Unformatted text preview: 7 ∗ 129/61 ∗ 55) = −1.312 log(θ23 ) = log(55 ∗ 570/489 ∗ 129) = −.699 log(θ34 ) = log(489 ∗ 431/(570 ∗ 475) = −.250 log(θ45 ) = log(475 ∗ 154/431 ∗ 293) = −.546 log(θ56 ) = log(293 ∗ 12/154 ∗ 38) = −.509 The 5 log local odds ratios were all negative, i.e., people who had the higher level of smoking always had the greater odds to be in the lung cancer group. It is more so for the comparison between people who never smoked and those who smoke fewer than 5 cigarettes per day. Further, we can ﬁnd that the log local odds ratio is the corresponding diﬀerence between each pair of adjacent log odds (cf. part(a.)). The nature of the association between level of smoking and lung cancer is not linear. Should the nature of the association be linear, we would have found the log local odds identical (consistent with the slope). c. [2 points]
We know the log local odds ratio for each pair of adjacent levels of smoking can be expressed as: Given log(oddsi ) = α + βi Ωi ) Ωi+1 = log Ωi − log Ωi+1 = −β Hence, if the log odds of lung cancer was linearly related to the level of smoking, all log local odds ratios would be identical as a constant value (−β), i.e., all local odds ratios are identical (exp−β ). You can also ﬁnd this relationship by graph. log θi,i+1 = log( = α + βi − (α + β(i + 1))...
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- Spring '12