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oddsratio - Odds Ratios Suppose we have 2 independent...

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Odds Ratios Suppose we have 2 independent random samples from Bernoulli Distributions: ) ( ~ ,..., ) ( ~ ,..., 1 1 Y n X m p Bernoulli Y Y p Bernoulli X X We wish to estimate p X and p Y by maximum likelihood (as well as some functions of them): ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 m x p x x m p p p x m p x p p x m p x p l p x m p x p x x L l p p p x x L m i i X m i i m i i X X X m i i X m i i X X m i i X m i i X X m i i X m i i X m x m X x X X m m i i m i i = - = - - - = - - - = - - + = = - = = = = = = = = = = - = = 1 ^ 1 1 ^ ^ ^ 1 ^ 1 ^ 1 1 1 1 1 1 1 1 for solving and 0 to equal this Setting 1 1 ln ln ; ,..., ln 1 ; ,..., 1 1 Similarly: n y p n i i Y = = 1 ^ Consider the odds of success ( P (Success)/ P (Failure) = P (S)/[1- P (S)]) = = = = - = - = - = - = m i i n i i Y Y Y m i i m i i X X X y n y p p x m x p p 1 1 ^ ^ ^ 1 1 ^ ^ ^ 1 odds 1 odds Due to distributional reasons, we often work with the(natural) logarithm of the odds (its sampling distribution approaches normality quicker wrt sample size). This measure is often called the logit : - = - = - = - = - = = = = = n i i n i i Y Y Y m i i m i i X X X X X X y n y p p x m x p p p p 1 1 ^ ^ ^ 1 1 ^ ^ ^ ln 1 ln logit ln 1 ln logit 1 ln logit
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