Sampling of Units Sampling and Elementary Units

123

Sampling of Units

Study population

Units of one kind or another are

sampled from the study population

• Clusters• Households

Your main interest, however, may not be in characteristics of the units being sampled

124

Reference 1

• Provides overview of WHO/EPI survey method One form of rapid survey (but not correct in all situations)

Download from: http://www.ph.ucla.edu/epi/faculty/frerichs/thai2007/whostatquarterly38_65_75_1985.pdf

125

• Better overview of rapid survey methodApplicable in most situations

Download from: http://www.ph.ucla.edu/epi/faculty/frerichs/thai2007/whostatquarterly44_98_106_1991.pdf

Reference 2

126

Sampling and Elementary UnitsSampling Units

Elementary Units

- the units sampled from the study population

- the units that are the subject of your analysis

127

Enumeration Rules

Sampling Units

Elementary Units

Link sampling units to elementary units

128

Still More on UnitsSampling Units

Elementary Units

- clusters sampled from the study population

- people who are the subject of your analysis

129

Enumeration RuleSampling Units

Elementary Units

130

Download from: http://www.ph.ucla.edu/epi/faculty/frerichs/thai2007/chap2rapid.pdf

Reference 3

131

Equal interval

Height

ObeseYes or No

Binomial(yes = 1, no = 0)

Binomial versus Equal Interval Data

Types of Data

132

Equal Interval

0

4

8

12

16

20

Education OpinionFavorable or unfavorable

Binomial (favorable = 1, unfavorable = 0)

Binomial versus Equal Interval Data

Types of Data

133

Proportion Percentage

Example0.10 10%

Range0 to 1 0 to 100

0.90 90%

Proportions and Percentages

134

Total IV drug injections

Shared IV drug injections

Equalintervals

Equalintervals

Ratio of Two Variables

Sampled…

Variable Two

Variable One

135

Ratio Estimator

Total IV drug injectionsVariable One

Shared IV drug injectionsVariable Two

Termed a “ratio estimator”

p r A ratio estimator looks like a proportion, acts like a proportion but is

not a proportion

136

Ratio of Two Variables

Sampled…

Variable TwoVariable One

137

Another Ratio Estimator

Total persons in HHVariable One

Total IV drug injections in past weekVariable Two

Also termed a “ratio estimator”

r The ratio estimator looks like a mean, acts like a mean but is not a meany

138

Population Sample

1

0

P pY_

y_

Equal interval data

Binomial data

Equal interval data

Binomial data

Population and Sample Means

1

0

139

Data

Standard Error

Confidence Interval

Mean Variance of Mean

Derivations from Data

140

Derivations from Equal Interval Data

Equal Interval Data

SE(y)

y V(y)

y +/- z x SE(y)

141

Binomial Data

SE(p)

p +/- z x SE(p)

p V(p)

Derivations from Binomial Data

142

Derivations from Ratio Estimator Data

Ratio Estimator

V(r)

SE(r)

r +/- z x SE(r)

r

143

y or p

Population Sample

un-biased

Estimate

Y or P

Confidence Interval

un-biased

v(y) or v(p)

se(y) or se(p)

Derivation of Confidence Interval

variability of data

144

r

Population Sample

Small bias

Estimate

Confidence Interval

Smallbias

v(r)

se(r)

Derivation of Confidence Interval for Ratio Estimator

variability of data

R( Y / X )

145

Survey Data

Information

Mean or Proportion

Knowledge of Population

Action

Moving from Data to Action

146

Y_

1

0

P

Action 1

Action 2

Action 3

Action 4

Population SampleEqual

interval dataEqual

interval dataBinomial

data1

0

p

Binomial data

Action Levels and Sampling

y

147

CurrentFP Users

(%)

No further action in community unless there is unmet FP need

Interview non-users and improvemanagement of local FP program

Community education and establish or improve local FP program

Action Level100

80

60

40

20

0

Example of Action Levels and Sampling

A small survey with wide confidence intervals would

probably be acceptable

148

7

YES NO

Using FP Method

p = 35%

13

Number of sampled women

A Survey of 20

7/20 = 0.35

149

Precision or Accuracy

A Survey of 20If done blindly over and over again…

Between sample variability

Depending on the presence or

absence of bias

Number of Sample Surveys

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

Using FP Method (%)25 40 45 50 55 60 65 70 7530 3520 80

P

150

with Attribute (%)with Attribute (%)

