STAT 572: Bootstrap Project

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STAT 572 Bootstrap Project

• Group Members
• Cindy Bothwell
• Erik Barry Erhardt
• Nina Greenberg
• Casey Richardson
• Zachary Taylor

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Histograms of Complex Population Distribution
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Histograms of Population Sampling Distribution of
the Median and Estimated Bootstrap Sampling
Distributions
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What is a Bootstrap

• A method of Resampling creating many samples
  from a single sample
• Generally, resampling is done with replacement
• Used to develop a sampling distribution of
  statistics such as mean, median, proportion,
  others.

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The Bootstrap and Complex Surveys

• Number of bootstrap samples
• n sample size, N population size
• Possible resamples nn (example n200,
  2002001.6x10460)
• Too many possibilities N!/n!(N-n)!, limit to B
  a large number, (example 1000) - the Monte
  Carlo approximation
• Determine sampling distribution with parameters
• Calculate variance in the normal way

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Advantages and Disadvantages

• Advantages
• Avoids the costs of taking new samples (Estimate
  a sampling distribution when only one sample is
  available)
• Checking parametric assumptions
• Used when parametric assumptions cannot be made
  or are very complicated
• Estimation of variance in quantiles
• Disadvantages
• Relies on a representative sample
• Variability due to finite replications (Monte
  Carlo)

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Computations

• With more computing power available, bootstrap is
  possible for a large number of resamples
• Possible programs
• Matlab
• Minitab
• SAS
• Excel
• S-Plus
• SPSS
• Fathom

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Bootstrap using SURVEY program

• Main parameter of interest is the median price
  that all households in Lockhart City are wiling
  to pay for cable.
• The price that a household is willing to pay for
  cable is positively correlated with
  average-district house value.
• Districts in Lockhart City are divided into
  strata based on average house value.
• Estimate the variance and create 95 CI

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Lockhart City Strata Characteristics

• Take a stratified random sample of size 200 using
  proportional allocation.
• Using the stratified random sample, implement the
  general bootstrap procedure, BWO, and
  mirror-match.

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Variations of the Bootstrap in Strata

• General Bootstrap
• Mimic the original sampling method
• BWO Bootstrap Without Replacement
• Grow the sample to the size of the population
• Mirror-Match
• Repeated miniature resamples

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BWO Bootstrap Without Replacement

• Grow the sample to the size of the population
• For each stratum L, create a pseudo-population by
  replicating the sample kL times.
• Resample nL units from each stratum without
  replacement to obtain a single bootstrap sample
  for stratum L.
• Repeat a large number of times

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BWO Variable Definitions
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Disadvantages of extended BWO

• NL must be known
• nL and kL are often non-integers
• Must bracket between integers if nL and kL are
  non-integer
• Computing time

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Mirror-Match

• Repeated miniature resamples
• Resample size is determined to match the
  proportion of the original sample size to the
  population sample size (nL/NL).
• Using the resample size nL, we resample nL
  units (SRSWOR) from each stratum L.
• Repeat previous step kL times with replacement to
  obtain a single bootstrap sample for stratum L.
• Repeat a large number times

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Mirror-Match Variable Definitions
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Mirror Match Disadvantages

• NL must be known
• kL is often non-integer
• Must bracket between integers when kL is
  non-integer
• Computing time

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Estimation of the Population Sampling
Distributions

• 100,000 independent stratified random samples.
• Medians computed and plotted to form empirical
  sampling distributions.
• Variables house value, cable price, and TV hours.

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Estimation of the Population Sampling
Distributions
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Simulations

• Matlab code General, BWO, and Mirror-match.
• Two independent stratified random samples from
  Lockhart City.
• Comparison of the sample bootstrap sampling
  distributions with the population sampling
  distributions.
• 95 confidence intervals were determined
  bootstrap 2.5 and 97.5 percentiles.

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Sampling Distributions 1
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Sampling Distributions 2
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Confidence Intervals
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The Empirical verses the Bootstrap Sampling
Distributions

• Bootstrap sampling distributions are expected to
  mimic actual sampling distributions.
• Bootstrap sampling is sensitive to individual
  samples.
• The shape of bootstrap sampling distributions may
  vary, but the statistic of interest and its
  variance are considered accurate.

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Comparison of Bootstrap Methods
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Empirical Coverages

• The empirical coverages were close to the
  expected 95. They differed very little between
  the different bootstrap procedures.

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Empirical Coverages

• Empirical coverages are dependent on the type of
  confidence interval that was originally selected.
  
• Our confidence intervals were calculated from the
  2.5 and 97.5 percentiles of each bootstrap
  distribution.
• There are many different types of bootstrap
  confidence intervals. The one we selected,
  although intuitive in design, is considered
  generally biased (Bedrick 2006).

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Computer Processing Times

• Computer processing times varied greatly.
• Mean processing time per sample in seconds.

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Computer Processing Times

• BWO took 381 times as long as general
  bootstrapping procedures.
• Mirror-match took 293 times as long as general
  bootstrapping procedures.
• For our study, the BWO and mirror-match conferred
  no advantage over general bootstrapping with
  regard to statistical estimates. However, their
  vastly greater processing times are a great
  disadvantage.

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CONCLUSIONS General Bootstrap verses BWO and
Mirror-Match

• BWO and Mirror-match procedures are designed to
  mimic complex sampling designs.
• We only analyzed stratified samples of 200 from a
  fictitious city.
• BWO and Mirror-match methods may be advantageous
  in other complex sampling scenarios.