Ratio estimation with stratified samples Consider the agriculture stratified sample. In addition to the data of 1992, we also have data of 1987. Suppose

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Ratio estimation with stratified samples

• Consider the agriculture stratified sample. In addition to the data of 1992, we also have data of 1987. Suppose that the population data of 1987 are available. How can we combine the two techniques?

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Method 1: combined ratio estimator

• Step 1: combine strata to estimate tx and ty

• Step 2: use ratio estimation

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Method 2: separate ratio estimators

• Step 1: use ratio estimation in each stratum

• Step 2: combine strata

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Method 1 vs Method 2

• If the ratios vary from stratum to stratum, use method 2

• If sample sizes are small, use method 1

• Poststratificatio is a special case of method 2

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Cluster Sampling

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A new sampling method

• Motivating example• Want to study the average amount water used

by per person• How would you design a survey?

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• Consider the two strategies– Sample person by person– Sample household by household

• Which one do you prefer and why?

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• In the water usage example, I would sample households, in other words, I would use household as the sampling unit.

• I do this for convenience. I am interested in average monthly usage per person, but I sample household

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• The example of water usage is an example of cluster sampling– Households are the primary sampling units (PSUs)

or clusters– Persons are the secondary sampling units (SSUs).

They are the elements in the population

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Definition of Cluster Sampling

• Take an SRS on clusters• Individual elements of the population are

allowed in the sample only if they belong to a cluster (primary sampling unit) that is included in the sample

• The sampling unit (psu) is not the same as the observation unit (ssu), and the two sizes of experimental units must be considered when calculating standard errors from cluster samples

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Stratified sampling vs Cluster sampling

• The two sampling methods look similar– A cluster is also a grouping of elements of the

population• But the sampling schemes are different– Stratified: SRS from each stratum– Cluster: SRS of the clusters. For each selected

cluster, we select all its elements– See the following two slides

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Stratified sampling

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Cluster sampling

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• Stratified sampling– Variance of the estimate of depends on the

variability of values within strata– For greater precision, individual elements within

each stratum should be similar values, but stratum means should differ from each other as much as possible

– Stratified sampling usually improves the precision of SRS

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• Cluster sampling– The cluster is the sampling unit– The more clusters we sample, the smaller the

variance– The variance of the estimate of depends

primarily on the variability between cluster means– For greater precision, individual elements with each

cluster should be heterogeneous and cluster means should be similar to one another

– Cluster sampling usually ??? the precision of SRS

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Why does cluster sampling tend to reduce precision?

• Elements of the same cluster tend to be more similar than elements selected at random from the whole population. E.g, – Elements of the same household tend to have similar political views– Fish in the same lake tend to have similar concentrations of mercury– Residents of the same nursing home tend to have similar opinions of

the quality of care

• The similarities arise because of some underlying factors that may or may not be measurable– Residents of the same nursing home may have similar opinions

because the care is poor– The concentration of mercury in the fish will reflect the concentration

of mercury in the lake

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Why does cluster sampling tend to reduce precision?

• Because of the similarities of elements within clusters, we do not obtain as much information

• By sampling everyone in the cluster, we partially repeat the same information instead of obtaining new information

• As a result, cluster sampling leads to less precision for estimates of population quantities

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Motivation of using cluster sampling

• A sampling frame list of observation units may be difficult, expensive, or unavailable– Cannot list all honeybees in a region

• The population may be widely distributed geographically or may occur in nature clusters– Nursing home residents cluster in nursing homes

• Cluster sampling leads to convenience and reduced cost

• Cluster sampling may result in more information per dollar spent

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Versions of cluster sampling: one-stage vs two-stage cluster sampling

• We will consider one-stage and two-stage sampling– One-stage sampling: every element within a

sampled cluster is included in the sample– Two-stage sampling: we subsample only some of

the elements of selected clusters

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One-stage cluster sampling(1)

(2) (3)

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Two-stage cluster sampling(1)

(2) (3)

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Notation for cluster sampling

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One-stage cluster sampling(1)

(2) (3)

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One-stage cluster sampling

• Every element within a cluster (PSU) is included in the sample

• Either “all” or “none” of the elements that compose a cluster (PSU) are in the sample

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Clusters of equal sizes

• – Most naturally occurring clusters do not fit into

this framework– Can occur in agricultural and industrial sampling– Estimating population means or totals is simple• We treat the cluster means or totals as the

observations and simply ignore the individual elements• We have an SRS of n observations , where ti

is the total for all the elements in PSU i. },{ Siti

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Nothing is new here

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Clusters of equal sizes: an example

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Clusters of equal sizes: an example

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Clusters of equal sizes: sampling weights

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Theory of Cluster sampling with equal sizes

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• In one-stage cluster sampling, the variability of the unbiased estimator of t depends entirely on the between-cluster part of the variability

• For cluster sampling

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• When MSB/MSW is large– MSB is relatively large: elements in different clusters

vary more than elements in the same cluster– cluster sampling is less precise than SRS