BIASED

Average value of all sample means

True value in population

UNBIASED



Concept of Bias

151

BIASED UNBIASED

Concept of Accuracy

with Attribute (%)



with Attribute (%)



Accuracy Accuracy

Measuring what you are intending to measure

Since we do not know

truth, we can only estimate accuracy

152

BIASED UNBIASED

Concept of Precision

with Attribute (%)



with Attribute (%)



Accuracy

Measuring statistical variation due to sampling

Precision PrecisionAccuracy

We can measure

precision with statistical

methods, but not accuracy

Precision is a statistical term for the

measurement of sample variation

If unbiased, accuracy is measured by

precision

153

Value of Statistical TestsMeasuring statistical variation due to sampling

BIASED UNBIASED

with Attribute (%)



with Attribute (%)



Accuracy

Precision Precision

AccuracySince knowledge of

accuracy is what we want, statistical tests

for precision are only valued when the

investigator can assure us that the sample survey was

derived in an unbiased manner.

154

Measuring PrecisionVariance and Standard Error of the Sample Mean

Use sample survey data to derive…

Variance of the sample mean

Equal interval data Binomial data

( yi – y )2

n (n – 1) i=1

n

Standard error of the sample mean

( yi – y )2

n (n – 1) i=1

n

v(y) =

se(y) =

p q

n – 1v(p) =

se(p) = p q

n – 1

155

Measuring Precision


Freq

uenc

y

Scale ( P or Y )

Normal distribution

Expected value The standard

error, estimated in a sample survey, is a measure of

precision of the underlying

distribution of all possible samples

1 se(p) or 1 se(y)


PRECISION

ACCURACYThe standard error may also

be a measure of the accuracyof the underlying distribution of

all possible samples

156

Change in ScaleUse of precision and the normal curve

The planned new scale on the x-axis

Using FP Method (%)

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P P

25 40 45 50 55 60 65 70 7530 35

Number of Sample Surveys P

P

P

P

P

20

P

P

P

P

P

80

Standard Errors-3 -2 -1 0 +1 +2 +3

Mean of the one survey

that was actually done

Distribution of sample means if the same

survey was done over and over again

157


Sample Value

Change in ScaleStep A

PPPP PP

25

Using FP Method (%)

35 45 55 65 75

P P P PPP P

158

Change in ScaleStep B

True value

PPPP PP

25

Using FP Method (%)

35 45 55 65 75

P P P PPP P

50

Deviation of all possible sample means from true value

-15-25 -5 +5 +15 +250

159

p qn-1 x 100se(p) =

Change in ScaleStep C

For the one survey that was done where p = 35% and n = 20…

.35 x .6520-1 x 100 = 10.9%se(p) =

160

Change in ScaleStep C

p qn-1 x 100se(p) =

9.9 11.4 11.4 10.9 9.911.5

PPPP PP

25

Using FP Method (%)

35 45 55 65 75

P P P PPP P

50True value

10.9

161

Change in ScaleStep D

A typical scale with equal units measured in inches

or centimeters

A new scale with equal units measured in

Frerichs’ walking shoes

To be measured

60 inches(152 cm.)

= 5 shoes

162

Change in ScaleStep D

Deviations from true valuese(p) of sample value

-259.9

+1510.9

+259.9

-511.4

+511.4

011.5

-2.5 -0.4 +0.4 +1.4 +2.50

PPPP PP

25

Using FP Method (%)

35 45 55 65 75

P P P PPP P

50True value

-1510.9

A new scale

-1.4Standard Errors

163

Change in ScaleDeviations in standard error units from the true value

Number of Sample Surveys

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

Standard Errors-3 -2 -1 0 +1 +2 +3

P

Deviation from the true value in standard error

units for the one survey that was

done

Distribution of sample means of replicated

sample surveys

164

MostSome SomeFew Few

Expected value of proportion or mean

Distribution of all Possible Samples

Low estimates

High estimates

-3 SE -2 SE -1 SE +3 SE+2 SE+1 SE0

Number of Possible Samples

_ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y y y

y y y

y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y

y y y y y y y y y y y y y y y y y y y_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _

_y

p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p

165

+1 +3

Standard Errors

0-1-3

Number of Possible Samples

-2 +2

Distribution of all Possible SamplesMean of Binomial Variable

99% of Possible Samples


-1.64 +1.64


-1.96 +1.96-2.58 +2.58

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p pp p p p p p p p p p p p p p p p p

p p p p p p p p p p p p pp p p p p p p p p

p p p p pp p p

p

166

0.50

1.00

0

0.25

0.75

Playing DartsHow confident are you at hitting the target?

