Similar question and discussion....

https://www.researchgate.net/post/What_is_the_difference_between_Odd_Ratios_OR_and_Adjusted_Odd_Ratios_AOR_Why_Adjusted_Odd_Ratios_AOR_are_calculated_and_how_interpreted

The main difference between a crude odds ratio and an adjusted odds ratio is that the adjusted odds ratio is adjusted according to the other variables within a model. By holding all other independent variables within a model constant, then you get the adjusted odds ratio for the variable of interest (whatever that may be). Hosmer and Lemeshow (2000) [older edition of the book], state that "adjusted odds ratios are obtained by comparing individuals who differ only in the characteristic of interest and have the values of all other variables constant."

For example, say you are using logistic regression to predict the risk of relapse of substance abuse after individuals received treatment at a drug rehabilitation center. The dependent variable (DV) would be a dichotomous variable (Relapse= yes|no), and you could have any number of independent variables (IV) to produce a model that can hopefully identify a group of individuals who possess specific characteristics who are at an increased risk of relapse. For the example, lets say the only IVs you have are, gender (male=0, female=1 [male= reference]), number of previous treatments (continuous variable [0=reference which can be changed to the mean or median]), and employment (no=0|yes=1; unemployed=reference category]).

Then let's say you main IV of interest is the number of previous treatment centers a patient has attended. If you created a model with just this variable as the IV for predicting the risk of relapse, you would have a crude odds ratio in which the exp(B)= the odds ratio when no other variables are taken into account. However, this coefficient is likely to change slightly or substantially if other variables (particularly relevant ones are also included within a model). So you redo your logistic regression to predict risk of relapse using number of previous treatment facilities, gender, and employment as your IVs. Now, when you exponentiate the coefficient of the 'prior number of treatments' variable the result you get is the Adjusted odds ratio holding gender and employment constant. But what does holding them constant mean? It means that the variables are set to the reference category for the two dichotomous variables. So for the example the exp(coefficient for number of prior treatments) would be the multiplicative odds of relapse per additional treatment for males who are unemployed.

I hope this clarifies any confusion. To briefly summarize: a crude odds ratio is just an odds ratio of one IV for predicting the DV. The adjusted odds ratio holds other relevant variables constant and provides the odds ratio for the potential variable of interest which is adjusted for the other IVs included in the model.

Sincerely,

Logan Netzer

References:

Hosmer DW, and Lemeshow S. (2000). Applied Logistic Regression, 2nd Edition. Wohn Wiley & Sons, Inc. Hoboken, New Jersey.

I'll explain you with an example.

You want to know the odds of male patients suffering from a specific cancer in comparison to female patients with that cancer. You calculate the odds ratio and the odds ratio will obtain 1.5. In this situation, you didn't control other variables. such as age, education, marital status and etc. however, if you control these variables you will obtain the adjusted odds ratio.

Great so exp B is the adjusted odds ratio in a logistics regression model

When adjusted odds ratio could greater than crude odds ratio?

Adjusted odds ratio is that the adjusted odds ratio is adjusted according to the other variables within a model.

Azimeraw Arega each added independent variable could be increase or decrease crude OR, producing higher or lower AOR. That's the point of model building.