STA304H1F/1003HF Summer 2015: Lecture 11 · STA304H1F/1003HF Summer 2015: Lecture 11 You should...

STA304H1F/1003HF Summer 2015: Lecture 11

You should know...

I What is one-stage vs two-stage cluster sampling?

I What are primary and secondary sampling units?

I What are the two types of estimation in cluster sampling?When are they equivalent?

I Unbiased and ratio; same when cluster sizes are equal

I Which s2’s represent between and within- cluster sums ofsquares?

I between-s2b or s2r ; within- s2iI What table can be used to display results from cluster

sampling?

I analysis of variance table

Lecture 11 June 18, 2015 1

STA304H1F/1003HF Summer 2015: Lecture 11

You should know...

I What is one-stage vs two-stage cluster sampling?

I What are primary and secondary sampling units?

I What are the two types of estimation in cluster sampling?When are they equivalent?

I Unbiased and ratio; same when cluster sizes are equal

I Which s2’s represent between and within- cluster sums ofsquares?

I between-s2b or s2r ; within- s2iI What table can be used to display results from cluster

sampling?

I analysis of variance table


Notation (§8.3, 9.3)One-stage:

I mi = cluster size

I n =number of clusters in the sample

I N =number of clusters in the population

I M =∑N

i=1mi = population size

I M =M

N=average cluster size for all clusters

m =

∑ni=1mi

n=average cluster size for the sample of clusters

Two-stage:

I No longer have the cluster totals, yiI Now Mi= size of cluster i , M =

∑Ni=1Mi=population size

I mi = number of elements in a SRS from cluster i

I yi · =1

mi

∑mij=1 yij ; estimate yi by Mi yi ·


Estimation of population mean in two-stage clustersampling §9.3-4

I Unbiased:

µ = (1/M)n∑

i=1

Mi yi ./n

needs the average cluster size in the colourredwholepopulation M =

∑Ni=1Mi/N, depends on M

I Ratio estimation:

µr =

∑ni=1Mi yi .∑ni=1Mi


... UNBIASED estimation, two-stage clustersampling

I

µ =1

M

∑ni=1Mi yi .

nI

V (µ) =

(1

M2

){(1− n

N

) s2bn

+1

nN

n∑i=1

M2i

(1− mi

Mi

)s2imi

}

where

s2b =

∑ni=1(Mi yi − Mµ)2

n − 1

and

s2i =

∑mij=1(yij − yi .)

2

mi − 1


... RATIO estimation, two-stage cluster sampling

I

µr =


I

V (µr ) =

(1

M2

){(1− n

N

) s2rn

+1

nN

n∑i=1

M2i

(1− mi

Mi

)s2imi

}

where

s2r =

∑ni=1(Mi yi − Mµr )2

n − 1

and

s2i =

∑mij=1(yij − yi .)

2

mi − 1

I if Mi are all equal, then ratio and unbiased estimate are thesame


Example

Exercise 9.2, 9.3: “A nurseryman wants to estimate the averageheight of seedlings in a large field...”

Number of Heights ofNumber of seedlings seedlings

Plot seedlings sampled (in inches)∑mi

j=1 yij1 52 5 12, 11 ,12 ,10 ,13 582 56 6 10, 9, 7, 9, 8, 10 533 60 6 6, 5, 7, 5, 6, 4 334 46 5 7, 8, 7, 7, 6 355 49 5 10, 11, 13, 12, 12 586 51 5 14, 15, 13, 12, 13 677 50 5 6, 7, 6, 8, 7 348 61 6 9, 10, 8, 9, 9, 10 559 60 6 7, 10, 8, 9, 9, 10 53

10 45 6 12, 11, 12, 13, 12, 12 72


... example

I N = 50 plots are the primary sampling units; n = 10 aresampled

I Mi seedlings in each plot; mi ≈ 10%

I yij height of jth sampled seedling in the ith plot

I∑N

i=1Mi is unknown, so we use ratio estimation

I

µr =


=4970.833

530= 9.38

I Exercise 9.3: Assume∑N

i=1Mi is known to be 2600

I

µ =1

M

∑ni=1Mi yi .