The special dart

Erratic but unbiased dart

thrower

167

0.50

1.00

0

0.25

0.75

Playing DartsHow confident are you at hitting the target?

Small interval

Medium interval

Large interval

168

0.50

1.00

0

0.25

0.75

Playing Confidence IntervalsHow confident are you at hitting the expected value?

Expected value of sample proportions

p

p

p

p

p

pp

p

p p

10 Tries with Repeated Sample Surveys

Knowing this, are you 80% confident that the next confidence interval will bracket the true value?

8 of 10 bracketed the

expected value

169

Binomial data

p q

n – 1v(p) =

se(p) = p q

n – 1

Deriving a Confidence IntervalProportion of 0.50 with 95% CI in a sample of 300

From earlier… For our sample…

se(p) = 0.000836 = 0.029

0.5 x 0.5

300 – 1v(p) =

0.25

299= = 0.000836

95% CI = p +/- 1.96 x se (p)

= 0.5 +/- 1.96 x 0.029 = 0.443, 0.557

170

Standard Errors

+1 +30-1-3-1.96 +1.96


True proportion in population

Confidence Intervals of all Possible Samples95% Confidence Limits

All samples are drawn from the same

underlying study population

For about 5% of the possible

samples, the 95% confidence

intervals will not bracket the

expected value

p

p p p p p

p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

p p p

p p p p p p p p p

p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp

p

p

p

p

171

4,54,5

1,2,3

Sample Survey

1

100

50

75

25


95% confidence

interval does not bracket expected value of sample

proportions

Confidence Intervals for 100 of all Possible SamplesArranged in Ascending Order

172

Confidence Intervals for 100 of all Possible SamplesArranged in Random Order

21

5

4

3Sample Survey

1

100

50

75

25


Five of the 100 repeated

surveys have CIs that do not

bracket the expected value

Since only one of all possible surveys was

actually done, it is not clear if it was one like the 5 or one like the 95

173

Probability Confidence

Before the Flips After the Flips

Flips of Five CoinsProbability versus Confidence

The true proportion or

mean in a survey is not

know

Yet by applying statistical principles, we can say with 95% confidence that our 95% CI brackets the expected value

(and true value if unbiased) in the study population

174

Download from: http://www.ph.ucla.edu/epi/faculty/frerichs/thai2007/chap3rapid.pdf

Reference 4

175

2

5

8

PopulationSelected random

numbersSelected sample

Simple Random SampleDraw 3 from a population of 9

2

3

4

5

6

7

8

9

1

2

5

8

Need to list the population

Select using random number table

176

The Listed PopulationNine Drug Addicts

Each person in the

population is identified

JOE HAL ROY JON SAM

BOB NAT TED BEN

177

IV Injections During Prior WeekNine Drug Addicts

The total number of

injections is identified for each addict

Total injections is an equal interval

variable

JOE HAL ROY JON SAM

BOB NAT TED BEN

8 9 1110 12

11 9 11 10 Mean IV injections in population during prior week = 10.1

178

Infected with HIVNine Drug Addicts

HIV status is identified for each addict

HIV status is a binomial variable

HIV+HIV-

JOE HAL ROY JON SAM

BOB NAT TED BEN

8 9 1110 12

11 9 11 10 Proportion infected with HIV in

population = 3/9 or 0.33

179

Shared IV Injections during Prior WeekNine Drug Addicts

Shared IV injections is tallied for

each addict in the population

Shared IV injections is an equal interval

variable

HIV+HIV-

JOE HAL ROY JON SAM

BOB NAT TED BEN

8 9 1110 12

11 9 11 10

6 2 5 3 0

2 2 25

Mean shared IV injections in

population during prior week = 3.0

180

First selection(1 in 9)

Second selection(1 in 9)

Third selection(1 in 9)

Sample of Three AddictsSelected With Replacement

Population(N=9)

Each addict has the same probability of being selected at the three steps

181

First selection(1 in 9)

Second selection(1 in 8)

Third selection(1 in 7)

Sample of Three AddictsSelected Without Replacement

Population(N=9)

