Weighting in Survey Sampling

Transcription

1 Weighting in Survey Sampling Geert Molenberghs Interuniversity Institute for Biostatistics and statistical Bioinformatics Universiteit Hasselt, Belgium Katholieke Universiteit Leuven, Belgium EUROSTAT, March 29, 2011

2 The Belgian Health Interview Survey Conducted in years: Commissioned by: Federal government Flemish Community French Community German Community Walloon Region Brussels Region EUROSTAT, March 29,

3 Design At-a-Glance Regional stratification: fixed a priori Provincial stratification: for convenience Three-stage sampling: Primary sampling units (PSU): Municipalities: proportional to size Secondary sampling units (SSU): Households Tertiary sampling units (TSU): Individuals Over-representation of German Community Over-representation of 4 (2) provinces in 2001 (2004): Limburg Hainaut Antwerpen Luxembourg Sampling done in 4 quarters: Q1, Q2, Q3, Q4 EUROSTAT, March 29,

4 Regional Stratification ( Weights) Region Goal Obt d Goal Obt d Goal Obt d Flanders = elderly +450= Wallonia = elderly +450= Brussels elderly +350= Belgium 10,000 10,221 10, =12,050 12,111 10, elderly +1250=12,600 12,945 EUROSTAT, March 29,

5 Provincial Stratification in 1997 Province sample # sample % pop. % Antwerpen Oost-Vlaanderen West-Vlaanderen Vlaams-Brabant Limburg Hainaut Liège Namur Brabant-Wallon Luxembourg Brussels 3051 EUROSTAT, March 29,

6 Multi-Stage Sampling: Primary Sampling Units Towns Within each province, order communities size Systematically sample in groups of 50 EUROSTAT, March 29,

7 Representation with certainty of larger cities. For 1997: Antwerpen: 6 groups Liège and Charlerloi: 4 groups each Gent: 3 groups Mons and Namur: 2 groups each All towns in Brussels Representation ensured of respondents, living in smaller towns EUROSTAT, March 29,

8 Multi-Stage Sampling: Secondary Sampling Units Households List of households, ordered following statistical sector age of reference person size of household clusters of 4 households selected households within clusters randomized twice as many clusters as households needed, to account for refusal and non-responders EUROSTAT, March 29,

9 Multi-stage Sampling: Tertiary Sampling Units ( Weights) Individual Respondents Households of size 4: all members Households of size 5: reference person and partner (if applicable) other households members selected on birthday rule in 1997 or by prior sampling from household members in 2001 and 2004 maximum of 4 interviews per household EUROSTAT, March 29,

10 W e i g h t s Stratification Region Province Age of reference person Household size Quarter Multi-stage sampling Selection probability of individual within household Taking this into account is relatively easy, even with standard software EUROSTAT, March 29,

11 Design Analysis Weights & selection probabilities Stratification Multi-stage sampling & clustering Incomplete data EUROSTAT, March 29,

12 Simple Random Sampling (SRS) We need the following information: Population P Population size N Sample size n Whether sampling is done with or without replacement The sample fraction: f = n N No need for weights EUROSTAT, March 29,

13 Stratification (STRAT) Population P Population size N Sample size n Whether sampling is done with or without replacement The strata indicators h = 1,...,H The number of subjects in stratum h: I = 1,...,N h, with N = H N h h=1 This defines the subpopulations, or population strata, P h EUROSTAT, March 29,

14 The way the sample of n units is allocated to the strata: n h, with n = H n h h=1 We can calculate the stratum-specific sample fraction: f h variable need for weights f h = n h N h EUROSTAT, March 29,

15 Multi-Stage Sampling: the Relative Approach Selection probabilities Stage 1: f 1 Stage 2: f 2 Total: f = f 1 f 2 a = 1 10 b = 1 10 c = 1 10 d = 1 10 EUROSTAT, March 29,

16 Multi-Stage Sampling: the Absolute Approach Assume N, n, and hence f are prespecified. Fix the number of SSU taken per PSU: n c. Construct a cumulative list of the number of SSU per PSU. Conduct systematic selection within the cumulative list, with jump g = 1 f n c = = 100 EUROSTAT, March 29,

