I've got a couple of non-significant logistic regression models, and am figuring out how to word it appropriately. Before that, I want to confirm the implication. Most questions/websites I have gone to don't go to the verbiage aspect or explain in-depth the implications, hence I'm clarifying here:
If my overall logistic regression model is non-significant:
Does this mean:
(a) All odds ratios are nonsignificant, or
(b) All odds ratios are equal to 1? (i.e., same as the base case)
Thanks for your clarification.
Best Answer
For some link function $g$, let the GLM model be $ g\left(\mathbb E\left[Y\vert X\right]\right) = \beta_0 + \beta_1x_1 +\cdots +\beta_px_p $.
When you get an "overall significance: of the model, the standard hypothesis test is:
$$ H_0: \beta_1 = \cdots = \beta_p = 0\\ H_a: H_0\text{ is false} $$
When you get an "insignificant" result of this hypothesis test, it means that there is insifficient evidence to reject $H_0$ in favor of $H_a$, so insufficient evidence to reject the null hypothesis of all non-intercept parameters being zero.
In your logistic regression, changing a feature by some amount makes no statistically significant change in the log-odds. This is equivalent to saying that the odds ratios are not statistically significantly different from $1$.
$$ \log\left(\dfrac{p_1}{1-p_1}\right) = \log\left(\dfrac{p_2}{1-p_2}\right)\\ \log\left(\dfrac{p_1}{1-p_1}\right) - \log\left(\dfrac{p_2}{1-p_2}\right)=0\\ \log\left(\dfrac{\dfrac{p_1}{1-p_1}}{\dfrac{p_2}{1-p_2}}\right) = 0 \\ \dfrac{\dfrac{p_1}{1-p_1}}{\dfrac{p_2}{1-p_2}} = 1 $$