n=

1

52

4970.833

10= 9.56


Exercise 9.4: Estimating p (§9.5)

“A supermarket chain has stores in 32 cities. A company officialwants to estimate the proportion of stores in the chain that do notmeet a specified cleanliness criterion. ... two-stage cluster samplecontaining 1/2 of the stores in each of four cities... Estimate theproportion of stores not meeting the cleanliness criterion and placea bound on the error of estimation”

Number of Number of Number of stores notCity stores in city stores sampled meeting criterion pi

1 25 13 3 0.232 10 5 1 0.203 18 9 4 0.444 16 8 2 0.25

p =∑4

i=1 Mi pi∑4i=1 Mi

= 25(3/13)+10(1/5)+18(4/9)+16(2/8)25+10+18+16 = 0.2865

V(ˆp) = 0.002723 + 0.003114 = 0.05582

See Eg. 9.4, formulas (9.11-9.13);∑N

i=1 Mi/N estimated by∑n

i=1 Mi/n


Cluster sizes equal (§9.6): Mi = M, mi = m

cluster elements (ssu’s)1 2 . . . m

1 y11 y12 . . . y1m y1.cluster 2 y12 y22 . . . y2m y2.

(psu)...

......

...n yn1 yn2 . . . ynm yn.

y..

I Unbiased:1

M

∑ni=1Mi yi .

n= y.. =


: Ratio

V(µ) =(

1− n

N

) MSB

nm+(

1− m

M

) 1

N

MSW

m

I 3 special cases (p.300):(i) N →∞ (ii) m = M (iii) n = N


Cluster sizes equal: Mi = M, mi = m

I

V(µ) =(

1− n

N

) MSB

nm+(

1− m

M

) 1

N

MSW

m(1)

(1) can be used to choose n and m; see Eg. 9.5; not on exam

I Efficiency of one-stage cluster sampling relative to SRS:

s2

MSB={(m − 1)MSW + MSB}/m

MSB

see §8.4; on exam


Sampling with unequal probabilities

I in Chapter 8, psus are chosen by SRS

I and population mean estimated directly (unbiased), or usinginformation on cluster sizes (ratio)

I in practice, cluster sizes often used in the sample design,instead of in the estimation

I using sampling with unequal probabilities

I refer back to §3.3: let δi be the probability of selection ofpopulation element i ,i = 1, . . . ,N, on any one draw. Forexample,

SRSWR : δi = 1/N


Sampling with unequal probabilities

I refer back to §3.3: let δi be the probability of selection ofpopulation element i ,i = 1, . . . ,N, on any one draw.Estimating τ ,

τ =1

n

n∑i=1

yiδi

where τ is unbiased

I want to choose δi to make the variance small

I find an auxiliary variable (something correlated with yi ); foreg., in cluster sampling we use cluster size mi (or Mi )


Probability Proportional to Size §8.9I special case of unequal probability sampling

I δi is probability of selection of population element on any onedraw

I choose this proportional to the cluster size, i.e., δi = mi/M

I for population mean, τpps = (1/n)∑n

i=1yi

(mi/M)

µpps =τppsM

=1

n

n∑i=1

yi . s.t. E (µpps) = µ

I what about variance? assume sampling with replacement tomake the calculation easier

V(µpps) =1

n(n − 1)

n∑i=1

(yi . − µpps)2

I Note, there is no finite population correctionLecture 11 June 18, 2015 15





i=1yi

(mi/M)

µpps =τppsM

=1

n

n∑i=1



V(µpps) =1

n(n − 1)

n∑i=1

(yi . − µpps)2


... probability proportional to size §9.7

I two-stage cluster sampling

I sample clusters (psu’s) with probability proportional to sizeMi = M

I sample units (ssu’s) by SRS

I leads to identical estimators as in §8.9; see (9.22)-(9.25)

