Odds ratio interpretation if no significant correlation between outcome and predictor Does it make sense to conduct a logistic regression (binary outcome vitamin D deficiency and predictor CRP) if there was no significant correlation between the two variables in the first place? 
However, there is a significant difference in the mean concentration between the binary outcome groups. 
The second question for the same case: if, for example, the OR is 0.224 and P-value is significant, what is the interpretation? 
 A: Yes. An odds ratio is a more powerful measure of association between a binary outcome and a continuous predictor. 
It is particularly powerful (relative to a simple correlation) when the overall prevalence of the outcome is either very low or very high. So if the prevalence of D deficiency was 10% or lower, you would not be surprised to see a non-significant correlation but a highly significant odds ratio from a logistic regression model.
An odds ratio of 0.244 is interpreted as a "0.244 fold difference in odds of Vitamin D deficiency comparing samples differing by 1mg/L of c-reactive protein".
I would hazard against making the mistake of interpreting it as a relative risk or additive risk, and not a causal interpretation unless coming from a randomized controlled design. For instance, do not use the word "1mg/L increase in c-reactive protein" and do not use the word "A 0.244 fold difference in risk of Vitamin D deficiency..." (this is estimated with a relative risk model) and especially not "A 0.756 decrease in risk..." (this is estimated with an additive risk model)
A: Yes, that makes sense: correlation is not a good measure of association for a binary variable, so doing something more appropriate for that type of data is a good idea.
An odds ratio of .224 means that if CRP (whatever that may be) increases by one unit (whatever the unit may be) then the odds of having a vitamin D deficientcy is multiplied by 0.224. This is equivalent to saying that the odds decreases by 77.6 % [(0.224 - 1)*100%=-77.6%]. 
Remember that odds and probabilities are different things: the probability of "success" (in this case vitamin D deficiency) is the expected proportion of successes, while the odds of success is the expected number of successes per failure.