All addicts do not have the same probability of being selected at each step

182

All possible samples = Nn

=729 Possible

Samples93

Sampling with Replacement (WR)

All Possible Samples

Sample 3 from Population of 9 Drug Addicts

183

ROY BOB TED

1112 11

Sample of 3 from the population of 9 drug addicts

Total IV Injections during Prior Week

Average total IV injections

= 11.312 + 11 + 11

3

Equal Interval Data

Sample mean (x) =

Population mean (X) = 10.1

184

0

25

50

75

100

125

150

Mean IV Drug Injections

Num

ber o

f Pos

sibl

e Sa

mpl

es

8 9 10 11 12

Total = 729 samples

All Possible SamplesSelected With Replacement

Selected was one

sample with a mean of

11.3

185

Example: assume n = 5

n x n-1 x n-2 x n-3 x … n-(n-1) = 5 x 4 x 3 x 2 x 1 =

N or n FactorialNotation for Deriving All Possible Samples Without Replacement

n !

N or n are positive numbers tallying the units in the study population and in the sample survey

120

186

Sample 3 from Population of 9 Drug Addicts

= = 84 possible samples

9 !

3 ! (9-3) !9 x 8 x 7 x 6!

3 x 2 x 1 x 6!

Cancel

N !

n ! (N-n) !=All possible

samples

Sampling Without Replacement (WOR)

All Possible Samples when Order is Disregarded

187

Num

ber o

f Po

ssib

le S

ampl

es


25

08 9 10 11 12

Total = 84 samples

Selected Without ReplacementAll Possible Samples

Selected was one

sample with a mean of

11.3

188

All Possible Samples


Num

ber o

f Pos

sibl

e Sa

mpl

es

25

08 9 10 11 12

Selected With Replacement (total = 729 samples)

0

25

50

75

100

125

150

Selected Without Replacement (total = 84 samples)

189

0

5

10

15

20

25


Perc

ent o

f All

Poss

ible

Sam

ples

WOR (84)WR (729)

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Comparison of Sampling With (WR) and Without Replacement (WOR)

Average WR sample mean = 10.1

Population mean = 10.1Average WOR sample mean = 10.1

Less variability among samples selected without

replacement (WOR)

190

Variance (WOR) = FPC x Variance (WR)

Finite Population Correct (FPC)Use when sampling without replacement (WOR)

FPC =Variance (WOR)Variance (WR)

Smaller

LargerLess than 1.0

FPC = N - nN

nN

= 1 -The magnitude of

the correction depends on the size

of the sample in relation to the size of the populationUsually omitted

with rapid surveys

FPC = 50,000 - 30050,000

= 1 - 30050,000

= 0.994

Example: population of 50,000 persons

191

Meanpopulation = 10.1

0

1

2

3

4

8 9 10 11 12Total IV Injections

No.

of D

rug

Add

icts

Total IV Injections during Prior WeekPopulation of Nine Drug Addicts

192

ROY BOB TED

1112 11



Average total IV injections

= 11.312 + 11 + 11

3

Equal Interval Data

Sample mean (x) =

Population mean (X) = 10.1

Sampled without replacement (WOR), disregarding order

193

ROY BOB TED

1112 11



Variance and Standard Error

( xi – x )2

n (n – 1) i=1

n

v(x) =

Equal Interval Data

FPC

N – nNse(x) =

9 - 39

se(x) = (12 - 11.3)2 + (11- 11.3)2 + (11 - 11.3)2

3 (3 - 1) = 0.27

194

0

10

20

30

40 Average = 10.1

Sample x

Num

ber o

f Po

ssib

le S

ampl

es

Mean

8 9 10 11 120

10

20

30

40 Average = 0.56

Sample se(x)

Standard Error of Mean

0 0.4 0.6 0.8 1.00.2

Mean and Standard Error of All Possible Samples of 3 Selected from a Population of 9 (WOR)

Total = 84 samples


195


Confidence Interval

Replace by t when samples are small

Equal Interval DataROY BOB TED

1112 11

95% CI = x +/- z x se(x)


196

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70 80 90 100 110 120

Sample Size minus One

t va

lue

z value = 1.96

Use t for Class Examples and later for Rapid SurveysRelation of t value to z value

t = 4.30 for sample of three

197

Based on the statistical calculations, you would be 95% confident that the interval of 10.16 to 12.50 brackets

the true value in the population

= 11.3 +/- 4.30 x 0.27 = 10.16, 12.50

Instead of 1.96 for 95% CI


Confidence Interval


Equal Interval DataROY BOB TED

1112 11

95% CI = x +/- z x se(x)


198

Confidence Intervals for All Possible Samples (WOR)

Note that 5 of the 84 confidence intervals do not bracket the true value. Thus the 95% confidence interval for this small

population and sample is actually a 94% confidence interval [i.e., (84-5)/84 = 0.94]

For the one sample that was done, you

would be 94% confident that the derived interval of

10.2 to 12.5 brackets the true value in the

population0

2

4

6

8

10

12

14

16

Tota

l Inj

ectio

ns.

