial derivatives of n wit respect to W, φ, and ψ depend on moments of order 0, 1, and of x β witin stratum. Tese moments are evaluated using te N x-values in te population. or instance, Applications of tis general metod are provided next. Wen using Seti s algoritm, one typically as L $ 3. Note owever tat it also works wen L. In tis case, te algoritm is searcing for te boundary between a take-all and a take-some stratum. Successive evaluations ofb β new presented above yield an optimal boundary. Wen one assumes tat te stratification and te study variable coincide, i.e., X Y, tis boundary is nearly identical to tat obtained wit te algoritm presented in Hidiroglou ( An Algoritm for Power Allocation or te log-linear model of section 3.1, te conditional expectation is E(Y*b Cφ /W wile te conditional variance is Var(Y*b $ X > b 1 C {e σ ψ /W (φ /W }, were C exp(ασ /. Under te power allocation rule, a,x φ p /3 φp k, and formula (5.7 for n becomes n NW L j φ p j (e σ W ψ φ /φp j x β i /N c j 1/ 1/ e σ ψ φ /W /N. Te partial derivatives needed to implement te stratification algoritm are easily calculated; for #,,

6 196 Rivest: A Generalization of te Lavallée and Hidiroglou Algoritm for Stratification in Business Surveys Aeσ ψ /φ p AB(φ /W /N MW A{pe(σ W ψ φ Mφ ABφ /(nw H /φp1 /φ p1 }pφ p1 B were and A j j x β i / N c j e σ ψ W φ 1/, e σ ψ φ /W N. e σ AW /φ p AB /N e σ, Mφ were and A j n NW L φ p, B j j x β i / N c j a,x j e σ W ψ φ φ p, e σ ψ W φ e σ ψ φ / W 5.3 An algoritm for Neyman allocation e σ ψ W φ 1/ 1/ j (e σ ψ W φ 1/ j x β i /N c j /N. Under Neyman allocation, allocation rule (.3 written in terms of W, φ, and is ψ and te formula for n is (e σ ψ φ /W /N. Te partial derivatives needed to implement Seti s (1963 iterative algoritm are, ψ Aeσ /(e σ ψ W φ 1/ A (φ /W /N MW Aφ /{e σ W ψ φ }1/ A φ /(W N Mφ e σ AW /{e σ W ψ φ }1/ A e /N σ, Mψ 6. NUMERICAL CONSIDERATIONS Slanta and Krenzke (1994, 1996 encountered numerical difficulties wen using te Lavallée and Hidiroglou algoritm wit Neyman allocation: convergence was slow and sometimes te algoritm did not converge to te true minimum value for n. Indeed Scneeberger (1979 and Slanta and Krenzke (1994 sowed tat, for a particular bimodal population, te problem as a saddle; tat is te partial derivatives are all null at boundaries b wic do not give a true minimum for n. Wen using te algoritms constructed in tis paper, we also experienced te numerical difficulties alluded to in Slanta and Krenzke (1994. Te algoritms constructed under power allocation were generally more stable tan tose using Neyman allocation; numerical difficulties were more frequent wen te number L of strata was large. urtermore, as te distribution for Y moved away from tat σ of X, i.e., as increases, non convergence of te algoritm and failure to reac te global minimum for n were more frequent. In tese situations, te stratification algoritm s starting values were of paramount importance. or instance, in Table, te design accounting for canges between Y and X obtained under Neyman allocation depends eavily on te starting values. Te one presented in Table uses te boundaries presented in Table for te power allocation as starting values. Starting te algoritm wit te boundaries obtained in Table 1 for te Lavallée Hidiroglou algoritm wit Neyman allocation yields a different sampling design aving n 9. A good numerical strategy is to run te stratification algoritm for several intermediate designs to get to a final sampling design, wit te stratum boundaries obtained at one step used as starting values for te algoritm at te next step. Te log-linear algoritm is always run in two steps; first run te Lavallée and Hidiroglou algoritm, setting σ0, and use tese boundaries as starting value for te algoritm wit a non null σ. Also use as starting value for Neyman allocation te corresponding boundaries found under power allocation wit a p value around 0.7.

7 Survey Metodology, December CONCLUSION Tis paper as proposed generalizations of te Lavallée and Hidiroglou stratification algoritm tat account for a difference between te stratification and te survey variables. Two statistical models ave been introduced for tis purpose. Te new class of algoritms uses te Cain Rule to derive partial derivatives and Seti s (1963 tecnique to find te optimal stratum boundaries. Te log-linear model stratification algoritm proposed in tis paper was used successfully in several surveys designed at te Statistical Consulting Unit of Université Laval. or estimating total maple syrup production in a year, te number of sap producing maples for a producer was a convenient size variable. Historical data was used to estimate te parameters of te log-linear model linking sap producing maples and production volume. Anoter example is te estimation of te total maintenance deficit of ospital buildings in Quebec. Te value of eac building was te known stratification variable. Te maintenance deficit was estimated to be in te range (0, 40 by experts. Solving 4σ log log(40 - log(0 gives σ log log(/ as a possible parameter value for te log-linear model of section 3.1. In tese two examples accounting for canges between te stratification and te survey variables increased te sample size n by a fair percentage and yielded survey estimators wose estimated CVs were close to te target CVs. Two SAS IML functions implementing te algoritm presented in tis paper, for power and Neyman allocation, are available on te autor s website at ttp: // ulaval.ca/pages/lpr/. Tey allow user specified starting values for te stratum boundaries; tey can be used to implement te numerical strategies presented in section 6. ACKNOWLEDGMENTS I am grateful to Natalie Vandal and to Gaétan Daigle for constructing SAS IML programs for te stratification algoritms used in te paper. 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8 198 Rivest: A Generalization of te Lavallée and Hidiroglou Algoritm for Stratification in Business Surveys SLANTA, J., and KRENZKE, T. (1994. Applying te Lavallée and WANG, M.C., and AGGARWAL, V. (1984. Stratification under a Hidiroglou metod to obtain stratification boundaries for te particular Pareto distribution. Communications in Statistics, Part Census Bureau s annual Capital Expenditure Survey. Proceedings A Teory and Metods. 13, of te Section on Survey Researc Metods, American Statistical Association SLANTA, J., and KRENZKE, T. (1996. Applying te Lavallée and Hidiroglou metod to obtain stratification boundaries for te Census Bureau s annual Capital Expenditure Survey. Survey Metodology., YAVADA, S., and SINGH, R. (1984. Optimum stratification for allocation proportional to strata totals for simple random sampling sceme. Communications in Statistics, Part A Teory and Metods. 13,