EMR 6500: Survey Research Dr. Chris L. S. Coryn Kristin A. Hobson Spring 2013.

EMR 6500:Survey Research

Dr. Chris L. S. CorynKristin A. Hobson

Spring 2013

Agenda

• Systematic sampling• Cluster sampling for means and

totals

Systematic Sampling

Systematic Sampling

• Systematic sampling simplifies the sample selection process compared to both simple random sampling and stratified random sampling

• In systematic sampling an interval (k) is used to select sample elements

• The starting point is (should be) selected randomly

Systematic Sampling

• Systematic sampling is a useful alternative to simple random sampling because:1. It is easier to perform in the field and

less subject to selection errors, especially if a good frame is not available

2. It can provide greater information per unit cost than simple random samples for populations with certain patterns in the arrangement of elements

1-in-k Systematic Sampling

• Divide the population size N by the desired sample size n

• Let k = N/n• k must be equal to or less than N/n

(i.e., k ≤ N/n)– If N = 15,000 and n = 100, then k ≤ 150

1-in-k Systematic Sampling

• If N were 1,000 and n were 100• k would equal 1,000/100 = 10• If k = 10, the start value would range

between 1 to 10 and all selections thereafter would be every 10th entry on the sampling frame– If the start value was 8, then the next

selection would be 18, followed by 28, and so forth

Random Population Elements

Ordered Population Elements

Periodic Population Elements

Estimation of a Population Mean and Total

Estimation of a Population Mean

*Note: This formula assumes a randomly ordered population

Estimation of a Population Total

*Note: This formula assumes a randomly ordered population

Estimation of a Population Proportion

Estimation of a Population Proportion

*Note: This formula assumes a randomly ordered population

Selecting the Sample Size

Sample Size for Estimating a Population Mean

Sample Size for Estimating a Population Proportion

Variance Estimation for Ordered and Periodic Distributions

Variance Estimates

• Repeated systematic sampling– Divides a systematic sample into smaller

systematic samples to approximate a random population

– Multiple 1-in-k systematic samples

• Successive difference method– A samples of size n yields n-1 successive

differences that are used to estimate variance

– Best choice when population elements are not randomly ordered

Cluster Sampling

Cluster Sampling

• Cluster sampling is a probability sampling method in which each sampling unit is a collection, or cluster, of elements

• Clusters can consist of almost any imaginable natural (and artificial) grouping of elements

Cluster Sampling

• Cluster sampling is an effective sampling design if:1. A good sampling frame listing

population elements is not available or is very costly to obtain, but a frame listing clusters is easily obtained

2. The cost of obtaining observations increases as the distance separating elements increases

Cluster Sampling

• Unlike stratified random sampling, in which strata are ideally similar within stratum and where stratum should differ from one another, clusters should be different within clusters and be similar between clusters

Take a simple random sample from every stratum Take a simple random sample of clusters; observe all elements within clusters in the sample

Each element of the population is in exactly one stratum

Each element of the population is in exactly one cluster

Variance of the estimate depends on the variability within strata

Variance of the estimate depends primarily on the variability between clusters

For greatest precision, individual elements within each stratum should have similar values, but stratum means should differ from each other as much as possible

For greatest precision, individual elements within each cluster should be heterogeneous, and cluster means should be similar to one another

Cluster Sampling Notation

Estimation of a Population Mean and Total

Estimation of a Population Mean*Note: takes the form of a ratio estimator, with taking the place of

*Note: can be estimated by if M is unknown

Example for a Population MeanCluster

Number of residents, mi

Total income per cluster, yi

ClusterNumber of residents,

Total income per cluster

1 8 $96,000 14 10 $49,000

2 12 $121,000 15 9 $53,000

3 4 $42,000 16 3 $50,000

4 5 $65,000 17 6 $32,000

5 6 $52,000 18 5 $22,000

6 6 $40,000 19 5 $45,000

7 7 $75,000 20 4 $37,000

8 5 $65,000 21 6 $51,000

9 8 $45,000 22 8 $30,000

10 3 $50,000 23 7 $39,000

11 2 $85,000 24 3 $47,000

12 6 $43,000 25 8 $41,000

13 5 $54,000

Example for a Population Mean

n M Med SD

Resident ( ) 25 6.040 6.000 2.371

Income ( ) 25 $51,360 $49,000 $21,784

25 0 993 25,189

Example for a Population Mean

*Note: Because M is not known, is estimated by

Example for a Population Mean

Estimation of a Population Total

Estimation of a Population Total

Estimation of a Population Total that Does not Depend on M

Example of Estimation of a Population Total that Does not Depend on M

Example of Estimation of a Population Total that Does not Depend on M

Equal Cluster Sizes

Equal Cluster Sizes for Estimating a Population Mean

• All mi values are equal to a common, or constant, value m

• In this case, M = Nm, and the total sample size is nm elements (n clusters of m elements each)

• When cluster sizes are equal m1 = m2 = mN

• Variance components analysis simplifies estimating the variance using ANOVA methods

Equal Cluster Sizes for Estimating a Population Mean

ANOVA Method

Cluster Number of Newspapers Total

1 1 2 1 3 3 2 1 4 1 1 19

2 1 3 2 2 3 1 4 1 1 2 20

3 2 1 1 1 1 3 2 1 3 1 16

4 1 1 3 2 1 5 1 2 3 1 20

• There are 4,000 households (elements)• There are 400 geographical regions

(clusters)• There are 10 households in each region

ANOVA Method

ANOVA Method

Source df SS MS

Factor 3 1.07 0.36

Error 36 43.30 1.20

Total 39 44.38

*Note: ‘Factor’ denotes between-cluster variation and ‘Error’ denotes within cluster variation

ANOVA Method

Selecting the Sample Size for Estimating Population Means and Totals

Sample Size for Estimating Population Means

Where is estimated by

Example of Sample Size for Estimating Population Means

• How large a sample should be taken to estimate the average per-capita income with a bound on the error of estimation of B = $500?

Example of Sample Size for Estimating Population Means

*Note: Because M is not known, is estimated by

Example of Sample Size for Estimating Population Means

Sample Size for Estimating Population Totals When M is Known

Where is estimated by

Example of Sample Size for Estimating Population Totals When M is Known• How large a sample should be taken to

estimate the total income of all residents with a bound on the error of estimation of B = $1,000,000? (M = 2,500)

Sample Size for Estimating Population Totals When M is Known

Sample Size for Estimating Population Totals When M is Unknown

Where is estimated by

Example of Sample Size for Estimating Population Totals When M is Unknown• How large a sample should be taken to

estimate the total income of all residents with a bound on the error of estimation of B = $1,000,000? (M = 2,500)

Sample Size for Estimating Population Totals When M is Unknown