Survey and Sampling Methods

Session 9

• Introduction• Nonprobability Sampling and Bias• Stratified Random Sampling• Cluster Sampling• Systematic Sampling• Nonresponse• Summary and Review of Terms

9-1 Sampling Methods

• Sampling methods that do not use samples with known probabilities of selection are knows as nonprobability sampling methods.

• In nonprobability sampling methods, there is no objective way of evaluating how far away from the population parameter the estimate may be.

• Frame - a list of people or things of interest from which a random sample can be chosen.

9-2 Nonprobability Sampling and Bias

In stratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.

7654321 Group

Nii

Population Distribution7654321 Group

ni

Sample Distribution

In proportional allocation, the relative frequencies in the sample (ni/n) are the same as those in the population (Ni/N) .

9-3 Stratified Random Sampling

True weight of stratum i:

Sampling fraction in stratum i:

True mean of population: True mean in stratum i: iTrue variance of the population: 2

True variance of stratum i: 2

Sample mean in stratum i:

Sample variance in stratum i: The in stratified random sampling:

st

WiNiN

finin

iXi

si

X Wi Xii

m

2

1

estimator of the population mean

Relationship Between the Population and a Stratified Random Sample

1. If the estimator of the mean in each stratum, Xi , is then the stratified estimator of the mean, Xst is an estimator of the population mean, .2. If the samples in the different strata are drawn independently of each other, then the variance of the stratified estimator of the population mean, Xst , is given by:

( ) = Xii=1

m

If sampling in all strata is random, then the variance of Xst is further equal to:

( ) =i=1

m

When the sampling fractions, , are small and may be ignored, we have:

unbiased unbiased,

( )

.

( )

V Xst Wi V

V Xst Wii

ni

fi

fi

2

3

2 21

V Xst Wii

ni

( ) =i=1

m 2 2

Properties of the Stratified Estimator of the Sample Mean

4. If the sample allocation is proportional for all i , then

( ) =1 - f

n i=1

m

which reduces to

( ) =1n i=1

m

when the sampling fraction is small. In addition, if the population variances in all strata are equal, then

( ) =2

n

when the sampling fraction is small.

ni

nN

iN

V X st Wi

V X st Wi

V X st

i

i

2

2

Properties of the Stratified Estimator of the Sample Mean (continued)

An unbiased estimator of the population variance of stratum i,12 , is:

i2

data in iIf sampling in each stratum is random:

2 ( ) = i2

i=1

m

SX X

ini

S X st

Wi S

nf

ii

( )

( )

2

1

1

2

When the Population Variance is Unknown

A (1 - )100% confidence interval for the population mean, , using stratifiedsampling: x

st

The effective degrees of freedom:

Effective df =

( )=

( ) /

z s Xst

Ni

Ni

ni

si

nii

m

Ni

Ni

ni

ni

si

nii

m

2

2

1

2

2 4

11

( )

( )

Confidence Interval for the Population mean in Stratified Sampling

Population True SamplingNumber Weights Sample Fraction

Group of Firms (Wi) Sizes (fi) 1. Diversified service companies 100 0.20 20 0.202. Commercial banking companies 100 0.20 20 0.203. Financial service companies 150 0.30 30 0.304. Retailing companies 50 0.10 10 0.105. Transportation companies 50 0.10 10 0.106. Utilities 50 0.10 10 0.10

N=500 n=100

StratumMeanVariance ni Wi Wixi 1 52.7 97650 20 0.2 10.54 156.240 2 112.6 64300 20 0.2 22.52 102.880 3 85.6 76990 30 0.3 25.68 184.776 4 12.6 18320 10 0.1 1.26 14.656 5 8.9 9037 10 0.1 0.89 7.230 6 52.3 83500 10 0.1 5.23 66.800

Estimated Mean: 66.12 532.582Estimated standard error of mean: 23.08

1 fn

Wi si2 95% Confdence Interval:

xst

66

z s Xst

212 1 96 23 08

66 12 45 2420 88 111 36

( )

. ( . )( . )

. .[ . , . ]

Example 9-1

Stratified estimator of the population proportion, ,

The approximate variance of

V(

When the finite - population correction factors, must be considered:

V(

When proportional allocation is used:

V(

p

Pst Wi Pii

m

Pst

Pst WiPi Qinii

m

f

PstN

Ni

Ni

ni

Pi QiN

inii

m

Pstf

nWi Pi Qii

m

i

,

)

,

) ( )

( )

)

1

21

12

211

11

Stratified Sampling for the Population Proportion

NumberGroup Wi ni fi InterestedMetropolitan 0.65 130 0.65 28 0.14 0.0005756Nonmetropolitan 0.35 70 0.35 18 0.09 0.0003099

Estimated proportion: 0.23 0.0008855Estimated standard error: 0.0297574

90% confidence interval:[0.181,0.279]

Wi piWi pi qi

n

90% Confdence Interval: p

st ( )

. ( . )( . )

. .[ . , . ]

z s Pst

20 23 1 645 0 2970 23 0 0490 181 0 279

Stratified Sampling for the Population Proportion: An Example

1. Preferably no more than 6 strata.2. Choose strata so that Cum f(x) is approximately constant for all strata (Cum f(x) is the cumulative square root of the frequency of X, the variable of interest).