17 Sample selection block # houses cumulative hits Selection probabilities block houses prob.(1) prob.(2) prob.(tot) /100 10/87 1/ /100 10/109 1/ /100 10/15 1/10 EUROSTAT, March 29,

18 Sample Sizes: The Belgian Health Interview Survey Allocations for Belgian Health Interview Survey Focus on Region N h population strata compromise Brussels 1,000, Flanders 6,000, Wallonia 3,000, EUROSTAT, March 29,

19 Simple Random Sampling: Estimators y = 1 n n i=1 y i ŷ = N n n i=1 y i Quantity Calculated Estimated Population variance S 2 Y = 1 N 1 N I=1 (Y I Y ) 2 ŝ 2 y = 1 n 1 n i=1 (y i y) 2 Total Average σ 2 ŷ = N2 n (1 f)s2 Y σ 2 y = 1 n (1 f)s2 Y σ 2 ŷ = N2 n (1 f)ŝ2 y σ 2 y = 1 n (1 f)ŝ2 y EUROSTAT, March 29,

20 Estimators: Stratification The total of the sub-sample within stratum h: y h = n h Estimator for the stratum-specific total: i=1 y hi ŷ h = N h n h n h y hi = N h y h i=1 n h Estimator for the population total: ŷ = H ŷ h h=1 It is the unweighted average of the stratum-specific totals. EUROSTAT, March 29,

21 Estimator for the stratum-specific average: y h = 1 n h n h Estimator for the population average: i=1 y hi = 1 n h y h y = 1 Nŷ = 1 N H h=1 ŷ h = H h=1 N h N y h The estimator for the population average is a weighted sum of the stratum-specific averages. EUROSTAT, March 29,

22 Estimators: Weighting y = n i=1 w iy i n i=1 w i ŷ = N n i=1 w iy i n i=1 w i EUROSTAT, March 29,

23 Analysis of Belgian Health Interview Survey Body Mass Index (BMI): Defined as: A continuous measure BMI = weight (kg) height 2 (m 2 ) Frequently analyzed on the log scale: ln(bmi) kg m 2 General Health Questionnaire 12 (GHQ-12): Comprises 12 questions, yielding a 13 category outcome The focus is on mental health Can be dichotomized as well EUROSTAT, March 29,

24 Vragenlijst voor Onderzoek naar de Ervaren Gezondheid (VOEG): Dutch instrument, leading to a sum score Questionnaire for Research Regarding Subjective Health Score translated into French for Belgium to obtain a more symmetric score, the analysis takes place on the log scale: ln(voeg + 1) Stable General Practioner (SGP): Do you have a steady general practitioner? (GP) Obviously a binary indicator EUROSTAT, March 29,

25 Logarithm of Body Mass Index Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) Logarithm of VOEG Score Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) EUROSTAT, March 29,

26 General Health Questionnaire 12 Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) Stable General Practitioner (0/1) Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) Weighting and clustering each increase the standard error, the combined analysis does more so. The point estimate is identical to the weighted one. EUROSTAT, March 29,

27 Design Effects Outcome Belgium Brussels Flanders Wallonia Design Effects for Clustering LNBMI LNVOEG GHQ SGP Design Effects for Weighting LNBMI LNVOEG GHQ SGP EUROSTAT, March 29,

28 Weighting: General Concepts and Design The concept of weighting Weighting in the context of stratification Weighting in the context of clustering Selection proportional to size (PPS) Self-weighting Examples EUROSTAT, March 29,

29 General Principles Weighting arises naturally in a variety of contexts: With stratification: different strata have different selection probabilities. With clustering: weights differ within and between clusters. Incomplete data: to correct for non-response. In general: units are given probabilities of selection, e.g., proportional to their size. We will consider the main ones in turn. EUROSTAT, March 29,

