# At what point in a logistic regression does a predictor become significant in predicting outcome?

Say we have a logistic regression with outcome equal to a women's pregnancy resulting in a birth =1 and loss=0. The predictor variable is age ranging from 22-50, the Odds ratio point estimate is .819 with CI (.782 - .858). How can we figure out the age in which this result become significant? I.e if this is interpreted as a 1 unit increase in age with a decreases in the chances of live birth vs pregnancy loss of 18%. However, this result is very unlikely if say a women ages from 22 to 23, so at what point in age does this result produce a significant outcome?

Younger subjects are more fertile, and this is reflected in the in that point estimate of $\small \text{OR} = 0.819$ which corresponds to the odds ratio:

\begin{align}\text{OR} &=\frac{ \text{odds(live birth})_{\text{age a +1}}}{\text{odds(live birth})_{\text{age a}}}=\frac{\left(\frac{\Pr(\text{live})}{\Pr(\text{loss})}\right)_{\text{age a + 1}}}{\left(\frac{\Pr(\text{live})}{\Pr(\text{loss})}\right)_{\text{age a}}}\\[2 ex] \text{OR} &= e^{\beta_{\text{age}}\times 1} = 0.819 \end{align}

As the age advances the odds ratio decreases, so that a difference in age of $\Delta_{\text{age}}$ will result in a change in the odds ratio of pregnancy of

$$0.819^{\Delta_{\text{age}}} = \left(e^{\beta_{\text{age}}}\right)^{\Delta_{\text{age}}}$$

In other words, we are exponentiating a value $<1$, and, hence, the relative odds keep on getting smaller and smaller. On the other hand, the odds of a live delivery would increase with time if the $\small \text{OR}>1.$

• Additionally, to more directly answe the quesiton: if you assume it's a linear function does not really match up with a question of whether there's a certain threshold age where there is a change (a linear function is linearly increasing throughout the whole data range - not that we necessarily believe this to be true). The OP could try splines to visualize things. – Björn May 1 '17 at 18:23
• @Bjorn Yes! I didn't touch on that because it seemed as though mulling through the definitions could make it apparent that the idea of a threshold simply doesn't apply... We'd quickly get into ROCs and cut-off points, which may have more contextual interest when trying to determine, say, normal levels of cholesterol in blood. – Antoni Parellada May 1 '17 at 18:26
• I calculated optimal cut off points using the J statistic....however I agree this question kind of asks whether there is a cutt off point when there really shouldn't be one when its a linear function? – Demetri Smith May 1 '17 at 19:04
• @usεr11852 Thank you very much! It's been an amazing learning experience. I plan on shifting to doing more reading and less writing. I wish you, on the other hand, did more writing - your posts are awesome! – Antoni Parellada May 6 '17 at 0:58
• I don't know if it was an amazing experience for you but it was an obvious one for an outside observer like me. The quality of your posts is getting better and better. (Thanks for the compliment! It is all part of my grand plan to become a one-man-Stats-army when I retire!) – usεr11852 May 6 '17 at 11:59