Case Study - Relative Risk and Odds Ratio

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Case Study - Relative Risk and Odds Ratio

• John Snows Cholera Investigations

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Population Information

• 2 Water Providers Southwark Vauxhall (SV) and
  Lambeth (L)
• SV Population 267625 Cholera Deaths 3706
• L Poulation 171528 Choleta Deaths 411

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Sampling Distribution of RR OR

• Goal Obtain Empirical Sampling Distributions of
  sample RR and OR and observe coverage rate of 95
  Confidence Intervals
• Process Take independent random samples of size
  nSV and nL from the 2 populations and observe XSV
  and XL deaths in sample. These XSV and XL are
  approximately distributed as Binomial random
  variables (approximate due to sampling from
  finite, but very large, populations)

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Binomial Distribution for Sample Counts

• Binomial Experiment
• Consists of n trials or observations
• Trials/observations are independent of one
  another
• Each trial/observation can end in one of two
  possible outcomes often labelled Success and
  Failure
• The probability of success, p, is constant across
  trials/observations
• Random variable, X, is the number of successes
  observed in the n trials/observations.
• Binomial Distributions Family of distributions
  for X, indexed by Success probability (p) and
  number of trials/observations (n). Notation
  XB(n,p)

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Binomial Distributions and Sampling

• Problem when sampling from a finite sample the
  sequence of probabilities of Success is altered
  after observing earlier individuals.
• When the population is much larger than the
  sample (say at least 20 times as large), the
  effect is minimal and we say X is approximately
  binomial
• Obtaining probabilities

Table C gives probabilities for various n and p.
Note that for p gt 0.5, use 1-p and you are
obtaining P(Xn-k)
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Simulating Binomial RVs

• Select n and p
• Obtain n random numbers distributed uniformly
  between 0 and 1 (any software package should have
  built-in random number generator) U1,,Un
• Let X be the number of Ui values that ? p
• XB(n,p)
• Finite population adjustments can be made by
  correcting p after each draw
• EXCEL has built in Function
• Tools --gt Data Analysis --gt Random Number
  Generation
• --gt Binomial --gt Fill in p and n

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Simulation Example

• Simulate by taking samples of nSVnL5000
  individuals from each population of customers
• Generate XSVB(5000,.013848) and
  XLB(5000,.002396)
• Compute sample relative risk, ln(RR), odds ratio,
  ln(OR), and estimated std. errors of ln(RR) and
  ln(OR)
• Obtain 95 CIs for RR, OR (based on ln(RR),ln(OR)
  
• Repeat for a large number of samples (1000
  samples)
• Obtain the empirical distribution of each
  statistic
• Obtain an indicator of whether the 95 CI for RR
  contains the population RR (5.78) and whether the
  95 CI for OR contains the population OR (5.85)

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Computations
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Note that the distribution of Relative Risks is
not normal
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Note that distribution of ln(RR) is approximately
normal