STA304H1F/1003HF Summer 2015: Lecture 11

Transcription

1 STA304H1F/1003HF Summer 2015: Lecture 11 You should know... What is one-stage vs two-stage cluster sampling? What are primary and secondary sampling units? What are the two types of estimation in cluster sampling? When are they equivalent? Unbiased and ratio; same when cluster sizes are equal Which s 2 s represent between and within- cluster sums of squares? between-s 2 b or s2 r ; within- s 2 i What table can be used to display results from cluster sampling? analysis of variance table Lecture 11 June 18,

2 STA304H1F/1003HF Summer 2015: Lecture 11 You should know... What is one-stage vs two-stage cluster sampling? What are primary and secondary sampling units? What are the two types of estimation in cluster sampling? When are they equivalent? Unbiased and ratio; same when cluster sizes are equal Which s 2 s represent between and within- cluster sums of squares? between-s 2 b or s2 r ; within- s 2 i What table can be used to display results from cluster sampling? analysis of variance table Lecture 11 June 18,

3 Notation ( 8.3, 9.3) One-stage: m i = cluster size n =number of clusters in the sample N =number of clusters in the population M = N m i = population size M = M =average cluster size for all clusters N n m = m i =average cluster size for the sample of clusters n Two-stage: No longer have the cluster totals, y i Now M i = size of cluster i, M = N M i=population size m i = number of elements in a SRS from cluster i ȳ i = 1 m i mi j=1 y ij; estimate y i by M i ȳ i Lecture 11 June 18,

4 Estimation of population mean in two-stage cluster sampling Unbiased: ˆµ = (1/ M) n M i ȳ i. /n needs the average cluster size in the colourredwhole population M = N M i/n, depends on M Ratio estimation: ˆµ r = M iȳ i. M i Lecture 11 June 18,

5 ... UNBIASED estimation, two-stage cluster sampling V (ˆµ) = where ˆµ = 1 M M iȳ i. ( ) { ( 1 M 2 1 n ) s 2 b N n + 1 nn s 2 b = (M iȳ i M ˆµ) 2 n 1 n n M 2 i ( 1 m ) } i s 2 i M i m i and s 2 i = mi j=1 (y ij ȳ i. ) 2 m i 1 Lecture 11 June 18,

6 ... RATIO estimation, two-stage cluster sampling ˆµ r = M iȳ i. M i V (ˆµ r ) = where ( ) { ( 1 M 2 1 n ) s 2 r N n + 1 nn n M 2 i ( 1 m ) } i s 2 i M i m i and sr 2 = (M iȳ i M ˆµ r ) 2 n 1 s 2 i = mi j=1 (y ij ȳ i. ) 2 m i 1 if M i are all equal, then ratio and unbiased estimate are the same Lecture 11 June 18,

7 Example Exercise 9.2, 9.3: A nurseryman wants to estimate the average height of seedlings in a large field... Number of Heights of Plot Number of seedlings seedlings seedlings sampled (in inches) mi j=1 y ij , 11,12,10, , 9, 7, 9, 8, , 5, 7, 5, 6, , 8, 7, 7, , 11, 13, 12, , 15, 13, 12, , 7, 6, 8, , 10, 8, 9, 9, , 10, 8, 9, 9, , 11, 12, 13, 12, Lecture 11 June 18,

8 ... example N = 50 plots are the primary sampling units; n = 10 are sampled M i seedlings in each plot; m i 10% y ij height of jth sampled seedling in the ith plot N M i is unknown, so we use ratio estimation ˆµ r = M iȳ i. M i = = 9.38 Exercise 9.3: Assume N M i is known to be 2600 ˆµ = 1 M M iȳ i. n = = Lecture 11 June 18,

9 ... example N = 50 plots are the primary sampling units; n = 10 are sampled M i seedlings in each plot; m i 10% y ij height of jth sampled seedling in the ith plot N M i is unknown, so we use ratio estimation ˆµ r = M iȳ i. M i = = 9.38 Exercise 9.3: Assume N M i is known to be 2600 ˆµ = 1 M M iȳ i. n = = Lecture 11 June 18,