30 Estimators for averages and total then take the form: y = n w iy i i=1 n w, i i=1 ŷ = N n w iy i i=1 n w. i i=1 The unweighted expressions result from setting all w i equal to a constant. Due to the division by the sum of weights, the actual constant is not important, but sensible choices are 1 or 1/n. EUROSTAT, March 29,

31 Weighting and Stratification There are two main reasons why selection probabilities are different between strata: A subgroup is of interest and not oversampling would lead to too small a sample size. Example: German Region in the Belgian HIS. Strata are given equal sample sizes for comparative purposes, but also an estimate for the entire population is required. Example: Brussels, Flanders, and Wallonia in the Belgian HIS. Units are then reweighted to ensure proper representativity. EUROSTAT, March 29,

32 Example Suppose a certain subgroup represents 10% of the population. With an unweighted scheme (SRS or stratified), this group will also contribute 10% to the sample, on average. If we need a sample which includes 100 individuals of the subgroup, then a total sample of 1000 individuals has to be selected. Enlarging the subgroup with 50% implies scaling up from 100 to 150, and hence 500 additional interviews for the entire sample are needed. It is perfectly possible that 50 extra interviews in the subgroup are essential, but that the other 450 are redundant. EUROSTAT, March 29,

33 A solution is to increase the selection probability for the subgroup, relative to the others. Quantity Majority Minority Population Percentage Sample portion 1/10 1/5 Number selected Unweighted percentage in sample Weight 1 1/2 Weighted number in sample Weighted percentage in sample EUROSTAT, March 29,

34 Unfortunately, it is not always possible to pre-determine whether a respondent belongs to the majority or to the minority. This implies that determining the weight is difficult. As a surrogate, entire quarters (or other geographical entities) which are known to have large minority populations can be oversampled. This procedure works, since the weighting is done at the quarter level, hence producing correct weights, such as in the example above. If one calculates the subsample selection probability carefully, then it can be ensured that the sample will contain a sufficient number of minority members. EUROSTAT, March 29,

35 Example: Artificial Population Consider classical stratification: P s1 = ( ) P s2 = ( ) Samples are selected proportional to the stratum size: 1 out of 2 units in each: n = (1, 1). Consider a third stratification: P s3 = ( ) Retain the sample size n = (1, 1) EUROSTAT, March 29,

36 The sampling mechanisms then are: P s Stratified s Sample SRS P s1 P s2 P s3 1 {1,2} 1/6 0 1/4 1/3 2 {1,3} 1/6 1/4 1/4 1/3 3 {1,4} 1/6 1/4 0 1/3 4 {2,3} 1/6 1/ {2,4} 1/6 1/4 1/4 0 6 {3,4} 1/6 0 1/4 0 EUROSTAT, March 29,

37 The corresponding estimators are: ŷ Stratified s Sample SRS P s1 P s2 P s3 1 {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} EUROSTAT, March 29,

38 The expectations for the total: P s1 : E(y) = 1 4 [ ] = 10 P s2 : E(y) = 1 4 [ ] = 10 P s2 : E(y) = 1 3 [ ] = 10 Hence, also the third stratification produces an unbiased estimator. EUROSTAT, March 29,

39 Very important: The estimates differ depending on the sampling mechanism. Indeed, the sample {1, 2} produces 6 in the unweighted case and 7 in this weighted case. This is because the weighted expression is used. For example: ŷ = 4 1 1/ /3 1 1/ /3. The weights are the inverse of the selection probability. EUROSTAT, March 29,

40 Weighting and Multi-Stage Sampling / Clustering In multi-stage sampling and clustering, subunits may be selected with differential probabilities. Example: Household members in the Belgian HIS. In addition, entire clusters may be selected with variable probabilities. Example: Towns in the Belgian HIS. Just like in the stratified case, this needs to be taken into account via weights. EUROSTAT, March 29,

41 Example Consider a selection of households from a population with two household types: person households of married couples person households of singles. Obviously: 50% of the households consist of married couples. 66.7% of the people are married. Select a sample of 100 households, and then one person per household. We expect, on average, in the sample: 50 married persons. 50 unmarried persons. EUROSTAT, March 29,

