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# Case Study - Relative Risk and Odds Ratio - PowerPoint PPT Presentation

Case Study - Relative Risk and Odds Ratio. John Snow’s Cholera Investigations. Population Information. 2 Water Providers: Southwark & Vauxhall (S&V) and Lambeth (L) S&V: Population: 267625 # Cholera Deaths: 3706 L: Poulation: 171528 # Choleta Deaths: 411.

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### Case Study - Relative Risk and Odds Ratio

John Snow’s Cholera Investigations

• 2 Water Providers: Southwark & Vauxhall (S&V) and Lambeth (L)

• S&V: Population: 267625 # Cholera Deaths: 3706

• L: Poulation: 171528 # Choleta Deaths: 411

• 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)

• 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: X~B(n,p)

• 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 > 0.5, use 1-p and you are obtaining P(X=n-k)

• 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 Uivalues that  p

• X~B(n,p)

• Finite population adjustments can be made by “correcting” p after each draw

• EXCEL has built in Function:

• Tools --> Data Analysis --> Random Number Generation

• --> Binomial --> Fill in p and n

• Simulate by taking samples of nSV=nL=5000 individuals from each population of customers

• Generate XSV~B(5000,.013848) and XL~B(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)