• If MSB>S^2, cluster sampling is less precise

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Measurements of correlation

• ICC (or ρ): Intraclass (or intracluster) Correaltion Coefficient– Describes how similar elements in the same

cluster are– Provides a measure of homogeneity within the

clusters• Definition:• It can be shown that

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Measurements of correlation

If SSB=0, then

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One-stage cluster sampling with equal sizes vs SRS

If N is large

1+(M-1)ICC SSU’s, taken in a one-stage cluster sample, giveThe same amount of information as one SSU from an SRSe.g, ICC=1/2, M=5, then 1+(M-1)ICC=3 → 300 SSUs in the cluster sample = 100 SSUs in an SRS

• If ICC<0, cluster sampling is more efficient than SRS • ICC is rarely negative in naturally occurring clusters

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The GPA example

The population ANOVA table (estimated)

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The GPA exampleThe population ANOVA table (estimated)

• The sample mean square total should not be used to estimate when n is small

• The data were collected as a cluster sample. They do not reflect enough of the cluster-to-cluster variability.

• Multiply the unbiased estimates of MSB and MSW by the df from the population ANOVA table to estimate the population sums of squares

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The GPA example

The population ANOVA table (estimated)

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The GPA example

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Clusters of unequal sizes

• The adjusted R2 measures the relative amount of variability in the population explained by the cluster means, adjusted fro the number of degrees of freedom

• If the clusters are homogeneous, then the cluster means are highly variable relative to the variation within cluster, and R2 will be high.

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An example

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An example

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The GPA example

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The GPA exampleThe population ANOVA table (estimated)

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Clusters of unequal sizes

• In social surveys, clusters are usually of equal sizes

• In a one-stage sample, we will introduce two methods to estimate the population total/mean– Unbiased estimation– Ratio estimation

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Unbiased estimation for cluster sampling with unequal sizes

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Unbiased estimation for cluster sampling with unequal sizes

• Nothing is different from cluster sampling with equal sizes

• The problem is that the between cluster variance is large when the sizes of clusters are quite different from each other, as we expect large total from clusters of large sizes

• Therefore, we consider another estimator

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Ratio estimation for cluster sampling with unequal sizes

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where

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Note, it is not difficult to find that

• The variance of the ratio estimator depends on the variability of the means per element in the clusters

• It can be much smaller than that of the unbiased estimator• The ratio estimator requires the total number of elements in the

population, K.• The unbiased estimator does not require K.

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Two-stage cluster sampling

• In one-stage cluster sampling, we – Examine all the SSU’s within the selected PSU’s– Obtain redundant information because SSU’s in a

PSU tend to be similar– Expensive

• An alternative: taking a subsample within each selected PSU – two stage cluster sampling

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Two-stage cluster sampling with equal probability

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• Compared with the one-stage cluster sampling, the two-stage uses one extra stage.

• The extra stage complicates the notation and estimators, as one needs to consider variability arising from both stages of data collection

• The points estimates are similar to those in one-stage, but variances are much more complicated

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Two-stage cluster sampling with equal probability: an unbiased estimator

• Since we do not observe every SSU in the sampled PSU’s, we need to estimate the totals for the sampled PSU’s

• An unbiased estimator of the population total is

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• The estimator is unbiased

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Two-stage cluster sampling with equal probability: a ratio estimator

As in one-stage cluster sampling with unequal sizes, the between-PSU variance can be very large since it is affected both by variations in the cluster sizes and by variation in y.

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where

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The egg volume example

• A study (Arnold 1991) on egg volume of American coot eggs in Minnesota. We looked at volumes of a subsample of eggs in clutches (nests of eggs) with at least two eggs.

• For each sampled clutch, two eggs were measured

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Using weights in cluster samples

• For estimating overall means and totals in cluster samples, most survey statisticians use sampling weights.

• Weights can be used to find a point estimate of almost any quantity of interest

• For cluster sampling:

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Using weights in cluster samples

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SRS : one-stage cluster: two-stage cluster

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Design a cluster survey

• It is worth spending a great deal of effort on designing the survey for an expensive and large-scale survey

• It can take several years to design and pre-test• For designing a cluster sample– What overall precision is needed?– What size should the PSU’s be?– How many SSU’s should be sampled in each sampled

PSU?– How many PSU’s should be sampled?

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Choosing the PSU size

• In many situations, the PSU size exists naturally. E.g, a clutch of eggs, a household

• In some situations, one needs to choose PSU sizes. E.g., area of a region, 1km2, 2km2,…

• Many ways to “try out” different PSU sizes• Pilot study, perform an experiment• The goal is get the most information for the

least cost and inconvenience

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Two-stage cluster design with equal cluster size and equal variance

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Summary of two-stage cluster

• Cluster sampling is widely used in large surveys

• Variances from cluster samples are usually greater than SRSs with the same SSUs

• Less expensive – the per-dollar information from cluster sampling might be greater than that of SRS

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Summary of two-stage cluster

Documents

Ratio estimation with stratified samples Consider the agriculture stratified sample. In addition to the data of 1992, we also have data of 1987. Suppose