10 Exercise 9.4: Estimating p ( 9.5) A supermarket chain has stores in 32 cities. A company official wants to estimate the proportion of stores in the chain that do not meet a specified cleanliness criterion.... two-stage cluster sample containing 1/2 of the stores in each of four cities... Estimate the proportion of stores not meeting the cleanliness criterion and place a bound on the error of estimation Number of Number of Number of stores not City stores in city stores sampled meeting criterion ˆp i ˆp = 4 M i ˆp i 4 M i = 25(3/13)+10(1/5)+18(4/9)+16(2/8) = V(ˆp) = = See Eg. 9.4, formulas ( ); N M i/n estimated by M i/n Lecture 11 June 18,

11 Cluster sizes equal ( 9.6): M i = M, m i = m cluster elements (ssu s) m 1 y 11 y y 1m ȳ 1. cluster 2 y 12 y y 2m ȳ 2. (psu).... n y n1 y n2... y nm ȳ n. ȳ.. Unbiased: 1 M iȳ i. M n = ȳ.. = M iȳ i. M i : Ratio V(ˆµ) = ( 1 n ) MSB (1 N nm + m M ) 1 MSW N m 3 special cases (p.300): (i) N (ii) m = M (iii) n = N Lecture 11 June 18,

12 Cluster sizes equal: M i = M, m i = m V(ˆµ) = ( 1 n ) MSB (1 N nm + m M ) 1 MSW N m (1) can be used to choose n and m; see Eg. 9.5; not on exam (1) Efficiency of one-stage cluster sampling relative to SRS: ŝ 2 MSB see 8.4; on exam = {(m 1)MSW + MSB}/m MSB Lecture 11 June 18,

13 Sampling with unequal probabilities in Chapter 8, psus are chosen by SRS and population mean estimated directly (unbiased), or using information on cluster sizes (ratio) in practice, cluster sizes often used in the sample design, instead of in the estimation using sampling with unequal probabilities refer back to 3.3: let δ i be the probability of selection of population element i,i = 1,..., N, on any one draw. For example, SRSWR : δ i = 1/N Lecture 11 June 18,

14 Sampling with unequal probabilities refer back to 3.3: let δ i be the probability of selection of population element i,i = 1,..., N, on any one draw. Estimating τ, where ˆτ is unbiased ˆτ = 1 n n y i δ i want to choose δ i to make the variance small find an auxiliary variable (something correlated with y i ); for eg., in cluster sampling we use cluster size m i (or M i ) Lecture 11 June 18,

15 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

16 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

17 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

18 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

19 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

20 Probability Proportional to Size 8.9 special case of unequal probability sampling δ i is probability of selection of population element on any one draw choose this proportional to the cluster size, i.e., δ i = m i /M for population mean, ˆτ pps = (1/n) y i (m i /M) ˆµ pps = ˆτ pps M = 1 n n ȳ i. s.t. E(ˆµ pps ) = µ what about variance? assume sampling with replacement to make the calculation easier V(ˆµ pps ) = 1 n(n 1) n (ȳ i. ˆµ pps ) 2 Note, there is no finite population correction Lecture 11 June 18,

21 ... probability proportional to size 9.7 two-stage cluster sampling sample clusters (psu s) with probability proportional to size M i = M sample units (ssu s) by SRS leads to identical estimators as in 8.9; see (9.22)-(9.25) not on exam HW: Exercises 9.1, 9.7, 9.8, 9.10 Lecture 11 June 18,

22 Use of weights in sample surveys 11.5 in one-stage sampling, let w i = 1 Pr(unit i is in sample) then ˆµ = w iy i w i in two-stage sampling (either cluster or stratified), let w ij = 1 Pr(unit j in group i is in sample) then ˆµ = j=1 w ijy ij j=1 w ij Lecture 11 June 18,

23 Use of weights in sample surveys 11.5 in one-stage sampling, let w i = 1 Pr(unit i is in sample) then ˆµ = w iy i w i in two-stage sampling (either cluster or stratified), let w ij = 1 Pr(unit j in group i is in sample) then ˆµ = j=1 w ijy ij j=1 w ij Lecture 11 June 18,

24 ...use of weights w i or w ij = N each sampled unit y i represents w i population units each sampled unit y ij represents w ij population units for example: SRS ˆµ = (N/n)y i (N/n) = w i = N/n; each sampled unit represents N/n population units for example: StRS ˆµ st = 1 N L N l ȳ l = l=1 L l l=1 L l=1 j=1 (N l/n l )y ij l j=1 (N l/n l ) Lecture 11 June 18,

