Inference Concerning Proportions

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Chapter 8

• Inference Concerning Proportions

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Inference for a Single Proportion (p)

• Goal Estimate proportion of individuals in a
  population with a certain characteristic (p).
  This is equivalent to estimating a binomial
  probability
• Sample Take a SRS of n individuals from the
  population and observe X that have the
  characteristic. The sample proportion is X/n and
  has the following sampling properties

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Large-Sample Confidence Interval for p

• Take SRS of size n from population where p is
  true (unknown) proportion of successes.
• Observe X successes
• Set confidence level C and choose z such that
  P(-z?Z ?z)C (C 90 ? z1.645 C 95 ?
  z1.96 C 99 ? z2.576)

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Example - Ginkgo and Azet for AMS

• Study Goal Measure effect of Ginkgo and
  Acetazolamide on occurrence of Acute Mountain
  Sickness (AMS) in Himalayan Trackers
• Parameter p True proportion of all trekkers
  receiving GinkgoAcetaz who would suffer from
  AMS.
• Sample Data n126 trekkers received GA, X18
  suffered from AMS

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Wilsons Plus 4 Method

• For moderate to small sample sizes, large-sample
  methods may not work well wrt coverage
  probabilities
• Simple approach that works well in practice
  (n?10)
• Pretend you have 4 extra individuals, 2
  successes, 2 failures
• Compute the estimated sample proportion in light
  of new data as well as standard error

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Example Listers Tests with Antiseptic

• Experiments with antiseptic in patients with
  upper limb amputations (John Lister, circa 1870)
• n12 patients received antiseptic X1 died

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Significance Test for a Proportion

• Goal test whether a proportion (p) equals some
  null value p0 H0 pp0

Large-sample test works well when np0 and n(1-p0)
gt 10
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Ginkgo and Acetaz for AMS

• Can we claim that the incidence rate of AMS is
  less than 25 for trekkers receiving GA?
• H0 p0.25 Ha p lt 0.25

Strong evidence that incidence rate is below 25
(plt0.25)
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Comparing Two Population Proportions

• Goal Compare two populations/treatments wrt a
  nominal (binary) outcome
• Sampling Design Independent vs Dependent Samples
• Methods based on large vs small samples
• Contingency tables used to summarize data
• Measures of Association Absolute Risk, Relative
  Risk, Odds Ratio

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Contingency Tables

• Tables representing all combinations of levels of
  explanatory and response variables
• Numbers in table represent Counts of the number
  of cases in each cell
• Row and column totals are called Marginal counts

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2x2 Tables - Notation
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Example - Firm Type/Product Quality

• Groups Not Integrated (Weave only) vs
  Vertically integrated (Spin and Weave) Cotton
  Textile Producers
• Outcomes High Quality (High Count) vs Low
  Quality (Count)

Source Temin (1988)
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Notation

• Proportion in Population 1 with the
  characteristic of interest p1
• Sample size from Population 1 n1
• Number of individuals in Sample 1 with the
  characteristic of interest X1
• Sample proportion from Sample 1 with the
  characteristic of interest
• Similar notation for Population/Sample 2

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Example - Cotton Textile Producers

• p1 - True proportion of all Non-integretated
  firms that would produce High quality
• p2 - True proportion of all vertically
  integretated firms that would produce High
  quality

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Notation (Continued)

• Parameter of Primary Interest p1-p2, the
  difference in the 2 population proportions with
  the characteristic (2 other measures given below)
• Estimator
• Standard Error (and its estimate)
• Pooled Estimated Standard Error when p1p2p

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Cotton Textile Producers (Continued)

• Parameter of Primary Interest p1-p2, the
  difference in the 2 population proportions that
  produce High quality output
• Estimator
• Standard Error (and its estimate)
• Pooled Estimated Standard Error when p1p2p

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Confidence Interval for p1-p2 (Wilsons Estimate)

• Method adds a success and a failure to each group
  to improve the coverage rate under certain
  conditions
• The confidence interval is of the form

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Example - Cotton Textile Production
95 Confidence Interval for p1-p2
Providing evidence that non-integrated producers
are more likely to provide high quality output
(p1-p2 gt 0)
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Significance Tests for p1-p2

• Deciding whether p1p2 can be done by
  interpreting plausible values of p1-p2 from the
  confidence interval
• If entire interval is positive, conclude p1 gt p2
  (p1-p2 gt 0)
• If entire interval is negative, conclude p1 lt p2
  (p1-p2 lt 0)
• If interval contains 0, do not conclude that p1 ?
  p2
• Alternatively, we can conduct a significance
  test
• H0 p1 p2 Ha p1 ? p2 (2-sided) Ha
  p1 gt p2 (1-sided)
• Test Statistic
• P-value 2P(Z?zobs) (2-sided) P(Z?
  zobs) (1-sided)

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Example - Cotton Textile Production
Again, there is strong evidence that
non-integrated performs are more likely to
produce high quality output than integrated firms