Notes 3 - ’ & ST3241 Categorical Data Analysis I...

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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Two-way Contingency Tables Odds Ratio and Tests of Independence 1 ’ & Inference For Odds Ratio (p. 24) • For small to moderate sample size, the distribution of sample odds ratio ˆ θ is highly skewed. • For θ = 1 , ˆ θ cannot be much smaller that θ , but it can be much larger than θ with nonnegligible probability. • Consider log odds ratio, log θ • X and Y are independent implies log θ = 0 . 2 ’ & Log Odds Ratio • Log odds ratio is symmetric about zero in the sense that reversal of rows or reversal of columns changes its sign only. • The sample log odds ratio, log ˆ θ has a less skewed distribution and can be approximated by the normal distribution well. • The asymptotic standard error of log ˆ θ is given by ASE (log ˆ θ ) = r 1 n 11 + 1 n 12 + 1 n 21 + 1 n 22 3 ’ & Confidence Intervals • A large sample confidence interval for log θ is given by log( ˆ θ ) ± z α/ 2 ASE (log ˆ θ ) • A large sample confidence interval for θ is given by exp[log( ˆ θ ) ± z α/ 2 ASE (log ˆ θ )] 4 ’ & Example: Aspirin Usage • Sample Odds Ratio = 1.832 • Sample log odds ratio, log ˆ θ = log(1 . 832) = 0 . 2629 • ASE of log ˆ θ r 1 89 + 1 10933 + 1 10845 + 1 104 = 0 . 123 • 95% confidence interval for log θ equals . 605 ± 1 . 96 × . 123 • The corresponding confidence interval for θ is ( e . 365 ,e . 846 ) or (1.44,2.33) . 5 ’ & Recall SAS Output Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) 1.8321 1.4400 2.3308 Cohort (Col1 Risk) 1.8178 1.4330 2.3059 Cohort (Col2 Risk) 0.9922 0.9892 0.9953 Sample Size = 22071 6 ’ & A Simple R Function For Odds Ratio > odds.ratio <- function(x, pad.zeros = FALSE, conf.level=0.95) { if(pad.zeros) { if(any(x==0)) x <-x+0.5 } theta <-x[1,1] * x[2,2]/(x[2,1] * x[1,2]) ASE <-sqrt(sum(1/x)) CI <-exp(log(theta) + c(-1,1) * qnorm(0.5 * (1+conf.level)) * ASE) list(estimator=theta, ASE=ASE, conf.interval=CI, conf.level=conf.level) } 7 ’ & Notes (p. 25) • Recall the formula for sample odds ratio ˆ θ = n 11 n 22 n 12 n 21 • The sample odds ratio is or ∞ if any n ij = 0 and it is undefined if both entries in a row or column are zero. • Consider the slightly modified formula ˜ θ = ( n 11 + 0 . 5)( n 22 + 0 . 5) ( n 12 + 0 . 5)( n 21 + 0 . 5) • In the ASE formula also, n ij ’s are replaced by n ij + 0 . 5 . 8 ’ & Observations • A sample odds ratio 1.832 does not mean that p 1 is 1.832 times p 2 . • A simple relation: OddsRatio = p 1 / (1- p 1 ) p 2 / (1- p 2 ) = RelativeRisk × 1- p 2 1- p 1 • If p 1 and p 2 are close to 0, the odds ratio and relative risk take similar values....
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This note was uploaded on 11/20/2011 for the course STATISTICS ST3241 taught by Professor Manwai's during the Spring '11 term at National University of Singapore.

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Notes 3 - ’ & ST3241 Categorical Data Analysis I...

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