I not on exam

I HW: Exercises 9.1, 9.7, 9.8, 9.10


Use of weights in sample surveys §11.5

I in one-stage sampling, let

wi =1

Pr(unit i is in sample)

then

µ =

∑ni=1 wiyi∑ni=1 wi

I in two-stage sampling (either cluster or stratified), let

wij =1

Pr(unit j in group i is in sample)

then

µ =

∑ni=1

∑nj=1 wijyij∑n

i=1

∑nj=1 wij


...use of weights

I∑

wi or∑

wij = N

I each sampled unit yi represents wi population units

I each sampled unit yij represents wij population units

I for example: SRS

µ =

∑ni=1(N/n)yi∑ni=1(N/n)

=⇒ wi = N/n; each sampled unit represents N/npopulation units

I for example: StRS

µst =1

N

L∑`=1

N`y` =

∑L`=1

∑n`j=1(N`/n`)yij∑L

`=1

∑n`j=1(N`/n`)


...use of weights

I∑

wi or∑

wij = N



I for example: SRS

µ =



I for example: StRS

µst =1

N

L∑`=1

N`y` =

∑L`=1


`=1

∑n`j=1(N`/n`)


...use of weights

I most complex surveys use combinations of cluster sampling(with unequal probabilities), stratified sampling, and simplerandom sampling

I survey data file includes responses y and associated weights w

I estimates of population totals and means computed using theweights

I often the weights also incorporate auxiliary information on oneor more xs

I and finally, weights are used to adjust for non-response §11.6

I much of the art in polling and Statistics Canada surveysderives from calculating appropriate weights for the sampledunits, so that they represent the population units

I using auxiliary information about non-respondents (e.g.,older/more rural/...)

I Eg: National Crime Victimization Survey (NCVS), Lohr §7.6Lecture 11 June 18, 2015 29

Topics covered in course

I planning of investigations, accuracy and precision(systematic and random errors)

I sample surveys: cross-sectional observational studies

I definitions: sampling frame, target population, etc.

I overview of types of sampling: SRS, StRS, Cluster,systematic, quota, convenience sampling

I biases; e.g. selection bias, non-response bias, questionnairebias, anchoring bias, recall bias

I the role of randomization

I margin of error


..topics

I model and finite population parameters

I sample mean and sample variance:y and s2

I distributions of y and s2 determined by (a) model (infinitepopulation sampling) or (b) sampling method (finitepopulation sampling)

I central limit theorem

I simple random sampling (without replacement): how tosample; inference for population mean, total, proportion

I stratified random sampling: design, inference, allocation ofobservations, dollar-unit sampling


..topics

I ratio and regression estimation: to improve precision usingauxiliary variables; model-assisted estimation

I systematic sampling: construction, inference, advantages anddisadvantages

I one- and two-stage cluster sampling: construction, psus, ssus,two types of estimates, variability within and between clusters

I equal cluster sizes: MSB and MSW

I relative efficiency of one-stage cluster estimation to SRS

I sampling with unequal probabilities; probability proportionalto size (usually used for clusters at stage 1)

I survey weights


Final Exam Coverage

I From Text:I Chapters 1 and 2I Chapter 3.1 3.5I Chapter 4.1 4.5I Chapter 5.1 5.9I Chapter 6.1 6.4, 6.6 6.8I Chapter 7.1 7.5I Chapter 8.1 8.7, 8.9I Chapter 9.1 9.7

I Slides from class

I Homework and assignment questions

I Materials covered in midterm test

I No R codes or output


Final Exam

I June 24, 7-9pm in Galbraith 3202 (A-LA) or SanfordFleming (LE-Z)

I One handwritten, 2-sided 8 1/2 X11 sheet of notes

I Non-programmable calculator

I Six questions. Do five. Choice between two proof-typequestions

I Office hours: TA: M 2-5, T 4-6 in SS1091SS: T 10:30-1, W 12-1:30 in SS6026


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