42 If the survey question is: Are your married? then a naive estimate would produce: ẑ = 50% are married, which is wrong. Weighting the answers by the relative selection probabilities: ẑ 1 = / / / /1 = = In case we want to assess the proportion of married households, then no weighting is necessary: ẑ 2 = = = 0.5 EUROSTAT, March 29,

43 Example: Artificial Population Consider three ways of clustering: P c1 = ({1, 3}, {2, 4}) P c2 = ({1, 2}, {3, 4}) P c3 = ({1, 4}, {2, 3}) Let us add another one: P c4 = ({1}, {2, 3, 4}) EUROSTAT, March 29,

44 The sampling mechanisms for the original clusterings are: P s Clustering s Sample SRS P 1 P 2 P 3 1 {1,2} 1/6 0 1/2 0 2 {1,3} 1/6 1/ {1,4} 1/ /2 4 {2,3} 1/ /2 5 {2,4} 1/6 1/ {3,4} 1/6 0 1/2 0 EUROSTAT, March 29,

45 We cannot merely add the new samples, since they have a different, and in fact differing sample size: S c4 = { {1}, {2, 3, 4} } Let us decide to change the selection probabilities so as to comply with selection proportional to size (PPS): s Sample P s ŷ 1 {1} 1/4 4 2 {2,3,4} 3/4 12 The expectation of the total: P c4 : E(y) = = 10 EUROSTAT, March 29,

46 Example: The Belgian Health Interview Survey Design-based estimation for LNBMI, LNVOEG, GHQ12, and SGP Regression-based estimation for the continuous LNBMI Logistic regression-based estimation for the binary SGP EUROSTAT, March 29,

47 Estimation of Means Taking weighting into account, the means are recomputed for LNBMI LNVOEG GHQ12 SGP The following program can be used: proc surveymeans data=m.bmi_voeg mean stderr; title weighted means - infinite population for Belgium and regions ; where (regionch^= ); domain regionch; weight wfin; var lnbmi lnvoeg ghq12 sgp; run; EUROSTAT, March 29,

48 The program includes the weights by means of the WEIGHT statement. While it would be possible to include a finite sample correction, as we have seen, the impact is so negligible that it has been omitted. The output takes the usual form, with weighting information listed: weighted means - infinite population for Belgium and regions The SURVEYMEANS Procedure Data Summary Number of Observations 8564 Sum of Weights Statistics Std Error Variable Mean of Mean LNBMI LNVOEG GHQ SGP EUROSTAT, March 29,

49 Domain Analysis: REGIONCH Std Error REGIONCH Variable Mean of Mean Brussels LNBMI LNVOEG GHQ SGP Flanders LNBMI LNVOEG GHQ SGP Walloonia LNBMI LNVOEG GHQ SGP Note that the weights were chosen so that they recombine the entire population. The fact that the sum is not around 10 million is due to empty strata. The sum of the weights does not matter for genuine survey procedures, such as the SURVEYMEANS procedure used here. EUROSTAT, March 29,

50 It does matter for some of the model-based procedures, as we will see further in this chapter. We summarize the results and compare them to SRS (and still foreshadow a bit): Logarithm of Body Mass Index Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) Logarithm of VOEG Score Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) EUROSTAT, March 29,

51 General Health Questionnaire 12 Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) Stable General Practitioner (0/1) Analysis Belgium Brussels Flanders Wallonia SRS ( ) ( ) ( ) ( ) Stratification ( ) ( ) ( ) ( ) Clustering ( ) ( ) ( ) ( ) Weighting ( ) ( ) ( ) ( ) All combined ( ) ( ) ( ) ( ) EUROSTAT, March 29,

52 Discussion Unlike with stratification and clustering, the impact is major and differential between outcomes. Recall that an unweighted analysis implicitly assumes the following incorrect facts: the Brussels, Flemish, and Walloon populations are roughly equal members within a household have roughly the same selection probability (other components of the weights are relatively unimportant) Weighting reduces precision: this is reflected throughout in larger standard errors. They all increase, roughly, by a factor 1.5. EUROSTAT, March 29,