10 20 30 40 50 60 70 80

Sample Number* * * * *

Total IV Injections During Prior Week

As it turns out, your confidence was not warranted

X = 10.1

True value

199

Ratio of Shared IV Injections to Total Injections during Prior WeekRatio of Two Variables

R = XY

=Xi

i=1

N

Yii=1

N The ratio of shared IV injections to total

injections in prior week is 0.297

JOE HAL ROY JON SAM

BOB NAT TED BEN

6 2 5 0

2 2 25

38 9 1110 12

11 9 11 10

Shared (Yi)Total (Xi)

Shared (Yi)Total (Xi)

200

Ratio of Shared to Total IV InjectionsPopulation of Nine Drug Addicts

0

20

40

60

80

100

Shared Total

IV Drug Injections

No.

of I

njec

tions

. Ratiopopulation = 27/91 = 0.297

201


Ratio of Shared IV Injections to Total Injections During Prior Week

Population ratio (R) = 0.297

Sample ratio (r) = = 0.3535 + 2 + 512 + 11 + 11

Ratio of Shared IV Injections to Total Injections during Prior Week

ROY BOB TED

5 2 512 11 11

Shared (Yi)

Total (Xi)

202

0

5

10

15

20

25

30

35

40

0.1 0.2 0.3 0.4 0.5 0.6

No.

of P

ossi

ble

Sam

ples

.

Mean for All Possible Samples

Average = 0.296

Distribution of 84 possible samples of 3 drawn WOR from a population of 9 drug

addicts

Sample r


Slightly different from population mean of 0.297 (i.e., a small bias)

203


Shared IV Injections during Prior Week

Variance and Standard ErrorRatio Data

FPC

9 – 39

se(r) = [5 – (.353 x 12)]2 + [2 – (.353 x 11)]2 + [5 – (.353 x 11)]2

3 (3 - 1) [(12 + 11 + 11)/3]2= 0.068

ROY BOB TED

5 2 512 11 11

n

v(r) = ( yi – r xi )2

n (n – 1) x2i=1N – n

Nse(r) =

Shared (Yi)

Total (Xi)

204

0

5

10

15

20

25

30

0.00 0.03 0.06 0.09 0.12 0.15 0.18

No.

of P

ossi

ble

Sam

ples

. Average = 0.078

Standard Errors for All Possible Samples

Distribution of 84 possible samples of 3 drawn WOR from a population of 9 drug addicts

Sample se(r)


205


Confidence Interval

Based on the statistical calculations, you would be 95% confident that the interval of 0.06 to 0.65 brackets the

true value in the population

= 0.353 +/- 4.30 x 0.068 = 0.06, 0.65


Instead of 1.96 for 95% CI

Ratio Data

95% CI = r +/- z x se(r)

Shared IV Injections during Prior Week

ROY BOB TED

5 2 512 11 11

Shared (Yi)

Total (Xi)

206

Confidence Intervals for All Possible Samples

Note that 5 of the 84 confidence intervals do not bracket the true value. Thus the 95% confidence interval for this small

population and sample is actually a 94% confidence interval [i.e., (84-5)/84 = 0.94]




population

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Rat

io (S

hare

d/To

tal).

10 20 30 40 50 60 70 80


Ratio of Shared to Total Injections

R = 0.297

True value

207

Confidence Intervals for All Possible Samples if Presented as a Proportion

By limiting the axis to 0 and 1, the lower and upper bounds of some confidence intervals are attenuated, but otherwise the

bracket relation to truth remains valid

Ratio of Shared to Total Injections




population

0.0

0.2

0.4

0.6

0.8

1.0

Rat

io (S

hare

d/To

tal).

10 20 30 40 50 60 70 80


R = 0.297

True value

Documents

Sampling of Units Sampling and Elementary Units