Age Frequency (fi) 20-25 1 126-30 16 4 531-35 25 5 536-40 4 241-45 9 3 5

f(x) Cum f(x)

Rules for Constructing Strata

For optimum allocation of effort in stratified random sampling, minimize thecost for a given variance, or minimize the variance for a given cost.

Total Cost = Fixed Cost + Variable Cost C = C0 Cini

Optimum Allocation: nin

(Wi i ) / Ci(Wi i ) / Ci

If the cost per unit sampled is the same for all strata (Ci = c):

Neyman Allocation: nin

(Wi i )

(Wi i )

Optimum Allocation

1 0.4 1 4 0.4 0.200 0.329 0.235 2 0.5 2 9 1.0 0.333 0.548 0.588 3 0.1 3 16 0.3 0.075 0.123 0.176

i W W i isi Ci si W

isi

Ci

OptimumAllocation

Neyman

Allocation

1.7 0.608

Optimum Allocation: An Example

7654321 Group

Population Distribution

In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).

Sample Distribution

In cluster sampling observations are drawn from m out of M areas or clusters of the population.

9-4 Cluster Sampling

Cluster sampling estimator of :

Estimator of the variance of the sample mean:

s

where

=

2

Xn X

n

X M mMmn

n X X

m

nn

m

cl

i ii

m

ii

m

cl

i i cli

m

ii

m

1

1

2

2 2

1

1

1( )( )

Cluster Sampling: Estimating the Population Mean

Cluster sampling estimator of :

Estimator of the variance of the sample proportion:

s

2

p

Pn P

n

P M mMmn

n P P

m

cl

i ii

m

ii

m

cl

i i cli

m

( )( )

1

1

2

2 2

1

1

Cluster Sampling: Estimating the Population Proportion

95% Confdence Interval: x

cl

z s Xcl

2

2183 1 96 15872183 2 4719 36 24 30

( )

. ( . )( . )

. .[ . , . ]

xi ni nixi xi-xcl (xi-xcl)2

21 8 168 -0.8333 0.694 0.0011822 8 176 0.1667 0.028 0.0000511 9 99 -10.8333 117.361 0.2526934 10 340 12.1667 148.028 0.3934828 7 196 6.1667 38.028 0.0495325 8 200 3.1667 10.028 0.0170618 10 180 -3.8333 14.694 0.0390624 12 288 2.1667 4.694 0.0179719 11 209 -2.8333 8.028 0.0258220 6 120 -1.8333 3.361 0.0032230 8 240 8.1667 66.694 0.1134626 9 234 4.1667 17.361 0.0373812 9 108 -9.8333 96.694 0.2081917 8 136 -4.8333 23.361 0.0397413 10 130 -8.8333 78.028 0.2074129 8 232 7.1667 51.361 0.0873824 8 192 2.1667 4.694 0.0079926 10 260 4.1667 17.361 0.0461518 10 180 -3.8333 14.694 0.0390622 11 242 0.1667 0.028 0.00009

3930 s2(Xcl)= 1.58691 xcl = 21.83

M mMmn

n X Xm

i i cl

2

2 2

1( )

Cluster Sampling: Example 9-2

Randomly select an element out of the first k elements in the population, and then select every kth unit afterwards until we have a sample of n elements.

Systematic sampling estimator of :

Estimator of the variance of the sample mean: s2

When the mean is constant within each stratum of k elements but different between strata:

s2

When the population is linearly increasing or decreasing with respect to the variable of interest:

s2

X sy

Xii

m

n

X syN n

NnS

X syN n

Nn

Xi Xi ki

n

n

X syN n

Nn

Xi Xi k Xi ki

n

n

1

2

21

2 1

2 22

16 2

( )

( )( )

( )

( )( )

( )

9-5 Systematic Sampling

s2

s2

A 95% confidence interval for the average price change for all stocks: s

X syXii

m

n

X syN n

NnS

X sy X sy

1 0 5 0 36

2 2100 1002100 100

0 36 0 0034

1 960 5 1 96 0 00340 5 0114

0 386 0 614

. .

( )( )( )

. .

( . ) ( ). ( . )( . ). .

[ . , . ]

Systematic Sampling: Example 9-3

Systematic nonresponse can bias estimates. Callbacks of nonrespondents. Offers of monetary rewards for

nonrespondents. Random-response mechanism.

9-6 Nonresponse