53 Let us discuss each of the four outcomes: LNBMI: The regional estimates are relatively stable. The Belgian estimate is stable, too. This is a coincidence, as can be seen from the following rounded computations: General: µbel = w Bru µbru + w Fla µfla + w Wal µwal Unweighted: µbel = = Weighted: µbel = = Hence, the weights shift a low between Flanders and Brussels, but these regions have the same average, as a coincidence. EUROSTAT, March 29,

54 LNVOEG: Here, the situation is rather different: General: µbel = w Bru µbru + w Fla µfla + w Wal µwal Unweighted: µbel = = Weighted: µbel = = Since the two smaller regions have a higher average, the unweighted Belgian average is higher than the weighted Belgian average. This also implies there is a larger impact on the standard error for Belgium. The standard errors for the regions increase with 35, 26, and 40%, while the standard error for Belgium increases with 48%, more than for each of the regions separately. This is because there are two sources of additional variation: (1) variability in the weights; (2) variability between the regional means. EUROSTAT, March 29,

55 GHQ-12: The phenomenon is similar to what was observed for LNVOEG. SGP: The phenomenon is not as extreme, since Brussels and Wallonia are rather different: they do not reinforce each other. But still, weighting downplays the low Brussels estimate and upgrades the high Flemish estimate, producing a higher Belgian average. EUROSTAT, March 29,

56 Regression-Based Estimation for LNMBI Like before, the procedures SURVEYREG and MIXED can be used to take weighting into account. PROC SURVEYREG code is: proc surveyreg data=m.bmi_voeg; title 15. Mean. Surveyreg, weighted, for Belgium ; weight wfin; model lnbmi = ; run; with straightforward syntax and output (for Belgium): Estimated Regression Coefficients Standard Parameter Estimate Error t Value Pr > t Intercept <.0001 PROC MIXED code is: EUROSTAT, March 29,

57 proc mixed data=m.bmi_voeg method=reml; title 25. Survey mean with PROC MIXED, for Belgium; title2 weighted ; where (regionch^= ); weight wfin; model lnbmi = / solution; run; There is no need for a RANDOM statement, since no clustering is taken into account. The relevant portion of the output for Belgium is: Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001 While the estimate is similar, the standard error is considerably smaller. EUROSTAT, March 29,

58 An overview of the results: Logarithm of Body Mass Index Analysis Procedure Belgium Brussels Flanders Wallonia SRS SURVEYMEANS (0.0018) (0.0034) (0.0030) (0.0032) SRS MIXED (0.0018) (0.0034) (0.0030) (0.0032) Stratification SURVEYMEANS (0.0018) (0.0034) (0.0030) (0.0032) Clustering SURVEYMEANS (0.0020) (0.0036) (0.0033) (0.0034) Clustering MIXED (0.0020) (0.0036) (0.0033) (0.0034) Weighting SURVEYMEANS (0.0027) (0.0046) (0.0039) (0.0042) Weighting MIXED (0.0018) (0.0034) (0.0030) (0.0032) All combined SURVEYMEANS (0.0040) (0.0048) (0.0043) (0.0044) Clust+Wgt MIXED (0.0023) (0.0039) (0.0036) (0.0038) EUROSTAT, March 29,

59 Advanced Topic: Analysis Selection Proportional to Size Self-weighting Horvitz-Thompson estimator Examples EUROSTAT, March 29,

60 Selection Proportional to Size and Self-Weighting Define an estimator of the cluster-specific total as: ŷ i = 1 f i n i j=1 y ij = 1 f i y i Define an estimator for the population total as: ŷ = m i=1 = m i=1 = m i=1 1 m 1 ŷ i π i 1 m 1 1 π i f i n i j=1 y ij 1 m 1 π i 1 f i y i EUROSTAT, March 29,

61 where f i is the sample fraction in selected cluster i π i is the probability to select cluster i y ij is the value of the survey variable for subject j in cluster i EUROSTAT, March 29,

