lecture_05 - Two-Way Contingency Tables Lecture-05...

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    Two-Way Contingency Tables Lecture -05
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            Comparing Proportions The difference of proportions  1 -  2 compares the  π π success of probabilities in the two rows. This difference falls between –1 and +1. It equals  zero when  1 =  2; that is , when the response is  π π independent of the group classification. 
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    An example (Difference of proportions)   The table below taken from the report on the relationship  between aspirin use and myocardial infarction (heart  attacks) by the Physician’s Health Study Research group at  Harvard Medical School.                              Myocardial Infarction Group Yes No Total Placebo 189 10845 11034 Aspirin 104 10933 11037
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               Relative Risk Example  -  Consider comparing 2 drugs on proportions  of subjects who have adverse reactions when using the  drug. The difference between .010 and 0.001 is the same as  between .410 and .401 which is .009. But the first  difference is more noteworthy. In such cases ratio of proportions plays a useful role. 
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               Relative Risk In a 2 x 2 tables, relative risk is the ratio of the success  probabilities for the two groups. Relative Risk (RR) =  π 1  /  π 2 The proportions .010 and 0.001 will have a RR of .010/.001  10  whereas the proportions of .410 and .401 will have a  RR of  1.02 Relative risk can be any nonnegative real number.
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    Relative Risk What happens if Relative Risk (RR) = 1.0 - This happens when response is     independent of the 2 groups. We can calculate Relative Risk of a sample  by taking the ratio of sample proportions of  p 1 / p
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    Exercise Calculate the sample Relative Risk for the  following aspirin and heart attack example:  Myocardial Infarction Group Yes No  Placebo n 11  = 189 n 12 = 10845 Aspirin n 21  = 104 n 22 = 10933 Total n 1+  = 11034 n 2+  = 11037 Find: 1) Relative Risk
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                 Relative Risk Remember!    Using the difference of proportions alone  to compare two groups can be misleading  when the proportions are both close to  zero.
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                   Odds Ratio Odds ratio is given by Relative Risk is the ratio of two probabilities whereas odds  ratio is a ratio of two odds . In a 2*2 tables, the probability of “success” is  π 1  in row 1 and  π in row 2. Within row 1, the odds of success are defines to  be       odds1 =  π 1  /(1-  π 1 ) Within row 2, the odds of success equal      odds2 =  π / (1-  π 2 )
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                     Odds Ratio When the conditional distributions are identical in the two  rows ( i.e.  π 1=  π ), the odds satisfy odds1 = odds2.
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