25 ...use of weights w i or w ij = N each sampled unit y i represents w i population units each sampled unit y ij represents w ij population units for example: SRS ˆµ = (N/n)y i (N/n) = w i = N/n; each sampled unit represents N/n population units for example: StRS ˆµ st = 1 N L N l ȳ l = l=1 L l l=1 L l=1 j=1 (N l/n l )y ij l j=1 (N l/n l ) Lecture 11 June 18,

26 ...use of weights w i or w ij = N each sampled unit y i represents w i population units each sampled unit y ij represents w ij population units for example: SRS ˆµ = (N/n)y i (N/n) = w i = N/n; each sampled unit represents N/n population units for example: StRS ˆµ st = 1 N L N l ȳ l = l=1 L l l=1 L l=1 j=1 (N l/n l )y ij l j=1 (N l/n l ) Lecture 11 June 18,

27 ...use of weights w i or w ij = N each sampled unit y i represents w i population units each sampled unit y ij represents w ij population units for example: SRS ˆµ = (N/n)y i (N/n) = w i = N/n; each sampled unit represents N/n population units for example: StRS ˆµ st = 1 N L N l ȳ l = l=1 L l l=1 L l=1 j=1 (N l/n l )y ij l j=1 (N l/n l ) Lecture 11 June 18,

28 ...use of weights w i or w ij = N each sampled unit y i represents w i population units each sampled unit y ij represents w ij population units for example: SRS ˆµ = (N/n)y i (N/n) = w i = N/n; each sampled unit represents N/n population units for example: StRS ˆµ st = 1 N L N l ȳ l = l=1 L l l=1 L l=1 j=1 (N l/n l )y ij l j=1 (N l/n l ) Lecture 11 June 18,

29 ...use of weights most complex surveys use combinations of cluster sampling (with unequal probabilities), stratified sampling, and simple random sampling survey data file includes responses y and associated weights w estimates of population totals and means computed using the weights often the weights also incorporate auxiliary information on one or more xs and finally, weights are used to adjust for non-response 11.6 much of the art in polling and Statistics Canada surveys derives from calculating appropriate weights for the sampled units, so that they represent the population units using auxiliary information about non-respondents (e.g., older/more rural/...) Eg: National Crime Victimization Survey (NCVS), Lohr 7.6 Lecture 11 June 18,

30 Topics covered in course planning of investigations, accuracy and precision (systematic and random errors) sample surveys: cross-sectional observational studies definitions: sampling frame, target population, etc. overview of types of sampling: SRS, StRS, Cluster, systematic, quota, convenience sampling biases; e.g. selection bias, non-response bias, questionnaire bias, anchoring bias, recall bias the role of randomization margin of error Lecture 11 June 18,

31 ..topics model and finite population parameters sample mean and sample variance:ȳ and s 2 distributions of ȳ and s 2 determined by (a) model (infinite population sampling) or (b) sampling method (finite population sampling) central limit theorem simple random sampling (without replacement): how to sample; inference for population mean, total, proportion stratified random sampling: design, inference, allocation of observations, dollar-unit sampling Lecture 11 June 18,

32 ..topics ratio and regression estimation: to improve precision using auxiliary variables; model-assisted estimation systematic sampling: construction, inference, advantages and disadvantages one- and two-stage cluster sampling: construction, psus, ssus, two types of estimates, variability within and between clusters equal cluster sizes: MSB and MSW relative efficiency of one-stage cluster estimation to SRS sampling with unequal probabilities; probability proportional to size (usually used for clusters at stage 1) survey weights Lecture 11 June 18,

33 Final Exam Coverage From Text: Chapters 1 and 2 Chapter Chapter Chapter Chapter , Chapter Chapter , 8.9 Chapter Slides from class Homework and assignment questions Materials covered in midterm test No R codes or output Lecture 11 June 18,

34 Final Exam June 24, 7-9pm in Galbraith 3202 (A-LA) or Sanford Fleming (LE-Z) One handwritten, 2-sided 8 1/2 X11 sheet of notes Non-programmable calculator Six questions. Do five. Choice between two proof-type questions Office hours: TA: M 2-5, T 4-6 in SS1091 SS: T 10:30-1, W 12-1:30 in SS6026 Lecture 11 June 18,