62 Self-Weighting Self-weighting is defined by requiring to be constant. f = n π i f i Hence, the estimator for the total reduces to: ŷ = m i=1 = m i=1 1 m 1 1 π i 1 n i f y ij j=1 f i n i j=1 y ij = 1 f y EUROSTAT, March 29,

63 For the Belgian Health Interview Survey: π i t i (town size) f i 50 t i n π i f i n t i 50 t i a constant Hence: the selection of respondents within towns is self-weighting. EUROSTAT, March 29,

64 Variances for PPS Quantity Expression Pop. var. 1 S 2 1Y = M I=1 π I Y I Mπ I Y 2 = 1 M 2 M Y I π I I=1 π I Y 2 Pop. var. 2 S 2 2Y = N2 N n M I=1 N I N NI n N I 1 N I 1 N I J=1 (Y IJ Y J ) 2 PPS (with) PPS (without) σ 2 ŷ = M2 m S2 1Y + M2 m N2 n σ 2 ŷ = M2 m M I=1 π I 1 nπ I 1 π I n 1 S 2 N 2Y Y I Mπ I Y + M2 m N n n 1 S 2 N 2Y EUROSTAT, March 29,

65 The Horvitz-Thompson Estimator The Horvitz-Thompson (HT) is general and broadly applicable. It can be a bit unstable at times. Alternatives, such as the Hansen-Hurwitz estimator exist. Let y i : total for cluster i (which can simply be an individual in the non-clustered case) π i : probability of selecting cluster i v: number of distinct clusters sampled Note that v m, with equality holding when sampling without replacement. EUROSTAT, March 29,

66 The Hovitz-Thompson estimator takes the form: ŷ HT = v i=1 y i π i The variance: σ 2 ŷ HT = M 1 π I YI 2 + M I=1 π I I=1 J I π IJ π I π J π I π J Y I Y J with now in addition = M 1 π I Y I=1 π I M 1 I I=1 M J=I+1 π IJ π I π J π I π J Y I Y J π IJ : probability of simultaneously selecting clusters I and J into the sample. EUROSTAT, March 29,

67 The Artificial Population and Horvitz-Thompson We will consider three situations SRS without replacement SRS with replacement Selection with unequal probabilities In all cases, n = 2 will be maintained. EUROSTAT, March 29,

68 SRS Without Replacement The clusters in the population are: P = {1}, {2}, {3}, {4} with samples: S = {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} The probability of selecting a 1 (or any other unit) is π I = 3 6 = 1 2 EUROSTAT, March 29,

69 The estimator: ŷ HT = y 1 1/2 + y 2 1/2 = 2(y 1 + y 2 ) = 2 y The variance: σ 2 ŷ HT = 4 I=1 1 π I π I Y 2 I I=1 4 J=I+1 π IJ π I π J π I π J Y I Y J = T 1 + T 2 with EUROSTAT, March 29,

70 T 1 = 4 I=1 1 1/2 1/2 Y I 2 = 4 YI 2 I=1 = = 30 π IJ = P(selecting two units simultaneously) = = 1 6 EUROSTAT, March 29,

71 π IJ π I π J π I π J = 1/6 1/2 1/2 1/2 1/2 = 1 3 T 2 = ( ) Hence, = σ 2 ŷ HT = T 1 + T 2 = = 20 3 = EUROSTAT, March 29,

72 Using the classical expressions: σ 2 ŷ = 1 S S s=1 ŷs 1 S 2 S ŷ s s=1 = (6.0 10)2 +( ) 2 +( ) 2 +( ) 2 +( ) 2 +( ) 2 6 = = EUROSTAT, March 29,

73 SRS With Replacement The clusters in the population are: P = {1}, {2}, {3}, {4} with samples: S = {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} {1, 1} {1}, {2, 2} {2}, {3, 3} {3}, {4, 4} {4} EUROSTAT, March 29,

