A Short Look at Several Esoteric Methods

1
A Short Look at Several Esoteric Methods

• Bootstrap
• Computational Models
• Nonlinear Systems
• Structural Equation Models
• LISREL
• AMOS
• Hierarchical Level Modeling
• There are others!

2
The Bootstrap

• Bootstrapping is a relatively new statistical
  technique (Bradley Efron, 1979)
• It asks the researcher a simple question
• Which do you have more confidence in
• The assumption that the Probability Density
  Function of the statistic is normally
  distributed,
• or
• Your Data

3
How to Bootstrap

• Bootstrapping is a technique that utilizes
  computational power in place of mathematical
  rigor.
• When we bootstrap a statistic, we treat the
  sample as if it were a population and draw a
  random sample, with replacement, from it.
• We then run the statistical test on that sample
  and record the results.
• We repeat the process a bunch of times say 1000.

4
The Central Limit Theorem

• By resampling, we are making the following claim
• My data set provides me a better estimate of the
  sampling distribution of the statistic in
  question than the assumption of normality.
• In large samples, the Central Limit Theorem says
  that the sampling distribution of the sample mean
  will approximate a normal distribution.
• How large is large?
• Statisticians often use ngt30
• That seems small to me.

5
When to bootstrap

• When you have reason to doubt that the sampling
  distribution of your statistic is normally
  distributed.
• When there is no theoretical sampling
  distribution
• E.g. difference between two sample medians
• When your statistic is inherently biased
• E.g ratio of two sample means

6
The Empirical Density Function

• The Central Limit Theorem says that, given this
  acceptably large sample, the probability density
  function (the sampling distribution) of the
  statistic will be normally distributed.
• Bootstrapping says that it doesnt want to assume
  anything about the sampling distribution of the
  statistic and would rather use the sample data to
  empirically estimate the sampling distribution.
• So what do you trust
• a general assumption that we universally apply
  without real inspection
• or your data?

7
How to Bootstrap

• Obtain sample, say n observations
• Using the sample, draw a resample of n
  observations from the n original values (using
  replacement)
• This will generate a sample with likely some
  observations more than once, and some others not
  at all.
• Calculate the statistic you are interested in.
• Repeat a large number of times I recommend 1000.

8
Bootstrap results

• Using the bootstrapped estimates, construct
  confidence intervals (CI) about the statistic you
  are interested.
• We use confidence intervals rather than test
  statistics
• So if 0.0 (or some other value) does not fall
  within out 95 CI, then we can conclude that the
  true ? is different from 0.0 (or the other value)
• E.g if we bootstrap a regression model and
  calculate a CI for the Bs, we can conclude that X
  is significant if 0.0 does not fall inside its
  95 CI.

9
Types of Confidence intervals

• There are several CIs of interest
• Normal approximation
• Percentile
• Percentile-t
• Bias Corrected
• Accelerated Bias Corrected

10
Normal approximation

• Assumes that the sampling distribution of the
  statistic is normal
• OK if it is normal, why use it?
• How about when there is no sampling distribution
  of the statistic
• OKHow about dont bother

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When to Use Normal Approximation CI

• OK if the EDF can be assumed to be Normal, why
  use it?
• How about when there is no sampling distribution
  of the statistic
• Why would you assume it is normal if you cant
  calculate it?
• OKHow about dont bother

12
Percentile

• Uses actual 2.5 and 97.5 points in empirical
  sampling distribution
• May perform poorly with small samples
• Also assumes tthat EDF is unbiased

13
Bias Corrected

• Uses the cumulative normal distribution of the
  sampling distribution to correct the endpoints
  based on bias in the EDF

14
Percentile-t

• Standardizes the estimates and adjusts each
  according to our confidence in it.
• Requires a double bootstrap!

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Accelerated Bias Corrected