Smoking has been shown to be a risk factor for LBW newborns with an OR of about $2\mathrm{.}0.$ Suppose that you want to engage in a nested case control study with $450$ cases of mothers with LBW newborns and $500$ controls $($mothers with normal weight infants$).$ Since your study is nested within an ongoing cohort study, you have access to both urine and serum samples from these women.

$1.$ Suppose that the true exposure status is $150$ smokers among the cases and $100$ smokers among the controls. what is the true crude OR$?$

$2.$ what would be the observed OR using the serum thiocyanate measure for smoking ststus with a sensativity of $86\%$ and specificity of $79\%?$

$3.$ So far we have been considering how the observed OR may be different from the true OR under confitions od an imperfect test to measure exposure. How might we do the opposite$-$adjust the observed OR to the true OR$,$ taking into consideration the characteristics of the test of exposure?