74 The probability of selecting a 1 (or any other unit) is π I = 1 4 P(sample with 1 element) P(sample with 2 elements) The estimator: = = 7 16 In a sample with one element: In a sample with two elements: ŷ HT = y 1 7/16 = 16 7 y 1 ŷ HT = y 1 7/16 + y 2 7/16 = 16 7 (y 1 + y 2 ) EUROSTAT, March 29,

75 Enumeration of the estimator: s Sample P s ŷ ŷ HT 1 {1,2} 2/ /7= {1,3} 2/ /7= {1,4} 2/ /7= {2,3} 2/ /7= {2,4} 2/ /7= {3,4} 2/ /7= {1,1} 1/ /7= {2,2} 1/ /7= {3,3} 1/ /7= {4,4} 1/ /7=9.14 EUROSTAT, March 29,

76 The expectation of the estimator: E(ŷHT) = = 70 7 = 10 Thus, the estimator is unbiased, but different from the classical one. The variance: σ 2 ŷ HT = 4 I=1 1 π I π I Y 2 I I=1 4 J=I+1 π IJ π I π J π I π J Y I Y J = T 1 + T 2 EUROSTAT, March 29,

77 with T 1 = 4 I=1 1 7/16 7/16 Y I 2 = 9 7 ( ) = π IJ = P(selecting two units simultaneously) = 2 16 EUROSTAT, March 29,

78 π IJ π I π J π I π J = 2/16 7/16 7/16 7/16 7/16 = Hence, T 2 = = 7 = ( ) σ 2 ŷ HT = T 1 + T 2 = = = EUROSTAT, March 29,

79 Using the conventional estimator: σ 2 ŷ = S s=1 P s ŷs S s=1 P sŷs 2 = 2 16 [( )2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 ] [( )2 + (8.0 10) 2 + ( ) 2 + ( ) 2 ] = = 10.0 Hence, the HT estimator is different and less efficient than the ordinary SRS estimator with replacement. EUROSTAT, March 29,

80 Selection With Unequal Probabilities Consider the following set of selection probabilities for the units: Unit p i 1 1/2 2 1/6 3 1/6 4 1/6 EUROSTAT, March 29,

81 Probability of selecting the various samples: Sample p s Sample p s {1,2} 1/2 1/3 = 1/6 {3,1} 1/6 3/5 = 1/10 {1,3} 1/2 1/3 = 1/6 {3,2} 1/6 1/5 = 1/30 {1,4} 1/2 1/3 = 1/6 {3,4} 1/6 1/5 = 1/30 {2,1} 1/6 3/5 = 1/10 {4,1} 1/6 3/5 = 1/10 {2,3} 1/6 1/5 = 1/30 {4,2} 1/6 1/5 = 1/30 {2,4} 1/6 1/5 = 1/30 {4,3} 1/6 1/5 = 1/30 EUROSTAT, March 29,

82 The probabilities of selecting the various units into the samples: π 1 = = 4 5 π 2 = π 3 = π 4 = = 2 5 EUROSTAT, March 29,

83 The estimator: Sample ŷ HT π IJ 1 {1,2} 4/ /5 = {1,3} 4/ /5 = {1,4} 4/ /5 = {2,3} 2/ /5 = {2,4} 2/ /5 = {3,4} 2/ /5 = = = = = = = 1 15 EUROSTAT, March 29,

84 The expectation of the estimator: E(ŷHT) = 4 15 = = The variance: σ 2 ŷ HT = 4 I=1 1 π I π I Y 2 I I=1 4 J=I+1 π IJ π I π J π I π J Y I Y J = T 1 + T 2 with EUROSTAT, March 29,

85 T 1 = 1 4/5 4/ /5 2/5 ( ) ( π1j π 1 π J T 2 = 2 π 1 π J = 2 = ) ( πij π I π J ( ) + 2 π I π J 4/15 4/5 2/5 ( ) + 2 4/5 2/5 = 2 ( ) = ) I,J 2 ( ) 1/15 2/5 2/5 ( ) 2/5 2/5 Hence, σ 2 ŷ HT = T 1 + T 2 = = = EUROSTAT, March 29,