Suppose if a dependent variable is having a disease (1) and not having a disease (0). similarly , independent variable is smoking and not smoking. here , not smoking as reference group. Again, if logistic coefficient of smoking is - 0.3. odd ratio = exp (-0.3) = 0.74 Can we interpret it as , reciprocal of odd ratio= 1/0.74 = 1.34 .. Smoker have 1.34 times odd of not having disease than non smoker. or is it " Non smoker have 1.34 times odd of having disease than smoker" is both interpretation correct or not ? I am just confused.
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3You really mean negative of the coefficient, not its reciprocal. The coefficient will be negated when you reverse your coding: use 0 for having the disease and 1 for not. Interpret the result accordingly. – whuber Jul 27 '23 at 13:26
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1is the changing sign of coefficient , reverses coding in dependent variables? – Prasanna Jul 29 '23 at 09:19
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1Yes. The math is simple: recoding your responses converts any $y$ to $1-y.$ Thus,you are changing from a model of the form $E[Y\mid X]=h^{-1}(X\beta)$ to $E[1-Y\mid X]=1-E[Y\mid X] = 1 - h^{-1}(X\beta).$ The link function $h$ is the logit, which enjoys the symmetry $h(-z)=1-h(z).$ Thus, $E[Y\mid X] = h^{-1}(X(-\beta)),$ demonstrating that recoding simply negates the coefficients. Consequently, the coefficient estimates are negated, too. – whuber Aug 01 '23 at 14:41
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1oh does it work in independent variables? – Prasanna Aug 24 '23 at 01:37
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1Yes, but it's a tiny bit more complicated. In a model where $h(E[y])=\alpha+\cdots+\beta x+\cdots$ and $x$ is a $0,1$ variable, if you switch the coding of $x,$ that replaces $x$ by $1-x.$ Plugging that in gives $$h(E[y])=\alpha+\cdots+\beta(1-x)+\cdots=(\alpha+\beta)+\cdots-\beta x+\cdots.$$ Thus, the coefficient of $x$ is negated and the intercept $\alpha$ is modified (in a predictable way). – whuber Aug 24 '23 at 12:19
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Thank you, it has been very useful for my thesis. Would you suggest any further reading for it? – Prasanna Aug 24 '23 at 14:50