Is it possible that in logistic regression model, one variable has high odds ratio and higly significant and other variable which has low odds ratio but is also higly significant.
2 Answers
$\begingroup$
$\endgroup$
Yes. An odds ratio is a (somewhat unintuitive) measure of how "large" an effect is. A p value is a measure of how confident we can be that the effect size is significantly different from "no effect at all" (which, if you are talking about odds ratios, corresponds to a value of 1). Those two things are related - the bigger an estimated effect is the more confident we can be that it exists at all....all else being equal. But in practice all else is often not equal. Sample size and standard deviation also play into the question of whether something is statistically significant. So a binary variable where only a small number of cases are "1" (or zero), or a continuous variable with a really big standard deviation might not be significant in a model, despite having bigger odds ratios than some other variables that are significant. This just means "we estimated a really big effect over here, but we're not sure it's real, while we estimated a smaller effect over there, and we're really confident that it exists."
$\begingroup$
$\endgroup$
Yes. "High" and "low" are not absolute terms but depend on the scale of the predictor. For example, if money affects the odds of the outcome, then having money measured in cents will yield a very different odds ratio from measuring money in thousands of dollars, and yet in both cases the p-value is the same. So unless it makes sense to compare odds ratios based on the scale of the predictors, one shouldn't look at the absolute value of the odds ratio to decide if it is "high" or "low" and one shouldn't compare two odds ratios and compare one is higher than the other.
Even if the variables are on the same scale, e.g., because they are both binary variables, the p-value of an odds ratio depends not only on the size of the odds ratio but also on the precision of the estimate. For example, the odds ratio for a binary predictor with high imbalance in the levels (e.g., 90% in one category and 10% in the other) will have less precision than the odds ratio for a binary predictor with balanced levels (e.g., close to 50% in one category and 50% in the other), even with the same odds ratio. The precision is estimated by the standard error of the log odds ratio, which is reported in most regression output.
The p-value is computed using the t-statistic, which is the ratio of the log odds ratio to its standard error. Many things can affect the standard error, including how correlated the predictor is with other predictors in the model and how imbalanced the predictor is (if it is binary). This can lead to cases where predictors measured on the same scale and with the same odds ratio effect on the outcome can have different precision and therefore different p-values. Similarly, this can also lead to scenarios with vastly different odds ratio and p-values that are similar.
